Lectures on Symplectic Geometry, Poisson Geometry ...

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Lectures on Symplectic Geometry, Poisson

Geometry, Deformation Quantization

and Quantum Field Theory

Nima Moshayedi

Institut fur Mathematik, Universitat Zurich, Winterthurerstrasse 190 CH-8057 ZurichEmail address, N. Moshayedi: [email protected]

http://arxiv.org/abs/2012.14662v1

Abstract. These are lecture notes for the course “Poisson geometry and deformation quan-tization” given by the author during the fall semester 2020 at the University of Zurich. Thefirst chapter is an introduction to differential geometry, where we cover manifolds, tensorfields, integration on manifolds, Stokes’ theorem, de Rham’s theorem and Frobenius’ theo-rem. The second chapter covers the most important notions of symplectic geometry suchas Lagrangian submanifolds, Weinstein’s tubular neighborhood theorem, Hamiltonian me-chanics, moment maps and symplectic reduction. The third chapter gives an introductionto Poisson geometry where we also cover Courant structures, Dirac structures, the localsplitting theorem, symplectic foliations and Poisson maps. The fourth chapter is aboutdeformation quantization where we cover the Moyal product, L∞-algebras, Kontsevich’sformality theorem, Kontsevich’s star product construction through graphs, the globalizationapproach to Kontsevich’s star product and the operadic approach to formality. The fifthchapter is about the quantum field theoretic approach to Kontsevich’s deformation quan-tization where we cover functional integral methods, the Moyal product as a path integralquantization, the Faddeev–Popov and BRST method for gauge theories, infinite-dimensionalextensions, the Poisson sigma model, the construction of Kontsevich’s star product througha perturbative expansion of the functional integral quantization for the Poisson sigma modelfor affine Poisson structures and the general construction. Any comments and remarks canbe send by email to the author.

Introduction

Poisson geometry appears naturally in physics in the context of the dynamics for classicalmechanical systems. Mathematically, it lies in the intersection of differential geometry andnoncommutative algebra. In physics, people are often interested in observables, which mathe-matically can be described as elements of the algebra of smooth functions on some underlyingmanifoldM endowed with certain structure, and their dynamics. In fact, the algebra C∞(M)can be endowed with an algebra structure given by a bracket , , called a Poisson bracket,satisfying certain properties, similarly as in the case of a Lie algebra g endowed with its Liebracket [ , ]. Time evolution of an observable O is described by the equation

dO

dt= H,O,

where H ∈ C∞(M) denotes the Hamiltonian of the system.

This structure is actually motivated by the symplectic structure appearing additionally withinthe natural manifold structure for the phase space of a classical system. The phase space isa way of expressing the dynamics in a classical system where each state is represented by aunique point. The most simple phase space is given by R2 and in higher dimension the localstructure looks like some R2n, for n ≥ 1. One can then induce a Poisson bracket , from aclosed, nondegenerate 2-form ω, the symplectic form. Such a 2-form also appears naturally inthe structure of a phase space containing the information for the base coordinates qi and thecorresponding fiber momentum-coordinates pi on the vector bundle given by the cotangentbundle T ∗M of the manifoldM . Instead of looking at the structure sheaf of smooth functionsC∞(M) on a manifold M which is endowed with a Poisson bracket, one can also consider thegeometric picture of the manifold endowed with a bivector field π encoding the informationof the bracket , geometrically in an equivalent way.

A quantum system is described by a complex Hilbert space H together with an operator H.A physical state of the system is then represented by an element in H whereas the physicalobservables are given by self-adjoint operators on H. Denote this space by L(H). Timeevolution is then given by the Heisenberg equation

dO

dt=

i

~

[H, O

],

where [ , ] denotes the commutator of operators. The parameter ~ is called the reducedPlanck constant which is a physical constant naturally appearing at the quantum level. Theconnection to classical mechanics is usually given by the introduction of position qi and

3

4 INTRODUCTION

momentum pj operators. They satisfy the following commutation relation:

[pi, qj] =i

~δij .

In this way, one can obtain classical mechanics from the quantum theory in the limit where~! 0. This is a general concept. An important question is whether there is a precise mathe-matical formulation of such a quantization procedure in terms of a well-defined map betweenclassical objects and their quantum picture. There are different ways of quantizing a classicalsystem. If one starts from the canonical quantization on R2n, one can consider the method ofgeometric quantization (see e.g. [Kir85; Woo97; Mos20]). There the idea is the quantizationof the classical phase space R2n to the corresponding Hilbert space H = L2(Rn) on which theSchrodinger equation is defined. Another quantization approach focuses on the observables ofthe classical system. There one tries to capture the noncommutativity structure of the spaceof operators from the commutative structure of C∞(R2n). A result of Groenewold [Gro46]states that it is impossible to quantize the Poisson algebra C∞(R2n) in a way where thePoisson bracket of two functions is sent onto the Lie bracket of the corresponding operators.To overcome this issue, one can instead consider a deformation of the pointwise product onC∞(R2n) to a noncommutative product.

Deformation quantization originated from the work of Weyl [Wey31], who gave an explicitformula for the operator Of on L(Rn) associated to a function f ∈ C∞(R2n):

Of :=

∫

R2n

f(ξ, η) exp

(i

~(Pξ +Qη)

)dnξdnη,

where f denotes the inverse Fourier transform of f . Here P = (Pi) and Q = (Qj) denoteoperators satisfying the canonical commutation relation. Moreover, the integral is consideredin the weak sense. An inverse map was later found by Wigner [Wig32], who gave a way torecover the corresponding classical observable by taking the symbol of the operator. Moyal[Moy49] interpreted the symbol of the commutator of two operators corresponding to thefunctions f and g as what is today called the Moyal bracket :

M(f, g) :=sinh(ǫP )

ǫ(f, g) =

∞∑

k=0

ǫ2k

(2k + 1)!P 2k+1(f, g),

where ǫ := i~2 and P k is the k-th power of the Poisson bracket on C∞(R2n). Already Groe-

newold had a similar formula for the symbol of a productOfOg which today can be interpretedas the first appearance of the Moyal star product ⋆. The Moyal bracket can then be rewrittenin terms of this star product as

M(f, g) =1

2ǫ(f ⋆ g − g ⋆ f).

It was Flato who recognized this star product as a deformation of the commutative pointwiseproduct on C∞(R2n). This was the beginning of deformation quantization. He conjecturedthe problem of giving a general recipe to deform the pointwise product on C∞(M) in such away that 1

2ǫ(f ⋆ g− g ⋆ f) still remains a deformation of the Poisson structure on M . Follow-ing this conjecture, a first way of formulating quantum mechanics as such a deformation ofclassical mechanics had been discovered. The work of Bayen, Flato, Fronsdal, Lichnerowicz

INTRODUCTION 5

and Sternheimer was essential for the formulation of the deformation problem for symplecticspaces and the physical applications [Bay+78a; Bay+78b]. DeWilde and Lecomte [DL83] haveproven the existence of a star product on a generic symplectic manifold by using Darboux’stheorem which tells that locally any symplectic manifold of dimension 2n can be identifiedwith R2n by a choice of Darboux charts. Using cohomological arguments, one can constructsuch a star product by a correct gluing of the locally defined Moyal product. Independentlyof the previous result, Fedosov [Fed94] gave an explicit construction of a star product ona symplectic manifold. The generalization to any Poisson manifold was given through theformality theorem of Kontsevich [Kon03]. He derived an explicit formula for the product onRd by using special graphs and configuration space integrals, which can be used to define itlocally on any Poisson manifold M . The globalization procedure was described by Cattaneo,Felder and Tomassini [CFT02b; CFT02a] by extending Fedosov’s construction to the Poissoncase where they used notions of formal geometry developed by Gelfand–Fuks [GF69; GF70],Gelfand–Kazhdan [GK71] and Bott [Bot10], which completed the program of Flato proposedthirty years before. Another approach to Kontsevich’s result was given by Tamarkin [Tam98;Tam03] who used the notion of operads in order to prove the formality theorem.

Moreover, Cattaneo and Felder [CF00] have shown that Kontsevich’s formula can actuallybe formulated in terms of a perturbative expression of the functional integral of a topologicalfield theory. This theory is called the Poisson sigma model. The Poisson sigma model wasdiscovered independently by Ikeda [Ike94] and Schaller–Strobl [SS94; SS95] by an attemptto combine 2-dimensional gravity with Yang–Mills theories. In fact, one can show that thegraphs, which have been constructed by Kontsevich, arise naturally in this context as theFeynman diagrams subject to the expectation of certain observables. Recently, Cattaneo,Moshayedi and Wernli gave a field-theoretic picture to obtain a global deformation quan-tization for constant Poisson structures (Moyal product) by cutting and gluing techniquesfor certain worldsheet manifolds [CMW17]. This is done by using a globalized version of thePoisson sigma model and methods for gauge theories on manifolds with boundary developedin [CMW20]. It is expected that the cutting and gluing construction can be extended forgeneral Poisson structures.

An approach regarding higher gauge theories is encoded in the setting of higher shifted sym-plectic and Poisson structures. A way of dealing with these concepts and constructing a highershifted deformation quantization in the setting of derived algebraic geometry was formulatedby Calaque, Pantev, Toen, Vaquie and Vezzosi [Pan+13; Cal+17].

Acknowledgements I would like to thank A. S. Cattaneo for comments on these notes.This research was supported by the NCCR SwissMAP, funded by the Swiss National ScienceFoundation, and by the SNF grant No. 200020 192080.

Contents

Introduction 3

Chapter 1. Foundations of Differential Geometry 111.1. Differentiable manifolds 111.1.1. Charts and atlases 111.1.2. Pullback and pushforward 131.1.3. Tangent space 131.2. Vector fields and differential 1-forms 151.2.1. Tangent bundle 151.2.2. Vector bundles 161.2.3. Vector fields 171.2.4. Flow of a vector field 181.2.5. Cotangent bundle 191.2.6. Differential 1-forms 191.3. Tensor fields 201.3.1. Tensor bundle 201.3.2. Multivector fields and differential s-forms 211.4. Integration on manifolds and Stokes’ theorem 231.4.1. Integration of densities 231.4.2. Integration of differential forms 251.4.3. Stokes’ theorem 271.5. de Rham’s theorem 291.5.1. Singular homology 291.5.2. de Rham cohomology and de Rham’s theorem 331.6. Distributions and Frobenius’ theorem 331.6.1. Plane distributions 331.6.2. Frobenius’ theorem 34

Chapter 2. Symplectic Geometry 372.1. Symplectic manifolds 372.1.1. Symplectic form 372.1.2. Symplectic vector spaces 382.1.3. Symplectic manifolds 402.1.4. Symplectomorphisms 402.2. The cotangent bundle as a symplectic manifold 412.2.1. Tautological and canonical forms in coordinates 41

7

8 CONTENTS

2.2.2. Coordinate-free construction 422.2.3. Symplectic volume 432.3. Lagrangian submanifolds 432.3.1. Lagrangian submanifolds of T ∗X 432.3.2. Conormal bundle 442.3.3. Applications to symplectomorphisms 452.4. Local theory 462.4.1. Isotopies and vector fields 462.4.2. Tubular neighborhood theorem 472.4.3. Homotopy formula 482.5. Moser’s theorem 492.5.1. Equivalences for symplectic structures 492.5.2. Moser’s trick 492.6. Weinstein tubular neighborhood theorem 522.6.1. Weinstein Lagrangian neighborhood theorem 522.6.2. Weinstein tubular neighborhood theorem 532.6.3. Application 542.7. Hamiltonian mechanics 562.7.1. Hamiltonian and symplectic vector fields 562.7.2. Classical mechanics 572.7.3. Brackets 582.7.4. Integrable systems 592.8. Moment maps 612.8.1. Smooth actions 612.8.2. Symplectic and Hamiltonian actions 612.8.3. Adjoint and coadjoint representations 622.8.4. Hamiltonian actions 632.9. Symplectic reduction 642.9.1. Orbit spaces 642.9.2. Principal bundles 642.9.3. The Marsden–Weinstein theorem 672.9.4. Noether’s theorem 682.10. The Duistermaat–Heckman theorems 682.10.1. Duistermaat–Heckman polynomial 682.10.2. Local form for reduced spaces 692.10.3. Variation of the symplectic volume 71

Chapter 3. Poisson Geometry 753.1. Poisson manifolds 753.1.1. Poisson structures and the Schouten–Nijenhuis bracket 753.1.2. Examples of Poisson structures 763.2. Dirac manifolds 773.2.1. Courant algebroids 773.2.2. Dirac structures 783.2.3. Dirac structures for constrained manifolds 79

CONTENTS 9

3.3. Symplectic leaves and local structure of Poisson manifolds 803.3.1. Local and regular Poisson structures 803.3.2. Local splitting and symplectic foliation 813.4. Poisson maps 823.4.1. Two definitions 823.4.2. Examples of Poisson maps 82

Chapter 4. Deformation Quantization 854.1. Star products 854.1.1. Formal deformations 854.1.2. Moyal product 864.1.3. Fedosov’s globalization approach 874.1.4. Equivalent star products 884.2. Formality 894.2.1. Some formal setup 894.2.2. Differential graded Lie algebras 904.2.3. L∞-algebras 914.2.4. The DGLA of multivector fields V 954.2.4.1. The case of Poisson bivector fields 964.2.5. The DGLA of multidifferential operators D 964.2.6. The Hochschild–Kostant–Rosenberg map 994.2.7. The dual point of view 1004.2.8. Formality of D and classification of star products on Rd 1044.3. Kontsevich’s star product 1054.3.1. Data for the construction 1054.3.1.1. Admissible graphs 1054.3.1.2. The multidifferential operators BΓ 1064.3.1.3. Weights of graphs 1074.3.1.4. Configuration spaces 1074.3.2. Proof of Kontsevich’s formula 1094.3.2.1. U1 coincides with U

(0)1 109

4.3.2.2. Checking the degrees 1104.3.2.3. Reformulation of the L∞-condition in terms of graphs 1104.3.2.4. The key is Stokes’ theorem 1114.3.2.5. Classification of boundary strata 1114.3.2.6. A trick using logarithms 1134.3.2.7. Last step: vanishing terms for type S2 strata 1144.4. Globalization of Kontsevich’s star product 1164.4.1. The multiplication, connection and curvature maps 1164.4.2. Construction of solutions for a Fedosov-type equation 1184.5. Operadic approach to formality and Deligne’s conjecture 1204.5.1. Operads and algebras 1204.5.2. Topological operads 1214.5.3. The little disks operad 1224.5.4. Deligne’s conjecture 123

10 CONTENTS

4.5.5. Formality of chain operads 124

Chapter 5. Quantum Field Theoretic Approach to Deformation Quantization 1255.1. Functional integrals 1255.1.1. Functional integrals and expectation values 1255.1.2. Gaussian integrals 1265.1.2.1. Infinite-dimensional case 1275.1.3. Integration of Grassmann variables 1285.2. The Moyal product as a path integral quantization 1295.2.1. The propagator 1305.2.2. Expectation values 1315.2.3. Divergence of vector fields 1325.2.4. Independence of evaluation points 1335.2.5. Associativity 1345.2.6. The evolution operator 1355.2.7. Perturbative evaluation of integrals 1355.2.8. Infinite dimensions 1365.2.9. A simple generalization 1375.2.9.1. Quantum mechanics 1385.3. Symmetries and the BRST formalism 1395.3.1. The main construction: Faddeev–Popov ghost method 1405.3.2. The BRST formalism 1415.3.2.1. The BRST operator and the proof of Theorem 5.3.4 1425.3.3. Infinite dimensions 1455.3.3.1. The trivial Poisson sigma model on the plane 1455.3.3.2. Expectation values 1485.3.3.3. The trivial Poisson sigma model on the upper half-plane 1485.3.3.4. Generalizations 1495.4. The Poisson sigma model 1505.4.1. Formulation of the model 1505.4.2. Observables 1535.5. Deformation quantization for affine Poisson structures 1535.5.1. Gauge-fixing and Feynman diagrams 1545.5.2. Independence of the evaluation point 1565.5.3. Associativity 1565.6. The general construction 1575.6.1. The theorem of Cattaneo–Felder 1575.6.2. Other similar constructions 157

Bibliography 159

CHAPTER 1

Foundations of Differential Geometry

We want to start by introducing the main concepts and notions of differential geometry whichare needed in order to be able to understand the discussions in the following chapters. Theexperienced reader might skip this chapter, but we will still refer to some parts of it in otherchapters and one can always come back in case there are any uncertainties. In this chapterwe cover differentiable manifolds, vector bundles, (multi)vector fields, differential forms, gen-eral tensor fields, integration on manifolds, Stokes’ theorem for manifolds, Stokes’ theoremfor chains, de Rham cohomology, singular homology, de Rham’s theorem, distributions andFrobenius’ theorem. This chapter is mainly based on [BT82; Cat18; Lee02].

1.1. Differentiable manifolds

1.1.1. Charts and atlases.

Definition 1.1.1 (Chart). A chart on a set M is a pair (U, φ) where U ⊂M is a subset andφ is an injective map U ! Rn for some n.

We call φ a chart map or a coordinate map. Sometime we also refer only to φ as a chart sinceU is already contained inside the definition of φ. Let (U, φU ) and (V, φV ) be charts on M .Then we can compose the bijections (φU )|U∩V : U ∩V ! φU (U ∩V ) and (φV )|U∩V : U ∩V !φV (U ∩ V ) to the bijection

φU,V : (φU )|U∩V ((φV )|U∩V )−1 : φU (U ∩ V )! φV (U ∩ V ).

We call this the transition map between the charts (U, φU ) and (V, φV ). Moreover, we referto n as being the dimension of M (we will give another definition for the dimension lateron). See Figure 1.1.1 for a visualization.

Definition 1.1.2 (Atlas). An atlas on a set M is a collection of charts (Uα, φα)α∈I , whereI is an index set such that

⋃α∈I Uα =M .

Remark 1.1.3. We will denote the transition maps between two charts (Uα, φα) and (Uβ , φβ)simply by φαβ .

If φα(Uα) is open for all α ∈ I, then the atlas A = (Uα, φα)α∈I induces a topology on M .This topology is given by

OA(M) := V ⊂M | φα(V ∩ Uα) is open ∀α ∈ I.Definition 1.1.4 (Open atlas). An atlas is said to be open if φα(Uα ∩ Uβ) is open for allα, β ∈ I.

Definition 1.1.5 (Differentiable atlas). An atlas is said to be differentiable if it is open andall transition functions are Ck-maps for k = 0, 1, . . . ,∞.

11

12 1. FOUNDATIONS OF DIFFERENTIAL GEOMETRY

M

Rn Rn

φV

φU

U V

φU (U) φV (V )

U ∩ V

φU,V

Figure 1.1.1. Example of charts and transition map on an n-dimensionalmanifold M .

Definition 1.1.6 (Smooth atlas). An atlas is said to be smooth if it is open and all transitionfunctions are C∞-maps.

Definition 1.1.7 (Ck-equivalence). Two Ck-atlases on the same set are said to be Ck-equivalent if their union is a Ck-atlas for k = 0, 1, . . . ,∞.

Definition 1.1.8 (Ck-manifold). A Ck-manifold is an equivalence class of Ck-atlases fork = 0, 1, . . . ,∞.

Definition 1.1.9 (Smooth manifold). A smooth manifold is an equivalence class of C∞-atlases.

Remark 1.1.10. From now on, if we call a set M a manifold, we will always mean a smoothmanifold. Moreover, all maps between manifolds will be regarded as smooth maps. We willcall a map of manifolds a diffeomorphism, if it is invertible with smooth inverse.

Definition 1.1.11 (Submanifold). Let N be an n-dimensional manifold. A k-dimensionalsubmanifold, with k ≤ n, is a subset M of N such that there is an atlas (Uα, φα)α∈I of Nwith the property that for all α with Uα ∩M 6= ∅ we have φα(Uα ∩M) = Wα × x withWα ⊂ Rk an open subset and x ∈ Rn−k.

1.1. DIFFERENTIABLE MANIFOLDS 13

Remark 1.1.12. Any chart with this property is called an adapted chart and an atlas con-sisting of adapted charts is called an adapted atlas. Moreover, by a diffeomorphism of Rn wecan always assume that x = 0.

Example 1.1.13 (Graphs). Let F be a smooth map from an open subset V ⊂ Rk to Rn−k

and consider its graph

M = (x, y) ∈ V × Rn−k | y = F (x).Then M is a submanifold of N := V × Rn−k. As an adapted atlas we may take the oneconsisting of a single chart (N, ι), where ι : N ! Rn denotes the inclusion map.

1.1.2. Pullback and pushforward. Let M and N be two manifolds and consider amap F : M ! N .

Definition 1.1.14 (Pullback). The R-linear map

F ∗ : C∞(N)! C∞(M),

f 7! f F.is called pullback by F .

Exercise 1.1.15. Show that for f, g ∈ C∞(N) we have

F ∗(fg) = F ∗(f)F ∗(g).

Moreover, if G : N ! Z is a map between manifolds N and Z, show that

(G F )∗ = F ∗ G∗.

Definition 1.1.16 (Pushforward). Using F as before, we define the pushforward to be theinverse of the pullback F ∗ which we denote by F∗. In fact, we get

F∗ : C∞(M)! C∞(N),

f 7! f F−1.

Exercise 1.1.17. Show thatF∗(fg) = F∗(f)F∗(g),

and

(G F )∗ = G∗ F∗.

1.1.3. Tangent space. Let M be a manifold.

Definition 1.1.18 (coordinatized tangent vector). A coordinatized tangent vector at q ∈Mis a triple (U, φU , v) where (U, φU ) is a chart with U ∋ q and v is an element of Rn.

We say that two coordinatized tangent vectors (U, φU , v) and (V, φV , w) are equivalent if

w = dφU (q)φU,V v.

Definition 1.1.19 (Tangent vector). A tangent vector at q ∈ M is an equivalence class ofcoordinatized tangent vectors at q.

Definition 1.1.20 (Tangent space). The tangent space of M at q ∈M is given by the set ofall tangent vectors at q.


Note that each chart (U, φU ) at q ∈M defines a bijection of sets

Φq,U : TqM ! Rn,

[(U, φU , v)] 7! v.

We will also just write ΦU when the base point q ∈M is understood. This bijection allows usto transfer the vector space structure of Rn to TqM and gives Φq,U the structure of a linearisomorphism.

Lemma 1.1.21. TqM has a canonical vector space structure for which Φq,U is a linear iso-morphism for every chart (U, φU ) containing q.

Proof. Given a chart (U, φU ), the bijection ΦU defines the linear structure

λ ·U [(U, φU , v)] = [(U, φU , λv)], ∀λ ∈ R,

[(U, φU , v)] +U [(U, φU , v′)] = [(U, φU , v + v′)], ∀v, v′ ∈ Rn

If (V, φV ) is another chart, we have

λ ·U [(U, φU , v)] = [(U, φU , λv)]

= [(V, φV ,dφU (q)φU,V λv)]

= [(V, φV , λdφU (q)φU,V v)]

= λ ·V [(V, φV ,dφU (q)φU,V v)]

= λ ·V [(U, φU , v)],

and so ·U = ·V . Similarly, we have

[(U, φU , v)] +U [(U, φU , v′)] = [(U, φU , v + v′)]

= [(V, φV ,dφU (q)φU,V (v + v′)]

= [(V, φV ,dφU (q)φU,V v + dφU (q)φU,V v′]

= [(V, φV ,dφU (q)φU,V v)] +V [(V, φV ,dφU (q)φU,V v′)]

= [(U, φU , v)] +V [(U, φU , v′)],

and so +U = +V .

Definition 1.1.22 (Dimension of a manifold). We define the dimension of a manifold M by

dimM := dimTqM, q ∈M.

Let F : M ! N be a differentiable map between two manifolds M and N . Given a chart(U, φU ) of M containing q ∈ M and a chart (V, ψV ) of N containing F (q) ∈ N , we have alinear map

dU,Vq F := Φ−1F (q),V dφU (q)FU,V Φq,U : TqM ! TF (q)N.

Lemma 1.1.23 (Differential/Tangent map). The map dU,Vq F does not depend on the choiceof charts, so we get a canonically defined linear map

dqF : TqM ! TF (q)N,

called the differential (or tangent map) of F at q ∈M .

1.2. VECTOR FIELDS AND DIFFERENTIAL 1-FORMS 15

Proof. Let (U ′, φU ′) be another chart of M containing q ∈ M and (V ′, ψV ′) anotherchart of N containing F (q) ∈ N . Then

dU,Vq F [(U, φU , v)] = [(V, ψV ,dφU (q)FU,V v)]

= [(V ′, ψV ′ ,dψV ′(F (q))ψV,V ′dφU (q)FU,V v)]

= [(V ′, ψV ′ ,dφU′(q)FU ′,V ′(dφU (q)φU,U ′)−1v)]

= dU′,V ′

q F [(U ′, φU ′ , (dφU (q)φU,U ′)−1v)]

= dU′,V ′

q F [(U, φU , v)],

so we get dU,Vq F = dU′,V ′

q F .

We immediately get the following lemma.

Lemma 1.1.24. Let F : M ! N and G : N ! Z be maps between manifolds M,N,Z. Then

dq(G F ) = dF (q)G dqF, ∀q ∈M.

1.2. Vector fields and differential 1-forms

1.2.1. Tangent bundle. We can glue all the tangent spaces of a manifold M togetherand obtain the following definition.

Definition 1.2.1 (Tangent bundle). The tangent bundle of a manifold M is given by

TM :=⊔

q∈M

TqM.

An element of TM is of the form (q, v), where q ∈M and v ∈ TqM . Consider the surjectivemap π : TM !M , (q, v) 7! q. Then the fiber TqM can be denoted by π−1(q).

Proposition 1.2.2. TM is a manifold.

Proof. Let (Uα, φα)α∈I be an atlas in the equivalence class definingM . We set Uα :=π−1(Uα) and

φα : Uα ! Rn × Rn,

(q, v) 7! (φα(q),Φq,Uαv).(1.2.1)

Note that the chart maps are linear in the fibers. The transition maps are then given by

φαβ(x,w) = (φαβ(x),dxφαβw).

We can then define the tangent bundle ofM to be the equivalence class of the atlas (Uα, φα)α∈I .

Remark 1.2.3. Note that dimTM = 2dimM .


1.2.2. Vector bundles.

Definition 1.2.4 (Vector bundle). A vector bundle of rank r over a manifoldM of dimensionn is a manifold E together with a surjection π : E !M such that

(1) Eq := π−1(q) is an r-dimensional vector space for all q ∈M .

(2) E possesses an atlas of the form (Uα, φα)α∈I with Uα = π−1(Uα) for an atlas(Uα, φα)α∈I of M and

φα : Uα ! Rn × Rr

(q, v ∈ Eq) 7! (φα(q), Aα(q)v),(1.2.2)

where Aα(q) is a linear isomorphism for all q ∈ Uα.

Remark 1.2.5. We usually call E the total space andM the base space. Moreover, we usuallycall Eq := π−1(q) the fiber at q ∈M .

The maps

Aαβ : Uα ∩ Uβ ! End(Rr),

q 7! Aαβ(q) := Aβ(q)Aα(q)−1 : Rr ! Rr.

(1.2.3)

are smooth for all α, β ∈ I. The transition maps

φαβ(x, u) = (φαβ(x), Aαβ(φ−1α (x))u)

are linear in the second factor Rr.

Example 1.2.6 (Tangent bundle). The tangent bundle TM of a manifold M is an exampleof a vector bundle.

Example 1.2.7 (Trivial bundle). Let M be a manifold. Then we can define a vector bundleof rank n with total space E = M × Rn and base space M . Note that π : E ! M is theprojection onto the first factor. We call this a trivial bundle over M .

Example 1.2.8 (Mobius band). We can view the Mobius band E as a vector bundle of rank1 over the circle S1. Each fiber at a point x ∈ S1 is given in the form U ×R, where U ⊂ S1 isan open arc on the circle including x. Note that the total space E is not given by the trivialbundle S1 × R which would be the cylinder.

Definition 1.2.9 (Line bundle). We call a vector bundle a line bundle if it has rank 1.

Remark 1.2.10. Example 1.2.8 is an example of a line bundle.

Definition 1.2.11 (Morphism of vector bundles). Let (E1,M1, π1) and (E2,M2, π2) be twovector bundles. A morphism between (E1,M1, π1) and (E2,M2, π2) is given by a pair ofcontinuous maps f : E1 ! E2 and g : M1 !M2 such that g π1 = π2 f and for all q ∈M1,the map π−1

1 (q)! π−12 (g(q)) induced by f is a homomorphism of vector spaces.

Definition 1.2.12 (Section). Let π : E ! M be a vector bundle. A section on an opensubset U ⊂M is a continuous map σ : U ! E such that π σ = idU .

Remark 1.2.13. We denote the space of sections of a vector bundle E !M by Γ(E).


1.2.3. Vector fields.

Definition 1.2.14 (Vector field). LetM be a manifold. A section X ∈ Γ(TM) of the tangentbundle TM is called a vector field.

In an atlas (Uα, φα)α∈I of M and the corresponding atlas (Uα, φα)α∈I of TM , a vectorfield X is represented by a collection of smooth maps Xα : φα(Uα)! Rn. All these maps arerelated by

Xβ(φαβ(x)) = dxφαβXα(x), ∀α, β ∈ I,∀x ∈ φα(Uα ∩ Uβ).Remark 1.2.15. The vector at q ∈ M defined by the vector field X is usually denoted byXq as well as by X(q). The latter notation is often avoided as one may apply a vector fieldto a function f , and in this case the standard notation is X(f). One also uses Xα to denotethe representation of X in the chart with index α, but this should not create confusion withthe notation Xq for X at the point q.

We can add and multiply vector fields. Let X,Y ∈ Γ(TM) be two vector fields on M , λ ∈ R,and f ∈ C∞(M). Then

(X + Y )q := Xq + Yq, ∀q ∈M(1.2.4)

(λX)q := λXq, ∀q ∈M(1.2.5)

(fX)q := f(q)Xq, ∀q ∈M.(1.2.6)

Remark 1.2.16. We denote the space of vector fields on a manifold M by X(M) := Γ(TM).

Let F : M ! N be a diffeomorphism between two manifolds. If X is a vector field on M ,then dqFXq is a vector in TF (q)N for each q ∈M . If F is a diffeomorphism, we can perform

this construction for each y ∈ N , by setting q = F−1(y), and define a vector field, denotedby F∗X on N :

(F∗X)F (q) := dqFXq, ∀q ∈M,

or, equivalently,(F∗X)y = dF−1(y)FXF−1(y), ∀y ∈ N.

Definition 1.2.17 (Push-forward of a vector field). For a map of manifolds F : M ! N , theR-linear map F∗ : X(M)! X(N) is called the push-forward of vector fields.

Remark 1.2.18. Note that if G : N ! Z is another diffeomorphism between manifolds, weimmediately have

(G F )∗ = G∗F∗.

Moreover, we have (F∗)−1 = (F−1)∗.

If U ⊂ Rn is an open subset, X a smooth vector field and f a smooth map, then we candefine

X(f) =

n∑

i=1

Xi ∂f

∂xi,

where on the right hand side we regard X as a map U ! Rn. Each Xi is given by a mapin C∞(M). Note that the map C∞(M) ! C∞(M), f 7! X(f), is R-linear and satisfies theLeibniz rule

X(fg) = X(f)g + fX(g), ∀f, g ∈ C∞(M).


This means, choosing local coordinates (xi) on M , we can write a vector field as

X =

n∑

i=1

Xi ∂

∂xi∈ X(M).

Definition 1.2.19 (Commutator of vector fields). One can define an anti-symmetric R-bilinear map on X(M) by

[ , ] : X(M) × X(M)! X(M)

(X,Y ) 7! [X,Y ] := XY − Y X,(1.2.7)

called the commutator of vector fields.

Exercise 1.2.20 (Jacobi identity). Show that for all X,Y,Z ∈ X(M)

[X, [Y,Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0.

Exercise 1.2.21 (Derivation). Show that [ , ] is a derivation, i.e. for X,Y ∈ X(M) andf ∈ C∞(M) we have

[X, fY ] = X(f)Y + f [X,Y ].

1.2.4. Flow of a vector field. To a vector field X ∈ X(M) we associate the ODE

(1.2.8) q = X(q).

Definition 1.2.22 (Integral curve). A solution of (1.2.8) is called an integral curve. It is givenby a path q : I ! M such that q(t) = X(q(t)) ∈ Tq(t)M for all t ∈ I. Here, I := [a, b] ⊂ R

denotes some interval.

Definition 1.2.23 (Maximal integral curve). An integral curve is called maximal if it is notthe restriction of a solution to a proper subset of its domain.

Definition 1.2.24 (Flow). Let M be a manifold and let X ∈ X(M) be a vector field. Theflow ΦXt of X is given as follows: For q ∈ M and t in a neighborhood of 0, ΦXt (q) is theunique solution at time t to the Cauchy problem with initial condition at q ∈M . Explicitly,

∂

∂tΦXt (q) = X(ΦXt (q))

ΦX0 (q) = q

Remark 1.2.25. One can then show that

(1.2.10) ΦXt+s(q) = ΦXt (ΦXs (q)), ∀q ∈M

and for all t, s ∈ R such that the flow is defined.

Definition 1.2.26 (Complete vector field). A vector field is called complete if all its integralcurves exist for all t ∈ R.

Definition 1.2.27 (Global flow). The flow of a complete vector field X ∈ X(M) is a diffeo-morphism ΦXt : M !M for all t ∈ R. We then call ΦXt the global flow of X.

Remark 1.2.28. The condition (1.2.10) for a global flow can then be written more compactlyas

ΦXt+s = ΦXt ΦXs


1.2.5. Cotangent bundle. If E is a vector bundle over M , then the union of the dualspaces E∗

q is also a vector bundle, called the dual bundle of E. Namely, let E∗ =⊔q∈M E∗

q .

We denote an element of E∗ as a pair (q, ω) with ω ∈ E∗q . We let π : E∗

! M to be such

that π(q, ω) = q (the projection onto the first factor). To an atlas (Uα, φα)α∈I of E we

associate the atlas (Uα, φα)α∈I of E∗ with Uα = π−1(Uα) =⊔q∈M E∗

q and

φα : Uα ! Rn × (Rr)∗

(q, ω ∈ E∗q ) 7! (φα(q), (Aα(q)

∗)−1ω),(1.2.11)

where we regard (Rr)∗ as the manifold Rr with its standard structure.

Definition 1.2.29 (Cotangent bundle). The dual bundle of the tangent bundle TM of amanifold M , denoted by T ∗M is called the cotangent bundle of M .

1.2.6. Differential 1-forms. Similarly as we have defined vector fields as sections ofthe tangent bundle, we can construct a differential 1-form to be a section of the cotangentbundle. We denote the space of 1-forms on a manifold M by

Ω1(M) := Γ(T ∗M).

Definition 1.2.30 (de Rham differential). The de Rham differential is defined as the map

d: C∞(M)! Ω1(M)

f 7! df(1.2.12)

which is R-linear and satisfies the Leibniz rule

d(fg) = dfg + fdg, ∀f, g ∈ C∞(M).

Note that a 1-form ω ∈ Γ(T ∗M) is dual to a vector field X ∈ Γ(TM), so there is a pairingbetween them. The pairing is often denoted by ιXω. The operation ι is often called thecontraction. If V ⊂ Rn is open, we can consider the differentials dxi of the coordinatefunctions xi. Note that we have

ι∂jdxi =

∂xi

∂xj= δij ,

where ∂j :=∂∂xj

. Thus we can write a 1-form as

ω =

n∑

i=1

ωidxi,

where ωi ∈ C∞(M) are uniquely determined functions for 1 ≤ i ≤ n. If f ∈ C∞(M), then

df =n∑

i=1

∂ifdxi.

One can define a new notion of derivative by using the notion of a flow of a vector field. Wedefine the Lie derivative of a 1-form by

LXω :=∂

∂t

∣∣∣∣t=0

(ΦX−t)∗ω =∂

∂t

∣∣∣∣t=0

(ΦXt )∗ω, ∀X ∈ X(M),∀ω ∈ Ω1(M).


1.3. Tensor fields

1.3.1. Tensor bundle. Let V be a vector space over a field K. We define the k-thtensor power by

V ⊗k := V ⊗ · · · ⊗ V︸︷︷︸k times

.

By convention we have V ⊗0 := K. Moreover, note that dimV ⊗k = (dimV )k.

Definition 1.3.1 (Tensor). We call an element of V ⊗k a tensor of order k.

Let (ei)i∈I be a basis of V . Then (ei1 ⊗ · · · ⊗ eik)i1,...,ik∈I is a basis of V ⊗k and a tensor T oforder k can be uniquely written by

T =∑

i1,...,ik∈I

T i1···ikei1 ⊗ · · · ⊗ eik , T i1···ik ∈ K,∀i1, . . . , ik ∈ I.

Definition 1.3.2 (Tensor algebra). The tensor algebra of a vector space V is defined as

T (V ) :=

∞⊕

k=0

V ⊗k.

Definition 1.3.3 (Tensor of type (k, s)). We define a tensor of type (k, s) for a vector spaceV as an element of

T ks (V ) := V ⊗k ⊗ (V ∗)⊗s.

Remark 1.3.4. Tensors of type (0, s) are called covariant tensors of order s and tensors oftype (k, 0) are called contravariant of order k.

If we pick a basis (ei)i∈I of V and consider the dual basis (ej)j∈I of V∗. Then we get a basis

of T ks (V ) as

(ei1 ⊗ · · · ⊗ eik ⊗ ej1 ⊗ · · · ⊗ ejs)i1,...,ik,j1,...,js∈I .

A tensor of type (k, s) can then be uniquely written as

T =∑

i1,...,ik,j1,...,js∈I

T i1···ikj1...jsei1 ⊗ · · · ⊗ eik ⊗ ej1 ⊗ · · · ⊗ ejs .

Definition 1.3.5 (Tensor bundle). If E is a vector bundle over M , we define T ks (E) as the

vector bundle whose fiber at q is T ks (Eq). Namely, to an adapted atlas (Uα, φα)α∈I of E

over the trivializing atlas (Uα, φα)α∈I of M , we associate the atlas (Uα, φα)α∈I of T ks (E)

with Uα = π−1(Uα) =⊔q∈Uα

T ks (Eq) and

φα : Uα ! Rn × T ks (Rr)

(q, ω ∈ T ks (Eq)) 7! (φα(q), (Aα(q))ksω),

(1.3.1)

where we identify T ks (Rr) with Rr(k+s).

We have the transition maps

φαβ(x, u) = (φαβ(x), (Aαβ(φ−1α (x)))ksu).

1.3. TENSOR FIELDS 21

1.3.2. Multivector fields and differential s-forms. The important case is whenE = TM is the tangent bundle of a n-dimensional manifold M . Denote by Altks the tensorbundle induced by the alternating tensor product (wedge product) denoted by ∧. Then acontravariant tensor field of order k is also called a multivector field and a covariant tensorfield of order s is called a differential s-form. In particular, the space of multivector fields oforder k on an n-dimensional manifold M is given by

Xk(M) := Γ(Altk0(TM)) = Γ

(k∧TM

).

Choosing local coordinates (xi) on M we can represent an element X ∈ Xk(M) as

X =∑

1≤i1<···<ik≤n

Xi1···ik∂i1 ∧ · · · ∧ ∂ik .

The space of differential s-forms is then given by

Ωs(M) := Γ(Alt0s(TM)) = Γ

(s∧T ∗M

).

Choosing local coordinates (xi) on M we can represent an element ω ∈ Ωs(M) as

ω =∑

1≤i1<···<is≤n

ωi1···isdxi1 ∧ · · · ∧ dxis .

We can define a product ∧ between differential forms. For ω ∈ Ωs(M), η ∈ Ωℓ(M) given by

ω ∧ η =∑

1≤i1<···<is≤n

∑

1≤j1<···<jℓ≤n

ωi1···isηj1···jℓdxi1 ∧ · · · ∧ dxis ∧ dxj1 ∧ · · · ∧ dxjℓ.

Note that is s+ ℓ > 0, then ω ∧ η = 0.

Exercise 1.3.6. Show that if ω ∈ Ωs(M) and η ∈ Ωℓ(M), then

ω ∧ η = (−1)s+ℓη ∧ ω.One can extend the de Rham differential to general s-forms

d: Ωs(M)! Ωs+1(M)

ω 7! dω(1.3.2)

and obtain a sequence(1.3.3)

0! C∞(M)d−! Ω1(M)

d−! Ω2(M)

d−! · · · d

−! Ωs(M)d−! Ωs+1(M)

d−! · · · d

−! Ωn(M)! 0

This sequence is called de Rham complex. Note that by construction Ω0(M) = C∞(M). Inparticular, for a differential form ω =

∑1≤i1<···<is≤n

ωi1···isdxi1 ∧ · · · ∧ dxis ∈ Ωs(M) we get

dω =∑

1≤i1<···<is≤n

n∑

j=1

∂jωi1···isdxj ∧ dxi1 ∧ · · · ∧ dxis .

Exercise 1.3.7. Show that d2 = 0. Hint: use that derivatives commute, i.e. ∂∂xi

∂∂xj

=∂∂xj

∂∂xi

.


Exercise 1.3.8. Show that if ω ∈ Ωs(M) and η ∈ Ωℓ(M) then

d(ω ∧ η) = dω ∧ η + (−1)sω ∧ dη.

We can extend the Lie derivative and the contraction to any differential s-forms.

Definition 1.3.9 (Lie derivative). We define the Lie derivative of an s-form ω with respectto a vector field X as

LXω := limt!0

(ΦXt )∗ω − ω

t,

where ΦXt denotes the flow of X at time t. Explicitly, we have

(1.3.4) LXω =∑

1≤i1<...<is≤n

X(ωi1···is)dxi1 ∧ · · · ∧ dxis+

+∑

1≤i1<...<is≤n

∑

1≤k,r≤n

(−1)k−1ωi1···is∂rXikdxr ∧ dxi1 ∧ · · · ∧ dxik ∧ · · · ∧ dxis ,

where means that this element is omitted.

Exercise 1.3.10 (Properties for Lie derivative). Let X,Y ∈ X(M). Show that

• LXf = X(f), ∀f ∈ C∞(M),• LX(ω ∧ η) = LXω ∧ η + ω ∧ LXη, ∀ω ∈ Ωs(M),∀η ∈ Ωℓ(M),• LXdω = dLXω, ∀ω ∈ Ωs(M),• LXLY ω − LY LXω = L[X,Y ]ω, ∀ω ∈ Ωs(M),

Definition 1.3.11 (Contraction). The contraction of a vector field X with a differentials-form ω is the differential (s− 1)-form given by

ιXω =∑

1≤i1<...<is≤n

n∑

k=1

(−1)kωi1...isXikdxi1 ∧ · · · ∧ dxik ∧ · · · ∧ dxis .

Remark 1.3.12. If ω ∈ Ω0(M), i.e. s = 0, then for any X ∈ X(M) we get that ιXω isautomatically zero.

Exercise 1.3.13. Let X ∈ X(M) and let ω ∈ Ωs(M), η ∈ Ωℓ(M). Show that

ιX(ω ∧ η) = ιXω ∧ η + (−1)sω ∧ ιXη.Moreover, show that if Y ∈ X(M) is another vector field, we have

ιX ιY ω = −ιY ιXω,and

ιXLY ω − LY ιXω = ι[X,Y ]ω.

Theorem 1.3.14 (Cartan’s magic formula). Let M be a manifold. Let ω ∈ Ωs(M) and letX ∈ X(M). Then

(1.3.5) LXω = ιXdω + dιXω.

Exercise 1.3.15. Prove Theorem 1.3.14.

1.4. INTEGRATION ON MANIFOLDS AND STOKES’ THEOREM 23

1.4. Integration on manifolds and Stokes’ theorem

1.4.1. Integration of densities. LetM be a manifold endowed with an atlas (Uα, φα)α∈I .The differential dxφαβ is a linear map

Txφα(Uα ∩ Uβ)! Tφαβ(x)φβ(Uα ∩ Uβ).Since the image of the charts are open subsets of Rn we can identify the tangent spaces withRn. Hence the linear map dxφαβ is canonically given by an n× n matrix. Let s ∈ R and

Aαβ(q) = |det dφα(q)φαβ|−s.Exercise 1.4.1. Show that this defines a line bundle |ΛM |s over M .

Definition 1.4.2 (s-density). A section of |ΛM |s is called an s-density.

Let σ ∈ Γ(|ΛM |s) be an s-density. We represent it in the chart (Uα, φα) by the smoothfunction σα. It satisfies the transition rule

(1.4.1) σβ(φαβ(x)) = |det dqφαβ |−sσα(x), ∀α, β ∈ I,∀x ∈ φα(Uα ∩ Uβ)For s = 1 one simply speaks of a density. Densities are the natural objects to integrate on amanifold. Let σ be a density on M and let (Uα, φα)α∈I be an atlas on M . One can showthat there always exists a finite partition of unity ρjj∈J subordinate to Uαα∈I , i.e. foreach j ∈ J we have an αj with suppρj ⊂ Uαj

. In fact, supp ρj is compact. Since φαjis

a homeomorphism, also φαj(supp ρj) is compact. The representation (ρjσ)αj

of the densityρjσ in the chart (Uα, φα)α∈I is smooth in φαj

(Uαj), so it is integrable on φαj

(supp ρj). Wedefine

(1.4.2)

∫

M ;(Uα,φα);ρjσ :=

∑

j∈J

∫

φαj(supp ρj)

(ρjσ)αjdnx,

where dnx denotes the Lebesgue measure on Rn (also denoted by dx1 · · · dxn).Lemma 1.4.3. The integral defined in (1.4.2) does not depend on the choice of atlas andpartition of unity.

Proof. Consider an atlas (Uα, φα)α∈I and a finite partition of unity ρjj∈J subordi-nate to it. From

∑j∈J ρj = 1, it follows that σ =

∑j∈J ρjσ, so we have

(1.4.3)

∫

M ;(Uα,φα);ρjσ =

∑

j∈J

∫

φαj(supp ρj)

(ρjσ)αjdnx =

∑

j∈J

∑

j∈J

∫

φαj(supp ρj)

(ρjρjσ)αjdnx,

where we have taken out the finite sum∑

j∈J . Observe that

(1.4.4) Sjj :=

∫

φαj(supp ρj)

(ρj ρjσ)αjdnx

=

∫

φαj(supp ρj∩supp ρj)

(ρj ρjσ)αjdnx

=

∫

φ−1αjαj

(φαj(supp ρj∩supp ρj))

(ρj ρjσ)αjdnx,


where φαjαjdenotes the transition map φαj

(Uαj) ! φαj

(Uαj). Since ρj ρjσ is a density, we

have

(ρj ρjσ)αj(x) = |det dxφαjαj

|(ρj ρjσ)αj(x),

with x := φαjαj(x) and x ∈ φαj

(supp ρj ∩ supp ρj). By change-of-variables we get

Sjj =

∫

φαj(supp ρj∩supp ρj)

(ρj ρjσ)αjdnx =

∫

φαj(supp ρj)

(ρj ρjσ)αjdnx.

Hence∑

j∈J

Sjj =

∫

φαj(supp ρj)

(ρjσ)αjdnx

and∫

M ;(Uα,φα);ρjσ =

∑

j∈J

∑

j∈J

Sjj =∑

j∈J

∫

φαj(supp ρj)

(ρjσ)αjdnx =

∫

M ;(Uα,φα);ρjσ.

We can drop the choice of atlas and partition of unity and simply define

(1.4.5)

∫

Mσ :=

∑

j∈J

∫

φαj(supp ρj)

(ρjσ)αjdnx

Exercise 1.4.4. Let M1 and M2 be disjoint open subsets of a manifold M such that M \(M1 ∪M2) has measure zero. Show that

∫

Mσ =

∫

M1

σ +

∫

M2

σ.

We can also pullback densities by diffeomorphisms F : M ! N . We define the pullback by

(F ∗σ)q := (dqF )∗σF (q), ∀q ∈M.

We can also formulate it in terms of coordinates. Let (Uα, φα)α∈I be an atlas of M and let(Vj , ψj)j∈J be an atlas of N . Let σj denote the representation of the density σ in theatlas of N and Fαj the representation of F with respect to the two atlases. Then we have

(F ∗σ)α(x) = |det dxFαj |sσj(Fαj(x)), ∀x ∈ φα(Uα).

Exercise 1.4.5. Show that F ∗ is linear and that

F ∗(σ1σ2) = F ∗σ1F∗σ2.

Exercise 1.4.6. Show that if F : M ! N is a diffeomorphism and σ is a density on M , then∫

Mσ =

∫

NF∗σ.


1.4.2. Integration of differential forms. Similarly as for densities we can define theintegral of a differential form for some manifold M as in (1.4.5). The difference is that wewant to consider the notion of orientation onM . This corresponds to the notion of an orientedatlas.

Definition 1.4.7. A diffeomorphism F : U ! V of open subsets of Rn is orientation preserv-ing (orientation reversing) with respect to the standard orientation if and only if detF > 0(detF < 0).

Definition 1.4.8. We call an atlas oriented if all its transition maps are orientation preserv-ing.

At first, one defines the notion of an orientable manifold.

Definition 1.4.9 (Top form). A differential n-form on an n-dimensional manifold is calleda top form. We will denote the space of top forms on a manifold M by Ωtop(M) withoutmentioning the dimension of M .

Definition 1.4.10 (Volume form). A volume form on a manifold M is a nowhere vanishingtop form.

Definition 1.4.11 (Orientable manifold). A manifold M is called orientable if it admits avolume form.

Let (Uα, φα)α∈I be an atlas of a manifold M . Consider a volume form v ∈ Ωtop(M) whoserepresentation in the given atlas is denoted by vα. Let vα = vαdx

1 ∧ · · · ∧ dxn. Then thefunctions vα transform as

vα(x) = det dxφαβvβ(φαβ(x)), ∀α, β ∈ I,∀x ∈ φα(Uα ∩ Uβ).This is almost the same as the transformation rule (1.4.1) for densities. To fix this, we cantake absolute values of the representations vα. We define the absolute value |v| as the densitywith representations |vα|. Moreover, we want to restrict everything to top forms that do notchange sign (at least locally). Choosing a volume form v onM , we can define a C∞(M)-linearisomorphism

φv : Ωtop(M)! Dens(M)

as follows: since v is nowhere vanishing, for every top form ω there is a uniquely definedfunction f such that ω = fv; the corresponding density is then defined to be f |v|. Formally,we may write

φvω = ω|v|v.

Two volume forms v1 and v2 yield the same isomorphism, i.e. φv1 = φv2 , if and only if thereis a positive function g such that v1 = gv2.

Exercise 1.4.12. Show that this defines an equivalence relation on the set of volume formson M .

Definition 1.4.13 (Orientation). An equivalence class of volume forms on an orientablemanifold M is called an orientation. An orientable manifold with a choice of orientation iscalled oriented.


We denote by [v] an orientation, by (M, [v]) the corresponding oriented manifold and by φ[v]the isomorphism given by φv for any v ∈ [v]. If M admits a partition of unity, we can definethe integral of a top form ω ∈ Ωtop(M) by

∫

(M,[v])ω :=

∫

Mφ[v]ω,

where we use the already defined integration of densities. More explicitly, for an atlas(Uα, φα)α∈I we have the representations ωα = ωαd

nx, for uniquely defined maps ωα, andwe get

(φ[v]ω)α = ωα|dnx|.Hence, the integral of ω on an oriented manifold M is given by

(1.4.6)

∫

(M,[v])ω =

∑

j∈J

∫

φαj(supp ρj)

(ρj)αjωαj

dnx,

where (Uα, φα)α∈I is an oriented atlas corresponding1 to [v] and ρjj∈J is a partitionof unity subordinate to Uαα∈I . If the identification of differential forms and densities onφαj

(Uαj) is understood, then we can write

∫

(M,[v])ω =

∑

j∈J

∫

φαj(supp ρj)

(ρjω)αj.

Lemma 1.4.14. Let (Uα, φα)α∈I be an oriented atlas of (M, [v]) and ρjj∈J a partition ofunity subordinate to it. Let ω ∈ Ωtop(M) with suppω ⊂ Uαk

for some k ∈ J . Then∫

(M,[v])ω =

∫

φαk(Uαk

)ωαk

.

Proof. We have∫

(M,[v])ω =

∑

j∈J

∫

φαj(supp ρj)

(ρjω)αj=∑

j∈J

∫

φαj(supp ρj∩suppω)

(ρjω)αj

=∑

j∈J

∫

φαj(Uαj

∩Uαk)(ρjω)αj

=∑

j∈J

∫

φαj(Uαj

∩Uαk)(ρjω)αk

=∑

j∈J

∫

φαk(Uαk

)(ρjω)αk

=

∫

φαk(Uαk

)

∑

j∈J

(ρjω)αk=

∫

φαk(Uαk

)ωαk

.

Lemma 1.4.15. A connected oriented manifold admits two orientations.

Exercise 1.4.16. Prove Lemma 1.4.15.

Definition 1.4.17 (Orientation preserving/reversing). A diffeomorphism F of connectedoriented manifolds (M, [vM ]) and (N, [vN ]) is called orientation preserving if F ∗[vN ] = [vM ]and orientation reversing if F ∗[vN ] = −[vM ].

1This means that it is an atlas in which any v ∈ [v] is represented by a positive volume form.


Proposition 1.4.18 (Change of variables). let (M, [vM ]) and (N, [vN ]) be connected orientedmanifolds, F : M ! N a diffeomorphism and ω a top form on N . Then∫

(M,[vM ])F ∗ω = ±

∫

(N,[vN ])ω,

with plus sign if F is orientation preserving and the minus sign if F is orientation reversing.

Exercise 1.4.19. Prove Proposition 1.4.18.

Typically, the chosen orientation is understood, so one simply writes∫

Mω.

1.4.3. Stokes’ theorem. We will denote the space of compactly supported differentials-forms on a manifold M by Ωsc(M). Moreover we define the n-dimensional upper half-spaceby

(1.4.7) Hn := (x1, . . . , xn) ∈ Rn | xn ≥ 0.The boundary of the upper half-space is given by

∂Hn = (x1, . . . , xn) ∈ Rn | xn = 0.On the boundary, we can take the orientation induced by the outward pointing vector field−∂n, i.e.

[i∗ι−∂ndnx] = (−1)n[dx1 ∧ · · · ∧ dxn−1],

where i : ∂Hn! Hn denotes the inclusion.

Lemma 1.4.20. Let ω ∈ Ωn−1c (Hn). Then, using the orientations defined as before, we get

∫

Hn

dω =

∫

∂Hn

ω.

Proof. We write

ω =

n∑

j=1

(−1)j−1ωjdx1 ∧ · · · ∧ dxj ∧ · · · ∧ dxn.

The components ωj are then related to the components ωi1···in by a sign. Then

dω =

n∑

j=1

∂jωjdnx.

Using the standard orientation, we get∫

Hn

dω =

n∑

j=1

∫

Hn

∂jωjdnx.

We use Fubini’s theorem to integrate the j-th term along the j-th axis. Then, since ω hascompact support, we get ∫ +∞

−∞∂jω

jdxj = 0, ∀j < n,


but for j = n we have ∫ +∞

0∂nω

ndxn = −ωn∣∣xn=0

.

Hence, we get ∫

Hn

dω = −∫

∂Hn

ωndn−1x.

On the other hand, we have

i∗ω = (−1)n−1ωn∣∣xn=0

dx1 ∧ · · · ∧ dxn−1.

Thus, using the orientation of ∂Hn as before, we get∫

∂Hn

ω = −∫

∂Hn

ωndn−1x,

which concludes the proof.

Similarly, we get the following lemma.

Lemma 1.4.21. Let ω ∈ Ωn−1c (Rn). Then∫

Rn

dω = 0.

Definition 1.4.22 (Manifold with boundary). An n-dimensional manifold with boundary isan equivalence class of atlases whose charts take values in Hn.

Remark 1.4.23. One considers Hn as a topological space with topology induced from Rn.

Remark 1.4.24. Let M be a manifold with boundary. For any q ∈ M one can show that ifthere is a chart map sending q to an interior point of Hn, then any chart map will send it toan interior point of Hn. On the other hand, if there is a chart map sending q to a boundarypoint of Hn, then any chart map will send q to a boundary point of Hn. Hence, one caninduce a manifold structure on the interior points M and boundary points ∂M of M out ofthe manifold structure of M , such that dim M = dimM = dim ∂M +1, by restricting atlasesof M . The manifold ∂M is called the boundary of M .

Theorem 1.4.25 (Stokes). Let M be an n-dimensional oriented manifold with boundary andlet ω ∈ Ωn−1

c (M). Then

(1.4.8)

∫

Mdω =

∫

∂Mω

where we use the induced orientation on ∂M .

Remark 1.4.26. If M has no boundary, we get∫M dω = 0.

Remark 1.4.27. Theorem 1.4.25 was never officially proved by Stokes but appeared (insome version) in Maxwell’s book on electrodynamics from 1873 [Max73] where he mentionsin a footnote that the idea comes from Stokes who used this theorem in the Smith’s PrizeExamination of 1854. This is the reason why we call it Stokes’ theorem today. A first proofof this theorem was given by Hermann in 1861 [Her61] who does not mention Stokes at all.See also [Kat79] for some more historical facts on Stokes’ theorem.

1.5. DE RHAM’S THEOREM 29

Proof of Theorem 1.4.25. Let (Uα, φα)α∈I be an orientable atlas ofM correspond-ing to the given orientation and let ρjj∈J be a partition of unity subordinate to it. First,observe that

dω = d

∑

j∈J

ρjω

=

∑

j∈J

d(ρjω).

Note that supp (ρjω) ⊂ Uαjand, by Lemma 1.4.14, we have∫

Md(ρjω) =

∫

φαj(Uαj

)d(ρjω)αj

.

If φαj(Uαj

) is contained in the interior of Hn, then we regard d(ρjω)αjas a compactly

supported top form on Rn by extending it by zero outside of its support. Hence, by Lemma1.4.21, we get ∫

φαj(Uαj

)d(ρjω)αj

= 0.

Otherwise, we regard d(ρjω)αjas a compactly supported top form on Hn by again extending

by zero outside of its support. Hence, by Lemma 1.4.20, we get∫

φαj(Uαj

)d(ρjω)αj

=

∫

∂(φαj(Uαj

))(ρjω)αj

.

Note that ∂(φαj(Uαj

)) = φαj(∂Uαj

) by definition and both are oriented by outward pointingvectors. Thus, again by Lemma 1.4.14, we get

∫

Md(ρjω) =

∫

∂Mρjω.

Summing over j yields the result.

1.5. de Rham’s theorem

1.5.1. Singular homology.

Definition 1.5.1 (p-simplex). The standard p-simplex is the closed subset given by

∆p :=

(x1, . . . , xp) ∈ Rp

∣∣∣∣p∑

i=1

xi ≤ 1, xi ≥ 0∀i

⊂ Rp.

The interior of ∆p is a p-dimensional manifold. A smooth differential form on ∆p is bydefinition the restriction to ∆p of a smooth differential form defined on an open neighborhoodof ∆p in Rp. Let ω ∈ Ωp−1(∆p). Explicitly, let

ω =

p∑

j=1

ωjdx1 ∧ · · · ∧ dxj ∧ · · · ∧ dxp.

Then

dω =

p∑

j=1

(−1)j+1∂jωjdpx


and ∫

∆p

dω =

p∑

j=1

(−1)j+1

∫

∆p

∂jωjdpx.

Using Fubini’s theorem and the fundamental theorem of analysis, we get∫

∆p

∂jωjdxj = ωj

∣∣xj=1−

∑i=1i6=j

xi− ωj

∣∣xj=0

.

Hence

(1.5.1)

∫

∆p

dω =

p∑

j=1

(−1)j+1

∫

∆p∩∑pi=1 x

i=1ωjdx1 ∧ · · · ∧ dxj ∧ · · · ∧ dxp+

+

p∑

j=1

(−1)j∫

∆p∩xj=0ωjdx1 ∧ · · · ∧ dxj ∧ · · · ∧ dxp.

We can rewrite this in another way if we regard the faces on which we integrate as images of(p− 1)-simplices. Namely, for i = 0, . . . , p, we define smooth maps

kp−1i : ∆p−1

! ∆p,

by

kp−10 (a1, . . . , ap−1) =

(1−

p−1∑

i=1

ai, a1, . . . , ap−1

)

and

kp−1j (a1, . . . , ap−1) = (a1, . . . , aj−1, 0, aj , . . . , ap−1), ∀j > 0.

The j-th integral in the second line of (1.5.1) is just the integral on ∆p−1 of the pullback of

ω by kp−1j . In fact,

(1.5.2) (kp−1j )∗ω = (kp−1

j )∗p∑

i=1

ωidx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxp

= ωj(a1, . . . , aj−1, 0, aj , . . . , ap−1)dp−1a

We integrate over ∆p−1 with the standard orientation and rename variables xi = ai for i < jand xi = ai+1 for i > j. Note that the j-th integral is given by the integral over ∆p−1 of the

pullback of (−1)j+1ωjdx1 ∧ · · · ∧ dxj ∧ · · · ∧ dxp by kp−10 . In particular,

(1.5.3) (kp−10 )∗ωjdx1 ∧ · · · ∧ dxj ∧ · · · ∧ dxp

= −ωj(1−

∑

i

ai, a1, . . . , ap−1

)∑

i

dai ∧ da1 ∧ · · · ∧ daj−1 ∧ · · · ∧ dap−1

= (−1)j+1ωj

(1−

∑

i

ai, a1, . . . , ap−1

)dp−1a.

1.5. DE RHAM’S THEOREM 31

Summing everything up, we get Stokes’ theorem for a simplex :

(1.5.4)

∫

∆p

dω =

p∑

j=0

(−1)j∫

∆p−1

(kp−1j )∗ω,

where the j = 0 term corresponds to the whole sum in the first line of (1.5.1) and each otherterm corresponds to a term in the second line. Consider a map

σ : ∆p!M,

where M is a manifold. For a p-form ω on M , we define∫

σω :=

∫

∆p

σ∗ω.

If we define σj := σ kp−1j : ∆p−1

!M , then, by (1.5.4), we get

∫

σdω =

p∑

j=0

(−1)j∫

σjω.

Definition 1.5.2 (p-chains). A p-chain with real coefficients in a manifold M is a finitelinear combination

∑k akσk, where ak ∈ R for all k, of maps σk : ∆

p! M . If ω is a p-form

on M , we define

(1.5.5)

∫∑

k akσk

ω :=∑

k

ak

∫

σk

ω.

Theorem 1.5.3 (Stokes’ theorem for chains). We have∫

σdω =

∫

∂σω

where

∂σ :=

p∑

j=0

(−1)jσj .

Let Ωp(M,R) denote the vector space of p-chains in M with real coefficients and extend ∂ toit by linearity. Then we get that ∂ is an endomorphism of degree −1 for the graded vectorspace

Ω•(M,R) :=∞⊕

j=0

Ωj(M,R),

i.e. a map for all 1 ≤ p ≤ dimM we have

∂ : Ωp(M,R)! Ωp−1(M,R)

Exercise 1.5.4. Show that ∂2 := ∂ ∂ = 0.

For σ ∈ Ωp(M,R) and ω ∈ Ωp(M) we can define

(1.5.6) 〈σ, ω〉 :=∫

σω.


Exercise 1.5.5. Show that 〈 , 〉 as defined in (5.3.6) is a bilinear map Ωp(M,R)×Ωp(M)!R.

By Stokes’ theorem for chains we get

〈σ,dω〉 = 〈∂σ, ω〉.

In particular we have a sequence

0 Ωn(M,R) · · · Ωp(M,R) · · · Ω0(M,R) 0∂ ∂ ∂ ∂

If we consider a map F : M ! N between manifolds, it induces a graded linear map

F∗ : Ω•(M,R)! Ω•(N,R),

σ 7! F σ.(1.5.7)

Hence, if dimM = dimN = n, we get a chain complex

0 Ωn(M,R) · · · Ωp(M,R) Ωp−1(M,R) · · · Ω0(M,R) 0

0 Ωn(N,R) · · · Ωp(N,R) Ωp−1(N,R) · · · Ω0(N,R) 0

∂

F∗

∂

F∗

∂ ∂

F∗

∂

F∗

∂ ∂ ∂ ∂ ∂

Exercise 1.5.6. Show that each square of the diagram commutes, i.e. ∂ F∗ = F∗ ∂.

Definition 1.5.7 (Singular homology groups). The p-th singular homology group on a man-ifold M with real coefficents is given by

Hp(M,R) := ker ∂(p)/im ∂(p+1),

where ∂(p) : Ωp(M,R)! Ωp−1(M,R).

Remark 1.5.8. Elements of ker ∂(p) := σ ∈ Ωp(M,R) | ∂(p)σ = 0 are usually called p-cyclesand elements of im ∂(p) := σ ∈ Ωp(M,R) | ∃τ ∈ Ωp+1(M,R), σ = ∂(p+1)τ are usually calledp-boundaries.

Exercise 1.5.9. Using Exercise 1.5.6, show that F∗ descends to a graded linear map

F∗ : H•(M,R)! H•(N,R).

Thus we get a chain complex on the level of homology

0 Hn(M,R) · · · Hp(M,R) Hp−1(M,R) · · · H0(M,R) 0

0 Hn(N,R) · · · Hp(N,R) Hp−1(N,R) · · · H0(N,R) 0

F∗ F∗ F∗ F∗

Exercise 1.5.10. Show that 〈F∗σ, ω〉 = 〈σ, F ∗ω〉 for all σ ∈ Ω•(M,R) and all ω ∈ Ω•(N).

1.6. DISTRIBUTIONS AND FROBENIUS’ THEOREM 33

1.5.2. de Rham cohomology and de Rham’s theorem. LetM be an n-dimensionalmanifold and recall its de Rham complex(1.5.8)

0! C∞(M)d−! Ω1(M)

d−! Ω2(M)

d−! · · · d

−! Ωs(M)d−! Ωs+1(M)

d−! · · · d

−! Ωn(M)! 0

Definition 1.5.11 (de Rham cohomology groups). We define the s-th de Rham cohomologygroup by

Hs(M) := ker d(s)/im d(s−1),

where d(s) : Ωs(M)! Ωs+1(M).

Remark 1.5.12. we call element of ker d(s) := ω ∈ Ωs(M) | d(s)ω = 0 closed forms and

elements of imd(s) := ω ∈ Ωs(M) | ∃η ∈ Ωs−1(M), ω = d(s−1)η exact forms.

Exercise 1.5.13. Using Stokes’ theorem for chains, show that the bilinear map 〈 , 〉 definedin (5.3.6) descends to a bilinear map

(1.5.9) Hp(M,R)×Hp(M)! R.

Theorem 1.5.14 (de Rham[Rha31]). The bilinear map (1.5.9) is nondegenerate. In partic-ular,

(Hp(M,R))∗ ∼= Hp(M), ∀p.

1.6. Distributions and Frobenius’ theorem

1.6.1. Plane distributions.

Definition 1.6.1 (Plane distribution). A k-plane distribution D, or simply a k-distributionor just a distribution, on a smooth n-dimensional manifold M is a collection Dqq∈M oflinear k-dimensional subspaces Dq ∈ TqM for all q ∈M .

Remark 1.6.2. The number k is called the rank of the distribution. Obviously, we wantk ≤ n.

Definition 1.6.3 (Distribution). A k-distribution D on M is called smooth if every q ∈ Mhas an open neighborhood U and smooth vector fields X1, . . . ,Xk defined on U such that

Dx = span(X1)x, . . . , (Xk)x, ∀x ∈ U.

The vector fields X1, . . . ,Xk are also called (local) generators for D on U .

Remark 1.6.4. In the case when U = M , we speak of global generators. The existence oflocal generators is required since there are certain interesting distributions which do not haveglobal generators.

We say that a vector field X ∈ X(M) is a tangent to a distribution D if Xq ∈ Dq for allq ∈M . Note that any linear combination of tangent vector fields are again tangent.

Definition 1.6.5 (Involutive distribution). A smooth distributionD onM is called involutiveif Γ(D) is a Lie subalgebra of X(M), i.e. when [X,Y ] ∈ Γ(D) for all X,Y ∈ Γ(D).


Remark 1.6.6. A distribution generated by vector fields X1, . . . ,Xk is involutive if and onlyif [Xi,Xj ] is a linear combination over C∞(M) of the generators. The only-if side follows fromthe definition. The if-implication can be obtained by observing that a vector field tangent tothe distribution is necessarily a linear combination of the generators. Moreover, we have

∑

i

fiXi,∑

j

gjYj

=

∑

i,j

((fiXi(gj)− giXi(fj)

)Xj + figj[Xi,Xj ]

).

Remark 1.6.7. Note that the previous discussion directly implies that any distribution ofrank 1 is involutive. Locally, it is generated by a single vector field X and by the skew-symmetry of the Lie bracket, we have [X,X] = 0.

Definition 1.6.8 (Push-forward of a distribution). Let D be a distribution on M and letF : M ! N be a diffeomorphism. We define the push-forward F∗D of D by

(F∗D)y := dF−1(y)DF−1(y), ∀y ∈ N.

1.6.2. Frobenius’ theorem.

Definition 1.6.9 (Integral manifold). An immersion ψ : N !M with N connected is calledan integral manifold for a distribution D on M if

dnψ(TnN) = Dψ(n), ∀n ∈ N.Remark 1.6.10. An integral manifold that is not a proper restriction of an integral manifoldis called maximal.

Remark 1.6.11. If ψ is an embedding, restricting ψ : N ! ψ(N) to its image allows us torewrite the above condition as

ψ∗(TN) = D∣∣ψ(N)

,

where the fact that D can be restricted to ψ(N), i.e. Dq ∈ Tqψ(N) for all q ∈ ψ(N), is partof the condition.

Definition 1.6.12 (Integrable distribution). A smooth distribution D on M is called inte-grable if for all q ∈M there is an integral manifold D passing through q.

Lemma 1.6.13. If a distribution D is integrable, then D is involutive.

Proof. For all q ∈M , we can find an integral manifold ψ : N !M with q ∈ ψ(N). If Xand Y are tangent to D, in a neighborhood of q in ψ(N) we can write them as push-forwards

of vector fields X and Y on N . Since the push-forward preserves the Lie bracket and TN is

involutive, we see that in this neighborhood Z := [X,Y ] is the push-forward of [X, Y ] andhence tangent to D. Note also that this is indeed the Lie bracket of X and Y , as they do nothave components transverse to ψ(N) by definition. We can compute Z by this procedure ateach point in M , which shows that D is involutive.

Proposition 1.6.14. Let X be a vector field on a Hausdorff manifold M . Let q ∈ M be apoint such that Xq 6= 0. Then there is a chart (U, φU ) with U ∋ q such that (φU )∗X

∣∣U

is theconstant vector field (1, 0, . . . , 0). As a consequence, if γ is an integral curve of X passing

through U , then φU γ is of the form x ∈ φU (U) | x1(t) = x10 + t; xj(t) = xj0, j > 1 wherethe xi0s are constants.

1.6. DISTRIBUTIONS AND FROBENIUS’ THEOREM 35

Theorem 1.6.15 (Frobenius[Fro77]). Let D be an involutive k-distribution on a smooth,Hausdorff n-dimensional manifold M . Then each point q ∈ M has a chart neighborhood(U, φ) such that φ∗D = span

∂∂x1 , . . . ,

∂∂xk

, where x1, . . . , xk are coordinates on φ(U).

Corollary 1.6.16. On a smooth, Hausdorff manifold a smooth distribution is involutive ifand only if it is integrable.

Proof of Theorem 1.6.15. We will use induction on the rank k of the distribution.For k = 1, This is the content of Proposition 1.6.14. We assume that we have proved thetheorem for rank k−1. Let (X1, . . . ,Xk) be generators of the distribution in a neighborhoodof q. In particular, they are all not vanishing at q. By Proposition 1.6.14, we can find a chartneighborhood (V, χ) of q with χ(q) = 0 and χ∗X1 =

∂∂y1

, where y1, . . . , yn are coordinates on

χ(V ). We can define new generators of χ∗D by

Y1 := χ∗X1 =∂

∂y1

and for i > 1Yi := χ∗Xi − (χ∗Xi(y

1))χ∗X1.

For i > 1 we have Yi(y1) = 0 and thus, for i, j > 1, we have [Yi, Yj](y

1) = 0. This means thatthe expansion of [Yi, Yj ] in the Yℓs does not contain Y1. Hence, the distribution D

′ defined onS := y ∈ χ(V ) | y1 = 0 as the span of Y2, . . . , Yk is involutive. By the induction assumption,we can find a neighborhood U of 0 in S and a diffeomorphism τ such that τ∗Yi =

∂∂wi , for

i = 2, . . . , w, where w2, . . . , wn are coordinates on τ(U). Let U := U × (−ε, ε) for some

ε > 0 such that U ⊂ χ(V ). We then have the projection π : U ! U . Finally, consider thediffeomorphism

τ : U ! τ(U)× (−ε, ε),(u, y1) 7! (τ(u), y1),

and write x1 = y1, xi = τ i(y2, . . . , yn) = wi for i > 1. We write Zi := τ∗Yi for i = 1, . . . , k

being the generators of the distribution D := τ∗χ∗D. Now since ∂xi

∂y1 is equal to one if i = 1

and zero otherwise, we get that Z1 =∂∂x1

. For i = 2, . . . , k and j > 1 we have

∂

∂x1(Zi(x

j)) = Z1(Zi(xj)) = [Z1, Zi](x

j) =

k∑

ℓ=2

cℓiZℓ(xj),

where the cℓi are functions that are guaranteed to exist by the involutivity of the distribution.For fixed j and fixed x2, . . . , xn we regard these identities as ODEs in the variable x1. Notethat, for i = 2, . . . , k and j > k, we have Zi(x

j) = 0 at x1 = 0 (since at x1 = 0 we haveZi = Yi). This means that Zi(x

j) = 0 for i = 2, . . . , k and j > k, is the unique solution with

this initial condition. These identities mean that D is the distribution spanned by the vectorfields ∂

∂x1 , . . . ,∂∂xk

.

CHAPTER 2

Symplectic Geometry

Symplectic geometry appears in the mathematical structure of the phase space M = R2n fora classical mechanical system. The dynamical information is usually encoded in a functionH ∈ C∞(M) called the Hamiltonian. In order to express the dynamics in terms of the flowlines, we want to extract a vector field XH ∈ X(M) out of H, i.e. consider a differentialequation with respect to the change of H, similarly as for the canonical equations of motionin Hamiltonian mechanics. This means that we want to consider a map ω : TM ! T ∗M , orequivalently an element of T ∗M ⊗ T ∗M , such that dH = ιXH

ω = ω(XH , ). Additionally,we want that the choice of XH for each H is unique in this way, which requires ω to benondegenerate. Moreover, we want that H doesn’t change along flow lines, i.e. dH(XH) = 0,meaning that ω(XH ,XH) = 0 and thus we require ω to be alternating, which means thatω has to be a 2-form. Note that this also implies that the underlying space has to be evendimensional since every skew-symmetric linear map for odd dimensions is singular. Finally,we want also that ω doesn’t change under flow lines. Mathematically, this is expressed as thevanishing of the Lie derivative LXH

ω = 0. Using Cartan’s magic formula (1.3.5), we get

LXHω = dιXH

ω + ιXHdω = d(dH) + ιXH

dω = dω(XH).

Hence, if we require dω(XH) = 0 for the vector field induced by different Hamiltonians, werequire ω to be closed, i.e. dω = 0. Such a 2-form is then called symplectic. In fact, it givesan area measure which induces a time-independent area by

A :=

∫

Σω

for a suitable surface Σ ⊂ M . We will generalize this concept to the definition of a volumeform on any 2n-dimensional manifold. This chapter is based on [Can08; Cat18; Arn78;Don96; DK90; MS95; EG98; DH82; Wei71; Wei77; Wei81; BGV92; AB84].

2.1. Symplectic manifolds

2.1.1. Symplectic form. Let V be an m-dimensional vector space over R and let

Ω: V × V ! R

be a bilinear map.

Definition 2.1.1 (Skew-symmetric). The map Ω is called skew-symmetric if Ω(u, v) =−Ω(v, u) for all u, v ∈ V .

37

38 2. SYMPLECTIC GEOMETRY

Theorem 2.1.2 (Standard form). Let Ω be a skew-symmetric bilinear map on V . Then thereis a basis u1, . . . , uk, e1, . . . , en, f1, . . . , fn of V such that

Ω(ui, v) = 0, 1 ≤ i ≤ k,∀v ∈ V,(2.1.1)

Ω(ei, ej) = Ω(fi, fj) = 0, 1 ≤ i, j ≤ n(2.1.2)

Ω(ei, fj) = δij , 1 ≤ i, j ≤ n.(2.1.3)

Remark 2.1.3. The basis in Theorem 2.1.2 is not unique even if it is historically calledcanonical basis.

Proof of Theorem 2.1.2. Let U := u ∈ V | Ω(u, v) = 0, ∀v ∈ V . Choose a basisu1, . . . , uk of U and choose a complementary space W to U in V , i.e. such that

V =W ⊕ U.

Let e1 ∈ W be a nonzero element. Then there is f1 ∈ W such that Ω(e1, f1) 6= 0. Nowassume that Ω(e1, f1) = 1. Let W1 be the span of e1 and f1 and let

WΩ1 := w ∈W | Ω(w, v) = 0, ∀v ∈W1.

We can then show that W1 ∩WΩ1 = 0. Indeed, suppose that v = ae1 + bf1 ∈ W1 ∩WΩ

1 .Then 0 = Ω(v, e1) = −b and 0 = Ω(v, f1) = a which together implies that v = 0. Moreover,we have W = W1 ⊕WΩ

1 . Indeed, suppose that v ∈ W has Ω(v, e1) = c and Ω(v, f1) = d.Then v = (−cf1 + de1) + (v + cf1 − de1), where −cf1 + de1 ∈ W1 and v + cf1 − de1 ∈ WΩ

1 .Now let e2 ∈ WΩ

1 be a nonzero element. Then there is f2 ∈ WΩ1 such that Ω(e2, f2) 6= 0.

Now assume that Ω(e2, f2) = 1. Moreover, let W2 be given by the span of e2 and f2. Thisconstruction can be continued until some point since dimV <∞ and thus we obtain

V = U ⊕W1 ⊕ · · · ⊕Wn.

where all the summands are orthogonal with respect to Ω and where Wi has basis ei, fi withΩ(ei, fi) = 1.

Remark 2.1.4. Note that the dimension of the subspace U ⊂ V does not depend on thechoice of basis and hence is an invariant on (V,Ω). Since dimU + 2n = dimV , we get thatn is an invariant of (V,Ω). We call the number 2n the rank of Ω.

2.1.2. Symplectic vector spaces. Let V be an m-dimensional real vector space andlet Ω: V × V ! R be a bilinear form. Define the map Ω : V ! V ∗ to be the linear mapdefined by

Ω(v)(u) := Ω(v, u).

Note that ker Ω = U .

Definition 2.1.5 (Linear symplectic fom). A skew-symmetric bilinear form Ω is called sym-

plectic (or nondegenerate) if Ω is bijective, i.e. U = 0.Remark 2.1.6. We sometimes also call a symplectic form Ω a linear symplectic structure.

Definition 2.1.7 (Symplectic vector space). We call a vector space V endowed with a linearsymplectic structure Ω symplectic.

2.1. SYMPLECTIC MANIFOLDS 39

Exercise 2.1.8. Let Ω be a symplectic structure. Check that the map Ω is a bijection andthat dimU = 0 so dimV is even. Moreover, check that a symplectic vector space (V,Ω) hasa basis e1, . . . , en, f1, . . . , fn satisfying

Ω(ei, fj) = δij , Ω(ei, ej) = Ω(fi, fj) = 0.

We will call such a basis symplectic.

Definition 2.1.9 (Symplectic subspace). A subspace W ⊂ V is called symplectic if Ω∣∣W

isnondegenerate.

Exercise 2.1.10. Show that the subspace given by the span of e1 and f1 is symplectic.

Definition 2.1.11 (Isotropic subspace). A subspace W ⊂ V is called isotropic if Ω∣∣W

= 0.

Exercise 2.1.12. Show that the subspace given by the span of e1 and e2 is isotropic.

Definition 2.1.13 (Symplectic orthogonal). LetW ⊂ V be a subspace of a symplectic vectorspace (V,Ω). The symplectic orthogonal of W is defined as

WΩ := v ∈ V | Ω(v, u) = 0, ∀u ∈W.Exercise 2.1.14. Show that (WΩ)Ω =W .

Exercise 2.1.15. Show that a subspace W ⊂ V is isotropic if W ⊆ WΩ. Moreover, showthat if W is isotropic, then dimW ≤ 1

2 dimV .

Definition 2.1.16 (Coisotropic subspace). A subspace W ⊂ V of a symplectic vector space(V,Ω) is called coisotropic if

WΩ ⊆W.

Exercise 2.1.17. Show that every codimension 1 subspace W ⊂ V is coisotropic.

Definition 2.1.18 (Lagrangian subspace). An isotropic subspace W ⊂ V of a symplecticvector space (V,Ω) is called Lagrangian if it is maximal, i.e. dimW = 1

2 dimV .

Exercise 2.1.19. Show that a subspace W ⊂ V of a symplectic vector space (V,Ω) isLagrangian if and only if W is isotropic and coisotropic if and only if W =WΩ.

Definition 2.1.20 (Standard symplectic vector space). The standard symplectic vector spaceis defined as the vector space R2n endowed with the linear symplectic structure Ω0 definedsuch that the basis

e1 = (1, 0, . . . , 0), . . . , en = (0, . . . , 0, 1︸︷︷︸n

, 0, . . . , 0),

f1 = (0, . . . , 0, 1︸︷︷︸n+1

, 0, . . . , 0), . . . , fn = (0, . . . , 0, 1)(2.1.4)

is a symplectic basis.

Remark 2.1.21. We can extend Ω0 to other vectors by using its values on a basis andbilinearity.

Definition 2.1.22 (Linear symplectomorphism). A linear symplectomorphism ϕ between

symplectic vector spaces (V,Ω) and (V ′,Ω′) is a linear isomorphism ϕ : V∼−! V such that

ϕ∗Ω′ = Ω.


Remark 2.1.23. By definition we have

(ϕ∗Ω′)(u, v) := Ω′(ϕ(u), ϕ(v)).

Definition 2.1.24 (Symplectomorphic spaces). We call two symplectic vector spaces (V,Ω)and (V ′,Ω′) symplectomorphic if there exists a symplectomorphism between them.

Exercise 2.1.25. Show that the relation of being symplectomorphic defines an equivalencerelation in the set of all even-dimensional vector spaces. Moreover, show that every 2n-dimensional symplectic vector space (V,Ω) is symplectomorphic to the standard symplecticvector space (R2n,Ω0).

2.1.3. Symplectic manifolds. Let ω be a 2-form on a manifold M . Note that for eachpoint q ∈M , the map

ωq : TqM × TqM ! R

is skew-symmetric bilinear on the tangent space TqM .

Definition 2.1.26 (Symplectic form). A 2-form ω is called symplectic if ω is closed and ωqis symplectic for all q ∈M .

Definition 2.1.27 (Symplectic manifold). A symplectic manifold is a pair (M,ω) where Mis a manifold and ω is a symplectic form.

Example 2.1.28. Let M = R2n with coordinates x1, . . . , xn, y1, . . . , yn. Then one can checkthat the 2-form

ω0 :=n∑

i=1

dxi ∧ dyi

is symplectic and that the set(∂

∂x1

)

q

, . . . ,

(∂

∂xn

)

q

,

(∂

∂y1

)

q

, . . . ,

(∂

∂yn

)

q

defines a symplectic basis of TqM .

Example 2.1.29. Let M = Cn with coordinates z1, . . . , zn. Then one can check that the2-form

ω0 :=i

2

n∑

k=1

dzk ∧ dzk

is symplectic. This is similar to Example 2.1.28 by the identification Cn ∼= R2n and zk =xk + iyk.

2.1.4. Symplectomorphisms.

Definition 2.1.30 (Symplectomorphism). Let (M1, ω1) and (M2, ω2) be 2n-dimensional sym-plectic manifolds, and let ϕ : M1 ! M2 be a a diffeomorphism. Then ϕ is called a symplec-tomorphism if

ϕ∗ω2 = ω1.

Remark 2.1.31. Note that by definition we have

(ϕ∗ω2)q(u, v) = (ω2)ϕ(q)(dqϕ(u),dqϕ(v)), ∀u, v ∈ TqM.

2.2. THE COTANGENT BUNDLE AS A SYMPLECTIC MANIFOLD 41

The classification of symplectic manifolds up to symplectomorphisms is an interesting prob-lem. The next theorem takes care of this locally. In fact, as any n-dimensional manifold lookslocally like Rn, one can show that any 2n-dimensional symplectic manifold (M,ω) is locallysymplectomorphic to (R2n, ω0). In particular, the dimension is the only local invariant ofsymplectic manifolds up to symplectomorphisms.

Theorem 2.1.32 (Darboux[Dar82]). Let (M,ω) be a 2n-dimensional symplectic manifold andlet q ∈M . Then there is a coordinate chart (U , x1, . . . , xn, y1, . . . , yn) centered at q such thaton U we have

ω =

n∑

i=1

dxi ∧ dyi.

Definition 2.1.33 (Darboux chart). A local coordinate chart (U , x1, . . . , xn, y1, . . . , yn) iscalled a Darboux chart.

Remark 2.1.34. The proof of Theorem 2.1.32 will be given later as an exercise (Exercise2.5.11) for a simple application of another important theorem (Moser’s relative theorem).

2.2. The cotangent bundle as a symplectic manifold

Let X be an n-dimensional manifold and let M := T ∗X be its cotangent bundle. Considercoordinate charts (U , x1, . . . , xn) on X with xi : U ! R. Then at any x ∈ U we have a basisof T ∗

xX defined by the linear maps

dxx1, . . . ,dxxn.

In particular, if ξ ∈ T ∗xX, then ξ =

∑ni=1 ξidxxi for some ξ1, . . . , ξn ∈ R. Note that this

induces a map

T ∗U ! R2n

(x, ξ) 7! (x1, . . . , xn, ξ1, . . . , ξn).(2.2.1)

The chart (T ∗U , x1, . . . , xn, ξ1, . . . , ξn) is a coordinate chart of T ∗X. The transition functionson the overlaps are smooth; given two charts (U , x1, . . . , xn) and (U ′, x′1 . . . , x

′n) and x ∈ U∩U ′

with ξ ∈ T ∗xX, then

ξ =

n∑

i=1

ξidxxi =∑

i,j

ξi

(∂xi∂xj

)dxx

′j =

n∑

j=1

ξ′jdxx′j,

where ξ′j =∑

i ξi

(∂xi∂xj

)is smooth. Hence, T ∗X is a 2n-dimensional manifold.

2.2.1. Tautological and canonical forms in coordinates. Consider a coordinatechart (U , x1, . . . , xn) forX with associated cotangent coordinates (T ∗U , x1, . . . , xn, ξ1, . . . , ξn).Define a 2-form ω on T ∗U by

ω :=

n∑

i=1

dxi ∧ dξi.


we want to show that this expression is independent of the choice of coordinates. Indeed,consider the 1-form

α :=

n∑

i=1

ξidxi,

and note that ω = −dα. Let (U , x1, . . . , xn, ξ1, . . . , ξn) and (U ′, x′1, . . . , x′n, ξ

′1, . . . , ξ

′n) be two

coordinate charts on T ∗X. As we have seen, on the intersection U ∩ U ′ they are related by

ξ′j =∑

i ξi

(∂xi∂xj

). Since dx′j =

∑i

(∂x′j∂xi

)dxi, we get

α =∑

i

ξidxi =∑

j

ξ′jdx′j = α′.

Hence, since α is intrinsically defined, so is ω. This finishes the claim.

Definition 2.2.1 (Tautological form). The 1-form α is called tautological form.

Remark 2.2.2. The tautological form is sometimes also called Liouville 1-form and ω = −dαis often called canonical symplectic form.

2.2.2. Coordinate-free construction. Let use denote by

π : M := T ∗X ! X,

q = (x, ξ) 7! x(2.2.2)

be the natural projection for ξ ∈ T ∗xX. We define the tautological 1-form α pointwise as

αq = (dqπ)∗ξ ∈ T ∗

qM.

where (dqπ)∗ denotes the transpose of dqπ, i.e. (dqπ)

∗ξ = ξ dqπ. We have the three maps

π : M := T ∗X ! X,(2.2.3)

dqπ : TqM ! TxX,(2.2.4)

(dqπ)∗ : T ∗

xX ! T ∗qM.(2.2.5)

In fact, we have

αq(v) = ξ ((dqπ)v) , ∀v ∈ TqM.

The canonical symplectic form is then defined by

ω = −dα.

Exercise 2.2.3. Show that the tautological form α is uniquely characterized by the propertythat, for every 1-form µ : X ! T ∗X we have

µ∗α = µ.

2.3. LAGRANGIAN SUBMANIFOLDS 43

2.2.3. Symplectic volume. Let V be a vector space of dimension dimV < ∞. Anyskew-symmetric bilinear map Ω ∈ ∧2 V ∗ is of the form

Ω = e∗1 ∧ f∗1 + · · ·+ e∗n ∧ f∗n,where u∗1, . . . , u

∗k, e

∗1, . . . , e

∗n, f

∗1 , . . . , f

∗n is a basis of V ∗ dual to the standard basis. Here we

have set dimV = k+2n. If Ω is also nondegenerate, i.e. a symplectic form on a vector spaceV with dimV = 2n, then the n-th exterior power of

Ωn := Ω ∧ · · · ∧ Ω︸︷︷︸n

does not vanish.

Exercise 2.2.4. Show that this also holds for the n-th exterior power ωn of a symplecticform ω on a 2n-dimensional symplectic manifold (M,ω). Deduce that it defines a volumeform on M .

Exercise 2.2.4 shows that any symplectic manifold (M,ω) can be canonically oriented by thesymplectic structure.

Definition 2.2.5 (Symplectic volume). Let (M,ω) be a symplectic manifold. Then the form

ωn

n!

is called the symplectic volume of (M,ω).

Remark 2.2.6. The symplectic volume is sometimes also called Liouville volume.

Exercise 2.2.7. Show that if, conversely, a given 2-form Ω ∈ ∧2 V ∗ satisfies Ωn 6= 0, then Ωis symplectic.

Exercise 2.2.8. Let (M,ω) be a 2n-dimensional symplectic manifold. Show that, if Mis compact, the de Rham cohomology class [ωn] ∈ H2n(M) is non-zero. Hint: Use Stokes’theorem. Conclude then that [ω] is not exact and show that for n > 1, there are no symplecticstructures on the sphere S2n.

2.3. Lagrangian submanifolds

2.3.1. Lagrangian submanifolds of T ∗X.

Definition 2.3.1 (Lagrangian submanifold). Let (M,ω) be a 2n-dimensional symplecticmanifold. A submanifold L ⊂M is called Lagrangian if for any point q ∈ L we get that TqLis a Lagrangian subspace of TqM .

Remark 2.3.2. Recall that this is equivalent to say that L ⊂M is Lagrangian if and only ifωq∣∣TqL

= 0 and dimTqL = 12 dimTqM for all q ∈ L. Equivalently, if i : L ! M denotes the

inclusion of L into M , then L is Lagrangian if and only if i∗ω = 0 and dimL = 12 dimM .

Let X be an n-dimensional manifold and let M := T ∗X be its cotangent bundle. considercoordinates x1, . . . , xn on U ⊆ X with cotangent coordinates x1, . . . , xn, ξ1, . . . , ξn on T ∗U ,then the tautological 1-form on T ∗X is given by

α =∑

i

ξidxi


and the canonical 2-form on T ∗X is

ω = −dα =∑

i

dxi ∧ dξi.

2.3.2. Conormal bundle. Let S be any k-dimensional submanifold of X.

Definition 2.3.3 (Conormal space). The conormal space at x ∈ S is defined by

N∗xS := ξ ∈ T ∗

xX | ξ(v) = 0, ∀v ∈ TxS.Definition 2.3.4 (Conormal bundle). The conormal bundle of S is

N∗S = (x, ξ) ∈ T ∗X | x ∈ S, ξ ∈ N∗xS.

Exercise 2.3.5. Show that the conormal bundle N∗S is an n-dimensional submanifold ofT ∗X.

Proposition 2.3.6. Let i : N∗S ! T ∗X be the inclusion, and let α be the tautological 1-formon T ∗X. Then

i∗α = 0.

Definition 2.3.7 (Adapted coordinate chart). A coordinate chart (U , x1, . . . , xn) on X issaid to be adapted to a k-dimensional submanfiold S ⊂ X if S ∩ U is described by xk+1 =· · · = xn = 0.

Proof of Proposition 2.3.6. Let (U , x1, . . . , xn) be a coordinate system on X cen-tered at x ∈ S and adapted to S, so that U ∩ S is described by xk+1 = · · · = xn = 0. Let(T ∗U , x1, . . . , xn, ξ1, . . . , ξn) be the associated cotangent coordinate system. The submanifoldN∗S ∩ T ∗U is then described by

xk+1 = · · · = xn = 0,

ξ1 = · · · = ξk = 0.

Since α =∑

i ξidxi on T∗U , we conclude that for p ∈ N∗S we get

(i∗α)p = αp∣∣Tp(N∗S)

=∑

i>k

ξidxi

∣∣∣∣span

(∂

∂xi

)i≤k

= 0.

Definition 2.3.8 (Zero section). The zero section of T ∗X is defined by

X0 := (x, ξ) ∈ T ∗X | ξ = 0 ∈ T ∗xX.

Exercise 2.3.9. Let i0 : X0 ! T ∗X be the inclusion of the zero section and let ω = −dα bethe canonical symplectic form on T ∗X. Show that i∗0ω and X0 are Lagrangian submanifoldsof T ∗X.

Corollary 2.3.10. For any submanifold S ⊂ X, the conormal bundle N∗S is a Lagrangiansubmanifold of T ∗X.

Exercise 2.3.11. prove Corollary 2.3.10.

Remark 2.3.12. If S = x ⊂ X, then the conormal bundle is given by a cotangent fiberN∗S = T ∗

xX. If S = X, then the conormal bundle is the zero section of T ∗X, i.e. N∗S = X0.

2.3. LAGRANGIAN SUBMANIFOLDS 45

2.3.3. Applications to symplectomorphisms. Let (M1, ω1) and (M2, ω2) be two 2n-

dimensional symplectic manifolds. Given a diffeomorphism ϕ : M1∼−! M2, when is it a

symplectomorphism? Equivalently, when do we have

ϕ∗ω2 = ω1?

Consider two projection maps

M1 ×M2

M1 M2

pr1 pr2

ϕ

Then

ω := (pr1)∗ω1 + (pr2)

∗ω2

is a 2-form on M1 ×M2 which is closed:

dω = (pr1)∗ dω1︸︷︷︸

=0

+(pr2)∗ dω2︸︷︷︸

=0

= 0,

and symplectic:

ω2n =

(2n

n

)((pr1)

∗ω1)n ∧ ((pr2)

∗ω2)n 6= 0.

Note that for all λ1, λ2 ∈ R we get that

λ1(pr1)∗ω1 + λ2(pr)

∗ω2

is a symplectic form on M1 ×M2. The twisted product form on M1 ×M2 is obtained bytaking λ1 = 1 and λ2 = −1. Namely,

ω = (pr1)∗ω1 − (pr2)

∗ω2.

For a diffeomorphism ϕ : M1∼−!M2, we define its graph by

(2.3.1) Γϕ := Graphϕ = (q, ϕ(q)) | q ∈M1 ⊂M1 ×M2.

Remark 2.3.13. Note that dimΓϕ = 2n and that Γϕ is the embedded image of M1 inM1 ×M2. The embedding is given by the map

γ : M1 !M1 ×M2,

q 7! (q, ϕ(q)).

Proposition 2.3.14. A diffeomorphism ϕ is a symplectomorphism if and only if Γϕ is aLagrangian submanifold of (M1 ×M2, ω).

Proof. The graph Γϕ is Lagrangian if and only if γ∗ω = 0. But we have

γ∗ω = γ∗(pr1)∗ω1 − γ∗(pr2)

∗ω2 = (pr1 γ)∗ω1 − (pr2 γ)∗ω2,

where pr1 γ is the identity map on M1 whereas pr2 γ = ϕ. Hence

γ∗ω = 0 ⇐⇒ ϕ∗ω2 = ω1.


2.4. Local theory

2.4.1. Isotopies and vector fields. Let M be a manifold and ρ : M × R!M a mapwith ρt(q) := ρ(q, t).

Definition 2.4.1 (Isotopy). The map ρ is an isotopy if each ρt : M !M is a diffeomorphismand ρ0 = idM .

Given an isotopy ρ we can construct a time-dependent vector field. This means that we geta family of vector fields Xt for t ∈ R, which at q ∈M satisfy

Xt(q) =d

dsρs(p)

∣∣∣∣s=t

, ∀p = ρ−1t (q).

Basically, this means

(2.4.1)dρtdt

= Xt ρt.

Conversely, let Xt be a time-dependent vector field and assume either that M is compact orXt is compactly supported for all t. Then there exists an isotopy ρ satisfying (2.4.1).Moreover, if M is compact, we have a one-to-one correspondence

isotopies of M ! time-dependent vector fields on M,(ρt)t∈R ! (Xt)t∈R.

Definition 2.4.2 (Exponential map). If a vector field Xt = X is time-independent, we callthe associated isotopy the exponential map of X and we denote it by exp tX.

Remark 2.4.3. Note that the family exp tX : M !M | t ∈ R is the unique smooth familyof diffeomorphisms satisfying the Cauchy problem

exp tX∣∣t=0

= idM ,

d

dt(exp tX)(q) = X(exp tX(q)).

Remark 2.4.4. The exponential map is the same as the flow of a vector field (see Section 1.2,Definition 1.2.24). The flow of a time-dependent vector field is given by the correspondingisotopy.

Exercise 2.4.5. Show that for a time-dependent vector field Xt and ω ∈ Ωs(M) we have

(2.4.2)d

dtρ∗tω = ρ∗tLXtω,

where ρ is the (local) isotopy generated by Xt.

Proposition 2.4.6. For a smooth family (ωt)t∈R of s-forms, we have

(2.4.3)d

dtρ∗tωt = ρ∗t

(LXtωt +

d

dtωt

).

Remark 2.4.7. Equation (2.4.3) will turn out to be very useful for the proof of Moser’stheorem (see Section 2.5, Theorem 2.5.6)

2.4. LOCAL THEORY 47

Proof of Proposition 2.4.6. If f(x, y) is a real function of two variables, we can usethe chain rule to get

d

dtf(t, t) =

d

dxf(x, t)

∣∣∣∣x=t

+d

dyf(t, y)

∣∣∣∣y=t

.

Hence, we get

d

dtρ∗tωt =

d

dxρ∗xωt

∣∣∣∣x=t︸︷︷︸

ρ∗xLXtωt

∣∣x=t

by (2.4.2)

+d

dyρ∗tωy

∣∣∣∣y=t︸︷︷︸

ρ∗tddyωy

∣∣y=t

= ρ∗t

(LXtωt +

d

dtωt

).

2.4.2. Tubular neighborhood theorem. Let M be an n-dimensional manifold andlet X ⊂M be a k-dimensional submanifold and consider the inclusion map

i : X !M.

By the differential of the inclusion dix : TxX ! TxM , we have an inclusion of the tangentspace of X at a point x ∈ X into the tangent space of M at the point x.

Definition 2.4.8 (Normal space). The normal space to X at the point x ∈ X is given bythe (n− k)-dimensional vector space defined by the quotient

NxX := TxM/TxX.

Definition 2.4.9 (Normal bundle). The normal bundle is then given by

NX := (x, v) | x ∈ X, v ∈ NxX.Remark 2.4.10. Using the natural projection, NX is a vector bundle over X of rank n− kand hence as a manifold it is n-dimensional. The zero section of NX

i0 : X ! NX,

x 7! (x, 0),

embeds X as a closed submanifold of NX.

Definition 2.4.11 (Convex neighborhood). A neighborhood U0 of the zero section X in NXis called convex if the intersection U0 ∩NxX with each fiber is convex.

Theorem 2.4.12 (Tubular neighborhood theorem). There exists a convex neighborhood U0

of X in NX, a neighborhood U of X in M , and a diffeomorphism ϕ : U0 ! U such that thefollowing diagram commutes:

NX ⊇ U0 U ⊆M

X

ϕ

i0 i

Remark 2.4.13. Restricting to the subset U0 ⊆ NX, we obtain a submersion U0π0−! X with

all fibers π−10 (x) convex. We can extend this fibration to U by setting π := π0 ϕ−1 such

that if NX ⊇ U0π0−! X is a fibration, then M ⊇ U π

−! X is a fibration. This is called thetubular neighborhood fibration.


2.4.3. Homotopy formula. Let U be a tubular neighborhood of a submanifoldX ⊂M .The restriction of de Rham cohomology groups

i∗ : Hs(U)! Hs(X)

by the inclusion map is surjective. By the tubular neighborhood fibration, i∗ is also injectivesince the de Rham cohomology is homotopy invariant. In fact, we have the following corollary:

Corollary 2.4.14. For any degree s we have

Hs(U) ∼= Hs(X).

Remark 2.4.15. Corollary 2.4.14 says that if ω is a closed s-form on U and i∗ω is exact onX, then ω is exact.

Proposition 2.4.16. If a closed s-form ω on U has restriction i∗ω = 0, then ω is exact, i.e.ω = dµ for some µ ∈ Ωs−1(U). Moreover, we can choose µ such that µx = 0 for all x ∈ X.

Proof. By using the map ϕ : U0∼−! U , we can work over U0. For t ∈ [0, 1], define a map

ρt : U0 ! U0,

(x, v) 7! (x, tv).

This is well-defined since U0 is convex. The map ρ1 is the identity and ρ0 = i0 π0. Moreover,each ρt fixes X, i.e. ρt i0 = i0. Hence, we say that the family (ρt)t∈[0,1] is a homotopy fromi0 π0 to the identity fixing X. The map π0 is called retraction because π0 i0 is the identity.The submanifold X is then called a deformation retract of U .A (de Rham) homotopy operator between ρ0 = i0 π0 and ρ1 = id is a linear map

Q : Ωs(U0)! Ωs−1(U0)

satisfying the homotopy formula

(2.4.4) id− (i0 π0)∗ = dQ+Qd.

When dω = 0 and i∗0ω = 0, the operator Q gives ω = dQω, so that we can take µ = Qω. Aconcrete operator Q is given by the formula

(2.4.5) Qω =

∫ 1

0ρ∗t (ιXtω)dt,

where Xt, at the point p = ρt(q), is the vector tangent to the curve ρs(q) at s = t. We claimthat the operator (2.4.5) satisfies the homotopy formula. Indeed, we compute

Qdω + dQω =

∫ 1

0ρ∗t (ιXtdω)dt+ d

∫ 1

0ρ∗t (ιXtω)dt =

∫ 1

0ρ∗t (ιXtdω + dιXtω︸︷︷︸

=LXtω

)dt.

Hence the result follows from (2.4.2) and the fundamental theorem of analysis:

Qdω + dQω =

∫ 1

0

d

dtρ∗tωdt = ρ∗1ω − ρ∗0ω.

This completes the proof since, for our case, we have that ρt(x) = x is a constant curve forall x ∈ X and for all t, so Xt vanishes at all x and all t. Hence, µx = 0.

2.5. MOSER’S THEOREM 49

2.5. Moser’s theorem

2.5.1. Equivalences for symplectic structures. Let M be a 2n-dimensional mani-fold with two symplectic forms ω0 and ω1, so that (M,ω0) and (M,ω1) are two symplecticmanifolds.

Definition 2.5.1 (Symplectomorphic). (M,ω0) and (M,ω1) are symplectomorphic if thereis a diffeomorphism ϕ : M !M with ϕ∗ω1 = ω0.

Definition 2.5.2 (Strongly isotopic). (M,ω0) and (M,ω1) are strongly isotopic if there isan isotopy ρt : M !M such that ρ∗1ω1 = ω0.

Definition 2.5.3 (Deformation-equivalent). (M,ω0) and (M,ω1) are deformation-equivalentif there is a smooth family ωt of symplectic forms joining ω0 to ω1.

Definition 2.5.4 (Isotopic). (M,ω0) and (M,ω1) are isotopic if they are deformation-equivalent with the de Rham cohomology classes [ωt] independent of t.

Remark 2.5.5. We have strongly isotopic =⇒ symplectomorphic, and isotopic =⇒ deformation-equivalent. Moreover, we have strongly isotopic =⇒ isotopic, because if ρt : M ! M is anisotopy such that ρ∗1ω1 = ω0, then the set ωt := ρ∗tω1 is a smooth family of symplecticforms joining ω1 to ω0 and [ωt] = [ω1] for all t by the homotopy invariance of the de Rhamcohomology.

2.5.2. Moser’s trick. Consider the following problem: Let M be a 2n-dimensionalmanifold and let X ⊂M be a k-dimensional submanifold. Moreover, consider neighborhoodsU0 and U1 of X and symplectic forms ω0 and ω1 on U0 and U1 respectively. Does thereexist a symplectomorphism preserving X? More precisely, does there exist a diffeomorphismϕ : U0 ! U1 with ϕ∗ω1 = ω0 and ϕ(X) = X?

We want to consider the extreme case when X = M and we consider M to be compactwith symplectic forms ω0 and ω1. So the question will change to: are (M,ω0) and (M,ω1)symplectomorphic, i.e. does there exist a diffeomorphism ϕ : M !M such that ϕ∗ω1 = ω0?Moser’s question was whether we can find such a ϕ which is homotopic to the identity on M .A necessary condition is

[ω0] = [ω1] ∈ H2(M)

because if ϕ ∼ idM then, by the homotopy formula (2.4.4), there exists a homotopy operatorQ such that

(idM )∗ω1 − ϕ∗ω1 = dQω1 +Q dω1︸︷︷︸=0

.

Thus, we getω1 = ϕ∗ω1 + dQω1

and hence[ω1] = [ϕ∗ω1] = [ω0].

So we ask ourselves whether, if [ω0] = [ω1], there exist a diffeomorphism ϕ homotopic to idMsuch that ϕ∗ω1 = ω0. In [Mos65], Moser proved that, with certain assumptions, this is true.Later, McDuff showed that, in general, this is not true by constructing a counterexample[MS95, Example 7.23].


Theorem 2.5.6 (Moser (version I)[Mos65]). Suppose that M is compact, [ω0] = [ω1] and thatthe 2-form ωt = (1 − t)ω0 + tω1 is symplectic for all t ∈ [0, 1]. Then there exists an isotopyρ : M × R!M such that ρ∗tωt = ω0.

Remark 2.5.7. In particular, ϕ := ρt : M !M satisfies ϕ∗ω1 = ω0.

The argument for the proof is known as Moser’s trick.

Proof of Theorem 2.5.6. Suppose that there exists an isotopy ρ : M × R ! M suchthat ρ∗tωt = ω0 for t ∈ [0, 1]. Let

Xt :=dρtdt

ρ−1t , ∀t ∈ R.

Then

0 =d

dt(ρ∗tωt) = ρ∗t

(LXtωt +

d

dtωt

)

which is equivalent to

(2.5.1) LXtωt +d

dtωt = 0.

Suppose conversely that we can find a smooth time-dependent vector field Xt for t ∈ R,such that (2.5.1) holds for t ∈ [0, 1]. Since M is compact, we can integrate Xt to an isotopyρ : M × R!M with

d

dt(ρ∗tωt) = 0,

which implies thatρ∗tωt = ρ∗0ω0 = ω0.

This means that everything reduces to solving (2.5.1) for Xt. First, note that, from ωt =(1− t)ω0 + tω1, we get

d

dtωt = ω1 − ω0.

Second, since [ω0] = [ω1], there is a 1-form µ such that

ω1 − ω0 = dµ.

Third, by Cartan’s magic formula (1.3.5), we have

LXtωt = dιXtωt + ιXt dωt︸︷︷︸=0

.

putting everything together, we need to find Xt such that

dιXtωt + dµ = 0.

Clearly, it is sufficient to solveιXtωt + µ = 0.

By nondegeneracy of ωt, we can solve this pointwise to obtain a unique (smooth) Xt.

Theorem 2.5.8 (Moser (version II)[Mos65]). Let M be a compact manifold with symplecticforms ω0 and ω1. Suppose that (ωt)t∈[0,1] is a smooth family of closed 2-forms joining ω0 andω1 and satisfying:

2.5. MOSER’S THEOREM 51

(1) (cohomology assumption) [ωt] is independent of t, i.e.

d

dt[ωt] =

[d

dtωt

]= 0,

(2) (nondegeneracy condition) ωt is nondegenerate for all t ∈ [0, 1].

Then there exists an isotopy ρ : M × R!M such that

ρ∗tωt = ω0, ∀t ∈ [0, 1].

Proof (Moser’s trick). We have the following implications: Condition (1) impliesthat there exists a family of 1-forms µt such that

d

dtωt = dµt, ∀t ∈ [0, 1].

Indeed, we can find a smooth family of 1-forms µt such that ddtωt = dµt. The argument

uses a combination of the Poincare lemma for compactly-supported forms and the Mayer–Vietoris sequence in order to use induction on the number of charts in a good cover of M (see[BT82] for a detailed discussion of the Poincare lemma and the Mayer–Vietoris construction).Condition (2) implies that there exists a unique family of vector fields Xt such that

(2.5.2) ιXtωt + µt = 0.

Equation (2.5.1) is called Moser’s equation. We can extend Xt to all t ∈ R. Let ρ be theisotopy generated by Xt (ρ exists by compactness ofM). Then, using Cartan’s magic formulaand Moser’s equation, we indeed have

d

dt(ρ∗tωt) = ρ∗t

(LXtωt +

d

dtωt

)= ρ∗t (dιXtωt + dµt) = 0.

Remark 2.5.9. Note that we have used compactness of M to be able to integrate Xt for allt ∈ R. If M is not compact, we need to check the existence of a solution ρt for the differentialequation

d

dtρt = Xt ρt, ∀t ∈ [0, 1].

Theorem 2.5.10 (Moser (relative version)[Mos65]). Let M be a manifold, X a compactsubmanifold of M , i : X ! M the inclusion map, ω0 and ω1 two symplectic forms on M .Then if ω0|q = ω1|q for all q ∈ X, we get that there exist neighborhoods U0,U1 of X in M anda diffeomorphism ϕ : U0 ! U1 such that ϕ∗ω1 = ω0 and the following diagram commutes:

U0 U1

X

ϕ

i i

Proof. Choose a tubular neighborhood U0 of X. The 2-form ω1 − ω0 is closed on U0

and (ω1 − ω0)q = 0 for all q ∈ X. By the homotopy formula on the tubular neighborhood,there exists a 1-form µ on U0 such that ω1 − ω0 = dµ and µq = 0 at all q ∈ X. Consider afamily ωt = (1 − t)ω0 + tω1 = ω0 + tdµ of closed 2-forms on U0. Shrinking U0 if necessary,


we can assume that ωt is symplectic for t ∈ [0, 1]. Then we can solve Moser’s equationιXtωt = −µ and note that Xt|X = 0. Shrinking U0 again if necessary, there is an isotopyρ : U0 × [0, 1] ! M with ρ∗tωt = ω0 for all t ∈ [0, 1]. Since Xt|X = 0, we have ρt|X = idX .Then we can set ϕ := ρ1 and U1 := ρ1(U0).

Exercise 2.5.11. Prove Darboux’s theorem (Theorem 2.1.32) by using the relative versionof Moser’s theorem for X = q.

2.6. Weinstein tubular neighborhood theorem

2.6.1. Weinstein Lagrangian neighborhood theorem.

Theorem 2.6.1 (Weinstein Lagrangian neighborhood theorem[Wei71]). Let M be a 2n-dimensional manifold, X a compact n-dimensional submanifold i : X ! M the inclusionmap, and ω0 and ω1 symplectic forms on M such that i∗ω0 = i∗ω1 = 0, i.e. X is a La-grangian submanifold of both (M,ω0) and (M,ω1). Then there exists neighborhoods U0 andU1 of X in M and a diffeomorphism ϕ : U0 ! U1 such that ϕ∗ω1 = ω0 and the followingdiagram commutes:

U0 U1

X

ϕ

i i

Theorem 2.6.2 (Whitney extension theorem[Whi34]). Let M be an n-dimensional manifoldand X a k-dimensional submanifold with k < n. Suppose that at each q ∈ X we are givena linear isomorphism Lq : TqM

∼−! TqM such that Lq|TqX = idTqX and Lq depends smoothly

on q. Then there exists an embedding h : N ! M of some neighborhood N of X in M suchthat h|X = idX and dqh = Lq for all q ∈ X.

Proof of Theorem 2.6.1. Let g be a Riemannian metric on M , i.e. at each q ∈ Mwe get that gq is a positive-definite inner product. Fix q ∈ X and let V := TqM , U := TqX

and W := U⊥ be the orthogonal complement of U in V relative to gq. Since i∗ω0 = i∗ω1 = 0,

the space U is a Lagrangian subspace of both (V, ω0|q) and (V, ω1|q). By the symplectic

linear algebra, we canonically get from U⊥ a linear isomorphism Lq : TqM ! TqM such thatLq|TqX = idTqX and L∗

qω1|p = ω0|q. Note that Lq varies smoothly with respect to q since theconstruction is canonical. By the Whitney extension theorem, there exists a neighborhoodN of X and an embedding h : N ! X with h|X = idX and dqh = Lq for q ∈ X. Hence, atany q ∈ X, we have

(h∗ω1)q = (dqh)∗ω1|q = L∗

qω1|q = ω0|q.applying the relative version of Moser’s theorem to ω0 and h∗ω1, we find a neighborhood U0

of X and an embedding f : U0 ! N such that f |X = idX and f∗(h∗ω1) = ω0 on U0. Thenwe can set ϕ := h f .

Theorem 2.6.3 (Coisotropic embedding theorem). Let M be a 2n-dimensional manifold,X a k-dimensional submanifold with k < n, i : X ! M the inclusion map and ω0 and ω1

two symplectic forms on M such that i∗ω0 = i∗ω1 with X being coisotropic for both (M,ω0)

2.6. WEINSTEIN TUBULAR NEIGHBORHOOD THEOREM 53

and (M,ω1). Then there exist neighborhoods U0 and U1 of X in M and a diffeomorphismϕ : U0 ! U such that ϕ∗ω1 = ω0 and the following diagram commutes:

U0 U1

X

ϕ

i i

Exercise 2.6.4. Prove Theorem 2.6.3. Hint: See [Wei77] for some inspiration and [GS77;Got82] for a proof.

2.6.2. Weinstein tubular neighborhood theorem.

Theorem 2.6.5 (Weinstein tubular neighborhood theorem[Wei71]). Let (M,ω) be a sym-plectic manifold, X a compact Lagrangian submanifold of M , ω0 the canonical symplecticform on T ∗X, i0 : X ! T ∗X the Lagrangian embedding as the zero section, and i : X ! Mthe Lagrangian embedding given by the inclusion. Then there exist neighborhoods U0 of X inT ∗X, U of X in M and a diffeomorphism ϕ : U0 ! U such that ϕ∗ω = ω0 and the followingdiagram commutes:

U0 U

X

ϕ

i0 i

Remark 2.6.6. A similar statement for isotropic submanifolds was also proved by Weinsteinin [Wei77; Wei81].

Proof of Theorem 2.6.5. The proof uses the tubular neighborhood theorem and theWeinstein Lagrangian neighborhood theorem. We first use the tubular neighborhood theo-rem. Since NX ∼= T ∗X, we can find a neighborhood N0 of X in T ∗X, a neighborhood N ofX in M and a diffeomorphism ψ : N0 ! N such that the following diagram commutes:

N0 N

X

ψ

i0 i

Let ω0 be the canonical symplectic form on T ∗X and ω1 := ψ∗ω. Note that ω0 and ω1 aresymplectic forms on N0. The submanifold X is Lagrangian for both ω0 and ω1. Now we usethe Weinstein Lagrangian neighborhood theorem. There exist neighborhoods U0 and U1 ofX in N0 and a diffeomorphism θ : U0 ! U1 such that θ∗ω1 = ω0 and the following diagramcommutes:

U0 U1

X

θ

i0 i0

Now we can take ϕ = ψ θ and U1 = ϕ(U0). It is easy to check ϕ∗ω = θ∗ ψ∗ω︸︷︷︸ω1

= ω0.


2.6.3. Application.

Definition 2.6.7 (C1-topology). Let X and Y be two manifolds. A sequence of C1 mapsfi : X ! Y is said to converge in the C1-topology to a map f : X ! Y if and only if thesequence (fi) itself and the sequence of the differentials dfi : TX ! TY converge uniformlyon compact sets.

Remark 2.6.8. Let (M,ω) be a symplectic manifold. Note that the graph of the identitymap, denoted by ∆ := ΓidM (recall Equation (2.3.1) for the definition), is a Lagrangiansubmanifold of (M ×M, (pr1)

∗ω − (pr2)∗ω) (see also Proposition 2.3.14). Moreover, we say

that f is C1-close to another map g, if f is in some small neighborhood of g in the C1-topology.

By the Weinstein tubular neighborhood theorem, there is a neighborhood U of

∆ ⊂ (M ×M, (pr1)∗ω − (pr2)

∗ω)

which is symplectomorphic to a neighborhood U0 of M in (T ∗M,ω0). Let ϕ : U ! U0 be thesymplectomorphism satisfying ϕ(q, q) = (q, 0) for all q ∈M .Denote by

Sympl(M,ω) := f : M ∼−!M | f∗ω = ω

and suppose that f ∈ Sympl(M,ω) is sufficiently C1-close to the identity, i.e. f is in somesufficiently small neighborhood of the identity idM in the C1-topology. Then we can assumethat the graph of f lies inside of U . Let j : M ! U be the embedding as Γf and i : M ! Uthe embedding as ΓidM = ∆. The map j is sufficiently C1-close to i. By the Weinsteintheorem, U ≃ U0 ⊆ T ∗M , so the above j and i induce two embeddings: j0 : M ! U0 wherej0 = ϕ j and i0 : M ! U0 embedding as 0-section. Hence, we have

U U0

M

ϕ

i i0

U U0

M

ϕ

j j0

where i(q) = (q, q), i0(q) = (q, 0), j(q) = (q, f(q)) and j0(q) = ϕ(q, f(q)) for q ∈ M . Themap j0 is sufficiently C1-close to i0. Thus, the image set j0(M) intersects each T ∗

qM at onepoint µq depending smoothly on q. The image of j0 is the image of a smooth intersectionµ : M ! T ∗M , that is, a 1-form µ = j0 (π j0)−1. Therefore,

(2.6.1) Γf ≃ (q, µq) | q ∈M, µq ∈ T ∗qM.

Vice-versa, if µ is a 1-form sufficiently C1-close to the zero 1-form, then we also get (2.6.1).In fact, we get

Γf is Lagrangian ⇐⇒ µ is closed.

2.6. WEINSTEIN TUBULAR NEIGHBORHOOD THEOREM 55

Hence, we can conclude that a small C1-neighborhood of the identity in Sympl(M,ω) ishomeomorphic to a C1-neighborhood of zero in the vector space of closed 1-forms on M .Thus, we have

TidM (Sympl(M,ω)) ≃ µ ∈ Ω1(M) | dµ = 0.In particular, TidM (Sympl(M,ω)) contains the space of exact 1-forms

µ = dh | h ∈ C∞(M) ≃ C∞(M)/locally constant functions.Theorem 2.6.9. Let (M,ω) be a compact symplectic manifold with H1(M) = 0. Then anysymplectomorphism of M which is sufficiently C1-close to the identity has at least two fixedpoints.

Proof. Suppose that any f ∈ Sympl(M,ω) is sufficiently C1-close to the identity idM .Then the graph Γf is homotopic to a closed 1-form on M . Now the fact that dµ = 0 andH1(M) = 0 imply that µ = dh for some h ∈ C∞(M). Since M is compact, h has at leasttwo critical points. Note that fixed points of f are equal to critical points of h. Moreover,fixed points of f are equal to the intersection of Γf with the diagonal ∆ ⊂ M ×M which isagain equal to S := q | µq = dqh = 0. Finally, note that the critical points of h are exactlypoints in S.

We say that a submanifold Y of M is C1-close to another submanifold X of M , when thereis a diffeomorphism X ! Y which is, as a map into M , C1-close to the inclusion X !M .

Theorem 2.6.10 (Lagrangian intersection theorem). Let (M,ω) be a symplectic manifold.Suppose that X is a compact Lagrangian submanifold of M with H1(X) = 0. Then everyLagrangian submanifold of M which is C1-close to X intersects X in at least two points.


Definition 2.6.12 (Morse function). A Morse function on a manifold M is a functionh : M ! R whose critical points are all nondegenerate, i.e. the Hessian at a critical point q

is not singular, det(

∂2h∂xi∂xj

∣∣q

)6= 0.

Let (M,ω) be a symplectic manifold and suppose that ht : M ! R is a smooth family offunctions which is 1-periodic, i.e. ht = ht+1. Let ρ : M ×R!M be the isotopy generated bythe time-dependent vector field Xt defined by ω(Xt, ) = dht. Then we say that f is exactlyhomotopic to the identity if f = ρ1 for such ht. Equivalently, using the notions which will beintroduced in Section 2.7, we have the following definition:

Definition 2.6.13 (Exactly homotopic to the identity). A symplectomorphism f ∈ Sympl(M,ω)is exactly homotopic to the identity when f is the time-1 map of an isotopy generated by somesmooth time-dependent 1-periodic Hamiltonian function.

Definition 2.6.14 (Nondegenerate fixed point). A fixed point q of a function f : M !M iscalled nondegenerate if dqf : TqM ! TqM is not singular.

Conjecture 2.6.15 (Arnold). Let (M,ω) be a compact symplectic manifold, and f : M !Ma symplectomorphism which is exactly homotopic to the identity. Then

|fixed points of f| ≥ min |critical points of a smooth function on M|.


Using the notion of a Morse function as in Definition 2.6.12, we obtain

|nondegenerate fixed points of f| ≥ min |critical points of a Morse function on M|

≥2n∑

i=0

dimH i(M,R).

Remark 2.6.16. The Arnold conjecture has been proven by Conley–Zehnder, Floer, Hofer–Salamon, Ono, Fukaya–Ono, Liu–Tian by using Floer homology. This is an infinite-dimensionalversion of Morse theory. Sharper bound versions of the Arnold conjecture are still open.

Exercise 2.6.17. Compute the estimates for the number of fixed points on the compactsymplectic manifolds S2, S2 × S2 and T 2 := S1 × S1.

2.7. Hamiltonian mechanics

2.7.1. Hamiltonian and symplectic vector fields. Let (M,ω) be a symplectic man-ifold and let H : M ! R be a smooth function. By nondegneracy of ω, there is a uniquevector field XH on M such that

ιXHω = dH.

Suppose that M is compact or that XH is complete. Let ρt : M ! M for t ∈ R be the1-parameter family of diffeomorphisms generated by XH , i.e. satisfying the Cauchy problem

ρ0 = idM ,

d

dtρt = XH ρt.

In fact, each diffeomorphism ρt preserves ω, i.e. ρ∗tω = ω for all t. Indeed, note that

d

dtρ∗tω = ρ∗tLXH

ω = ρ∗t (d ιXHω︸︷︷︸

=dH

+ιXHdω︸︷︷︸=0

) = 0

Definition 2.7.1 (Hamiltonian vector field and Hamiltonian function). A vector field XH

as above is called the Hamiltonian vector field with Hamiltonian function H.

Exercise 2.7.2. Let X be a vector field on some manifold M . Then there is a unique vectorfield X♯ on the cotangent bundle T ∗M whose flow is the lift of the flow of X. Let α be thetautological 1-form on T ∗M and let ω = −dα be the canonical symplectic form on T ∗M .Show that X♯ is a Hamiltonian vector field with Hamiltonian function H := ιX♯

α.

Remark 2.7.3. If XH is Hamiltonian, we get

LXHH = ιXH

dH = ιXHιXH

ω = 0.

Hence, Hamiltonian vector fields preserve their Hamiltonian functions and each integral curve(ρt(x))t∈R of XH must be contained in a level set of H, i.e.

H(x) = (ρ∗tH)(x) = H(ρt(x)), ∀t.Definition 2.7.4 (Symplectic vector field). A vector field X on a symplectic manifold (M,ω)preserving ω, i.e. LXω = 0, is called symplectic.

2.7. HAMILTONIAN MECHANICS 57

Remark 2.7.5. Note that X is symplectic if and only if ιXω is closed and X is Hamiltonianif and only if ιXω is exact.

Remark 2.7.6. Locally, on every contractible set, every symplectic vector field is Hamilton-ian. If H1(M) = 0, then globally every symplectic vector field is Hamiltonian. In general,H1(M) measures the obstruction for symplectic vector fields to be Hamiltonian.

2.7.2. Classical mechanics. Consider the Euclidean space R2n with coordinates(q1, . . . , qn, p1, . . . , pn) and ω0 =

∑j dqj ∧ dpj. If the Hamilton equations

dqidt

(t) =∂H

∂pi,

dpidt

(t) = −∂H∂qi

(2.7.1)

are satisfied, then the curve ρt = (q(t), p(t)) is an integral curve for XH . Indeed, let

XH =n∑

i=1

(∂H

∂pi

∂

∂qi− ∂H

∂qi

∂

∂pi

).

Then

ιXHω =

n∑

j=1

ιXH(dqj ∧ dpj) =

n∑

j=1

[(ιXHdqj) ∧ dpj − dqj ∧ (ιXH

dpj)]

=n∑

j=1

(∂H

∂pjdpj +

∂H

∂qjdqj

)= dH.

Consider the case when n = 3. By Newton’s second law, a particle with mass m ∈ R>0

moving in configuration space R3 with coordinates q := (q1, q2, q3) in a potential V (q) movesalong a curve q(t) satisfying

md2q

dt2= −∇V (q).

One then introduces momentum coordinates pi := mdqidt for i = 1, 2, 3 and a total energy

function

H(q, p) = kinetic energy + potential energy

=1

2m‖p‖2 + V (q)

The phase space is then given by T ∗R3 = R6 with coordinates (q1, q2, q3, p1, p2, p3). Newton’ssecond law in R3 is equivalent to the Hamilton equations in R6:

dqidt

=1

mpi =

∂H

∂pi,

dpidt

= md2qidt2

= −∂V∂qi

= −∂H∂qi

.

Remark 2.7.7. The total energy H is conserved by the physical motion, i.e. ddtH = 0.


2.7.3. Brackets. Recall that vector fields are differential operators on functions. Inparticular, if X is a vector field and f ∈ C∞(M), with df being the corresponding 1-form,then

X(f) = df(X) = LXf.

For two vector fields X,Y , there is a unique vector field Z such that

LZf = LX(LY f)− LY (LXf).

The vector field Z is called the Lie bracket of the vector fields X and Y and is denoted byZ = [X,Y ], since LZ = [LX , LY ] is the commutator (see also Definition 1.2.19).

Proposition 2.7.8. If X and Y are symplectic vector fields on a symplectic manifold (M,ω),then [X,Y ] is a Hamiltonian vector field with Hamiltonian function ω(Y,X).

Proof. We have

ι[X,Y ]ω = LXιY ω − ιY LXω

= dιXιY ω + ιX dιY ω︸︷︷︸=0

−ιY dιXω︸︷︷︸=0

−ιY ιX dω︸︷︷︸=0

= d(ω(Y,X)).

Definition 2.7.9 (Lie algebra). A Lie algebra is a vector space g together with a Lie bracket[ , ], i.e. a bilinear map [ , ] : g× g! g such that

(1) [X,Y ] = −[Y,X], ∀X,Y ∈ g (antisymmetry)(2) [X, [Y,Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0, ∀X,Y,Z ∈ g (Jacobi identity)

Denote by XH(M) the space of ]Hamiltonian vector fields and by XS(M) the space of sym-plectic vector fields on a manifold M .

Corollary 2.7.10. The inclusions

(XH(M), [ , ]) ⊆ (XS(M), [ , ]) ⊆ (X(M), [ , ])

are inclusions of Lie algebras.

Definition 2.7.11 (Poisson bracket). The Poisson bracket of two functions f, g ∈ C∞(M)on a symplectic manifold (M,ω) is defined by

(2.7.2) f, g := ω(Xf ,Xg).

Remark 2.7.12. Note that we have Xf,g = −[Xf ,Xg] since Xω(Xf ,Xg) = [Xg,Xf ].

Exercise 2.7.13. Show that the Poisson bracket , satisfies the Jacobi identity, i.e.

f, g, h + g, h, f + h, f, g = 0, ∀f, g, h ∈ C∞(M).

Definition 2.7.14 (Poisson algebra). A Poisson algebra is a commutative associative algebraA with a Lie bracket , : A×A! A satisfying the Leibniz rule

f, gh = f, gh + gf, h, ∀f, g, h ∈ A.

2.7. HAMILTONIAN MECHANICS 59

Exercise 2.7.15. Show that the Poisson bracket defined as in (2.7.2) is satisfies the Leibnizrule and deduce that if (M,ω) is a symplectic manifold, then (C∞(M), , ) is a Poissonalgebra.

Remark 2.7.16. Note that we have a Lie algebra anti-homomorphism

C∞(M)! X(M),

H 7! XH

such that the Poisson bracket , will correspond to −[ , ].

2.7.4. Integrable systems.

Definition 2.7.17 (Hamiltonian system). A Hamiltonian system is a triple (M,ω,H), where(M,ω) is a symplectic manifold and H ∈ C∞(M) is a Hamiltonian function.

Theorem 2.7.18. We have f,H = 0 if and only if f is constant along integral curves ofXH .

Proof. Let ρt be the flow of XH . Then

(2.7.3)d

dt(f ρt) = ρ∗tLXH

f = ρ∗t ιXHdf = ρ∗t ιXH

ιXfω = ρ∗tω(Xf ,XH) = ρ∗t f,H.

Remark 2.7.19. Given a Hamiltonian system (M,ω,H), a function f satisfying f,H = 0is called an integral of motion or a constant of motion. In general, Hamiltonian system donot admit integrals of motion which are independent of the Hamiltonian function.

Definition 2.7.20 (Independent functions). We say that functions f1, . . . , fn on a manifoldM are independent if their differentials dqf1, . . . ,dqfn are linearly independent at all pointsq ∈M in some open dense subset of M .

Definition 2.7.21 ((Completely) integrable system). A Hamiltonian system (M,ω,H) is(completely) integrable if it has n = 1

2 dimM independent integrals of motion f1 = H, f2 . . . , fn,which are pairwise in involution with respect to the Poisson bracket, i.e.

fi, fj = 0, ∀i, j.Let (M,ω,H) be an integrable system of dimension 2n with integrals of motion f1 =H, f2, . . . , fn. Let c ∈ Rn be a regular value of f := (f1, . . . , fn). Note that the corre-sponding level set, f−1(c), is a Lagrangian submanifold, because it is n-dimensional and itstangent bundle is isotropic.

Lemma 2.7.22. If the Hamiltonian vector fields Xf1 , . . . ,Xfn are complete on the level setsf−1(c), then the connected components of f−1(c) are homogeneous spaces for Rn, i.e. are ofthe form Rn−k × T k for some k with 0 ≤ k ≤ n, where T k denotes the k-torus.

Exercise 2.7.23. Prove Lemma 2.7.22. Hint: follow the flows to obtain coordinates.

Note that any compact component of f−1(c) must be a torus. These components, when theyexist, are called Liouville tori.


Theorem 2.7.24 (Arnold–Liouville[Arn78; Lio55]). Let (M,ω,H) be an integrable system ofdimension 2n with integrals of motion f1 = H, f2, . . . , fn. Let c ∈ Rn be a regular value off := (f1, . . . , fn). The corresponding level set f−1(c) is a Lagrangian submanifold of M .

(1) If the flows of Xf1 , . . . ,Xfn starting at a point q ∈ f−1(c) are complete, then theconnected component of f−1(c) containing q is a homogeneous space for Rn. Withrespect to this affine structure, that component has coordinates φ1, . . . , φn, known asangle coordinates, in which the flows of the vector fields Xf1 , . . . ,Xfn are linear.

(2) There are coordinates ψ1, . . . , ψn, known as action coordinates, complementary tothe angle coordinates such that the ψis are integrals of motion and

φ1, . . . , φn, ψ1, . . . , ψn

form a Darboux chart.

Exercise 2.7.25 (Pendulum). The pendulum is a mechanical system consisting of a masslessrod of length ℓ, where one end is fixed and the other end has some mass m attached to itwhich can oscillate in a vertical plane. Assume that gravity is constant pointing verticallydownwards, and that it is the only external force acting on this system.

(1) Let θ be the oriented angle between the rod and the vertical direction. Let ξ be thecoordinate along the fibers of T ∗S1 induced by the standard angle coordinate on S1.Show that the function

H : T ∗S1! R,

(θ, ξ) 7!ξ2

2mℓ2︸︷︷︸=:K

+mℓ(1− cos θ)︸︷︷︸=:V

is an appropriate Hamiltonian function to describe the pendulum. More precisely,check that gravity corresponds to the potential energy V (θ) = mℓ(1−cos θ) (univer-

sal constants are omitted), and that the kinetic energy is given by K(θ, ξ) = ξ2

2mℓ2.

(2) For simplicity, assume thatm = ℓ = 1. draw the level curves of H in the (θ, ξ)-plane.Show that there is a real number c ∈ R such that for some real number h ∈ R with0 < h < c the level curve H = h is a disjoint union of closed curves. Show that theprojection of each of these curves onto the θ-axis is an interval of length less thanπ. Show that neither of these assertions is true if h > c. What types of motion aredescribed by these two types of curves? What about the case H = c?

(3) compute the critical points of H. Show that, modulo 2π in θ, there are exactly twocritical points: a critical point s where H vanishes, and a critical point u whereH equals c. These points are called stable and unstable points of H, respectively.Justify this terminology, i.e., show that a trajectory of the Hamiltonian vector fieldof H whose initial point is close to s stays close to s forever, and show that this isnot the case for u. What is happening physically?

2.8. MOMENT MAPS 61

2.8. Moment maps

2.8.1. Smooth actions.

Definition 2.8.1 (Lie group). A Lie group is a manifold G equipped with a group structurewhere the group operations multiplication and taking the inverse are smooth maps.

Definition 2.8.2 (Representation). A representation of a Lie group G on a vector space Vis a group homomorphism

G! GL(V ).

Definition 2.8.3 (Action). Let M be a manifold and denote by

Diff(M) := ϕ : M ∼−!M | ϕ diffeomorphism

the diffeomorphism group of M . An action of a Lie group G on M is a group homomorphism

Ψ: G! Diff(M),

g 7! Ψg.

Remark 2.8.4. We will only consider left actions where Ψ is a Homomorphism. A rightaction is defined with Ψ being an anti-homomorphism.

Definition 2.8.5 (Evaluation map). The evaluation map associated with an action

Ψ: G! Diff(M)

is

evΨ : M ×G!M,

(q, g) 7! Ψg(q).

Remark 2.8.6. The action Ψ is smooth if evΨ is smooth.

2.8.2. Symplectic and Hamiltonian actions. Let (M,ω) be a symplectic manifold,and G a Lie group. Let Ψ: G! Diff(M) be a (smooth) action.

Definition 2.8.7 (Symplectic action). The action Ψ is a symplectic action if

Ψ: G! Sympl(M,ω) ⊂ Diff(M),

i.e., G acts by symplectomorphisms.

Remark 2.8.8. It is easy to show that there is a one-to-one correspondence between completevector fields on a manifoldM and smooth actions of R onM . One can show this by associatingto a complete vector field X its exponential map exp tX and, vice-versa, to a smooth action

Ψ its derivative dΨt(q)dt

∣∣t=0

=: Xq. Thus, we also get a one-to-one correspondence betweencomplete symplectic vector fields on M and symplectic actions of R on M .

Definition 2.8.9 (Hamiltonian action). A symplectic action Ψ of S1 or R on (M,ω) isHamiltonian if the vector field generated by Ψ is Hamiltonian. Equivalently, an action Ψ ofS1 or R on (M,ω) is Hamiltonian if there is a function H : M ! R with ιXω = dH, whereX is the vector field generated by Ψ.

Remark 2.8.10. When G is not a product of copies of S1 or R, the solution is to use anupgraded Hamiltonian function, which is called a moment map.


2.8.3. Adjoint and coadjoint representations. Let G be a Lie group. For g ∈ G,let

Lg : G! G,

a 7! g · abe left multiplication by g.

Definition 2.8.11 (left-invariant vector field). A vector field X on G is called left-invariantif

(Lg)∗X = X, ∀g ∈ G.

Remark 2.8.12. There are similar right notions.

Let g be the vector space of all left-invariant vector fields on G. Together with the Lie bracket[ , ] of vector fields, g forms a Lie algebra. It is called the Lie algebra of the Lie group G.

Exercise 2.8.13. Show that the map

g! TeG,

X 7! Xe,

where e is the identity element in G, is an isomorphism of vector spaces.

Note that any Lie group G acts on itself by conjugation:

G! Diff(M),

g 7! Ψg

where Ψg(a) = g · a · g−1. The derivative at the identity of Ψg is an invertible linear map

Adg : g! g.

Note that we have identified the Lie algebra g with the tangent space TeG.

Definition 2.8.14 (Adjoint representation). The adjoint representation (or adjoint action)of G on g is

Ad: G! GL(g),

g 7! Adg .

Exercise 2.8.15. Check that for matrix Lie groups

d

dtAdexp tX Y

∣∣∣∣t=0

= [X,Y ], ∀X,Y ∈ g.

Hint: use that for a matrix group G (i.e. a subgroup of GLn(R) for some n) we have

Adg(Y ) = gY g−1, ∀g ∈ G,∀Y ∈ g,

and

[X,Y ] = XY − Y X, ∀X,Y ∈ g.

2.8. MOMENT MAPS 63

Let 〈 , 〉 be the natural pairing between g∗ and g defined as

〈 , 〉 : g∗ × g! R,

(ξ,X) 7! 〈ξ,X〉 := ξ(X).

For ξ ∈ g∗, we define Ad∗g ξ by the property

〈Ad∗g ξ,X〉 = 〈ξ,Adg−1 X〉, ∀X ∈ g.

Definition 2.8.16 (Coadjoint representation). The collection of maps Ad∗g forms the coad-joint representation (or coadjoint action) of G on g∗:

Ad∗ : G! GL(g∗),

g 7! Ad∗g .

Exercise 2.8.17. Show that for all g, h ∈ G, we have

Adg Adh = Adgh, Ad∗g Ad∗h = Ad∗gh .

2.8.4. Hamiltonian actions. Let (M,ω) be a symplectic manifold, G a Lie group, andΨ: G ! Sympl(M,ω) a (smooth) symplectic action, i.e. a group homomorphism such thatthe evaluation map evΨ(g, q) := Ψg(q) is smooth. Moreover, let g be the Lie algebra of Gand g∗ its dual space.

Definition 2.8.18 (Hamiltonian action). The action Ψ is called Hamiltonian if there existsa map

µ : M ! g∗

such that

(1) for each X ∈ g, let• the map

µX : M ! R,

q 7! µX(q) := 〈µ(q),X〉,be the component of µ along X,

• X# be the vector field onM generated by the 1-parameter subgroup (exp tX)t∈R ⊆G.

Then

dµX = ιX#ω,

i.e., µX is a Hamiltonian function for the vector field X#.(2) µ is equivariant with respect to the given action Ψ of G on M and the coadjoint

action Ad∗ of G on g∗:

µ Ψg = Ad∗g µ, ∀g ∈ G.

Definition 2.8.19 (Moment map). A map µ as in Definition 2.8.18 is called a moment map.

Definition 2.8.20 (Hamiltonian G-space). The quadruple (M,ω,G, µ) is called a Hamilton-ian G-space.


Definition 2.8.21 (Comoment map). Let G be a connected Lie group. A comoment map isa map

µ∗ : g! C∞(M),

such that

(1) µ∗(X) := µX is a Hamiltonian function for the vector field X#,(2) µ∗ is a Lie algebra homomorphism:

µ∗[X,Y ] = µ∗(X), µ∗(Y ),where , denotes the Poisson bracket on C∞(M).

Remark 2.8.22. The condition for an action to be Hamiltonian can be equivalently rephrasedby using the notion of a comoment map instead of a moment map.

2.9. Symplectic reduction

2.9.1. Orbit spaces. Let Ψ: G! Diff(M) be any action.

Definition 2.9.1 (Orbit). The orbit of G through q ∈M is

Oq := Ψg(q) | g ∈ G.Definition 2.9.2 (Stabilizer). The stabilizer (or isotropy) of q ∈M is the subgroup

Gq := g ∈ G | Ψg(q) = q.Exercise 2.9.3. Show that if p is in the orbit of q, then Gp and Gq are conjugate subgroups.

Definition 2.9.4 (Transitive/Free/Locally free). The action Ψ is called

• transitive if there is just one orbit,• free if all stabilizers are trivial e,• locally free if all stabilizers are discrete.

Let ∼ be the orbit equivalence relation, i.e. for q, p ∈M , we define q ∼ p if and only if q andp are on the same orbit.

Definition 2.9.5 (Orbit space). The space M/∼ =:M/G is called the orbit space.

Remark 2.9.6. We can endowM/G with the quotient topology with respect to the projection

π : M !M/G,

q 7! Oq.

2.9.2. Principal bundles. Let G be a Lie group and let B be a manifold.

Definition 2.9.7 (Principal G-bundle). A principal G-bundle over B is a manifold P witha smooth map π : P ! B satisfying:

(1) G acts freely on P (from the left),(2) B is the orbit space for this action and π is the point-orbit projection, and(3) there is an open covering of B such that to each set U in that covering corresponds

a map ϕU : π−1(U)! U ×G with

ϕU (q) = (π(q), sU (q)), sU (g · q) = g · sU(q), ∀q ∈ π−1(U).

2.9. SYMPLECTIC REDUCTION 65

The G-valued maps sU are determined by the corresponding ϕU . Condition (3) is called theproperty of being locally trivial.

Remark 2.9.8. The manifold B is usually called the base, the manifold P is called the totalspace, the Lie group G is called the structure group, and the map π is called the projection.We can represent a principal G-bundle by the following diagram:

G P

B

π

Theorem 2.9.9. If a compact Lie group G acts freely on a manifold M , then M/G is amanifold and the map π : M !M/G is a principal G-bundle.

Proof. First, we will show that, for any q ∈ M , the G-orbit through q is a compactembedded submanifold of M diffeomorphic to G. Note that the evaluation map

ev : G×M !M,

(g, q) 7! ev(g, q) := g · qis smooth since the action is smooth. We claim that, for q ∈ M , the map evq provides thedesired embedding. Note that the image of evq is the G-orbit through q. Since the actionof G is free, we get that evq is injective. Clearly, the map evq is proper, since a compact(and hence closed) subset N of M has inverse image (evq)

−1(N) being a closed subset of acompact Lie group G, hence compact. We still have to show that evq is an immersion. Notethat for X ∈ g ∼= TeG we have

de evq(X) = 0 ⇐⇒ X#q = 0 ⇐⇒ X = 0,

since the action is free. Hence, we can conclude that de evq is injective. Thus, at any pointg ∈ G, for X ∈ TgG, we have

dg evq(X) = 0 ⇐⇒ de(evq Rg) dgRg−1(X) = 0,

where Rg : G! G denotes right multiplication by g. On the other hand, evq Rg = evg·q hasan injective differential at the identity e, and dgRg−1 is an isomorphism. This implies thatdg evq is always injective.In fact, one can show that even if the action is not free, the G-orbit through q is a compactembedded submanifold of M . In that case, the orbit is diffeomorphic to the quotient of G bythe isotropy of q, i.e.

Oq∼= G/Gq.

Let S be a transverse section to Oq at q. We call S a slice. Choose coordinates x1, . . . , xncentered at q such that

Oq∼= G : x1 = · · · = xk = 0,

S : xk+1 = · · · = xn = 0.

Let Sǫ := S ∩Bǫ(0,Rn), where Bǫ(0,Rn) denotes the ball of radius ǫ centered at 0 in Rn. Letη : G × S ! M , η(g, s) = g · s. Then we can apply the following equivariant version of thetubular neighborhood theorem:


Theorem 2.9.10 (Slice theorem). Let G be a compact Lie group G acting on a manifold Msuch that G acts freely at q ∈ M . For sufficiently small ǫ, η : G × Sǫ ! M maps G × Sǫdiffeomorphically onto a G-invariant neighborhood U of the G-orbit through q.

We will use the following corollaries of the Slice theorem:

Corollary 2.9.11. If the action of G is free at q, then the action is free on U .

Corollary 2.9.12. The set of points where G acts freely is open.

Corollary 2.9.13. The set G× Sǫ ∼= U is G-invariant. Hence, the quotient

U/G ∼= Sǫ

is smooth.

Now we can conclude the proof that M/G is a manifold and π : M !M/G is a smooth fibermap. For q ∈ M , let p = π(q) ∈ M/G. Choose a G-invariant neighborhood U of q as inthe slice theorem: U ∼= G × Sǫ. Then π(U) = U/G =: V is an open neighborhood of p in

M/G. By the slice theorem, we get that Sǫ∼−! V is a homeomorphism. We will use such

neighborhoods V as charts onM/G. We want to show that the transition functions associatedwith these charts are smooth. For this, consider two G-invariant open sets U1,U2 in M andcorresponding slices S1, S2 of the G-action. Then S12 := S1 ∩U2 and S21 := S2 ∩U1 are bothslices for the G-action on U1 ∩ U2. To compute the transition map S12 ! S21, consider thediagram

S12 id× S12 G× S12

U1 ∩ U2

S21 id× S21 G× S21

∼

∼

∼

∼

Then the composition

S12 U1 ∩ U2 G× S21 S21∼ pr2

is smooth. Finally, we need to show that π : M ! M/G is a smooth fiber map. For q ∈ Mwith p := π(q) ∈M/G, choose a G-invariant neighborhood U of the G-orbit through q of the

form η : G× Sǫ∼−! U . Then V = U/G ≃ Sǫ is the corresponding neighborhood of p ∈M/G:

M ⊇ U G× Sǫ G× V

M/G ⊇ V Vπ

η−1∼

pr2

id

since the projection pr2 is smooth. it is then easy to check that the transition maps for thebundle defined by π are smooth. We leave this as an exercise.

2.9. SYMPLECTIC REDUCTION 67

2.9.3. The Marsden–Weinstein theorem.

Theorem 2.9.14 (Marsden–Weinstein[MW74]). Let (M,ω,G, µ) be a Hamiltonian G-spacefor a compact Lie group G. Let i : µ−1(0) ! M be the inclusion map. Assume that G actsfreely on µ−1(0). Then

• the orbit space Mred := µ−1(0)/G is a manifold,• π : µ−1(0)!Mred is a principal G-bundle, and• there is a symplectic form ωred on Mred satisfying i∗ω = π∗ωred.

Definition 2.9.15 (Reduction). The pair (Mred, ωred) is called the reduction of (M,ω) withrespect to G and µ.

Remark 2.9.16. The reduction is sometimes also called reduced space, symplectic quotientor the Marsden–Weinstein quotient.

Lemma 2.9.17. Let (V, ω) be a symplectic vector space. Suppose that I is an isotropic subspaceof V . Then ω induces a canonical symplectic form Ω on Iω/I, where Iω denotes the symplecticorthocomplement of I.

Exercise 2.9.18. Prove Lemma 2.9.17.

Proof of Theorem 2.9.14. Note that since G acts freely on µ−1(0), we get that dqµis surjective for all q ∈ µ−1(0) since

imdqµ = Ann(gq) := ξ ∈ g∗ | 〈ξ,X〉 = 0, ∀X ∈ gq,

where Ann denotes the annihilator and gq the Lie algebra of the stabilizer Gq which in thiscase is trivial and hence imdqµ = g∗. Thus, 0 is a regular value and therefore µ−1(0) isa closed submanifold of codimension dimG. The first part of Theorem 2.9.14 is just anapplication of Theorem 2.9.9 to the free action of G on µ−1(0).Lemma 2.9.17 gives a canonical symplectic structure on the quotient Tqµ

−1(0)/TqOq since,by the fact that G acts freely on µ−1(0), we have that Oq

∼= G and thus

Tqµ−1(0) = ker dqµ = (TqOq)

ωq .

One can indeed check that TqOq is an isotropic subspace of TqM . The point [q] ∈ Mred =µ−1(0)/G has tangent space T[q]Mred

∼= Tqµ−1(0)/TqOq. Thus Lemma 2.9.17 defines a non-

degenerate 2-form ωred on Mred. This is well-defined because ω is G-invariant.By construction, we have i∗ω = π∗ωred, where

µ−1(0) M

Mred

i

π

Hence, π∗dωred = dπ∗ωred = di∗ω = i∗dω = 0. The closedness of ωred follows from theinjectivity of π∗.


2.9.4. Noether’s theorem. Let (M,ω,G, µ) be a Hamiltonian G-space.

Theorem 2.9.19 (Noether[Noe18]). A function f : M ! R is G-invariant if and only if µ isconstant along the trajectories of the Hamiltonian vector field of f .

Proof. Let Xf be the Hamiltonian vector field of f . Moreover, let X ∈ g and

µX = 〈µ,X〉 : M ! R.

Then we have

(2.9.1) LXfµX = ιXf

dµX = ιXfιX#ω = −ιX#ιXf

ω = −ιX#df = −LX#f = 0.

because f is G-invariant.

Definition 2.9.20 (Integral of motion/Symmetry). A G-invariant function f : M ! R iscalled an integral of motion of the Hamiltonian G-space (M,ω,G, µ). If µ is constant alongthe trajectories of a Hamiltonian vector field Xf , then the corresponding 1-parameter groupof diffeomorphisms (exp tXf )t∈R is called a symmetry of (M,ω,G, µ).

Remark 2.9.21. The Noether principle asserts that there is a one-to-one correspondencebetween symmetries and integrals of motion.

2.10. The Duistermaat–Heckman theorems

2.10.1. Duistermaat–Heckman polynomial. Let (M,ω) be a 2n-dimensional sym-plectic manifold and consider its Liouville volume ωn

n! .

Definition 2.10.1 (Liouville measure). The Liouville measure (or symplectic measure) of aBorel subset U ⊂M is given by

mω(U) =∫

U

ωn

n!.

Let G be a torus and suppose that (M,ω,G, µ) is a Hamiltonian G-space such that themoment map µ is proper. Moreover denote by g the Lie algebra of G.

Definition 2.10.2 (Duistermaat–Heckman measure). The Duistermaat–Heckman measure,denoted by mDH, on g∗ is the pushforward of mω by µ : M ! g∗. That is,

mDH(U) := (µ∗mω)(U) =

∫

µ−1(U)

ωn

n!

for any Borel subset U ⊂ g∗.

Remark 2.10.3. For a compactly supported function h ∈ C∞(g∗), we can define its integralwith respect to the Duistermaat–Heckman measure by∫

g∗hdmDH :=

∫

M(h µ)ω

n

n!.

On g∗, regarded as a vector space Rn, there is also the Lebesgue measure m0. The relationbetween mDH and m0 is given through the Radon–Nikodym derivative, denoted by dmDH

dm0,

which is a generalized function such that∫

g∗hdmDH =

∫

g∗

dmDH

dm0dm0.

2.10. THE DUISTERMAAT–HECKMAN THEOREMS 69

Theorem 2.10.4 (Duistermaat–Heckman [DH82]). The Duistermaat–Heckman measure is apiecewise polynomial multiple of the Lebesgue measure m0 on g∗ ∼= Rn. That is, the Radon–Nikodym derivative

f :=dmDH

dm0

is piecewise polynomial. More precisely, for any Borel subset U ⊂ g∗, we have

mDH(U) =

∫

Uf(x)dx,

where dx := dm0(x) denotes the Lebesgue volume form on U and f : g∗ ∼= Rn ! R ispolynomial on any region consisting of regular values of µ.

Remark 2.10.5. The Radon–Nikodym derivative f is also called the Duistermaat–Heckmanpolynomial.

Example 2.10.6. Consider the Hamiltonian S1-space (S2, ω = dθ ∧ dh, S1, µ = h). Theimage of µ is given by the interval [−1, 1]. The Lebesgue measure of [a, b] ⊆ [−1, 1] is givenby

m0([a, b]) = b− a.

The Duistermaat–Heckman measure of [a, b] is given by

mDH([a, b]) =

∫

(θ,h)∈S2 | a≤h≤bdθ ∧ dh = 2π(b− a).

Remark 2.10.7. As a consequence, one can show that the spherical area between two hori-zontal circles depends only on the vertical distance between them. This result was actuallyalready known by Archimedes.

Corollary 2.10.8. For the standard Hamiltonian action of S1 on (S2, ω), we have

mDH = 2πm0.

2.10.2. Local form for reduced spaces. Let (M,ω,G, µ) be a Hamiltonian G-space,where G is an n-torus. Assume that µ is proper. If G acts freely on µ−1(0), then is also actsfreely on nearby levels µ−1(t) for t ∈ g∗ and t near to 0 (which we will denote by t ≈ 0).Consider the reduced spaces

Mred := µ−1(0)/G, Mt := µ−1(t)/G

with reduced symplectic forms ωred and ωt. We want to see what the relation between thesereduced spaces when regarded as symplectic manifolds. For simplicity, we want to assumethat G is the circle S1. Let Z := µ−1(0) and let i : Z ! M be the inclusion. Fix a formα ∈ Ω1(Z) for the principal bundle

S1 Z

Mred

π


which means that LX#α = 0 and ιX#α = 1, where X# is the infinitesimal generator for theS1-action. Using α, we construct a 2-form on the product manifold Z × (−ε, ε) by

σ := π∗ωred − d(xα),

where x is a linear coordinate on the interval (−ε, ε) ⊂ R ∼= g∗. We want to abuse notation toshorten the symbols for forms on Z × (−ε, ε) which are given by pullback via the projectiononto each factor.

Lemma 2.10.9. The 2-form σ is symplectic for ε small enough.

Proof. Clearly, σ is closed. Note that at x = 0, we have

σ∣∣x=0

= π∗ωred + α ∧ dx,

which satisfies

σ∣∣x=0

(X#,

∂

∂x

)= 1,

and thus σ is nondegenerate along Z × 0. Since nondegeneracy is an open condition, wecan conclude that σ is nondegenerate for x in a sufficiently small neighborhood of 0.

Remark 2.10.10. Note that σ is invariant with respect to the S1-action on the first factor ofZ×(−ε, ε). Actually, this S1-action is Hamiltonian with moment map given by the projectiononto the second factor

x : Z × (−ε, ε)! (−ε, ε),which can be easily shown by

ιX#σ = −ιX#d(xα) = −LX#(xα) + dιX#(xα) = dx.

Lemma 2.10.11. There is an equivariant symplectomorphism between a neighborhood of Z inM and a neighborhood of Z × 0 in Z × (−ε, ε), intertwining the two moment maps, for εsmall enough.

Proof. The inclusion i : Z ! Z×(−ε, ε) as Z×0 and the natural inclusion i : Z !Mare S1-equivariant coisotropic embeddings. They actually satisfy i∗0σ = i∗ω since both sidesare equal to π∗ωred, and the moment maps coincide on Z since i∗0x = 0 = i∗µ. If we replaceε by a smaller positive number whenever necessary, the result follows from an equivariantversion of the coisotropic embedding theorem (Theorem 2.6.3).

Hence, in order to compare the reduced spaces

Mt = µ−1(t)/S1, t ≈ 0,

we work with Z × (−ε, ε) and compare instead the reduced spaces

x−1(t)/S1, t ≈ 0.

Proposition 2.10.12. The reduced space (Mtωt) is symplectomorphic to

(Mred, ωred − tβ),

where β is the curvature of the connection 1-form α.


Proof. Using Lemma 2.10.11, we can see that (Mt, ωt) is symplectomorphic to the re-duced space at level t for the Hamiltonian S1-space (Z × (−ε, ε), σ, S1, x). Since x−1(t) =Z × t, where S1 acts on the first factor, all manifolds x−1(t)/S1 are diffeomorphic toZ/S1 =Mred. For the symplectic forms, let ιt : Z ×0 ! Z × (−ε, ε) be the inclusion map.The restriction of σ to Z × t is

ι∗tσ = π∗ωred − tdα.

By definition of the curvature, we have dα = π∗β. Hence, the reduced symplectic form onx−1(t)/S1 is given by ωred − tβ.

Remark 2.10.13. Informally, Proposition 2.10.12 says that the reduced forms ωt vary linearlyin t, for t ≈ 0. On the other hand, the identification of Mt with Mred as abstract manifoldsis not natural. Nevertheless, any two such identifications are isotopic. Using the homotopyinvariance of the de Rham cohomology classes, we can obtain the following theorem.

Theorem 2.10.14 (Duistermaat–Heckman[DH82]). The cohomology class of the reduced sym-plectic form [ωt] varies linearly in t. More specifically, we have

[ωt] = [ωred] + tc,

where c = [−β] ∈ H2(Mred) is the first Chern class of the S1-bundle Z !Mred.

Remark 2.10.15. A connection on a principal bundle is given by a Lie algebra-valued 1-form.We can identify the Lie algebra S1 with 2πiR by using the exponential map exp: g ∼= 2πiR!S1, ξ 7! exp(ξ). If we consider a principal S1-bundle, this identification leads to the fact thatthe infinitesimal action maps the generator 2πi of 2πiR to the generating vector field X#.If we have a connection 1-form A, it is then an imaginary-valued 1-form on the total spacesatisfying LX#A = 0 and ιX#A = 2πi. Its curvature 2-form B is then an imaginary-valued2-form on the base satisfying π∗B = dA. By the Chern–Weil isomorphism [Che52; Wei49;BT82], the first Chern class of the principal S1-bundle is c =

[i2πB

]. However, we want to

identify the Lie algebra S1 with R and implicitly use the exponential map exp: g ∼= R! S1,t 7! exp(2πit). Thus, if we consider a principal S1-bundle, the infinitesimal action maps thegenerator 1 of R to X#, where now a connection 1-form α is an ordinary 1-form on the totalspace satisfying LX#α = 0 and ιX#α = 1. The curvature β is now an ordinary 2-form onthe base satisfying π∗β = dα. This implies that we have A = 2πiα, B = 2πiβ and the firstChern class c = [−β].

2.10.3. Variation of the symplectic volume. Consider a Hamiltonian S1-space

(M,ω, S1, µ)

of dimension 2n and let (Mx, ωx) be its reduced space at level x. Proposition 2.10.12 andTheorem 2.10.14 tell us that for x in a sufficiently narrow neighborhood of 0, the symplecticvolume of Mx given by

vol(Mx) :=

∫

Mx

ωn−1x

(n− 1)!=

∫

Mred

(ωred − xβ)n−1

(n− 1)!,

is a polynomial in x of degree n− 1. This volume can be also expressed as

vol(Mx) =

∫

Z

π∗(ωred − xβ)n−1

(n− 1)!∧ α.


Here, α is a chosen connection 1-form on the S1-bundle Z ! Mred and β is its curvature2-form. The Duistermaat–Heckman measure for a Borel subset U ⊂ (−ε, ε) is

mDH(U) =

∫

µ−1(U)

ωn

n!.

Note that (µ−1((−ε, ε)), ω) is symplectomorphic to (Z × (−ε, ε), σ) and moreover they areisomorphic as Hamiltonian S1-spaces. Thus, we can obtain

mDH(U) =

∫

Z×U

σn

n!.

Since σ = π∗ωred − d(xα), we can express its n-th power by

σn = n(π∗ωred − xdα)n−1 ∧ α ∧ dx.

Using Fubini’s theorem, we get

mDH(U) =

∫

U

(∫

Z


(n− 1)!∧ α)∧ dx.

Hence, the Radon–Nikodym derivative of mDH with respect to the Lebesgue measure dx isgiven by

f(x) =

∫

Z


(n− 1)!∧ α = vol(Mx).

Remark 2.10.16. Note that the previous discussion, for x ≈ 0, f(x) is indeed polynomial inx. This actually also holds for a neighborhood of any other regular value of µ, since we canchange the moment map µ by an arbitrary additive constant.

Exercise 2.10.17 (Equivariant cohomology). Let M be a manifold with an S1-action andX# the vector field on M generated by S1. The algebra of S1-equivariant forms on M is thealgebra of S1-invariant forms on M tensored with complex polynomials in x, i.e.

Ω•S1(M) := (Ω•(M))S

1 ⊗R C[x].

Note that the product ∧ on Ω•S1(M) combines the wedge product on Ω•(M) with the product

of polynomials on C[x].

(1) We grade Ω•S1(M) by adding the usual grading on Ω•(M) to a grading on C[x] where

the monomial x has degree 2. Show that (Ω•S1(M),∧) is a supercommutative graded

algebra, i.e.

α ∧ β = (−1)deg αdeg ββ ∧ α, ∀α, β ∈ Ω•S1(M).

(2) On Ω•S1(M) we define an operator

dS1 := d⊗ 1− ιX# ⊗ x.

This means that for an element α = α⊗ p(x) we have

dS1α = dα⊗ p(x)− ιX#α⊗ xp(x).


The operator dS1 is called Cartan differentiation. Show that dS1 is a subderivationof degree 1, i.e. show that it increases the degree by 1 and that it satisfies the superLeibniz rule

dS1(α ∧ β) = (dS1α) ∧ β + (−1)deg αα ∧ dS1β.

(3) Show that (dS2)2 = 0. Hint: Use Cartan’s magic formula.

We can hence deduce that the sequence

0! Ω0S1(M)

dS1−−! Ω1

S1(M)dS1−−! Ω2

S1(M)dS1−−! · · ·

is a graded complex whose cohomology is usually called equivariant cohomology. The equivari-ant cohomology of a topological space M endowed with a continuous action of a topologicalgroup G is the cohomology of the diagonal quotient (M × EG)/G, where EG denotes theuniversal bundle of G, i.e. EG is a contractible space where G acts freely. Cartan showedthat, for the action of a compact Lie group G on some manifold M , the de Rham complex(Ω•

G(M),dG) computes the equivariant cohomology, where Ω•G(M) denotes the G-equivariant

forms on M . For equivariant cohomology in the symplectic context see also [AB84]. Thek-th equivariant cohomology group is given by

HkS1(M) :=

ker dS1 : ΩkS1(M)! Ωk+1S1 (M)

imdS1 : Ωk−1S1 (M)! Ωk

S1(M)

(4) What is the equivariant cohomology of a point?(5) What is the equivariant cohomology of S1 with its multiplication action on itself?(6) Show that the equivariant cohomology of a manifold M with free S1-action is iso-

morphic to the ordinary cohomology of the quotient space M/S1.Hint: Consider the projection π : M !M/S1.Show that π∗ : H•(M/S1)! H•

S1(M), [α] 7! [π∗α⊗1] is a well-defined isomorphism.

Choose a connection on the principal S1-bundle M !M/S1, i.e. a 1-form θ on Msuch that LX#θ = 0 and ιX#θ = 1. Recall also that a form β on M is of type π∗αfor some α if and only if it is basic, i.e. LX#β = 0 and ιX#β = 0.

(7) Let (M,ω) be a symplectic manifold with an S1-action. Moreover, let µ ∈ C∞(M)be a real function. Consider the equivariant form

ω := ω ⊗ 1 + µ⊗ x.

Show that ω is equivariantly closed, i.e. dS1ω = 0, if and only if µ is a moment map.The equivariant form ω is called the equivariant symplectic form.

(8) Let M be a 2n-dimensional compact oriented manifold with an S1-action. Suppose

that the set MS1consisting of fixed points for the S1-action is finite. Consider an

S1-invariant form α(2n) which is the top degree part of an equivariantly closed form

of even degree, i.e. α(2n) ∈ Ω2n(M)S1is such that there is some α ∈ Ω•

S1(M) with

α = α(2n) + α(2n−2) + · · ·+ α(0), α(2k) ∈ (Ω2k(M))S1 ⊗R C[x], dS1α = 0.

(a) Show that the restriction of α(2n) to M \MS1is exact.

Hint: The generator X# of the S1-action does not vanish on M \MS1. Thus,

we can define a connection on M \MS1by θ(Y ) = 〈Y,X#〉

〈X#,X#〉, where 〈 , 〉 is


some S1-invariant metric on M . Use θ ∈ Ω1(M \MS1) to get the primitive of

α(2n) by starting at α(0).(b) Compute the integral of α(2n) over M .

Hint: Use Stokes’ theorem to localize the answer near the fixed points.This exercise is a very particular case of the Atiyah–Bott localization theorem forequivariant cohomology [AB84].

(9) What is the integral of the symplectic form ω on a surface with a Hamiltonian S1-action which is free outside a finite set of fixed points?Hint: Use (7) and (8).

CHAPTER 3

Poisson Geometry

Poisson geometry combines methods of differential geometry and noncommutative geometry.Motivated by the dynamical structure induced from the setting of classical mechanics, it con-nects to the notion of symplectic geometry and provides the actual mathematical structure totalk about deformation quantization. In fact, as we will see, the phase space can be regardedas a Poisson manifold with Poisson structure coming from the canonical symplectic formsince every symplectic manifold naturally induces a Poisson manifold. In this chapter wewill introduce Poisson manifolds, Lie algebroids, Courant algebroids, Dirac manifolds, thelocal behaviour of Poisson manifolds and the induced symplectic foliation. Finally, we willintroduce Poisson maps in the regular sense. We will not cover the notion of Morita equiv-alence [Mor58], which gives another construction for morphisms between Poisson manifolds,but we refer to [Xu91; Xu04; GL92; JSW02; Sch98] for the interested reader. This chapteris mainly based on [Wei83; GW92; BC05; Con95; Cat04; GG01; DZ05; Cou90a; Cou90b;BW04; GRS05]. Let P denote a manifold for the entire chapter.

3.1. Poisson manifolds

3.1.1. Poisson structures and the Schouten–Nijenhuis bracket.

Definition 3.1.1 (Poisson structure). A Poisson structure on P is an R-bilinear Lie bracket , on C∞(P ) satisfying the Leibniz rule

f, gh = f, gh+ gf, h, ∀f, g, h ∈ C∞(P ).

Definition 3.1.2 (Casimir function). A function f ∈ C∞(P ) is called Casimir if the Hamil-tonian vector field Xf = f, of f vanishes.

Remark 3.1.3. By the Leibniz rule of the Poisson bracket , , there exists a bivector field

π ∈ X2(P ) := Γ(∧2 TP ) such that

f, g = π(df,dg).

Definition 3.1.4 (Schouten–Nijenhuis bracket). The Schouten–Nijenhuis bracket is a uniqueextension of the Lie bracket of vector fields to a graded bracket of multivector fields. It isdefined by

(3.1.1)

[X1 ∧ · · · ∧Xm, Y1 ∧ · · · ∧ Yn] :=∑

1≤i≤m1≤j≤m

(−1)i+j [Xi, Yj]X1 ∧ · · · ∧Xi−1 ∧Xi+1 ∧ · · · ∧Xm∧

∧ Y1 ∧ · · · ∧ Yj−1 ∧ Yj+1 ∧ · · · ∧ Yn75

76 3. POISSON GEOMETRY

for vector fields X1, . . . ,Xm, Y1, . . . , Yn and by

[f,X1 ∧ · · · ∧Xm] := −ιX1∧···∧Xmdf

for a function f .

Exercise 3.1.5. Show that the Schouten–Nijenhuis bracket satisfies the graded Jacobi iden-tity:

(−1)(|X|−1)(|Z|−1)[X, [Y,Z]] + (−1)(|Y |−1)(|X|−1)[Y, [Z,X]] + (−1)(|Z|−1)(|Y |−1)[Z, [X,Y ]] = 0,

where | | denotes the degree operation, i.e. |X| = k if X ∈ Xk(P ).

Exercise 3.1.6. Show that the Jacobi identity for , is equivalent to the condition

[π, π] = 0.

See Section 4.2.4.1 for the computation.

In local coordinates (x1, . . . , xn) we can determine the components of the bivector field π as

πij(x) = xi, xj.Definition 3.1.7 (Poisson manifold). A pair (P, π), where π is a Poisson bivector field onP , is called a Poisson manifold.

Definition 3.1.8 (Symplectic Poisson structure). If the bivector field π is invertible at eachpoint x, it is called nondegenerate or symplectic.

Remark 3.1.9. Note that, if π is symplectic, then we can define a symplectic form on P .Indeed, we can locally define the matrices

(ωij) = (−πij)−1,

which defines globally a 2-form ω ∈ Ω2(P ). Moreover, the condition [π, π] = 0 implies thatdω = 0.

3.1.2. Examples of Poisson structures.

Example 3.1.10. Constant Poisson structure Let P = Rn and suppose that πij(x) is constant.Then we can find coordinates on P

(q1, . . . , qk, p1, . . . , pk, e1, . . . , eℓ), 2k + ℓ = n,

such that

π =∑

1≤i≤k

∂

∂qi∧ ∂

∂pi.

Expressing it in terms of a bracket, we get

f, g =∑

1≤i≤k

(∂f

∂qi

∂g

∂pi− ∂f

∂pi

∂g

∂qi

).

Note that this agrees with the usual Poisson bracket on C∞(T ∗Rn) in Hamiltonian mechanics(see Section 2.7.2). Note that all coordinates ej for 0 ≤ j ≤ ℓ are Casimirs with respect to , .

3.2. DIRAC MANIFOLDS 77

Example 3.1.11 (Poisson structures on R2). Let P = R2 and consider a smooth functionf ∈ C∞(P ). Such a function f induces a Poisson structure on P by

x1, x2 := f(x1, x2).

Moreover, any Poisson structure on R2 is of this form.

Example 3.1.12 (Lie-Poisson structure). Let P be a finite-dimensional vector space V withcoordinates (x1, . . . , xn). We can define a linear Poisson structure by

xi, xj :=∑

1≤k≤n

ckijxk,

where ckij are determined constants with ckij = −ckji. Such a Poisson structure is usually

called Lie-Poisson structure, since the Jacobi identity of , implies that the ckij are thestructure constants of a Lie algebra g, which might be identified naturally with V ∗. Thus,we might also identify V ∼= g∗. Conversely, any Lie algebra g with structure constants ckijdefines through , a linear Poisson structure on g∗.

Remark 3.1.13. Deformation quantization of a Lie-Poisson structure on g∗ leads to thedefinition of the universal enveloping algebra U(g). The elements in the center of U(g)are usually called Casimir elements. These exactly correspond to the center of the Poissonalgebra of functions on g∗, and hence, by extension, the Casimir functions for the center ofany Poisson algebra.

3.2. Dirac manifolds

3.2.1. Courant algebroids. We want to formulate a generalization of Poisson struc-tures and closed 2-forms.

Definition 3.2.1 (Presymplectic form). A closed 2-form is called presymplectic.

Note that each 2-form ω on P corresponds to a bundle map

ω : TP ! T ∗P,

v 7! ω(v) := ω(v, ).(3.2.1)

We can define a similar bundle map for a bivector field π ∈ X2(P ) by

π : T ∗P ! TP,

α 7! π(α) := π( , α).(3.2.2)

such that ιπ(α)β = β(π(α)) = π(β, α). The matrix representing π in the basis (dxi) and ( ∂∂xi

)

for local coordinates (x1, . . . , xn) of P , is (up to a sign) given by

πij(x) = x1, xj.Thus, bivector fields (or 2-forms) are nondegenerate if and only if the associated bundle mapsare invertible. Using these bundle maps, we can describe closed 2-forms and Poisson bivectorfields as a subbundles of TP ⊕ T ∗P . Indeed, consider the graphs

Lω := Γω = graph ω,

Lπ := Γπ = graph π.


Let us introduce the following canonical structure on TP ⊕ T ∗P :

(1) The symmetric bilinear form

〈 , 〉+ : TP ⊕ T ∗P × TP ⊕ T ∗P ! R,

((X,α), (Y, β)) 7! 〈(X,α), (Y, β)〉+ := α(Y ) + β(X).

(2) The bracket

[[ , ]] : Γ(TP ⊕ T ∗P )× Γ(TP ⊕ T ∗P )! Γ(TP ⊕ T ∗P ),

((X,α), (Y, β)) 7! [[(X,α), (Y, β)]] := ([X,Y ], LXβ − ιY dα).

Definition 3.2.2 (Lie algebroid). A Lie algebroid is a vector bundle E endowed with a Liebracket [ , ] on the space of sections Γ(E) together with an anchor map ρ : E ! TP suchthat for all X,Y ∈ Γ(E) and f ∈ C∞(P )

(1) [X, fY ] = ρ(X)(f) · Y + f [X,Y ],(2) ρ([X,Y ]) = [ρ(X), ρ(Y )].

Here ρ(X)(f) = Lρ(X)f denotes the derivative of f along the vector field ρ(X).

Definition 3.2.3 (Courant algebroid). A Courant algebroid is a vector bundle E equippedwith a nondegenerate symmetric bilinear form 〈 , 〉 : E×E ! R, a bilinear bracket [ , ] : Γ(E)×Γ(E)! Γ(E) and a bundle map ρ : E ! TP satisfying the following properties:

(1) [X, [Y,Z]] = [[X,Y ], Z] + [Y, [X,Z]] (left Jacobi identity),(2) ρ([X,Y ]) = [ρ(X), ρ(Y )],(3) [X, fY ] = f [X,Y ] + ρ(X)(f) · Y , (Leibniz rule)(4) [X,X] = 1

2D〈X,X〉, where

D := ρ∗d: C∞(P )d−! Ω1(P )

ρ∗−! E∗ ∼= E.

(5) ρ(X)〈Y,Z〉 = 〈[X,Y ], Z〉+ 〈Y, [X,Z]〉, (self-adjoint)Definition 3.2.4 (Split Courant algebroid). The bundle E := TP ⊕ T ∗P together with thebracket 〈 , 〉+ and [[ , ]] is an example of a Courant algebroid which we call the split Courantalgebroid.

3.2.2. Dirac structures.

Proposition 3.2.5. A subbundle L ⊂ TP ⊕ T ∗P is of the form Lπ (resp. Lω) for a bivectorfield π (resp. 2-form ω) if and only if

(1) TP ∩ L = 0 (resp. L ∩ T ∗P = 0) at all points of P ,(2) L is maximal isotropic with respect to 〈 , 〉+,

Moreover, [π, π] = 0 (resp. dω = 0) if and only if

(3) Γ(L) is closed under the Courant bracket [[ , ]], i.e.

[[Γ(L),Γ(L)]] ⊂ Γ(L).

Definition 3.2.6 (Dirac structure). A Dirac structure on P is a subbundle L ⊂ TP ⊕ T ∗Pwhich is maximal isotropic with respect to 〈 , 〉+ and whose sections are closed under theCourant bracket [[ , ]].

3.2. DIRAC MANIFOLDS 79

Definition 3.2.7 (Dirac manifold). A tuple (P,L), where L is a Dirac structure on P iscalled a Dirac manifold.

Remark 3.2.8. Dirac structures are exactly those which satisfy conditions (2) and (3) ofProposition 3.2.5, but do not necessarily appear as the graph of some bivector field or 2-form.

Definition 3.2.9 (Almost Dirac structure). If L only satisfies condition (2) of Proposition3.2.5, it is called an almost Dirac structure.

Remark 3.2.10. Usually, condition (3) of Proposition 3.2.5 is referred to as the integrabilitycondition of a Dirac structure.

Example 3.2.11 (Regular foliations). Let F ⊆ TP be a subbundle, and let Ann(F ) ⊂ T ∗Pbe its annihilator. Then L = F ⊕Ann(F ) is an almost Dirac structure. It is a Dirac structureif and only if F satisfies the Frobenius condition (see Theorem 1.6.15)

[Γ(F ),Γ(F )] ⊂ Γ(F ).

Hence, regular foliations are examples of Dirac structures.

Example 3.2.12 (Linear Dirac structure). If V is a finite-dimensional real vector space, thena linear Dirac structure on V is a subspace L ⊂ V ⊕ V ∗ which is maximal isotropic withrespect to the symmetric pairing 〈 , 〉+.Let L be a linear Dirac structure on V . Let pr1 : V ⊕ V ∗

! V and pr2 : V ⊕ V ∗! V ∗ be the

canonical projections, and consider the subspace

W := pr1(L) ⊂ V.

Then L induces a skew-symmetric bilinear form θ on W defined by

(3.2.3) θ(X,Y ) := α(Y ),

where X,Y ∈W and α ∈ V ∗ such that (X,α) ∈ L.

Exercise 3.2.13. Show that θ is well-defined, i.e. (3.2.3) is independent of the choice of α.

Conversely, any pair (W, θ), where W ⊆ V is a subspace and θ is a skew-symmetric bilinearform W , defines a linear Dirac structure by

L := (X,α) | X ∈W, α ∈ V ∗ such that α|W = ιXθ.Exercise 3.2.14. Check that this L is a linear Dirac structure on V with associated subspaceW and bilinear form θ.

Note that Example 3.2.12 induces a simple way in which linear Dirac structures can berestricted to subspaces.

3.2.3. Dirac structures for constrained manifolds.

Example 3.2.15 (Restriction of Dirac structures to subspaces). Let L be a linear Diracstructure on V , let U ⊆ V be a subspace and consider the pair (W, θ) associated to L. ThenU inherits the linear Dirac structure LU from L defined by

WU :=W ∩ U, θU := i∗θ,

where i : U ! V denotes the inclusion.


Exercise 3.2.16. Show that there is a a canonical isomorphism

LU ∼= L ∩ (U ⊕ V ∗)

L ∩Ann(U).

Remark 3.2.17. Let (P,L) be a Dirac manifold, and let i : N ! P be a submanifold. Theconstructions of Example 3.2.15, when applied to TxN ⊆ TxP for all x ∈ P , defines a maximalisotropic subbundle LN ⊂ TN ⊕ T ∗N . However, LN might not be a continuous family ofsubspaces. When LN is a continuous family, it is a smooth bundle which then directlysatisfies the integrability condition (condition (3) of Proposition 3.2.5). Hence, LN defines aDirac structure.

Example 3.2.18 (Moment level sets). Let µ : P ! g∗ be the moment map for a Hamiltonianaction of a Lie group G on a Poisson manifold (P, π). Let ξ ∈ g∗ be a regular value for µ andlet Gξ be the isotropy group at ξ with respect to the coadjoint action, and consider

Q = µ−1(ξ) ! P.

At each point x ∈ Q, we have a linear Dirac structure on TxQ given by

(LQ)x :=Lx ∩ (TxQ⊕ T ∗

xP )

Lx ∩Ann(TxQ).

One can verify that LQ defines a smooth bundle by verifying that Lx∩Ann(TxQ) has constantdimension. In fact, one can show that Lx ∩Ann(TxQ) has constant dimension if and only ifthe stabilizer groups of the Gξ-action on Q have constant dimension, which happens if theGξ-orbits on Q have constant dimension. In this case, LQ is a Dirac structure on Q.

3.3. Symplectic leaves and local structure of Poisson manifolds

3.3.1. Local and regular Poisson structures. Let π be a symplectic Poisson struc-ture on P . Then Darboux’s theorem asserts that, around each point of P , one can findcoordinates (q1, . . . , qk, p1, . . . , pk) such that

π =∑

1≤i≤k

∂

∂qi∧ ∂

∂pi.

The symplectic form is then given by

ω =∑

1≤i≤k

dqi ∧ dpi.

In general, the image of π : T ∗P ! TP defined as in (3.2.2) induces an integrable singulardistribution on P . In fact, P is a disjoint union of leaves O satisfying

TxO = π(T ∗xP ), ∀x ∈ P.

Definition 3.3.1 (Regular Poisson structure). A Poisson structure π is called regular if πhas locally constant rank.

Remark 3.3.2. If the considered Poisson structure is regular, then it defines a foliation inthe ordinary sense. Note that one can always find an open dense subset of P where this is infact the case. We call this the regular part of P .

3.3. SYMPLECTIC LEAVES AND LOCAL STRUCTURE OF POISSON MANIFOLDS 81

Locally, by using the Darboux theorem, if π has constant rank k around a given point, thereexist coordinates (q1, . . . , qk, p1, . . . , pk, e1 . . . , eℓ) such that

qi, pj = δij , qi, qj = pi, pj = qi, ej = pi, ej = 0.

3.3.2. Local splitting and symplectic foliation.

Theorem 3.3.3 (Local splitting theorem). Around any point x0 of a Poisson manifold (P, π)there exist local coordinates

(q1, . . . , qk, p1, . . . , pk, e1, . . . , eℓ), (q, p, e)(x0) = (0, 0, 0)

such that

(3.3.1) π =∑

1≤i≤k

∂

∂qi∧ ∂

∂pi+

1

2

∑

1≤i,j≤ℓ

ηij(e)∂

∂ei∧ ∂

∂ej

with ηij(0) = 0.

Remark 3.3.4. We have a symplectic factor in the splitting (3.3.1) associated to the co-ordinates (qi, pi) and a factor where all Poisson brackets vanish at e = 0 associated to thecoordinates (ej). The latter factor is often called the totally degenerate factor. We can iden-tify the symplectic factor with an open subset of the leaf Ox0 through x0. If we consider thefoliation given by the collection of all leaves

P =⊔

x∈P

Ox

we see that π canonically defines a singular foliation of P by symplectic leaves, since π inducesa symplectic structure on each leaf. One usually refers to the totally degenerate factor, whichis locally well-defined up to isomorphism, as the transverse structure to π along a given leaf.

Example 3.3.5 (Symplectic leaves of Poisson structures on R2). Let f : R2! R be a smooth

function, and consider the Poisson structure on R2 defined by

x1, x2 := f(x1, x2).

The connected components of the set Sf := (x1, x2) ∈ R2 | f(x1, x2) 6= 0 are the 2-dimensional symplectic leaves. Note that in the set Sf , each point is a symplectic leaf.

Example 3.3.6 (Symplectic leaves of Lie-Poisson structures). Consider a Lie algebra g withdual g∗ equipped with its Lie-Poisson structure as in Example 3.1.12. The symplectic leavesare then the coadjoint orbits for any connected Lie group with Lie algebra g. Since 0 isalways an orbit, a Lie-Poisson structure is not regular unless g is commutative.

Exercise 3.3.7. Describe the symplectic leaves in the duals of the Lie algebras su(2) andsl(2,R).

Remark 3.3.8 (Linearization problem). Linearizing the functions ηij as in Theorem 3.3.3 atx0, we can write

ei, ej =∑

1≤k≤ℓ

ckijek +O(e2).

Thus, it turns out that ckij defines a Lie-Poisson structure on the normal space to the sym-plectic leaf at x0. The linearization problem consists of determining whether one can choose


suitable transverse coordinates (e1, . . . , eℓ) with respect to which O(e2) vanishes. If the Liealgebra structure on the conormal bundle to a symplectic leaf determined by linearization ofπ at a point x0 is semi-simple and of compact type, then π is linearizable around x0 througha smooth change of coordinates.

3.4. Poisson maps

3.4.1. Two definitions.

Definition 3.4.1 (Poisson map (version I)). Let (P1, π1) and (P2, π2) be Poisson manifolds.A smooth map ψ : P1 ! P2 is a Poisson map if ψ∗ : C∞(P2)! C∞(P1) is a homomorphismof Poisson algebras, i.e.

ψ∗f, g2 = ψ∗f, ψ∗g1, ∀f, g ∈ C∞(P2).

Equivalently, we can reformulate Definition 3.4.1 in terms of Poisson bivector fields andHamiltonian vector fields.

Definition 3.4.2 (Poisson map (version II)). Let (P1, π1) and (P2, π2) be Poisson manifolds.A smooth map ψ : P1 ! P2 is a Poisson map if and only if either of the following twoequivalent conditions hold:

(1) ψ∗π1 = π2, i.e. π1 and π2 are ψ-related.(2) Xf = ψ∗(Xψ∗f ) for all f ∈ C∞(P2).

Remark 3.4.3. Condition (2) of Definition 3.4.2 shows that trajectories of Xψ∗f project tothose of Xf if ψ is a Poisson map. However, Xf being complete does not imply that Xψ∗f iscomplete. Hence, one define a Poisson map ψ : P1 ! P2 to be complete if for all f ∈ C∞(P2)such that Xf is complete, then Xψ∗f is complete.

3.4.2. Examples of Poisson maps.

Example 3.4.4 (Complete functions). Consider R as a Poisson manifold endowed with thezero Poisson structure. Then any map f : P ! R is a Poisson map, which is complete if andonly if Xf is a complete vector field.

Exercise 3.4.5. Find the Poisson manifolds for which the set of complete functions is closedunder addition.

Example 3.4.6 (Open subsets of symplectic manifolds). Let (P, π) be a symplectic manifold,and let U ⊆ P be an open subset. Then the inclusion U ! P is complete if and only if U isclosed. More, generally, the image of a complete Poisson map is a union of symplectic leaves.

Exercise 3.4.7. Show that the inclusion of every symplectic leaf in a Poisson manifold is acomplete Poisson map.

Exercise 3.4.8. let P1 be a Poisson manifold and let P2 be symplectic. Show that then anyPoisson map ψ : P1 ! P2 is a submersion. Furthermore, if P2 is connected and ψ is complete,then ψ is surjective (assuming that P1 is nonempty).

Remark 3.4.9. Exercise 3.4.8 gives a first hint to the fact that complete Poisson maps withsymplectic target must be fibrations. In fact, if P1 is symplectic and dimP1 = dimP2, thena complete Poisson map ψ : P1 ! P2 is a covering map. In general, a complete Poisson

3.4. POISSON MAPS 83

map ψ : P1 ! P2, where P2 is symplectic, is a locally trivial symplectic fibration with a flatEhresmann connection: the horizontal lift in TxP1 of a vector X ∈ Tψ(x)P2 is defined as

π1((dxψ)∗π−1

2 (X)).

The horizontal subspaces define a foliation whose leaves are coverings of P2, and P1 and ψare completely determined, up to isomorphism, by the holonomy

(3.4.1) π1(P2, x)! Aut(ψ−1(x)),

where π1(P2, x) in (3.4.1) denotes the fundamental group of P2 with base point x.

CHAPTER 4

Deformation Quantization

In this chapter we will consider the mathematical framework needed to understand Kontse-vich’s construction of deformation quantization and more generally the formality theorem.We will start by introducing the general concept of deformation quantization (star products)and answer the existence problem first for the local symplectic case (including a global ap-proach). Then we will move to the more general Poisson case where we build everythingup for the formality theorem which implies the existence of a deformation quantization asan application for a special case. Then we will discuss the construction of the explicit starproduct provided by Kontsevich, using the notion of graphs. At the end, we will also mentionthe formality result using the notion of operads. This chapter is mainly based on [Moy49;Fed94; Kon99; Kon03; Tam03; CI05; GRS05; Cat+05].

4.1. Star products

4.1.1. Formal deformations. Given a commutative ring k, the set of formal powerseries k[[X]] can be thought of polynomials with coefficients in k in some formal parameter Xwith infinite terms, not concerning any convergence issue. One can show that k[[X]] carriesin fact a ring structure. More precisely, one can define k[[X]] as the completion of k[X] byusing a particular metric, which endows k[[X]] with the structure of a topological ring.Let A be a commutative associative algebra with unit over some commutative base ring kand let ~ denote a formal parameter1.

Definition 4.1.1 (Formal deformation). A formal deformation of A is the algebra A[[~]] offormal power series over the ring k[[~]] of formal power series.

Remark 4.1.2. Elements of the deformed algebra A[[~]] are of the form

C =∑

i≥0

ci~i, ci ∈ A

and the product between such elements is given by the Cauchy formula∑

i≥0

ai~i

•~

∑

j≥0

bj~j

=

∑

r≥0

∑

ℓ≥0

ar−ℓbℓ

~r.

1In the introduction we have denoted the formal parameter by ǫ. We want to use ~ instead to emphasize thereduced Planck constant from the physics literature, which takes the role of a (small) deformation parameterin this theory. More precisely, the deformation parameter is usually given by i~

2.

85

86 4. DEFORMATION QUANTIZATION

Definition 4.1.3 (Star product (general)). A star product is a k[[~]]-linear associative product⋆ on A[[~]] which deforms the trivial extension •~ : A[[~]]⊗k[[~]] A[[~]]! A[[~]] in the sense thatfor any two v,w ∈ A[[~]] we have

v ⋆ w = v •~ w, mod ~.

Remark 4.1.4. In fact, the star product is a formal noncommutative deformation of theusual pointwise product for the case where A = C∞(P ) for some Poisson manifold (P, π) andk = R.

Definition 4.1.5 (Star product). A star product on a Poisson manifold (P, π) is an R[[~]]-bilinear map

⋆ : C∞(P )[[~]] × C∞(P )[[~]]! C∞(P )[[~]]

(f, g) 7! f ⋆ g(4.1.1)

such that

(1) f ⋆ g = fg +∑

i≥1Bi(f, g)~i,

(2) (f ⋆ g) ⋆ h = f ⋆ (g ⋆ h), ∀f, g, h ∈ C∞(P ),(3) 1 ⋆ f = f ⋆ 1 = f , ∀f ∈ C∞(P ),

Remark 4.1.6. The Bi are bidifferential operators on C∞(P ) of globally bounded order. Wecan write

Bi(f, g) =∑

K,L

βKLi ∂Kf∂Lg,

where we sum over all multi-indices K = (k1, . . . , km) and L = (ℓ1, . . . , ℓn) for some lengthsm,n ∈ N. The βKLi are smooth functions which are non-zero only for finitely many choicesof the multi-indices K and L.

Remark 4.1.7. A star product on A[[~]] is sometimes also called a formal deformation ofA ⊂ A[[~]].

4.1.2. Moyal product. Consider the standard symplectic manifold (R2n, ω0) endowedwith the canonical symplectic form ω0 regarded as a Poisson manifold with Poisson structureinduced by ω0. Moreover, choose Darboux coordinates (q, p) = (q1, . . . , qn, p1, . . . , pn) on R2n.

Definition 4.1.8 (Moyal product[Moy49]). TheMoyal product is the star product on C∞(R2n)defined by

(4.1.2) f ⋆ g := f(q, p) exp

(i~

2

( −

∂ q−!

∂ p − −

∂ p−!

∂ q

))g(q, p), f, g ∈ C∞(R2n),

where −

∂ denotes the derivative on f and−!

∂ the derivative on g.

Remark 4.1.9. Note that in (4.1.2) the formal parameter is actually replaced by ~ ! i~2 ,

which is typical in the physics literature, especially in quantum field theory.

If we consider Rd regarded as a Poisson manifold endowed with a constant Poisson structureπ, we can define a star product similarly to (4.1.2) by

(4.1.3) f ⋆ g(x) = exp

(i~

2πij

∂

∂xi∂

∂yj

)f(x)g(y)

∣∣∣∣y=x

, f, g ∈ C∞(Rd).

4.1. STAR PRODUCTS 87

Proposition 4.1.10. The star product defined in (4.1.3) is associative for any choice of πij .

Proof.

((f ⋆ g) ⋆ h)(x) = exp

(i~

2πij

∂

∂xi∂

∂zj

)(f ⋆ g)(x)h(z)

∣∣∣∣x=z

= exp

(i~

2πij(∂

∂xi+

∂

∂yi

)∂

∂zj

)exp

(i~

2πkℓ

∂

∂xk∂

∂yℓ

)f(x)g(y)h(z)

∣∣∣∣x=y=z

= exp

(i~

2

(πij

∂

∂xi∂

∂zj+ πkℓ

∂

∂yk∂

∂zℓ+ πmn

∂

∂xm∂

∂yn

))f(x)g(y)h(z)

∣∣∣∣x=y=z

= exp

(i~

2πij

∂

∂xi

(∂

∂yj+

∂

∂zj

))exp

(i~

2πij

∂

∂yk∂

∂zℓ

)f(x)g(y)h(z)

∣∣∣∣x=y=z

= (f ⋆ (g ⋆ h))(x).

4.1.3. Fedosov’s globalization approach. The Moyal product as in (4.1.2) is onlydefined locally on R2n. In [Fed94] Fedosov showed how to obtain a star product on anysymplectic manifold (M,ω) by using a symplectic connection.

Definition 4.1.11 (Symplectic connection). A symplectic connection on a symplectic man-ifold (M,ω) is a torsion-free connection ∇ preserving the tensor (ωij), i.e. ∇iωjℓ = 0 where

∇i denotes the covariant derivative with respect to ∂∂xi

for local coordinates (xi) on M .

Theorem 4.1.12 (Fedosov[Fed94]). Let (M,ω) be a 2n-dimensional symplectic manifold.

Consider the bundle2 W := Sym(T ∗M) and let ∇ be a symplectic connection of the Weylbundle W[[~]] and denote by ⋆ the Moyal product on (R2n, ω0). Then one can define a globalstar product ⋆M on C∞(M) by

f ⋆M g = σ(σ−1(f) ⋆ σ−1(g)),

where3 σ : H0D(W)

∼−! Z with D a flat connection (D2 = 0) on W induced by the symplectic

connection ∇ and Z the center of W.

Remark 4.1.13. Note that taking the center Z of W coincides with the center Z~ of W[[~]]which is given by C∞(M).

Remark 4.1.14. Another approach to globalize the Moyal product was given by DeWilde andLecomte [DL83] using Darboux’s theorem. However, it turns out that Fedosov’s approach ismore important in the sense that it provides a natural extension to the general Poisson case(see Section 4.4).

2We will consider the completed symmetric algebra, denoted by Sym. The completion of the underlyingtensor product is necessary for certain topological reasons. See also Section 4.2.7.

3We have denoted by H0D(W) the space of D-flat sections on W.


4.1.4. Equivalent star products.

Exercise 4.1.15. Check that the Bi of a star product ⋆ are strict bidifferential operators,i.e. there is no term in order zero and

Bi(1, f) = Bi(f, 1) = 0, ∀i ∈ N.

Moreover, check that the skew-symmetric part B−1 of the first bidifferential operator, defined

as

B−1 (f, g) :=

1

2(B1(f, g) −B1(g, f))) ,

satisfies:

• B−1 (f, g) = −B−

1 (g, f),• B−

1 (f, gh) = gB−1 (f, h) +B−

1 (f, g)h,• B−

1 (B−1 (f, g), h) +B−

1 (B−1 (g, h), f) +B−

1 (B−1 (h, f), g) = 0.

Deduce that f, g := B−1 (f, g) is a Poisson structure.

Remark 4.1.16. Exercise 4.1.15 shows that given a star product ⋆, we can always deduce aPoisson structure by the formula

(4.1.4) f, g := lim~!0

f ⋆ g − g ⋆ f

~.

On the other hand we want to consider the following problem: Given a Poisson manifold(P, π), can we define an associative, but possibly noncommutative, product ⋆ on the algebraof smooth functions, which is the pointwise product and such that (4.1.4) is satisfied.

Definition 4.1.17 (Equivalent star products). Two star products ⋆ and ⋆′ on C∞(P ) aresaid to be equivalent if and only if there is a linear operator D : C∞(P )[[~]] ! C∞(P )[[~]] ofthe form

Df := f +∑

i≥1

Dif~i,

such that

(4.1.5) f ⋆′ g = D−1(Df ⋆ Dg),

where D−1 denotes the inverse in the sense of formal power series.

Remark 4.1.18. Note that by the definition of a star product, the Di have to be differentialoperators vanishing on constants.

Lemma 4.1.19. For any equivalence class of star products, there is a representative whosefirst term B1 in the ~-expansion is skew-symmetric.

Proof. Given any star product

f ⋆ g = fg +B1(f, g)~+B2(f, g)~2 + · · ·

one can define an equivalent star product ⋆′ as in (4.1.5) by using a formal differential operator

D = id +D1~+D2~2 + · · ·

4.2. FORMALITY 89

The skew-symmetric condition for B′1 implies that

(4.1.6) D1(fg) = D1fg + fD1g +1

2(B1(f, g) +B1(g, f))

︸︷︷︸=:B+

1 (f,g)

,

which can be checked for polynomials and further be completed to hold to any smoothfunctions on P . If we choose D1 to vanish on linear functions, Equation (4.1.6) describesuniquely how D1 acts on quadratic terms, given by the symmetric part B+

1 of B1. Namely,we get

D1(xixj) = B+

1 (xi, xj) :=

1

2

(B1(x

i, xj) +B1(xj , xi)

),

where (xi) are local coordinates on P . By the associativity of ⋆, we get a well-defined operatorsince it does not depend on the way how we order the factors.

Exercise 4.1.20. Check that D1((xixj)xℓ) = D1(x

i(xjxℓ)).

Remark 4.1.21. A natural problem appearing in this setting concerns classification of starproducts. In fact, it was shown that equivalence classes of star products on a symplecticmanifold (M,ω) are in one-to-one correspondence with elements in H2(M)[[~]] [Del95; GR99].The general construction was formulated by Kontsevich [Kon03]. He constructed an explicitformula for a star product on Rd endowed with any Poisson structure. However, we wantto start with a much more general statement, formulated and also proved by Kontsevich[Kon03], which provides a solution to the existence of deformation quantization for a specialcase as we will see. This general result is called formality.

4.2. Formality

4.2.1. Some formal setup. For a Poisson manifold (P, π), one can define a bracket

(4.2.1) f, g~ :=∑

m≥0

~m∑

0≤i,j,ℓ≤mi+j+ℓ=m

πi(dfj,dgℓ),

where

f =∑

j≥0

fj~j, g =

∑

ℓ≥0

gℓ~ℓ.

Definition 4.2.1 (Formal Poisson structure). We call

π~ := π0 + π1~+ π2~2 + · · ·

a formal Poisson structure if , ~ defined as in (4.2.1) is a Lie bracket on C∞(P )[[~]].

The gauge group in this case is given by formal diffeomorphisms, i.e. formal power series ofthe form

(4.2.2) Ψ~ := exp(~X),

where X :=∑

i≥0Xi~i ∈ X(P )[[~]] is a formal vector field, i.e. a formal power series where

each coefficients Xi ∈ X(P ) are vector fields on P .


Definition 4.2.2 (Baker–Campbell–Hausdorff formula). For two vector fields X and Y ona manifold we have the Baker–Campbell–Hausdorff (BCH) formula for solving the equationexp(X) exp(Y ) = exp(Z) for a vector field Z:

exp(X) · exp(Y ) = exp

(X +

(∫ 1

0ψ (exp(adX) exp(t adY )) dt

)Y

)

= exp

(X + Y +

1

2[X,Y ] + · · ·

),

(4.2.3)

where ψ(x) := x log xx−1 = 1−∑n≥0

(1−x)n

n(n+1) and adX = [X, ].

Exercise 4.2.3. Check that on the set of formal vector fields X(P )[[~]] on P there is a groupstructure regarding the exponential induced by the BCH formula, i.e. we have a groupstructure on exp(~X) | X ∈ X(P )[[~]] by

exp(~X) · exp(~Y ) := exp

(~X + ~Y +

1

2~2[X,Y ] + · · ·

).

The group action is given by

(4.2.4) exp(~X)∗π~ :=∑

m≥0

~m∑

0≤i,j,ℓ≤mi+j+ℓ=m

(LXi)jπℓ

Remark 4.2.4. Kontsevich’s result was to find an identification between the set of starproducts modulo the action of the differential operators D as in Definition 4.1.17 and the setof formal Poisson structures modulo the gauge group consisting of formal diffeomorphisms as(4.2.2).

4.2.2. Differential graded Lie algebras.

Definition 4.2.5 (Graded Lie algebra). A graded Lie algebra (GLA) is a Z-graded vectorspace g =

⊕i∈Z g

i endowed with a bilinear operation

[ , ] : g⊗ g! g

satisfying

(1) [a, b] ∈ gα+β (homogeneity).(2) [a, b] = −(−1)αβ [b, a] (graded skew-symmetry),(3) [a, [b, c]] = [[a, b], c] + (−1)αβ [b, [a, c]] (graded Jacobi identity)

for all a ∈ gα, b ∈ gβ and c ∈ gγ .

Definition 4.2.6 (Shift). Given any graded vector space g =⊕

i∈Z gi, we can obtain a new

graded vector space g[n] by shifting each component by n, i.e.

g[n] :=⊕

i∈Z

g[n]i, g[n]i := gi+n.

Definition 4.2.7 (Differential graded Lie algebra). A differential graded Lie algebra (DGLA)is a GLA g together with a differential d : g! g, i.e. a linear operator of degree +1 (d: gi !gi+1) which satisfies the Leibniz rule

d[a, b] = [da, b] + (−1)α[a,db], a ∈ gα, b ∈ gβ

4.2. FORMALITY 91

and squares to zero (d d = 0).

Example 4.2.8. Note that one can turn any Lie algebra into a DGLA concentrated in degreezero by using the trivial zero differential d = 0.

Given a DGLA g, we can consider its cohomology

H i(g) := ker(d : gi ! gi+1)/im (d: gi−1! gi).

Remark 4.2.9. Note that H(g) :=⊕

iHi(g) has a natural structure of a graded vector space

and further has a GLA structure defined on equivalence classes [a], [b] ∈ H(g) by

[[a], [b]]H(g) := [[a, b]g],

where [ , ]g denotes the Lie bracket on g. Moreover, extending the GLA H(g) by the zerodifferential will turn it into a DGLA.

Remark 4.2.10. Note that each morphism ϕ : g1 ! g2 between two DGLAs induces a mor-phism H(ϕ) : H(g1) ! H(g2) between the cohomologies. In particular, we are interested inquasi-isomorphisms between DGLAs.

Definition 4.2.11 (Quasi-isomorphism). A quasi-isomorphism is a morphism between DGLAswhich induces an isomorphism on the level of cohomology.

Remark 4.2.12. The notion of quasi-isomorphism naturally holds more generally for any(co)chain complex and the induced (co)homology theory.

Definition 4.2.13 (Quasi-isomorphic). Two DGLAs g1 and g2 are called quasi-isomorphicif there is a quasi-isomorphism ϕ : g1 ! g2.

Remark 4.2.14. The existence of a quasi-isomorphism ϕ : g1 ! g2 does not imply that thereis also a quasi-isomorphism inverse ϕ−1 : g2 ! g1 and therefore they do not directly define anequivalence relation. We will deal with this issue by considering the category of L∞-algebras.

4.2.3. L∞-algebras. The notion of an L∞-algebra, first introduced in [Sta92], gives ageneralization of a DGLA. It has a lot of applications in modern constructions of quantumfield theory and homotopy theory. In this section, we will define what an L∞-algebra is andshow how one can extract an L∞-structure out of any DGLA. Moreover, we briefly explainits representation in terms of higher brackets. We refer to [Sta92; LS93; LM95] for moredetails.

Definition 4.2.15 (Formal). A DGLA g is called formal if it is quasi-isomorphic to itscohomology, regarded as a DGLA with zero differential and the induced bracket.

Remark 4.2.16. Kontsevich’s formality theorem [Kon03] consists of the result that the DGLAof multidifferential operators (see later) is formal.

Definition 4.2.17 (Graded coalgebra). A graded coalgebra (GCA) on a base ring k is aZ-graded vector space h =

⊕i∈Z h

i endowed with a comultiplication, i.e. a graded linear map

∆: h! h⊗ h

such that∆(hi) ⊂

⊕

j+ℓ

hj ⊗ hℓ


and which also satisfies the coassociativity condition

(∆⊗ id)∆(a) = (id⊗∆)∆(a), ∀a ∈ h.

Definition 4.2.18 (GCA with counit). We say that a GCA h has a counit if there is amorphism

ε : h! k

such that ε(hi) = 0 for all i > 0 and

(ε⊗ id)∆(a) = (id⊗ ε)∆(a) = a, ∀a ∈ h.

Definition 4.2.19 (Cocommutative GCA). We say that a GCA h is cocommutative if thereis a twisting map defined on homogeneous elements x, y ∈ h by

T : h⊗ h! h⊗ h,

x⊗ y 7! T(x⊗ y) := (−1)|x||y|y ⊗ x,

where |x| ∈ Z denotes the degree of x ∈ h, and extended linearly, such that

T ∆ = ∆.

Example 4.2.20 (Tensor algebra). Let V be a (graded) vector space over k and consider itstensor algebra T (V ) =

⊕∞n=0 V

⊗n. Denote by 1 the unit of k. Then, T (V ) can be endowedwith a coalgebra structure. We define a comultiplication ∆T (V ) on homogeneous elements by

∆T (V )(v1⊗· · ·⊗vn) := 1⊗(v1⊗· · ·⊗vn)+n−1∑

j=1

(v1⊗· · ·⊗vj)⊗(vj+1⊗· · ·⊗vn)+(v1⊗· · ·⊗vn)⊗1.

In particular, we have a counit εT (V ) as the canonical projection εT (V ) : T (V ) ! V ⊗0 := k.

Note that if we consider the reduced tensor algebra T (V ) :=⊕∞

n=1 V⊗n, then the projection

π : T (V )! T (V ) and the inclusion i : T (V ) ! T (V ) induce a comultiplication on T (V ), thusa coalgebra structure but without counit.

Example 4.2.21 (Symmetric and exterior algebra). Let V be a (graded) vector space over k.Then we can also consider the Symmetric algebra Sym(V ) and exterior algebra

∧V . Note

that for the graded case we have to take the quotients with respect to the two-sided idealsgenerated by homogeneous elements of the form v⊗w−T(v⊗w) and v⊗w+T(v⊗w). Theycan be endowed with a coalgebra structure (without counit if constructed for the reducedtensor algebra). Indeed, e.g. we can define a comultiplication ∆Sym(V ) on homogeneouselements v ∈ V by

∆Sym(V )(v) := 1⊗ v + v ⊗ 1,

Definition 4.2.22 (Coderivation). A coderivation of degree ℓ on some GCA h is a gradedlinear map δ : hi ! hi+ℓ which satisfies the co-Leibniz identity

∆δ(v) = (δ ⊗ id)∆(v) + ((−1)ℓ|v|id⊗ δ)∆(v), ∀v ∈ h|v|.

Definition 4.2.23 (Codifferential). A codifferential Q on a coalgebra is a coderivation ofdegree +1 which squares to zero (Q2 = 0).

Definition 4.2.24 (L∞-algebra). An L∞-algebra is a graded vector space g over k endowedwith a degree +1 coalgebra differential Q on the reduced symmetric space Sym(g[1]).

4.2. FORMALITY 93

Definition 4.2.25 (L∞-morphism). An L∞-morphism F : (g, Q)! (g, Q) is a morphism

F : Sym(g[1])! Sym(g[1]).

of graded coalgebras, which also commutes with the differentials, i.e.

FQ = QF.

Remark 4.2.26. As in the dual case an algebra morphism f : Sym(A) ! Sym(A) (resp. aderivation δ : Sym(A) ! Sym(A)) is uniquely determined by its restriction to the algebraA = Sym1(A) because of the homomorphism condition f(a⊗ b) = f(a)⊗ f(b) (resp. Leibnizrule), an L∞-morphism F (resp. a coderivation Q) is uniquely determined by its projectiononto the first component F 1 (resp. Q1).

Let us introduce the generalized notation F ij (resp. Qij) for the projection to the i-th com-ponent of the target vector space restricted to the j-th component of the domain. We canthen rephrase the condition for F (resp. Q) to be an L∞-morphism (resp. a codifferential).

Indeed, using this notation, we get that QQ, FQ, and QF are coderivations. It is sufficientto show this on the projection to the first factor for each term.

Definition 4.2.27. A coderivation Q is a codifferential if and only if

(4.2.5)∑

1≤i≤n

Q1iQ

in = 0, ∀n ∈ N.

Definition 4.2.28. A morphism F of graded coalgebras is an L∞-morphism if and only if

(4.2.6)∑

1≤i≤n

F 1i Q

in =

∑

1≤i≤n

Q1iF

in, ∀n ∈ N.

Example 4.2.29. For the case when n = 1, we get

Q11Q

11 = 0, F 1

1Q11 = Q1

1F11 .

Therefore, every coderivation Q induces the structure of a cochain complex of vector spaceson g and every L∞-morphism restricts to a cochain map F 1

1 .

Remark 4.2.30. We can generalize the definitions for a DGLA to this case. In particular,we can define a quasi-isomorphism of L∞-algebras to be an L∞-morphism F such that F 1

1 isa quasi-isomorphism of cochain complexes. Similarly, one can extend the notion of formality.

Lemma 4.2.31. Let F : (g, Q) ! (g, Q) be an L∞-morphism. If F is a quasi-isomorphism it

admits a quasi-inverse, i.e. there is an L∞-morphism G : (g, Q) ! (g, Q) which induces theinverse isomorphism in the corresponding cohomologies.

Remark 4.2.32. Lemma 4.2.31 implies an equivalence relation defined by L∞-quasi-isomorphisms,i.e. two L∞-algebras are L∞-quasi-isomorphic if and only if there is an L∞-quasi-isomorphismbetween them. This solves the problem that one has to face regarding DGLAs, where theequivalence relation only holds on the level of quasi-isomorphisms.

Let us see how we can extract an L∞-structure from any DGLA g which indeed implies thatan L∞-algebra is a generalization of a DGLA. Define Q1

1 to be a multiple of the differential.For n = 2, we can write condition (4.2.5) as

(4.2.7) Q11Q

12 +Q1

2Q22 = 0.


We identify Q11 with the differential d (up to some sign) and note that (4.2.7) suggests that

Q12 should be expressed in terms of the bracket [ , ] since (4.2.7) has the same form as the

compatibility condition between the bracket and the differential. In particular, we get

Q11(a) := (−1)αda, a ∈ gα,(4.2.8)

Q12(b⊗ c) := (−1)β(γ−1)[b, c], b ∈ gβ, c ∈ gγ ,(4.2.9)

Q1n = 0, ∀n ≥ 3.(4.2.10)

For n = 3, we get

(4.2.11) Q11Q

13 +Q1

2Q23 +Q1

3Q33 = 0.

If we insert the definition of the bracket (4.2.9) and expand Q23 in terms of Q1

2 in (4.2.11), weget

(4.2.12) (−1)(α+β)(γ−1)[(−1)α(β−1)[a, b], c

]

+ (−1)(α+γ)(β−1)(−1)(γ−1)(β−1)[(−1)α(γ−1)[a, c], b

]

+ (−1)(β+γ)(α−1)(−1)(β+γ)(α−1)[(−1)β(γ−1)[b, c], a

]= 0.

Remark 4.2.33. Note that, by rearranging signs, (4.2.12) is the same as the graded Jacobiidentity.

Similar constructions hold for a DGLA morphism F : g! g. It induces an L∞-morphism Fwhich is completely determined by its first component F 1

1 := F . The nontrivial conditionsfor n = 1 and n = 2 on F , induced by (4.2.6), are given by

F 11Q

11(f) = Q1

1F11 (f) ⇔ F (df) = dF (f),

F 11Q

12(f ⊗ g) + F 2

1Q22(f ⊗ g) = Q1

1F21 (f ⊗ g) + Q1

2F22 (f ⊗ g) ⇔ F ([f, g]) = [F (f), F (g)].

Remark 4.2.34. For n = 3, where the Q13 do not vanish, one can show that (4.2.12) would

have been satisfied up to homotopy, i.e. up to a term of the form

dρ(g, h, k) ± ρ(dg, h, k) ± ρ(g,dh, k) ± ρ(g, h,dk),

where ρ :∧3

g ! g[−1]. In this case, one says that g has the structure of a homotopy Liealgebra.

Remark 4.2.35. The concept of Remark 4.2.34 can be generalized by introducing the canon-ical isomorphism between the symmetric and exterior algebra (usually called declage isomor-phism) given by

decn : Symn(g[1])

∼−!

n∧g[n],

x1 ⊗ · · · ⊗ xn 7! (−1)∑

1≤i≤n(n−i)(|xi|−1)x1 ∧ · · · ∧ xn,to define for each n a multibracket of degree 2− n

[ , . . . , ]n :

n∧g! g[2− n]

4.2. FORMALITY 95

by starting from the corresponding Q1n. In fact, (4.2.5) induces an infinite family of conditions

on these multibrackets. A graded vector space g together with such a family of operators iscalled a strong homotopy Lie algebra (SHLA) (see also [LM95]).

Definition 4.2.36 (Maurer–Cartan equation). The Maurer–Cartan equation of a DGLA g

is given by

(4.2.13) da+1

2[a, a] = 0, a ∈ g1.

Remark 4.2.37. It is easy to show that the set of solutions to the Maurer–Cartan equation(4.2.13) is preserved under the action of a morphism between DGLAs. Moreover, the groupaction by the gauge group that is canonically defined through the degree zero part of anyformal DGLA also preserves the set of solutions to (4.2.13). This can be extended to theformal setting4. For a DGLA g, we will denote by MC(g) the set of solutions to (4.2.13) in g.

4.2.4. The DGLA of multivector fields V. Recall from Section 1.3.2 that a multi-vector field of degree j ≥ 1 on some manifold M is an element

X ∈ Xj(M) := Γ

(j∧TM

).

In local coordinates, we can write

X =∑

1≤i1<···<ij≤dimM

Xi1···ij∂i1 ∧ · · · ∧ ∂ij .

Hence, if we consider the collection of these vector spaces in all degrees, we naturally get agraded vector space structure

V(M) :=⊕

j≥0

Vj(M), Vj(M) :=

C∞(M), j = 0

Xj(M), j ≥ 1

Using the Schouten–Nijenhuis bracket as defined in (3.1.1), we can endow V(M) with a GLAstructure. We will write [ , ]SN to indicate that we mean the Schouten–Nijenhuis bracket.We note/recall the following identities for the Schouten Nijenhuis bracket:

(1) [X,Y ]SN = −(−1)(|X|+1)(|Y |+1)[Y,X]SN,

(2) [X,Y ∧ Z]SN = [X,Y ]SN ∧ Z + (−1)(|Y |+1)|Z|Y ∧ [X,Z]SN,(3) [X, [Y,Z]SN]SN = [[X,Y ]SN, Z]SN + (−1)(|X|+1)(|Y |+1)[Y, [X,Z]SN]SN.

4Note that we can set g[[~]] := g⊗ k[[~]] and show that this is again a DGLA. As we have seen before, thegauge group is then formally defined as G := exp(~g0[[~]]), where g0[[~]] denotes the Lie algebra given as thedegree zero part of g[[~]]. Note that the action of ~g1[[~]] is defined by generalizing the adjoint action. Namely,we have

exp(~X)a :=∑

n≥0

(adX)n

n!(a)−

∑

n≥0

(adX)n

(n+ 1)!(dX) = a+ ~[X, a]− ~dX +O(~2), ∀X ∈ g

0[[~]], a ∈ g1[[~]].


In order, to obtain the correct signs, we have to shift everything by +1. Thus, we define the

GLA of multivector fields to be V(M) := V(M)[1]. Namely, we have

V(M) :=⊕

j≥−1

Vj(M), Vj(M) := Vj+1(M).

Finally, using the zero differential, we can turn it into a DGLA (V(M), [ , ]SN,d = 0).4.2.4.1. The case of Poisson bivector fields. Consider a manifold M together with a Pois-

son bivector field π ∈ V1(M). Then we can consider its induced Poisson bracket , onC∞(M). Recall that, by Exercise 3.1.6, we can translate the Jacobi identity of the Poissonbracket to the condition [π, π]SN = 0. Indeed, we have

f, g, h + g, h, f + h, f, g = 0 ⇔πij∂jπ

kℓ∂if∂kg∂ℓh+ πij∂jπkℓ∂ig∂kh∂ℓf + πij∂jπ

kℓ∂ih∂k∂ℓg = 0 ⇔πij∂jπ

kℓ∂i ∧ ∂k ∧ ∂ℓ = 0 ⇔ [π, π]SN = 0.

Note that, since we have endowed V(M) with the zero differential, we get that Poissonbivector fields are exactly solutions to the Maurer–Cartan equation on the DGLA V(M):

dπ +1

2[π, π]SN = 0, π ∈ V1(M).

In particular, formal Poisson structures , ~ are associated to formal bivectors π~ ∈~V1(M)[[~]].

4.2.5. The DGLA of multidifferential operators D. Note that to any associativealgebra A over a field k, we can assign the complex of multilinear maps given by

C :=∑

j≥−1

Cj , Cj := Homk(A⊗(j+1),A).

Define a family of composition i such that for an (m+ 1)-linear operator φ ∈ Cm and an(n+ 1)-linear operator ψ ∈ Cn we have

(φ i ψ)(f0 ⊗ · · · ⊗ fm+n) := φ(f0 ⊗ · · · ⊗ fi−1 ⊗ ψ(fi ⊗ · · · ⊗ fi+n)⊗ fi+n+1 ⊗ · · · ⊗ fm+n).

If we sum over all the different ways of composition, we get

(4.2.14) φ ψ :=∑

0≤i≤m

(−1)inφ i ψ,

which we can use to endow C with a GLA structure.

Definition 4.2.38 (Gerstenhaber bracket). The Gerstenhaber bracket on the graded vectorspace C is defined by

[ , ]G : Cm ⊗ Cn ! Cm+n,

(φ,ψ) 7! [φ,ψ]G := φ ψ − (−1)mnψ φ.(4.2.15)

Remark 4.2.39. There is a more general notion of a Gerstenhaber algebra [Ger63], which de-scribes the connection between supercommutative rings [Var04] and graded Lie superalgebras[Kac77]. In particular, it consists of a graded commutative algebra endowed with a Poissonbracket of degree −1. This structure plays also a fundamental role in other constructions of

4.2. FORMALITY 97

theoretical physics, such as in DeDonder–Weyl theory [DeD30; Wey35] (a generalization ofthe Hamiltonian formalism to field theory) and in the Batalin–Vilkovisky formalism [BV77;BV81; BV83] (a way of treating perturbative quantization of gauge theories).

Proposition 4.2.40. The graded vector space C together with the Gerstenhaber bracket(4.2.15) is a GLA, which we call Hochschild GLA.

Proof. Clearly, the Gerstenhaber bracket is linear. Note that the sign (−1)mn ensuresthat it is also (graded) skew-symmetric, since

[φ,ψ]G = −(−1)mn(ψ φ− (−1)mnφ ψ

)= −(−1)mn[ψ, φ]G, ∀φ ∈ Cm, ψ ∈ Cn.

Next, we need to make sure that [ , ]G satisfies the (graded) Jacobi identity

(4.2.16) [φ, [ψ,χ]G]G = [[φ,ψ]G]G + (−1)mn[ψ, [φ, χ]G]G, ∀φ ∈ Cm, ψ ∈ Cm, χ ∈ Cp.

Note that the left hand side of (4.2.16) gives

(4.2.17)

(φ ψ − (−1)mnψ φ

) χ− (−1)(m+n)pχ

(φ ψ − (−1)mnψ φ

)

=∑

0≤i≤m0≤k≤m+n

(−1)in+kp(φ i ψ) k χ−∑

0≤j≤n0≤k≤m+n

(−1)m(j+n)+kp(ψ j φ) k χ

−∑

0≤i≤m0≤k≤p

(−1)(m+n)(k+p)+inχ k (φ i ψ) +∑

0≤j≤n0≤k≤p

(−1)(m+n)(k+p)+m(j+n)χ k (ψ j φ).

We can decompose the first sum according to the following rule

(φ i ψ) k χ =

(φ k χ) i ψ, k < i

φ i (ψ k−i χ), i ≤ k ≤ i+ n

(φ k−n χ) i ψ, i+ n < k

into a term of the form

(4.2.18)∑

0≤i≤mi≤k≤i+n

(−1)in+kpφ i (ψ k−i χ) =∑

0≤i≤m0≤k≤n

(−1)(n+p)i+kpφ i (ψ k χ).

Note that the sign of (4.2.18) is the same as the one in the corresponding term comingfrom (φ ψ) χ on the left hand side of (4.2.17) plus the terms in which the i-th andk-th composition commute. These then cancel the corresponding terms coming from theexpansion of the second term of the right hand side of (4.2.16). Using the same approach forthe remaining terms, we get the proof.

Remark 4.2.41. Note that associative multiplications on A are elements in C1. In particular,if we denote by m ∈ C1 a multiplication map, the associativity condition is given by

m(m(f ⊗ g)⊗ h)−m(f ⊗m(g ⊗ h)) = 0.


This is in fact equivalent to the condition that the Gerstenhaber bracket of m with itselfvanishes. Indeed, we have

[m,m]G(f ⊗ g ⊗ h) =∑

0≤i≤1

(−1)i(m i m)(f ⊗ g ⊗ h)− (−1)1∑

0≤i≤1

(−1)i(m i m)(f ⊗ g ⊗ h)

= 2

(m(m(f ⊗ g)⊗ h)−m(f ⊗m(g ⊗ h))

).

Now, for an element φ of a (DG) Lie algebra g (of degree ℓ), we get that

adφ := [φ, ]

is a derivation (of degree ℓ). Indeed, by the Jacobi identity, we have

adφ[ψ, ξ] = [adφψ, ξ] + (−1)ℓm[ψ, adφξ], ∀ψ ∈ gm, ξ ∈ gn.

Definition 4.2.42 (Hochschild differential). The Hochschild differential is given by

dm : Ci ! Ci+1

ψ 7! dmψ := [m, ψ]G.(4.2.19)

Proposition 4.2.43. The GLA C endowed with the Hochschild differential dm is a DGLA.

Proof. The only thing we need to check is that the Hochschild differential indeed squaresto zero (i.e. dm dm = 0). This follows immediately from the Jacobi identity and theassociativity condition of m expressed in terms of the Gerstenhaber bracket:

(dm dm)ψ = [m, [m, ψ]G]G = [[m,m]G, ψ]G − [m, [m, ψ]G]G

= −[m, [m, ψ]G]G ⇔ dm dm = 0.

Remark 4.2.44. In fact, we explicitly have

(dmψ)(f0 ⊗ · · · ⊗ fn+1) =∑

0≤i≤n

(−1)i+1ψ(f0 ⊗ · · · ⊗ fi−1 ⊗m(fi ⊗ fi+1)⊗ · · · ⊗ fn+1)

+m(f0 ⊗ ψ(f1 ⊗ · · · ⊗ fn+1)) + (−1)n+1m(ψ(f0 ⊗ · · · ⊗ fn)⊗ fn+1).

Let now A = C∞(M) and consider the and consider the subalgebra D(M) ⊂ C, which is the(graded) vector space given as the collection

Di :=⊕

i

Di

of subspaces Di(M) ⊂ Ci which consist of differential operators on C∞(M). One can check

that D(M) is closed with respect to the Gerstenhaber bracket [ , ]G and the differential dmand hence is a DGL subalgebra5 of C.

5Note that we will also allow differential operators of degree zero, i.e. operators which do not “derive”

anything. Thus, the associative product m is also an element of D1(M).

4.2. FORMALITY 99

Moreover, we want to restrict everything to differential operators which vanish on constant

functions. They will in fact form another DGL subalgebra D(M) ⊂ D(M). Note that wewant to have this restriction because of the fact that the coefficients in the expansion of astar product vanish on constant functions, i.e. Bi(1, f) = 0 for all i ∈ N and f ∈ C∞(M).

Note however that the Hochschild differential dm is not inner derivation anymore for D(M)since the multiplication does not vanish on constants.

Consider an element B ∈ D1(M). Then we want to regard m + B as a deformation of theoriginal product. The associativity condition for m+B implies

[m+B,m+B]G = 0,

and by associativity of m we get

[m, B]G = [B,m]G = dmB.

This does exactly lead to the Maurer–Cartan equation:

(4.2.20) dmB +1

2[B,B]G = 0.

Remark 4.2.45. If we consider the formal version of D(M), we can see that the deformedproduct does exactly describe a star product since B ∈ ~D1(M)[[~]]. Moreover, the gaugegroup is given by formal differential operators and the action on the star product is given by(4.1.5) since the adjoint action, by the definition of the Gerstenhaber bracket, is exactly thecomposition of Di with Bj .

4.2.6. The Hochschild–Kostant–Rosenberg map. In [HKR62], Hochschild–Kostant–Rosenberg have shown that the cohomology of the algebra of multidifferential operators andthe algebra of multivector fields, which is equal to its cohomology, are isomorphic. Theyestablished an isomorphism

(4.2.21) HKR: H(D(M))∼−! V(M) ∼= H(V(M)).

In particular, this isomorphism is induced by the map

U(0)1 : V(M)! D(M),

ξ0 ∧ · · · ∧ ξn 7!(f0 ⊗ · · · ⊗ fn 7!

1

(n+ 1)!

∑

σ∈Sn+1

sign(σ)ξσ(0)(f0) · · · ξσ(n)(fn))

(4.2.22)

Remark 4.2.46. This is extended to vector fields of order zero by the identity map. However,the map (4.2.22) fails to preserve the Lie structure. This can be easily checked for order two.Indeed, given homogeneous bivector fields χ1 ∧ χ2 and ξ1 ∧ ξ2, we get

U(0)1

([χ1 ∧ χ2, ξ1 ∧ ξ2]SN

)6=[U

(0)1 (χ1 ∧ χ2), U

(0)1 (ξ1 ∧ ξ2)

]G.


The left hand side gives

U(0)1

([χ1, ξ1]∧ χ2 ∧ ξ2 − [χ1, ξ2]∧ χ2 ∧ ξ1 − [χ2, ξ1]∧ χ1 ∧ ξ2 + [χ2, ξ2]∧χ1 ∧ ξ1

)(f ⊗ g⊗ h)

=1

6

(χ1

(ξ1(f)χ2(g)ξ2(h)

)− ξ1

(χ1(f)χ2(g)ξ2(h)

)

− χ1

(ξ2(f)χ2(g)ξ1(h)

)+ ξ2

(χ1(f)χ2(g)ξ1(h)

)− χ2

(ξ1(f)χ1(g)ξ2(h)

)+ ξ1

(χ2(f)χ1(g)ξ2(h)

)

+ χ2

(ξ2(f)χ1(g)ξ1(h)

)+ ξ2

(χ2(f)χ1(g)ξ1(h)

))+ permutations.

The right hand side is given by

1

4

(χ1

(ξ1(f)ξ2(g)

)χ2(h) + · · ·

).

One can, however, still show that the difference of the two terms is given by the image of aclosed term in the cohomology of D(M). Thus, there is some hope to control everything andstill extend it to a Lie algebra morphism whose first order approximation is given by (4.2.22).Again, to resolve this problem, we will consider an L∞-morphism U .

4.2.7. The dual point of view. Let V be a vector space and note that we can naturallyidentify polynomials on V as the symmetric functions on the dual space V ∗

f(v) :=∑

j

1

j!fj(v ⊗ · · · ⊗ v), ∀v ∈ V,

where fj ∈ Symj(V ∗). We want to extend this to the case when V is a graded vector space.In this case, we need to consider the exterior algebra instead. Note that, if we consider theinjective limit of the Symj(V ∗) (resp.

∧j V ∗) endowed with the induced topology, we can

define the corresponding completion Sym(V ∗) (resp.∧V ∗). Now we can define a function

in a formal neighborhood of 0 to be given by the formal Taylor expansion in the ~

f(~v) :=∑

j

~j

j!fj(v ⊗ · · · ⊗ v), ∀v ∈ V.

Thus, a vector field X on V can be regarded as a derivation on∧V ∗ and the Leibniz rule

makes sure that X is completely determined by its restriction on V ∗. Similarly, if we havean algebra morphism

ϕ :∧W ∗!

∧V ∗,

it induces a map

f := ϕ∗ :∧V !

∧W

whose components fj are completely defined by their projection on W as the ϕj are definedby their restriction on W ∗.

4.2. FORMALITY 101

Definition 4.2.47 ((Formal) pointed map). A (formal) pointed map is an algebra morphismbetween the reduced exterior algebras

ϕ :∧W ∗!

∧V ∗,

where the overline means that we are considering the reduced tensor algebra T (W ∗) (resp.T (V ∗)) as in Example 4.2.21.

Definition 4.2.48 (Pointed vector field). A pointed vector field X on a manifold M is avector field in X(M) which fixes the point zero, i.e.

X(f)(0) = 0, ∀f ∈ C∞(M).

Definition 4.2.49 (Cohomological pointed vector field). A pointed vector field X is calledcohomological (or Q-field) if and only if it commutes with itself, i.e. X2 = 1

2 [X,X] = 0.

Definition 4.2.50 (Pointed Q-manifold). A pointed Q-manifold is a (formal) pointed man-ifold together with a cohomological vector field.

Consider now a Lie algebra g. Note that the bracket [ , ] :∧2

g! g induces a linear map

[ , ]∗ : g∗ !

2∧g∗.

This can be extended to the whole exterior algebra by a map

δ :∧

g∗ !∧

g∗[1]

such that it satisfies the Leibniz rule and δ∣∣g∗

= [ , ]∗.

Remark 4.2.51. We can regard the exterior algebra as an odd analogue of a manifold onwhich δ takes the role of a (pointed) vector field. Note also that, since [ , ] satisfies theJacobi identity, we have δ2 = 0 and thus δ defines a cohomological (pointed) vector field.

Consider two Lie algebras g and h together with the corresponding cohomological vector fieldsδg and δh on the respective exterior algebras. Then a Lie algebra morphism ϕ : g ! h willcorrespond to a chain map ϕ∗ : h∗ ! g∗ since

(4.2.23) ϕ

([ , ]g

)= [ϕ( ), ϕ( )]h ⇐⇒ δg ϕ∗ = ϕ∗ δh.

Remark 4.2.52. Equation (4.2.23) gives a first hint to the correspondence between L∞-algebras and pointed Q-manifolds. Since a Lie algebra is a particular case of a DGLA, whichagain can be endowed with an L∞-structure, we can see that the map ϕ satisfies exactly thesame condition for the first component of an L∞-morphism as in (4.2.6) for n = 1.

Let us extend this picture to the case of a graded vector space Z over a field k. Let usdecompose the graded space Z into its even and odd part, i.e.

Z = Z[0] ⊕ Z[1]

where Z[0] denotes the even part, i.e. Z[0] :=⊕

i∈2Z Zi and Z[1] :=

⊕i∈2Z+1 Z

i. Here we havedenoted by [0], [1] the equivalence classes in Z/2. For a vector space W , denote by ΠW the(odd) space defined by a parity reversal on W , which we can also denote by W [1] using the


notation of Definition 4.2.6. Let us define V := Z[0] and ΠW := Z[1]. Then Z = V ⊕ ΠWand functions can be identified by elements of

Sym(Z∗) := Sym(V ∗)⊗∧W ∗.

We can express the condition for a vector field δ : Sym(Z∗)! Sym(Z∗)[1] to be cohomologicalin terms of its coefficients

δj : Symj(Z∗)! Symj+1(Z∗)

by expanding the equation δ2 = 0. Hence, we get an infinite family of equations

δ0δ0 = 0,(4.2.24)

δ1δ0 + δ0δ1 = 0,(4.2.25)

δ2δ0 + δ1δ1 + δ0δ2 = 0,(4.2.26)

...

Let now mj := (δj |Z∗)∗ be the dual coefficients and let 〈 , 〉 : Z∗ ⊗ Z ! k. Then we can

rewrite these conditions in terms of mj : Symj(Z)! Z. Note that the first Equation (4.2.24),

which is now given by m0m0 = 0, implies that m0 is a differential on Z and hence induces acohomology Hm0(Z). For j = 1, i.e. for the second Equation (4.2.25), we get

(4.2.27) 〈δ1δ0f, x⊗ y〉 = 〈δ0f,m1(x⊗ y)〉 = 〈f,m0(m1(x⊗ y))〉and

(4.2.28) 〈δ0δ1f, x⊗ y〉 = 〈δ1f,m0(x)⊗ y〉+ (−1)|x|〈δ1f, x⊗m0(y)〉= 〈f,m1(m0(x)⊗ y)〉+ (−1)|x|〈f,m1(x⊗m0(y))〉.

If we put (4.2.27) and (4.2.28) together, we get that m0 is a derivation with respect to themultiplication defined by m1.

Remark 4.2.53. If we consider Z := g[1] and consider the exterior algebra by the declage

isomorphism (Symn(g[1])∼−!

∧ng[n]), we can interpret m1 as a bilinear skew-symmetric

operator on g.

Note that Equation (4.2.26) implies that m1 is indeed a Lie bracket for which the Jacobiidentity holds up to terms containing m0. Since m0 is a differential, this means that m1 is aLie bracket up to homotopy. Hence, considering all equations, we will get a strong homotopyLie algebra structure on g.

Remark 4.2.54. Note that this will lead to a one-to-one correspondence between pointedQ-manifolds and SHLAs. Since SHLAs are equivalent to L∞-algebras, we get a one-to-onecorrespondence between pointed Q-manifolds and L∞-algebras.

Definition 4.2.55 (Q-map). A Q-map is a (formal) pointed map between two Q-manifolds

Z and Z which commutes with the Q-fields, i.e. a map

ϕ : Sym(Z∗)! Sym(Z∗)

such that ϕ δ = δ ϕ.(4.2.29)

4.2. FORMALITY 103

We want to express the condition (4.2.6) more explicitly in this setting by using Q-maps.

Consider the vector field δ and its restriction to Z. We define the coefficients of the dual mapas

Uj := (ϕj |Z∗)∗ : Symj(Z)! Z.

Then we can express the condition (4.2.29) of a Q-map on the dual coefficients. The firstequation gives

〈ϕ(δf), x〉 = 〈δϕ(f), x〉 ⇒ 〈δ0f, U1(x)〉 = 〈ϕ(f),m0(x)〉⇒ 〈f, m0(U1(x))〉 = 〈f, U1(m0(x))〉.

This tells us that the first coefficient U1 is a chain map with respect to the differential definedby the first coefficient of the Q-structure:

H(U1) : Hm0(Z)! Hm0(Z).

Remark 4.2.56. Similarly, we can consider the equation for the next coefficient:

(4.2.30) m1(U1(x)⊗ U1(y)) + m1(U2(x⊗ y))

= U2(m0(x)⊗ y) + (−1)|x|U2(x⊗m0(y)) + U1(m1(x⊗ y)).

This shows that U1 preserves the Lie structure induced by m1 and m1 up to terms containing

m0 and m0, i.e. up to homotopy. Recall that this solves the problem that the map U(0)1 ,

defined in (4.2.22), is a chain map but not a DGLA morphism. Namely, a Q-map U (or

equivalently an L∞-morphism) induces a map U1 which has the same property as U(0)1 .

Definition 4.2.57 (Koszul sign). Let V be a vector space endowed with a graded commu-tative product ⊗. The Koszul sign ε(σ) of a permutation σ is the sign defined by

x1 ⊗ · · · ⊗ xn = ε(σ)xσ(1) ⊗ · · · ⊗ xσ(n), xi ∈ V.

Definition 4.2.58 (Shuffle permutation). An (ℓ, n− ℓ)-shuffle permutation is a permutationσ of (1, . . . , n) such that σ(1) < · · · < σ(ℓ) and σ(ℓ+ 1) < · · · < σ(n).

Remark 4.2.59. The shuffle permutation associated to a partition I1 ⊔ · · · ⊔ Ij = 1, . . . , nis the permutation that takes at first all the elements index by the subset I1 in the givenorder, then those of I2 and so on.

The n-th coefficient of U satisfies the following condition:

(4.2.31) m0(Un(x1 ⊗ · · · ⊗ xn)) +1

2

∑

I⊔J=1,...,nI,J 6=∅

εx(I, J)m1(U|I|(xI)⊗ U|J |(xJ))

=∑

1≤j≤n

εjxUn(m0(xj)⊗ x1 ⊗ · · · ⊗ xj ⊗ · · · ⊗ xn)+

+1

2

∑

j 6=ℓ

εjℓx Un−1(m1(xj ⊗ xℓ)⊗ x1 ⊗ · · · ⊗ xj ⊗ · · · ⊗ xℓ ⊗ · · · ⊗ xn),

where εx(I, J) is the Koszul sign associated to the (|I|, |J |)-shuffle permutation for the par-

tition I ⊔ J = 1, . . . , n and εjx (resp. εjℓx ) for the particular case I = j (resp. I = j, ℓ).Moreover, we have set xI :=

⊗i∈I xi.


Remark 4.2.60. We will specialize (4.2.31) to an L∞-morphism where Z is chosen to be the

DGLA V of multivector fields and Z the DGLA D of multidifferential operators.

4.2.8. Formality of D and classification of star products on Rd. We want toanalyze the relation between the formality of D and the classification problem of all starproducts on Rd. Recall that the associativity of the star product and the Jacobi identity fora bivector field are equivalent to certain Maurer–Cartan equations.

Definition 4.2.61 (Maurer–Cartan equation on (formal) L∞-algebras). The Maurer–Cartanequation on a (formal) L∞-algebra (g[[~]], Q) is given by

(4.2.32) Q(exp(~x)) = 0, x ∈ g1[[~]],

where the exponential exp maps an element of degree +1 to a formal power series in ~g[[~]].

Remark 4.2.62. Note that, from the dual point of view, condition (4.2.32) tells us that x isa fixed point of the cohomological vector field δ. This means that for each f ∈ Sym(g∗[[~]][1])we have

(4.2.33) δf(~x) = 0.

Moreover, using that (δf)j = δj−1f , we can expand Equation (4.2.33) into a formal Taylorseries. In fact, using the pairing

〈δj−1f, x⊗ · · · ⊗ x〉 = 〈f,mj−1(x⊗ · · · ⊗ x)〉,we can write the Maurer–Cartan equation (4.2.32) as

(4.2.34)∑

j≥1

~j

j!mj−1(x⊗ · · · ⊗ x) = ~m0(x) +

~2

2m1(x⊗ x) +O(~3) = 0.

For two DGLAs g and h, an L∞-morphism ϕ : Sym(h∗[[~]][1]) ! Sym(g∗[[~]][1]) preserves theMaurer–Cartan equation (4.2.34), similarly as a morphism of DGLAs preserves the Maurer–Cartan equation (4.2.13). In particular, if x is a solution of (4.2.34) on g[[~]], we get that

U(~x) =∑

j≥1

~j

j!Uj(x⊗ · · · ⊗ x)

is also a solution of (4.2.34) on h[[~]].

Remark 4.2.63. The action of the gauge group on solutions to the Maurer-Cartan equation(4.2.13) can be generalized to the case of L∞-algebras. Namely, if x and x′ are equivalentmodulo this generalized action, then their images under U are still equivalent solutions.

Remark 4.2.64. Again, we are especially interested in the case where g = V and h = D.Thus, we get an L∞-morphism U which gives as a formula of how to construct an associativestar product out of any (formal) Poisson bivector π, given by

(4.2.35) U(π) =∑

j≥0

~j

j!Uj(π ∧ · · · ∧ π),

where we insert the coefficient of degree zero to be the original non-deformed product. Notethat if U is a quasi-isomorphism, the correspondence between (formal) Poisson structures on

4.3. KONTSEVICH’S STAR PRODUCT 105

a manifold M and formal deformations of the pointwise product on C∞(M) are one-to-one.In particular, as soon as we have a formality map, we have solved the problem of existenceand classification of star products on M .

4.3. Kontsevich’s star product

We want to construct an explicit expression of the formality map U . The idea is to introducegraphs and to rewrite things in those terms. An easy example is given by the Moyal productf ⋆ g for two smooth functions f and g on (R2n, ω0). We can write f ⋆ g as the sum of graphsin the following way:

f g f g f g f g

+ + + + . . .

Figure 4.3.1. The Moyal product f ⋆ g represented in terms of graphs. Thegray vertices represent the Poisson tensor π induced by ω0. The term withn Poisson vertices in the sum represents the n-th term in the formal powerseries of the Moyal product. For each gray vertex, the two outgoing arrowsrepresent the derivatives ∂i and ∂j acting on f and g.

Remark 4.3.1. This construction can be generalized by allowing more than two outgoingarrows for gray vertices when considering general multivector fields and allowing incomingarrows for gray vertices (i.e. derivations of the tensor coefficients for the assigned multivectorfields). Note that there are no incoming arrows for the Moyal product since the Poisson struc-ture is constant. In Kontsevich’s star product, we assign to each graph Γ a multidifferentialoperator BΓ and a weight wΓ such that the map U that sends an n-tuple of multivector fieldsto the corresponding weighted sum over all possible graphs in this set of multidifferentialoperators is an L∞-morphism.

4.3.1. Data for the construction.

4.3.1.1. Admissible graphs.

Definition 4.3.2 (Admissible graphs). The set Gn,n of admissible graphs consists of allconnected graphs Γ which satisfy the following properties:

• The set of vertices V (Γ) is decomposed in two ordered subsets V1(Γ) and V2(Γ)isomorphic to 1, . . . , n respectively 1, . . . , n whose elements are called verticesof first type respectively second type;

• The following inequalities involving the number of vertices of the two types aresatisfied: n ≥ 0, n ≥ 0 and 2n + n− 2 ≥ 0;

• The set of edges E(Γ) is finite and does not contain short loops, i.e. edges startingand ending at the same vertex;

• All edges E(Γ) are oriented and start from a vertex of the first type;• The set of edges starting at a given vertex v ∈ V1(Γ), which will be denoted byStar(v), is ordered.


See Figure 4.3.2 and 4.3.3 for examples of admissible and non-admissible graphs respectively.

1 2 1 2 3

1

2

3 1

2

Figure 4.3.2. Examples of admissible graphs.

1 2 1 2

1

2

12

Figure 4.3.3. Examples of non-admissible graphs.

4.3.1.2. The multidifferential operators BΓ. Let us consider pairs (Γ, ξ1 ⊗ · · · ⊗ ξn) con-sisting of a graph Γ ∈ Gn,n with 2n + n − 2 edges and of a tensor product of n multivector

fields on Rd. We want to understand how we can associate to such a pair a multidifferentialoperator BΓ ∈ Dn−1. This is done in the following way:

• Associate to each vertex v of first type with k outgoing arrows the skew-symmetric

tensor ξj1···jki corresponding to a given ξi via the natural identification.• Place a function at each vertex of second type.• Associate to the ℓ-th arrow in Star(v) a partial derivative with respect to the coor-dinate labeled by the ℓ-th index of ξi acting on the function or the tensor appearingat its endpoint.

• Multiply such elements in the order prescribed by the labeling of the graph.

Example 4.3.3. Denote by Γ1 the first graph in Figure 4.3.2. Then, to a triple of bivec-

tor fields (ξ1, ξ2, ξ3) with ξ1 =∑

i1<i2ξi1i21 ∂i1 ∧ ∂i2 , ξ2 =

∑j1<j2

ξj1j22 ∂j1 ∧ ∂j2 and ξ3 =∑ℓ1<ℓ2

ξℓ1ℓ23 ∂ℓ1 ∧ ∂ℓ2 , we associate the bidifferential operator

UΓ1(ξ1 ∧ ξ2 ∧ ξ3)(f ⊗ g) := ξj1j22 ∂j2ξi1i21 ∂j2ξ

ℓ1ℓ23 ∂i1∂ℓ1f∂i2∂ℓ2g.

Example 4.3.4. Denote by Γ2 the second graph in Figure 4.3.2. Then, to a tuple of multi-

vector fields (χ1, χ2) with χ1 =∑

i1<i2χi1i21 ∂i1 ∧∂i2 and χ2 =

∑j1<j2<j3

χj1j2j32 ∂j1 ∧∂j2 ∧∂j3 ,we associate the tridifferential operator

UΓ2(χ1 ∧ χ2)(f ⊗ g ⊗ h) := χi1i21 ∂i1χj1j2j32 ∂j1f∂j2g∂j3∂i2h.


Remark 4.3.5. In particular, for each admissible graph Γ ∈ Gn,n, we get a linear mapUΓ :

∧n V ! D which is equivariant with respect to the action of the symmetric group.Kontsevich’s main construction was to choose weights wΓ ∈ R such that the linear combina-tion

U :=∑

n,n≥0

∑

Γ∈Gn,n

wΓBΓ

becomes an L∞-morphism.

4.3.1.3. Weights of graphs. The weight wΓ associated to an admissible graph Γ ∈ Gn,nis defined by an integral of a differential form ωΓ over (a suitable compactification of) someconfiguration space C+

n,n. Namely, it is given by

(4.3.1) wΓ :=∏

1≤k≤n

1

|Star(k)|!1

(2π)2n+n−2

∫

C+n,n

ωΓ.

This holds for the case when Γ has exactly 2n+ n− 2 edges otherwise the weight is just zero.Let us look at the integral (4.3.1) more carefully.

4.3.1.4. Configuration spaces. Consider an embedding of the graph Γ into the 2-dimensionalupper half-space H2 (also called upper half-plane) as defined in (1.4.7) where we want toidentify R2 ∼= C. Moreover, denote by H2

+ := z ∈ H | ℑ(z) > 0 where ℑ(z) denotes theimaginary part of z.

Definition 4.3.6 (Open configuration space). Define the open configuration space of thedistinct n+ n vertices of an admissible graph Γ ∈ Gn,n as the smooth manifold(4.3.2)Confn,n :=

(z1, . . . , zn, z1, . . . , zn) ∈ Cn+n | zi ∈ H2

+, zi ∈ R, zi 6= zj for i 6= j, zi 6= zj for i 6= j

Remark 4.3.7. More precisely, we need to consider everything up to scaling and translation.Hence, we consider the Lie group G consisting of translations in the horizontal direction andrescaling such that the action on z ∈ H2 is given by

z 7! az + b, a ∈ R+, b ∈ R.

The action of G is free whenever the number of vertices is 2n+ n− 2 ≥ 0. Thus the quotientspace of Confn,n with respect to the G-action, which we will denote by Cn,n, is again asmooth manifold of (real) dimension 2n + n− 2. If there are no vertices of second type (i.e.n = 0), one can define the open configuration space as a subset of Cn instead of Hn and onecan consider the Lie group which consists of rescaling and translation in any direction. Thecorresponding quotient space Cn for n ≥ 2 still a smooth manifold but now of dimension2n− 3.

Definition 4.3.8 (Connected configuration space). Define the connected configuration spaceto be the subset of Cn,n given by

(4.3.3) C+n,n :=

(z1, . . . , zn, z1, . . . , zn) ∈ Cn,n | zi < zj for i < j

Remark 4.3.9. One can show that C+n,n is again a smooth manifold which is now connected.

Let us define an angle mapφ : C2,0 ! S1,


which associates to each pair of distinct points z1, z2 ∈ H2 the angle between the geodesicswith respect to the Poincare metric connecting z1 to i∞ and to z2, in the counterclockwisedirection (see Figure 4.3.4). Formally, we have

(4.3.4) φ(z1, z2) := arg

(z2 − z1z2 − z1

)=

1

2ilog

((z2 − z1)(z2 − z1)

(z2 − z1)(z2 − z1)

).

z1

z2H2

i∞

φ

Figure 4.3.4. Illustration of the angle map φ.

The differential of φ is then a 1-form on C2,0 which we can pull-back to the configurationspace including points in R by the projection pe associated to each edge e = (zi, zj) of Γdefined by

pe : Cn,n ! C2,0,

(z1, . . . , zn, z1, . . . , zn) 7! (zi, zj).

Hence we get a 1-form dφe := p∗edφ ∈ Ω1(Cn,n). The differential form ωΓ in (4.3.1) is thendefined as

ωΓ :=∧

e∈E(Γ)

dφe.

Remark 4.3.10. Note that the ordering one these 1-forms is induced by the ordering on theset of edges by the first-type vertices and the ordering on Star(v). Moreover, note that weindeed obtain a top-form on Cn,n (resp. C+

n,n) as long as we consider graphs with exactly

2n+ n− 2 edges since dimCn,n = dimC+n,n = 2n+ n− 2.

Remark 4.3.11. In order to make sense of the integral (4.3.1), we need to make sure that itconverges. Note that, by construction of φ, we get that dφ is not defined as soon two pointsapproach each other. This is the reason why we in fact need a suitable compactification C+

n,n

of the connected configuration space C+n,n.

Theorem 4.3.12 ([FM94; AS94]). For any configuration space Cn,n (resp. Cn) there existsa compactification Cn,n (resp. Cn) whose interior is the open configuration space and suchthat the projections pe, the angle map φ (and thus the differential form ωΓ) extend smoothlyto the corresponding compactifications.

Remark 4.3.13. This compactification is usually called FMAS compactification for Fulton–MacPherson who proved a first result of this in the algebro-geometrical setting for config-uration spaces of points in non-singular algebraic varieties [FM94] and Axelrod–Singer who


proved this in the smooth-geometrical setting for configuration spaces of points in smoothmanifolds [AS94].

Remark 4.3.14. The compactified configuration space is a compact smooth manifold withcorners.

Definition 4.3.15 (Manifold with corners). A manifold with corners of dimension m is atopological Hausdorff space M which is locally homeomorphic to Rm−n × Rn+ with n =0, . . . ,m.

Remark 4.3.16. The points x ∈ M of a manifold with corners of dimension m whose localexpression in some (and hence in any) chart has the form

(x1, . . . , xm−n, 0, . . . , 0)

are said to be of type n and form submanifolds of M which are called strata of codimensionn.

4.3.2. Proof of Kontsevich’s formula. To show that U is indeed an L∞-morphism,we need to show the following points:

(1) The first component of the restriction of U to V is up to a shift in degrees given by

the natural map U(0)1 as defined in (4.2.22).

(2) U is a graded linear map of degree zero.(3) U satisfies the equations (4.2.6) for an L∞-morphism.

4.3.2.1. U1 coincides with U(0)1 .

Lemma 4.3.17. The mapU1 : V ! D

is the natural map U(0)1 that identifies each multivector field with the corresponding multidif-

ferential operator.

Proof. Consider the set G1,n of admissible graphs with one vertex of first type and nvertices of second type. It is easy to see that this set only contains one element Γn whichhas 2 · 1 + n − 2 = n arrows outgoing from the single vertex of first type and each arrow isincoming to a vertex of second type (see Figure 4.3.5).

1 2 3 n

· · ·

1

Figure 4.3.5. The single element Γn ∈ G1,n.

Hence, for a multivector field ξ of degree k we associate the multidifferential operator

UΓn(ξ) : f1 ⊗ · · · ⊗ fn 7! wΓnξi1···in∂i1f1 · · · ∂infn.


One can easily check that ∫

C1,n

ωΓn =

∫

C1,n

∧

e∈E(Γn)

dφe = (2π)n.

Using (4.3.1), this will give us

wΓn =1

n!.

This shows that U1 is indeed the analogue for U(0)1 in (4.2.22) which induces the HKR

isomorphism (4.2.21).

4.3.2.2. Checking the degrees.

Lemma 4.3.18. The n-th component

Un :=∑

n≥1

∑

Γ∈Gn,n

wΓBΓ

has the correct degree for U to be an L∞-morphism.

Proof. Note that to a vertex vi with |Star(vi)| outgoing arrows, we associate an element

in V |Star(vi)| = V |Star(vi)|+1. On the other hand, each graph Γ with n vertices of second typeand n multivector fields ξ1, . . . , ξn induce a multidifferential operator Un(ξ1 ∧ · · · ∧ ξn) ofdegree s = n − 1. Since we are only considering graphs with exactly 2n + n − 2 edges andsince

|E(Γ)| =∑

1≤i≤n

|Star(vi)|,

we can write the degree of Un(ξ1 ∧ · · · ∧ ξn) ass = (2n + n− 2) + 1− n = |E(Γ)|+ 1− n =

∑

1≤i≤n

|Star(vi)|+ 1− n.

This is exactly the degree for the n-th component of an L∞-morphism.

4.3.2.3. Reformulation of the L∞-condition in terms of graphs. Next we discuss the geo-metric proof of the formality statement. We have to extend the morphism U with a degreezero component which represents the usual multiplication between smooth maps. Therefore,we can consider the special case of the L∞-condition (4.2.31) where m0, m0 are given by theTaylor coefficients Un:

(4.3.5)∑

0≤ℓ≤n

∑

−1≤k≤m

∑

0≤i≤m−k

εkim∑

σ∈Sℓ,n−ℓ

εξ(σ)Uℓ

(ξσ(1) ∧ · · · ∧ ξσ(ℓ)

)

(f0 ⊗ · · · ⊗ fi−1 ⊗ Un−ℓ

(ξσ(ℓ+1) ∧ · · · ∧ ξσ(n)

)(fi ⊗ · · · ⊗ fi+k

)⊗ fi+k+1 ⊗ · · · ⊗ fm

)

=

n∑

i 6=j=1

εijξ Un−1

(ξi ξj ∧ ξ1 ∧ · · · ∧ ξi ∧ · · · ∧ ξj ∧ · · · ∧ ξn

)(f0 ⊗ · · · ⊗ fn

),

where (ξj)j=1,...,n are multivector fields, f0, . . . , fm are smooth maps on which the multidiffer-ential operator is acting, Sℓ,n−ℓ is the subset of Sn consisting of (ℓ, n−ℓ)-shuffle permutations,


the product ξi ξj is defined in a way such that the Schouten–Nijenhuis bracket can be ex-pressed in terms of this composition by a formula similar to the one relating the Gerstenhaberbracket to the analogous composition on D in (4.2.14) and the signs involved are definedas follows: εkim := (−1)k(m+i), εξ(σ) is the Koszul sign associated to the permutation σ and

εijξ is defined as in (4.2.31).

Remark 4.3.19. Note that (4.3.5) carries the formality condition since the left hand sidecorresponds to the Gerstenhaber bracket of multidifferential operators and the right handside to (a part of) the Schouten–Nijenhuis bracket (since the differential on V is identical tozero). Recall that the differential on D is is given by the Hochschild differential [m, ]G whichin (4.3.5) is replaced by U0.

We can now rewrite (4.3.5) in terms of admissible graphs and weights to prove that it holds.Note that the difference between the left hand side and right hand side of (4.3.5) can beformulated as a linear combination

(4.3.6)∑

Γ∈Gn,n

cΓUΓ(ξ1 ∧ · · · ∧ ξn)(f0 ⊗ · · · ⊗ fn).

Recall that we want to consider admissible graphs Γ with exactly 2n + n− 2 edges. In fact,(4.3.5) is satisfied if cΓ = 0 for all such Γ. We will show that this is indeed the case.

4.3.2.4. The key is Stokes’ theorem. In order to prove that all these coefficients cΓ vanish,we will use Stokes’ theorem for manifolds with corners. Similarly as for the usual Stokes’theorem, we can replace an integral of an exact form over some manifold M as the integralof its primitive form over the boundary ∂M . In particular, we have

(4.3.7)

∫

∂C+n,n

ωΓ =

∫

C+n,n

dωΓ = 0, ∀Γ ∈ Gn,n

since dφe is closed and C+n,n is compact. Let us expand the left hand side of (4.3.7) in order

to show that it is exactly given by the coefficients cΓ. Therefore, we want to take a closer lookon the manifold ∂C+

n,n. Recall that we have restricted the weights in (4.3.5) to be equal tozero if the form degree does not match the dimension of the underlying manifold over whichwe integrate. Hence, we only have to consider the codimension 1 strata of ∂C+

n,n which have

dimension 2n+ n− 3. Note that the dimension of ∂C+n,n is equal to the number of edges and

thus of the 1-forms dφe.

Remark 4.3.20. Intuitively, one can think of the boundary of C+n,n to be represented by the

degenerate configuration in which some of the n+ n points collide with each other.

4.3.2.5. Classification of boundary strata. Using Remark 4.3.20, we can classify the codi-mension 1 strata of ∂C+

n,n as follows:

• (Strata of type S1) These are strata in which i ≥ 2 points in H2+ collide together to

a point which still lies in H2+. Points in such a stratum can be locally described by

(4.3.8) Ci × Cn−i+1,n.

The first term represents the relative position of the colliding points when we look atthem under a magnifying glass. The second term is the space of all remaining points


plus the point which occurs as the point on which the first i points have collapsedto.

• (Strata of type S2) These are strata in which i > 0 points in H2+ and j > 0 points

in R with 2i+ j − 2 ≥ 0 collide to a single point on R. Points in such a stratum canbe locally represented by

(4.3.9) Ci,j × Cn−i,n−j+1.

The two terms are similarly given as in (4.3.8).

See Figure 4.3.6 and 4.3.7 for an illustration of a stratum of type S1 and type S2 respectively.

1 2 3 n

ℓ1 n

2

H2

Figure 4.3.6. Example of a stratum of type S1. Note that here i = 3 pointshave collapsed together to a point ℓ ∈ H2

+.

1 2 3 nℓ

1 n

2

H2

Figure 4.3.7. Example of a stratum of type S2. Note that here i = 3 andj = 2 points have collapsed together to a point ℓ ∈ R.

Now we can split the integral on the left hand side of (4.3.7) into the sum of strata of type S1and type S2. For the strata of type S1, we will distinguish two subcases for the i collapsing


vertices. Since the integral vanishes if the form degree is not equal to the dimension of theunderlying manifold, one can show that the only contributions come from graphs Γ whosesubgraphs Γ1, spanned by the collapsing vertices, contain exactly 2i− 3 edges.When i = 2, there is only one edge e, an hence in the first integral of the decompositionof (4.3.8) the differential dφe is integrated over C2

∼= S1 and thus we get a factor of 2πwhich cancels the coefficient in (4.3.1). The remaining integral represents the weight of thecorresponding quotient graph Γ2 which is obtained from Γ after the contraction of the edgee. In particular, to the vertex j of first type, which results from this contraction, we associatethe j-composition of the two multivector fields that were associated to the endpoints of e.Hence, summing over all graphs and all strata of this subtype, we get the right hand side of(4.3.5).

4.3.2.6. A trick using logarithms. When i ≥ 3, the integral corresponding to this stratuminvolves the product of 2i − 3 angle forms over Ci. According to a Lemma of Kontsevich,this integral vanishes.

Lemma 4.3.21 (Kontsevich[Kon03]). The integral over the configuration space Cn of n ≥ 3points in the upper half-plane of any dimCn := 2n − 3 angle forms dφei with i = 1, . . . , nvanishes for n ≥ 3.

1 n

H2

Figure 4.3.8. Example of a non-vanishing term.

Proof of Lemma 4.3.21. First we need to restrict the integral to an even number ofangle forms. This can be done by identifying Cn with the subset of Hn, where one of theendpoints of e1 is set to be the origin and the second is forced to lie on the unit circle (notethat this can always be done by considering the action of the Lie group in the definitionof Cn). Then the integral decomposes into a product of integrals of dφe over S1 and theremaining 2n−4 =: 2N forms integrated over the resulting complex manifold U given by theisomorphism

Cn ∼= S1 × U.


1 n

H2

Figure 4.3.9. Example of a vanishing term.

Then the claim follows by the following computation:

∫

U

2N∧

j=1

d arg(fj) =

∫

U

2N∧

j=1

d log(|fj |) =∫

UI(d log(|f1|)

2N∧

j=2

d log(|fj|))

=

∫

Ud

(I(log(|f1|)

2N∧

j=2

d log(|fj|)))

= 0.

(4.3.10)

Let us see what is happening here. First, note that the angle function φej was expressed interms of the argument of the (holomorphic) function fj . In particular, fj is just the differenceof the coordinates of the endpoints of ej . The first equality follows by the decompositions

d arg(fj) =1

2i

(d log(fj)− d log(fj)

),

d log(|fj|) =1

2

(d log(fj) + d log(fj)

).

Thus, the product of 2N of these expressions is a linear combination of products of k holo-morphic and 2N − k anti-holomorphic forms. By a result of complex analysis, one can showthat in the integration over the complex manifold U , the only terms which do not vanishare the ones where k = N . Then one can observe that the terms coming from the firstdecomposition are equal to those coming from the second decomposition.In the second equality the integral of the differential form is replaced by an integral over asuitable 1-form with values in the space of distributions over the compactification U of U .One can show (this is another Lemma) that such a map I, which sends usual 1-forms todistributional ones, commutes with the differential and thus by Stokes’ theorem we get theclaim.

4.3.2.7. Last step: vanishing terms for type S2 strata. Finally, let us consider the strataof type S2. There we will have a similar dimensional argument, i.e. it analogously restrictsthe possible non-vanishing terms to the condition that the subgraph Γ1, spanned by i + jcollapsing vertices of first and second type respectively, contains exactly 2i + j − 2 edges.Similarly as before, for the quotient graph Γ2 obtained by contracting Γ1, the claim is that


the only non-vanishing contributions come from the graphs for which both graphs obtainedfrom a given Γ are admissible. In this case we get a decomposition of the weight wΓ into theproduct wΓ1 · wΓ2 which in general, by the conditions on the number of edges of Γ and Γ1,does not vanish.The only remaining thing to check is that we do not have bad edges by contraction (seeFigure 4.3.3). The only such possibility appears when Γ2 contains an edge which starts froma vertex of second type. In this case the corresponding integral vanishes because it containsthe differential of an angle map evaluated on the pair (z1, z2) where z1 is constrained to liein R and such maps vanish for every z2 because the angle is measured with respect to thePoincare metric (recall Figure 4.3.4 for an intuitive picture).

1 ℓ ℓ+ 1 n

H2

Figure 4.3.10. Example of an admissible quotient.

1 ℓ ℓ+ 1 n

bad edge

H2

Figure 4.3.11. Example of a non-admissible quotient. Such a term vanishes.

This means that the only non-vanishing terms correspond to the case when we plug thedifferential operator associated to Γ1 as the k-th argument of the one associated to Γ2, wherek is the vertex of the second type emerging from the collapse. Summing over all thesepossibilities, the strata of type S2 exactly corresponds to the left hand side of (4.3.5).


We have thus proven that U is indeed an L∞-morphism and since the first coefficients U1 are

given by U(0)1 it is also a quasi-isomorphism and hence determines uniquely a star product

for any bivector field π on Rd by the formula (4.2.35).

4.4. Globalization of Kontsevich’s star product

Kontsevich’s formula for the star product only gives a quantization for the case whenM = Rd

for a general Poisson structure π, and thus only describes a local description for the generalcase. Already in [Kon03], Kontsevich gave a globalization method which was very brieflymentioned, but it was explicitly constructed by Cattaneo, Felder and Tomassini in [CFT02b;CFT02a] in a similar way as Fedosov did for the symplectic case of the Moyal product [Fed94](see also Section 4.1.3). Their approach uses a flat connection D on a vector bundle over Msuch that the algebra of the horizontal sections with respect to D is a quantization of C∞(M).Consider the vector bundleE0 !M of infinite jets of functions endowed with a flat connectionD0. The fiber (E0)x over x ∈M is naturally a commutative algebra and carries the Poissonstructure induced fiberwise by the Poisson structure on C∞(M). The map which associates toany globally defined function its infinite jet at each point x ∈M is a Poisson isomorphism ontothe Poisson algebra of horizontal sections of E0 with respect to D0. Since the star productgives a deformation of the pointwise product on C∞(M), we want to have a quantum versionof the vector bundle and the flat connection in order to get an similar isomorphism. Thevector bundle E ! M is defined in terms of a section φaff of the fiber bundle Maff

! M ,whereMaff denotes the quotient of the manifold M coor of jets of coordinate systems onM by

the action of the group GL(d,R) of linear diffeomorphims given by E := (φaff )∗E, where Eis the bundle of R[[~]]-modules M coor×GL(d,R)R[[y

1, . . . , yd]][[~]]!Maff . Note that the section

φaff can be realized explicitly by a collection of jets at 0 of maps φx : Rd! M such that

φx(0) = x for all x ∈M (modulo the action of GL(d,R)). Thus, we can assume for simplicitythat we have fixed a representative φx of the equivalence class for each open set of a givencovering, hence realizing a trivialization of the bundle E. So from now on we will identify Ewith the trivial bundle with fiber given by R[[y1, . . . , yd]][[~]]. Therefore, E realizes the desiredquantization, since it is isomorphic (as a bundle of R[[~]]-modules) to the bundle E0[[~]] whoseelements are formal power series with infinite jets of functions as coefficients.

4.4.1. The multiplication, connection and curvature maps. To define the starproduct and the connection on E, one has to introduce new objects whose existence andproperties will be byproducts of the formality theorem. For a Poisson structure π and twovector fields ξ and η on Rd, we define

P (π) :=∑

j≥0

~j

j!Uj(π ∧ · · · ∧ π),(4.4.1)

A(ξ, π) :=∑

j≥0

~j

j!Uj+1(ξ ∧ π ∧ · · · ∧ π),(4.4.2)

F (ξ, η, π) :=∑

j≥0

~j

j!Uj+2(ξ ∧ η ∧ π ∧ · · · ∧ π).(4.4.3)

4.4. GLOBALIZATION OF KONTSEVICH’S STAR PRODUCT 117

It is easy to show that the degree of the multidifferential operators on the right hand sidesof (4.4.1), (4.4.2) and (4.4.3) induce that P (π) is a (formal) bidifferential operator, A(ξ, π)is a differential operator and F (ξ, η, π) is a function. Indeed, P (π) is actually just the starproduct associated to π. In particular, the maps P,A and F are elements of degree 0, 1 and 2respectively of the Lie algebra cohomology complex of (formal) vector fields with values in thespace of local polynomial maps, i.e. multidifferential operators which depend polynomially onπ. An element of degree j of this complex is a map that sends ξ1∧· · ·∧ξj to a multidifferentialoperator S(ξ1 ∧ · · · ∧ ξj ∧ π). The differential δ of this complex can then be defined as

δS(ξ1 ∧ · · · ∧ ξj+1 ∧ π) :=∑

1≤i≤j+1

(−1)i∂

∂t

∣∣∣t=0

S(ξ1 ∧ · · · ∧ ξi ∧ · · · ∧ ξj+1 ∧ (Φξit )∗π

)

+∑

i<ℓ

(−1)ℓS([ξi, ξℓ] ∧ ξ1 ∧ · · · ∧ ξi ∧ · · · ∧ ξℓ ∧ · · · ∧ ξj+1 ∧ π

),

where Φξt denotes the flow of the vector field ξ. The associativity of the star product can nowbe expressed by

P (P ⊗ id− id⊗ P ) = 0.

This follows from the formality theorem and hence the following equations do hold in the asimilar way:

(4.4.4) P (π) (A(ξ, π) ⊗ id + id⊗A(ξ, π)) = A(ξ, π) P (π) + δP (ξ, π),

(4.4.5)P (π) (F (ξ, η, π)⊗ id− id⊗F (ξ, η, π)) = −A(ξ, π) A(η, π) +A(η, π) A(ξ, π) + δA(ξ, η, π),

(4.4.6) −A(ξ, π) F (η, ζ, π) −A(η, π) F (ζ, ξ, π)−A(ζ, π) F (ξ, η, π) = δF (ξ, η, ζ, π)

Equation (4.4.4) describes the fact that under coordinate transformation induced by thevector field ξ, the star product P (π) is changed to an equivalent one up to higher orderterms. Equations (4.4.5) and (4.4.6) will be used for the construction of the relations betweena connection 1-form A and its curvature FA. For the explicit computation of the configurationspace integrals, which arise for the weight computation in the Taylor coefficients of Uj , wecan also describe the lowest order terms in the expansion of P,A and F and their action onfunctions:

(1) P (π)(f ⊗ g) = fg + ~π(df,dg) +O(~2),(2) A(ξ, π) = ξ + O(~), where we identify ξ with a first order differential operator on

the right hand side,(3) A(ξ, π) = ξ, if ξ is a linear vector field,(4) F (ξ, η, π) = O(~),(5) P (π)(1⊗ f) = P (π)(f ⊗ 1) = f ,(6) A(ξ, π)1 = 0.

Equations (1) and (5) have already been introduced before as the defining conditions for astar product. The equations involving the connection A are used to construct a connectionD on sections on E. A section f ∈ Γ(E) is locally given by a map x 7! fx, where for everyy, fx(y) is a formal power series with coefficients given by infinite jets. On this space wecan introduce a deformed product ⋆ which will be the desired star product on C∞(M) after


we have identified horizontal section with ordinary functions. Let us denote analogously byπx the push-forward by φ−1

x of the Poisson bivector field π on Rd. Then we can define thedeformed product by the formal bidifferential operator P (πx) similarly as we did for the usualproduct by P (π):

(f ⋆ g)x(y) := fx(y)gx(y) + ~πijx (y)∂fx(y)

∂yi∂gx(y)

∂yj+O(~2).

One can define the connection D on Γ(E) by

(Df)x := dxf +AMx f,

where dxf denotes the de Rham differential of f regarded as a function with values inR[[y1, . . . , yd]][[~]] and the formal connection 1-form is given through its action on a tangentvector ξ by

AMx (ξ) := A(ξx, πx),

where A is the operator defined as in (4.4.2) evaluated on the multivector fields ξ and πgiven through the local coordinate system defined by φx. Note that since the coefficientsUj of the formality map that appear in the definition of P and A are polynomial in thederivatives of the coordinate of the arguments ξ and π, all results which hold for P (π) andA(ξ, π) are inherited by their counterparts. In fact, Equations (1) and (5) above ensure that⋆ is an associative deformation of the pointwise product on sections and Equations (2) and(3) can be used to prove that D is independent of the choice of φ and hence induces a globalconnection on E.

4.4.2. Construction of solutions for a Fedosov-type equation. We can extendD and ⋆ by the (graded) Leibniz rule to the whole complex of formal differential formsΩ•(E) := Ω•(M)⊗C∞(M) Γ(E) and use (4.4.5) to get the following lemma.

Lemma 4.4.1. Let FM be the E-valued 2-form given by x 7! FMx where FMx (ξ, η) := F (ξx, ηx, πx)for any pair of vector fields ξ, η. Then FM represents the curvature of D and the two arerelated to each other and to the star product by the usual identities:

(1) D(f ⋆ g) = D(f) ⋆ g + f ⋆ D(g),(2) D2 = [FM , ]⋆,(3) DFM = 0.

Proof. We can deduce these identities from the relations (4.4.4), (4.4.5) and (4.4.6), inwhich the star commutator [f, g]⋆ := f ⋆g−g⋆f is already implicitly defined, after identifyingthe complex of formal multivector fields endowed with the differential δ with the complex offormal multidifferential operators with the de Rham differential. Such an isomorphism wasexplicitly given in [CFT02b].

Remark 4.4.2. A connection D which satisfies the above relation on a bundle E of associativealgebras is called Fedosov connection with Weyl curvature F . It is the kind of connectionFedosov introduced in order to give a global construction for the symplectic case (Section4.1.3) [Fed94].

The final step towards a globalization is to deform the connection D into a new connectionD which has the same properties and moreover has vanishing Weyl curvature, i.e. D2 = 0.

4.4. GLOBALIZATION OF KONTSEVICH’S STAR PRODUCT 119

Hence, we can define the complex HjD(E) and thus the (sub)algebra of horizontal sections

H0D(E).

Lemma 4.4.3. Let D be a Fedosov connection on E with Weyl curvature F and γ an E-valued1-form. Then

D := D + [γ, ]⋆is also a Fedosov connection with Weyl curvature F = F +Dγ + γ ⋆ γ.

Proof. Let f be a section in Γ(E). Then

D2f = [F , f ]⋆ +D[γ, f ]⋆ + [γ,Df ]⋆ + [γ, [γ, f ]⋆]⋆

= [F, f ]⋆ + [Dγ, f ]⋆ + [γ, [γ, f ]⋆]⋆

=

[F +Dγ +

1

2[γ, γ]⋆, f

]

⋆

,

where the last equality follows from the Jacobi identity of the star commutator [ , ]⋆, sinceeach associative product induces a Lie bracket given by the commutator. if we apply D tothe obtained curvature, we get

D

(F +Dγ +

1

2[γ, γ]⋆

)= D2γ +

1

2[Dγ, γ]⋆ −

1

2[γ,Dγ]⋆ + [γ, F +Dγ]⋆

= [F, γ]⋆ + [γ, F ]⋆

= 0,

where we again used the (graded) Jacobi identity and the (graded) skew-symmetry of [ , ]⋆.

Lemma 4.4.4. Let D be a Fedosov connection on a bundle E = E0[[~]] and F its Weylcurvature and let

D = D0 + ~D1 + · · · , F = F0 + ~F1 + · · ·be their expansion as formal power series. If F0 = 0 and the second cohomology of E0 withrespect to D0 is trivial, then there exists a 1-form γ such that D has vanishing Weyl curvature.

Proof. By Lemma 4.4.3, we can equivalently say that there exists a solution to theequation

F = F +Dγ +1

2[γ, γ]⋆ = 0.

We can explicitly construct a solution by induction on the order of ~. We start from γ0 = 0and assume that γ(j) is a solution modulo ~(j+1). We can add to F (j) = F + Dγ(j) +12

[γ(j), γ(j)

]⋆the next term ~j+1D0γj+1 to get F j+1 modulo higher terms. From the equation

DF (j)+[γ(j), F (j)

]⋆= 0 and the induction hypothesis F j = 0 modulo ~(j+1), we get D0F

(j) =

0. Now since H2D0

(E0) = 0, we can invert D0 in order to define γj+1 in terms of the lower

order terms F (j) in a way such that F (j+1) = 0 modulo ~(j+2). This completes the inductionstep.

Remark 4.4.5. Note that in our case D is a deformation of the natural flat connection D0

on sections of the bundle of infinite jets. Thus, the hypothesis of Lemma 4.4.4 is indeedsatisfied and we can find a flat connection D which is still a good deformation of D0. In


the setting of formal geometry, a flat connection as D0 is called a Grothendieck connection[Gro68; Kat70] which is a certain generalization of the Gauss–Manin connection [Man58].Another more technical lemma, which uses the notion of homological perturbation theory,actually gives an isomorphism between the algebra of horizontal sections H0

D(E) and its non-

deformed counterpartH0D0

(E0) which is isomorphic to C∞(M). This implies the globalizationprocedure.

Remark 4.4.6. In [Dol05], Dolgushev gave another proof of Kontsevich’s formality theoremfor general manifolds by using covariant tensors instead of ∞-jets of multidifferential opera-tors and multivector fields which is intrinsically local. In particular, he also formulated thedeformation quantization construction on Poisson orbifolds.

4.5. Operadic approach to formality and Deligne’s conjecture

4.5.1. Operads and algebras.

Definition 4.5.1 (Algebraic operad). An (algebraic) operad (of vector spaces) consists ofthe following;

(1) A collection of vector spaces P (n), n ≥ 0,(2) An action of the symmetric group Sn on P (n) for all n,(3) An identity element idP ∈ P (1),(4) Compositions mn1,...,nk

:

P (k)⊗ (P (n1)⊗ · · · ⊗ P (nk))! P (n1 + · · · + nk),

for all k ≥ 0 and n1, . . . , nk ≥ 0 satisfying a list of axioms.

Remark 4.5.2. We want to obtain the list of axioms for an operad by looking at someexamples.

Example 4.5.3 (Endomorphism operad). Consider the very simple operad P (n) := Hom(V ⊗n, V )for some vector space V . The action of the symmetric group and the identity element areobvious. The compositions are defined by

(mn1...,nk(φ⊗ (ψ1 ⊗ · · · ⊗ ψk)))(v1 ⊗ · · · ⊗ vn1+···+nk

)

:= φ(ψ1(v1 ⊗ · · · ⊗ vn1)⊗ · · · ⊗ ψk(vn1 + · · ·+ vnk−1+1 ⊗ · · · ⊗ vn1+···+nk)),

where φ ∈ P (k) = Hom(V ⊗k, V ), ψi ∈ P (ni) = Hom(V ⊗ni , V ) for i = 1, . . . , k. This operadis called the endomorphism operad of a vector space V .

Remark 4.5.4. We can see from Example 4.5.3 that there should be an assoicativity axiom formultiple compositions, various compatibilities for actions of symmetric groups, and evidentrelations for compositions including the identity element.

Example 4.5.5 (Associative operad). Consider the operad P = Assoc1. The n-th componentP (n) = Assoc1(n) for n ≥ 0 is defined as the collection of all universal (functorial) n-linearoperations A⊗n

! A of associative algebras A with unit. The space Assoc1(n) has dimensionn!, and is spanned by the operations

a1 ⊗ · · · ⊗ an 7! aσ(1) · · · aσ(n), σ ∈ Sn.

4.5. OPERADIC APPROACH TO FORMALITY AND DELIGNE’S CONJECTURE 121

We can identify Assoc1(n) with the subspace of free associative unital algebras in n generatorsconsisting of expressions which are multilinear in each generator.

Definition 4.5.6 (Algebra over an operad). An algebra over an operad P consists of a vectorspace A and a collection of multilinear maps fn : P (n) ⊗ A⊗n

! A for all n ≥ 0 satisfyingthe following axioms:

(1) The map fn is Sn-equivariant for any n ≥ 0,(2) We have f1(idP ⊗ a) = a for all a ∈ A,(3) All compositions in P map to compositions of multilinear operations on A.

Remark 4.5.7. We often also call an algebra A over an operad P a P -algebra.

Example 4.5.8. The algebra over the operad Assoc1 is an associative unital algebra. If wereplace the 1-dimensional space Assoc1(0) by the zero space 0, we obtain an operad Assoc

describing associative algebras possibly without unit.

Example 4.5.9 (Lie operad). There is an operad Lie such that Lie-algebras are Lie algebras.The dimension of the n-th component Lie(n) is (n− 1)! for n ≥ 0 and zero for n = 0.

Theorem 4.5.10. Let P be an operad and V a vector space. Then the free P -algebra FreeP (V )generated by V is naturally isomorphic as a vector space to

⊕

n≥0

(P (n)⊗ V ⊗n)Sn ,

where the subscript Sn denotes the quotient space of coinvariants for the diagonal action ofthe symmetric group.

Remark 4.5.11. The free algebra FreeP (V ) is defined by the usual categorical adjunctionproperty, i.e. the set HomP -algebras(FreeP (V ),A) (i.e. homomorphisms in the category ofP -algebras) is naturally isomorphic to the set Homvector spaces(V,A) for any P -algebra A.

4.5.2. Topological operads. We can replace vector spaces by topological spaces in thedefinition of an operad. The tensor product is then replaced by the Cartesian product.

Definition 4.5.12 (Topological operad). A topological operad consists of the following:

(1) A collection of topological spaces P (n) for n ≥ 0,(2) A continuous action of the symmetric group Sn on P (n) for all n,(3) An identity element idP ∈ P (1),(4) Compositions mn1,...,nk

:

P (k)× (P (n1)× · · · × P (nk))! P (n1 + · · · + nk),

which are continuous maps for all k ≥ 0 and n1, . . . , nk ≥ 0 satisfying a list of axiomssimilarly to the ones in Definition 4.5.1.

Example 4.5.13 (Topological version of endomorphism operad). The analog version to theendopmorphism operad is the operad P such that for any n ≥ 0, we get a topological spaceP (n) which is the space of continuous maps from Xn

! X, where X is some compacttopological space.


Remark 4.5.14. In general, one can define an operad and an algebra over an operad in anyarbitrary symmetric monoidal category C⊗, i.e. a category endowed with the functor

⊗ : C × C ! C ,

the identity element 1C ∈ Obj(C ) and various coherence conditions for associativity, com-mutativity of ⊗ and so on.In particular, we want to consider the symmetric monoidal category Complexes of Z-gradedcochain complexes of abelian groups (or vector spaces over some field). Such operads arecalled dg-operads. Note that each component P (n) is a cochain complex, i.e. a vectorspace decomposed into a direct sum P (n) =

⊕i∈Z P (n)

i and endowed with a differential

d : P (n)i ! P (n)i+1 which is of degree +1 satisfying d2 = 0.

Remark 4.5.15. We can construct an operad of cochain complexes out of a topological operadby considering a version of the singular chain complex. For a topological space X we denoteby Chains(X) the complex concentrated in negative degrees, whose (−k)-th component fork ≥ 0 consists of the formal finite additive combinations

∑

1≤i≤N

nifi, ni ∈ Z, N ∈ Z≥0.

of continuous maps fi : [0, 1]k! X (singular cubes in X) modulo the following relations:

(1) f σ = sign(σ)f for any σ ∈ Sk acting on the standard curbe [0, 1]k by permutationsof coordinates,

(2) f ′ prk!k−1 = 0, where prk!k−1 : [0, 1]k! [0, 1]k−1 is the projection onto the first

(k − 1) coordinates and f ′ : [0, 1]k−1! X is a continuous map.

The boundary operator is defined similarly as for singular chains. Note that, in contrast tosingular chains, for cubical chains we have an external product map

⊗

i∈I

Chains(Xi)! Chains

(∏

i∈I

Xi

).

If P is a topological operad, then Chains(P (n)) for n ≥ 0 has a natural operad structure inthe category of complexes of abelian groups. The compositions are then given in terms of theexternal tensor product of cubical chains. On the level of cohomology, we obtain an operadH(P ) of Z-graded abelian groups which are complexes endowed with the zero differential.This is called the homology of the operad P .

4.5.3. The little disks operad. Let d ≥ 1 and denote by Gd the (d + 1)-dimensionalLie group acting on Rd by affine transformations u 7! λu+ v where λ > 0 is a real numberand v ∈ Rd. This group acts simply and transitively on the space of closed disks in Rd

endowed with the usual Euclidean metric. The disk with center v and radius λ is given by atransformation of Gd with parameters (λ, v) of the standard disk

D0 := (x1, . . . , xd) ∈ Rd | x21 + · · ·+ x2d ≤ 1.Definition 4.5.16 (Little disks operad). The little disks operad Cd is a topological operadwith the following structure:

(1) Cd(0) = ∅,

4.5. OPERADIC APPROACH TO FORMALITY AND DELIGNE’S CONJECTURE 123

(2) Cd(1) = idCd,

(3) The space Cd(n) is the space of configurations of n disjoint disks (Di)1≤i≤n insidethe standard disk D0 for n ≥ 2.

The composition

Cd(k)× (Cd(n1)× · · · × Cd(nk))! Cd(n1 + · · ·+ nk)

is obtained by applying elements from Gd associated with disks (Di)1≤i≤n in the configurationin Cd(k) to configurations in all Cd(ni), i = 1, . . . , k and putting the resulting configurationstogether. The action of the symmetric group Sn on Cd(n) is given by renumerations of indicesof disks (Di)1≤i≤n (see Figure 4.5.1).

1

2

3

3

4

1

2

Figure 4.5.1. Example for the compositions of little disks.

Remark 4.5.17. Note that Cd(n) is homotopy equivalent to the configuration space Confn(Rd)

of n pairwise distinct points in Rd. Obviously, we can consider the map Cd(n)! Confn(Rd)

which takes a collection of disjoint disks to the collection of the centers. Note that in par-ticular, we have that Conf2(R

d) (and hence C2(n)) is homotopy equivalent to the (d − 1)-dimensional sphere Sd−1. The homotopy equivalence is given by the map

(v1, v2) 7!v1 − v2‖v1 − v2‖

∈ Sd−1 ⊂ Rd.

Remark 4.5.18. Since we are abusing notation, one should not confuse the operad Cd withthe configuration space Cn as in Remark 4.3.7.

4.5.4. Deligne’s conjecture. Denote the Hochschild complex of an associative algebraA concentrated in positive degree by

C•(A,A) := Homvector space(A⊗n,A), n ≥ 0

and the Hochschild differential by dH, given by the formula

(dHφ)(a1⊗· · ·⊗an+1) := a1φ(a2⊗· · ·⊗an)+∑

1≤i≤n

(−1)iφ(a1⊗· · ·⊗aiai+1⊗· · ·⊗an+1)+ · · ·

· · ·+ (−1)n+1φ(a1 ⊗ · · · ⊗ an)an+1, ∀φ ∈ Cn(A,A).


Conjecture 4.5.19 (Deligne). There exists a natural action of the operad Chains(C2) onthe Hochschild complex C•(A,A) for any associative algebra A.

Remark 4.5.20. The proof of this conjecture was given by a combination of results of [Tam98;Tam03] and (a higher version) also in [Kon99]. See also [MS99; KS00].

4.5.5. Formality of chain operads.

Theorem 4.5.21 (Kontsevich–Tamarkin[Tam03; Kon99]). The operad Chains(Cd) ⊗ R ofcomplexes of real vector spaces is quasi-isomorphic to its cohomology operad endowed withthe zero differential.

CHAPTER 5

Quantum Field Theoretic Approach to Deformation

Quantization

Kontsevich’s formula is in its nature a pure algebraic construction. However, the conceptof deformation quantization should give a physical quantization procedure which opens thequestion whether it is possible to naturally extract the star product out of a quantum fieldtheory, i.e. a perturbative Feynman path integral quantization [Fey42; Fey49; Fey50; FH65;Pol05]. The appearance of graphs in Kontsevich’s construction might already inspire to re-gard them as Feynman diagrams of a certain field theory. Indeed, using the Poisson sigmamodel [Ike94; SS94; CF01], one can show that its perturbative quantization produces thedesired outcome. It is however necessary to formulate everything in a suitable formalismsince it appears to be a gauge theory (i.e. the theory has symmetries). It turns out thatthe Batalin–Vilkovisky (BV) formalism [BV77; BV81; BV83] (see [Sch93] for a more math-ematical formulation and [Mne19] for a good introduction) is needed in order to deal withthis particular sigma model. In this chapter we will describe the mathematical frameworkfor functional integrals, construct the Moyal product out of a path integral quantization,describe the Faddeev–Popov [FP67] and BRST [BRS74; BRS75; BRS76; Tyu76] method todeal with gauge theories and introduce the Poisson sigma model. Moreover, we will explicitlyshow how the perturbative quantization of the Poisson sigma model for linear Poisson struc-tures produces Kontsevich’s star product (this only requires the BRST formalism). Finally,we state the theorem for general Poisson structures on Rd (this construction needs the BVformalism) [CF00; CF01]. This chapter is based on [Zin94; Pol05; Cat+05; CF00; CF01;Kon94; Mne19; Fed96].

5.1. Functional integrals

5.1.1. Functional integrals and expectation values. For a field theory constructionwe want to consider a space of fields M which in most cases is given by the space of sectionsfor some vector bundle and an action function S : M ! R. The action is usually givenlocally (see also Definition 5.2.14), i.e., roughly, as an integral over some Lagrangian densityS(φ) =

∫L (φ, ∂φ, . . . , ∂Nφ) for N ∈ Z>0.

Definition 5.1.1 (Expectation value). For a function O on M we define the expectationvalue

〈O〉 :=

∫

Mexp (iS/~)O

∫

Mexp (iS/~)

.

125

126 5. QUANTUM FIELD THEORETIC APPROACH TO DEFORMATION QUANTIZATION

Definition 5.1.2 (Observables). An observable is a function O on M whose expectationvalue is well-defined.

Remark 5.1.3. Integrals as in Definition 5.1.1 are called functional integrals or path inte-grals. One can consider a perturbative evaluation of such integrals by expanding S arounda nondegenerate critical point and defining the integral as a formal power series in ~ withcoefficients given by Gaussian expectation values. If S carries certain symmetries, the criticalpoints will always be degenerate. In this case we speak of a gauge theory. There are differ-ent methods to deal with such theories such a the Faddeev–Popov ghost method [FP67], theBRST method [BRS74; BRS75; Tyu76] and the Batalin–Vilkovisky method [BV77; BV81;BV83]. Interestingly, it was shown that the field theoretic construction of Kontsevich’s starproduct for general Poisson structures requires the Batalin–Vilkovisky formalism to deal withthe gauge theory given by the Poisson sigma model [CF00].

5.1.2. Gaussian integrals. Let A be a positive-definite symmetric matrix on Rn whichwe assume to be endowed with the Lebesgue measure dnx and the Euclidean inner product〈 , 〉 (note that n has to be even). Then

I(λ) :=

∫

Rn

exp

(−λ2〈x,Ax〉

)dnx =

(2π)n2

λn2

1√detA

, λ > 0.

If we continue I to the whole complex plane without the negative real axis, we get

I(−i) = (2π)n2 exp

(iπn

4

)1√detA

, I(i) = (2π)n2 exp

(−iπn

4

)1√detA

.

Thus, when A is negative-definite, we can define the integral by

∫

Rn

exp

(i

2〈x,Ax〉

)dnx =

∫

Rn

exp

(− i

2(−〈x,Ax〉)

)dnx = (2π)

n2 exp

(−iπn

4

)1√

|detA|.

For the case when A is nondegenerate (not necessarily positive- or negative-definite), we get

∫

Rn

exp

(i

2〈x,Ax〉

)dnx = (2π)

n2 exp

(iπ signA

4

)1√

|detA|,

where signA denotes the signature of A.We want to compute expectation values with respect to a Gaussian distribution. Let usdenote such an expectation value by 〈 , 〉0. Define first the generating function

Z(J) :=

∫

Rn

exp

(i

2〈x,Ax〉+ i

2〈J, x〉

)dnx

= (2π)n2 exp

(iπ signA

4

)1√

|detA|exp

(i

2〈J,A−1J〉

).

5.1. FUNCTIONAL INTEGRALS 127

Then we get

〈xi1 · · · xik〉0 =

∫

Rn

exp

(i

2〈x,Ax〉

)xi1 · · · xikdnx

∫

Rn

exp

(i

2〈x,Ax〉

)dnx

=∂

∂Ji1· · · ∂

∂JikZ(J)|J=0

Z(0)=

∂

∂J i1· · · ∂

∂J ikexp

(i

2〈J,A−1J〉

) ∣∣∣∣J=0

.

Remark 5.1.4. Note that 〈xi1 · · · xik〉0 vanishes if k is odd and is a sum of products of matrixelements of the inverse of A if k is even. For example, if k = 2 we have 〈xixj〉0 = i(A−1)ij

and if k = 2s, we get

〈xi1 · · · xi2s〉0 = is∑

σ∈S2s

1

2ss!(A−1)iσ(1)iσ(2) · · · (A−1)iσ(2s−1)iσ(2s) .

Theorem 5.1.5 (Wick). Denote by P (s) the set of pairings, i.e. permutations σ ∈ S2s withthe property that σ(2i − 1) < σ(2i) for i = 1, . . . , s and σ(1) < σ(3) < · · · < σ(2s − 3) <σ(2s − 1). Then we have

(5.1.1) 〈xi1 · · · xi2s〉0 = is∑

σ∈P (s)

(A−1)iσ(1)iσ(2) · · · (A−1)iσ(2s−1)iσ(2s) .

5.1.2.1. Infinite-dimensional case. By (5.1.1), the infinite-dimensional case only makessense if A is invertible. However, usually A will be a differential operator and thus G := A−1

will denote the distributional kernel of its inverse, i.e. its Green function. So if e.g. A is adifferential operator on functions on some manifold Σ, we get

(5.1.2)

〈φ(x1) · · · φ(x2s)〉0 =

∫exp

(i

2~

∫

ΣφAφ

)φ(x1) · · · φ(x2s)D [φ]

∫exp

(i

2~

∫

ΣφAφ

)D [φ]

:= (i~)s∑

σ∈P (s)

G(xσ(1), xσ(2)

)· · ·G

(xσ(2s−1), xσ(2s)

),

where φ denotes a function on Σ, D [φ] denotes the formal Lebesgue measure on the space offunctions and x1, . . . , x2s are distinct points in Σ.

Remark 5.1.6 (Normal ordering). Equation (5.1.2) is usually extended to the singular casewhen points coincide by restricting the sum to pairings with the property that xσ(2i−1) 6=xσ(2i) for all i. This is called normal ordering since it corresponds to the usual normalordering in the operator formulation of Gaussian field theories.

Remark 5.1.7 (Propagator). In Gaussian integrals all expectation values are given in terms ofthe expectation value of the quadratic monomials. These are usually called 2-point functionsor propagators. For the case of a Gaussian field theory defined by a differential operator,propagator is just another name for the Green function.


Remark 5.1.8 (Derivatives). One can extend the definition of expectation values to deriva-tives of fields by linearity. In particular, for multi-indices I1, . . . , I2s we set(5.1.3)

〈∂I1φ(x1) · · · ∂I2sφ(x2s)〉0 = (i~)s∂|I1|

∂xI11· · · ∂

|I2s|

∂xI2s2s

∑

σ∈P (s)

G(xσ(1), xσ(2)

)· · ·G

(xσ(2s−1), xσ(2s)

),

where the derivatives on the right hand side are meant in the distributional sense.

5.1.3. Integration of Grassmann variables. Let V be a vector space. Recall thatthe exterior algebra

∧V ∗ is regarded as the algebra of functions on the odd vector space

ΠV . Choosing a basis and orientation, we can identify∧top V ∗ with R. The composition of

this isomorphism with the projection∧V ∗!

∧top V ∗ gives a map∧V ∗! R which will be

denoted by∫ΠV and we call it the integral on ΠV .

Let B ∈ End(V ) and regard it as an element of V ∗ ⊗ V and thus as a function on

ΠV ∗ ×ΠV := Π(V ∗ ⊕ V ).

There is a natural identification of∧top(V ∗ ⊕ V ) with R. Thus, we have

∫

ΠV ∗×ΠVexp(B) = detB.

If B is nondegenerate, we define the expectation value of a function f on ΠV ∗ ×ΠV by

〈f〉0 :=

∫

ΠV ∗×ΠVexp(B)f

∫

ΠV ∗×ΠVexp(B)

.

Let ei be a basis of V and denote by ei the corresponding dual basis. Then∧(V ∗ ⊕ V )

can be identified with the Grassmann algebra generated by the anticommuting coordinatefunctions ei and ej . Functions on ΠV ∗ × ΠV are then linear combinations of monomialsej1 · · · ejr ei1 · · · eis . To B we associate the function

〈e, Be〉 = ejBijei,

where 〈 , 〉 denotes the canonical pairing between V ∗ and V . Hence, we can write∫

exp (〈e, Be〉) = detB,

and

〈ej1 · · · ejr ei1 · · · eis〉0 =

∫exp (〈e, Be〉) ej1 · · · ejr ei1 · · · eis

∫exp

(ejBi

jei) .

It is easy to see that 〈ej1 · · · ejr ei1 · · · eis〉0 vanishes if r 6= s and that

〈ej1 · · · ejs ei1 · · · eis〉0 =∑

σ∈Ss

sign(σ)(B−1)iσ(1)

j1· · · (B−1)

iσ(s)

js.

5.2. THE MOYAL PRODUCT AS A PATH INTEGRAL QUANTIZATION 129

Remark 5.1.9 (Odd vector fields). A vector field on ΠV is by definition a graded derivationof∧V ∗. In particular, an endomorphism X of

∧V ∗ is a vector field of degree |X| if

X(fg) = X(f)g + (−1)|X|rfX(g), ∀f ∈r∧V ∗,∀g ∈

∧V ∗,∀r.

A right vector field X of degree |X| is an endomorphism f 7! (f)X of∧V ∗ that satisfies

(fg)X = f(g)X + (−1)|X|s(f)Xg, ∀f ∈∧V ∗,∀g ∈

s∧V ∗,∀s.

We can identify the vector space of all vector fields on ΠV with∧V ∗ ⊗ V . Note that the

elements of V can be regarded as constant vector fields. Integration gives us∫

ΠVX(f) = 0, ∀f,

if X is a constant vector field. In general, one can define the divergence divX of X by∫

ΠVX(f) =

∫

ΠVdivXf, ∀f.

If X = g ⊗ v for g ∈ ∧s V ∗ and v ∈ V , we get

divX = (−1)s+1ιvg.

5.2. The Moyal product as a path integral quantization

Consider the symplectic manifold T ∗Rn endowed with the canonical symplectic form ω0 =∑1≤i≤n dpi ∧ dqi, where (q1, . . . , qn, p1, . . . , pn) ∈ T ∗Rn ∼= R2n. Denote by α =

∑1≤i≤n pidq

i

the Liouville 1-form such that ω0 = dα. Consider a path γ : I ! T ∗Rn, where I is a 1-dimensional manifold, and define an action S(γ) =

∫I γ

∗α. Let us write γ(t) = (Q(t), P (t))for t ∈ I. Then we can rewrite the action as

(5.2.1) S(Q,P ) =

∫

t∈IPi

d

dtQidt.

Given a Hamiltonian function H, one can deform the action to

(5.2.2) SH(Q,P ) =

∫

t∈I

(Pi

d

dtQi +H(Q(t), P (t), t)

)dt.

Let now I = S1 and, in order to make the quadratic form nondegenerate, we choose abasepoint ∞ ∈ S1. Let

M := (Q,P ) ∈ C∞(S1, T ∗Rn)and

M(q, p) := (Q,P ) ∈ C∞(S1, T ∗Rn) | Q(∞) = q, P (∞) = p.The path integral can then be defined by imposing Fubini’s theorem:

(5.2.3)

∫

M(· · · ) =

∫

(q,p)∈T ∗Rn

µ(q, p)

∫

M(q,p)(· · · )


where µ denotes a measure on T ∗Rn. One can check that the quadratic form in S is nonde-generate when restricted to M(q, p). Hence, we can compute

〈O〉0(q, p) :=

∫

M(q,p)exp (iS/~)O

∫

M(q,p)exp (iS/~)

,

where O is some function on M that is either polynomial or a formal power series in Q andP . Using the fact that the denominator is constant (infinite though), we can use (5.2.3) towrite

〈O〉0 :=

∫

(q,p)∈T ∗Rn

µ(q, p)〈O〉0(q, p)∫

(q,p)∈T ∗Rn

µ(q, p)

whenever the functions 〈O〉0(q, p) and 1 are integrable. The latter condition prevents us from

choosing µ to be the Liouville volumeωn0n! . However, at this point one can also forget about

the denominator and define

〈O〉′0 :=∫

(q,p)∈T ∗Rn

µ(q, p)〈O〉0(q, p)

such that the Liouville volume is allowed. In fact, we could also choose µ to be the deltameasure peaked at a point (q, p) ∈ T ∗Rn. In this case we have

〈O〉0 = 〈O〉′0 = 〈O〉0(q, p).If we have fixed (q, p) ∈ T ∗Rn, we can can use the change of variables Q = q + Q and

P = p+ P , where (Q, P ) is a map from S1 to T ∗Rn which vanishes at ∞. This is the sameas a map R! T ∗Rn which vanishes at ∞. The action is then given by

S(Q,P ) = S(q + Q, p+ P ) =

∫

R

Pid

dtQidt.

Expectation values are given by

〈O(Q,P )〉0(q, p) =⟨O(q + Q, p+ P )

⟩∼0:=

∫exp (iS/~)O(q + Q, p + P )D [P ]D [Q]

∫exp (iS/~)D [P ]D [Q]

.

5.2.1. The propagator. We need to find the Green function for the differential operatorddt . It is given in terms of the sign function:

(d

dt

)−1

(u, v) = θ(u− v) =1

2sig(u− v) :=

12 , u > v

−12 , u < v

As a consequence, we get ⟨Pi(u)Q

j(v)⟩∼0= i~θ(v − u)δji


and more generally⟨Pi1(u1) · · · Pis(us)Qj1(v1) · · · Qjs(vs)

⟩∼0= (i~)s

∑

σ∈Ss

θ(vσ(1)−u1) · · · θ(vσ(s)−us)δjσ(1)

i1· · · δjσ(s)

is.

The normal ordering can be implemented by setting θ(0) = 0 which is compatible with theskew-symmetry of d

dt .

5.2.2. Expectation values. Let us consider the observable on M which is given byevaluation of a smooth function f on T ∗Rn at some point u in the path. Define

Of,u(Q,P ) := f(Q(u), P (u)), f ∈ C∞(T ∗Rn), u ∈ S1 \ ∞.in order to compute the expectation value of this observable, we need to introduce somenotation. Let I = (i1, . . . , ir) be a multi-index and set |I| = r, pI := pi1 · · · pir , qI := qi1 · · · qir(similarly for PI and QI). Moreover, define

∂I :=∂

∂qi1· · · ∂

∂qir, ∂I :=

∂

∂pi1· · · ∂

∂pir.

We extend everything to the case of |I| = 0 by setting pI = qI = 1 and ∂I = ∂I = id. If weuse the change of variables as before and use the Taylor expansion of f , we get

f(Q(u), P (u)) = f(q, Q(u), p + P (u)) =∑

r,s≥0

1

r!s!

∑

|I|=r|J|=s

PI(u)QJ (u)∂I∂Jf(q, p).

Hence〈Of,u〉0(q, p) =

⟨Of,u(q + Q, p + P )

⟩∼0= f(q, p).

If f is integrable we can also define

〈Of,u〉′0 =∫

T ∗Rn

µ(q, p)f(q, p).

Remark 5.2.1. Note that these expectation values do not depend on the point u.

Consider now the observable given by(5.2.4)

Of,g;u,v = f(Q(u), P (u))g(Q(v), P (v)), f, g ∈ C∞(T ∗Rn), u, v ∈ S1 \ ∞ ∼= R, u < v.

Its expectation value can then be computed as

〈Of,g;u,v〉0(q, p) =⟨f(q + Q(u), p + P (u))g(q + Q(v), p + P (v))

⟩∼0

=∑

r1,s1,r2,s2≥0

1

r1!s1!r2!s2!

∑

|I1|=r1|J1|=s1

∑

|I2|=r2|J2|=s2

⟨PI1(u)Q

J1(u)PI2(v)QJ2(v)

⟩∼0∂I1∂J1f(q, p)∂

I2∂J2g(q, p)

=∑

r,s≥0

1

r!s!

(i~

2

)r+s(−1)s

∑

|I|=r

|J|=s

∂I∂Jf(q, p)∂J∂Ig(q, p) = f ⋆ g(q, p).

Here we have denoted by ⋆ the Moyal product induced by the Poisson bivector field ∂∂pi

∧ ∂∂qi

.

Remark 5.2.2. Note also here that the expectation is independent of u and v.


If f ⋆ g is integrable, we can compute

〈Of,g;u,v〉′0 =∫

T ∗Rn

µf ⋆ g.

If µ is given by the Liouville volumeωn0n! , we can define a trace (see also [Fed94; NT95; Fed96])

Tr(f ⋆ g) :=

∫

T ∗Rn

ωn0n!f ⋆ g =

∫

T ∗Rn

ωn0n!fg,

where the second equality follows from using integration by parts in the correction terms tothe commutative product (see also [CF10] for a similar trace formula for the Poisson manifoldRd endowed with any Poisson structure and [Mos19] for the formulation on general Poissonmanifolds). More generally, we can define an observable (see Figure 5.2.1)

Of1,...,fk;u1,...,uk = f1(Q(u1), P (u1)) · · · fk(Q(uk), P (uk)),

f1, . . . , fk ∈ C∞(T ∗Rn), u1, . . . , uk ∈ S1 \ ∞ ∼= R, u1 < · · · < uk.

Exercise 5.2.3. Show that

〈Of1,...,fk;u1,...,uk〉0(q, p) = f1 ⋆ · · · ⋆ fk(q, p).

u1

u2 u3

u4

...

uk

∞

S1

Figure 5.2.1. The points u1, . . . , uk on S1

5.2.3. Divergence of vector fields. The normal ordering condition implies that localvector fields are in fact divergence free. Consider a vector at (Q, P ) ∈ M(q, p) which can beidentified with a smooth map R! T ∗Rn that vanishes at ∞. Note that a vector field X onM(q, p) is an assignment of a map X(Q, P ) to each path (Q, P ). The vector field is said to

be local if X(Q, P )(t) is a function of Q(t) and P (t) for all t ∈ R.Formally, we want to express the divergence of a vector field in terms of path integrals similarto the description using ordinary integrals. We would like to have

∫

M(q,p)X (exp (iS/~)O) =

∫

M(q,p)divX exp (iS/~)O,

for all observables O.


Definition 5.2.4 (Divergence). If there exists an observable divX such that for any observ-able O we have

〈X(S)O〉0(q, p) = i~〈X(O)〉0(q, p)− i~〈divXO〉0(q, p),

then we say that divX is the divergence of X.

Lemma 5.2.5. If X is a local vector field, then divX = 0.

Proof. Write X(Q, P )(t) = (Xq(Q(t), P (t)),Xp(Q(t), P (t))). Then

X(S) =

∫ (Xp

d

dtQ−Xq

d

dtP

)dt.

By the normal ordering and the locality of X, when computing 〈X(S)O〉0(q, p) we only

have to contract the P ’s (Q’s) in O with the Q’s (P ’s) in X(S) and replace each pair by a

propagator. Whenever a P (Q) in O is contracted with the ddtQ ( d

dt P ) in X(S), we get theidentity operator times i~ (−i~). Summing up the various terms, we get that this is the sameas taking the expectation value of X(O) multiplied with i~. This completes the proof.

We can also consider the divergence of vector fields on M. In fact, a local vector field Xon M can be uniquely written as the sum of a vector field X∞ on T ∗Rn and a section Xof local vector fields (i.e. an assignment of a local vector field X(q, p) on M(q, p) to each(q, p) ∈ T ∗Rn). By Lemma 5.2.5 we get

〈X(S)O〉′0 = i~〈X(O)〉′0 − i~〈divµX∞O〉′0,

where divµX∞ denotes the ordinary divergence of the vector field X∞ with respect to themeasure µ. One can then define divµX∞ to be the divergence of the local vector field X onM.

Exercise 5.2.6. Using the local vector field X(Q, P )(t) := (P (t), 0), show that

~d

d~(f ⋆ g) = (pi∂

if) ⋆ g + f ⋆ (pi∂ig)− pi∂

i(f ⋆ g).

5.2.4. Independence of evaluation points. Note that in the previous computationswe have considered observables whose definition depends on the choice of points in S1 \∞.However, computing the expectation values we have seen that they are in fact independentof these points. In particular, we have the following Proposition:

Proposition 5.2.7. The expectation values are invariant under (pointed) diffeomorphism ofthe source manifold S1 on which the field theory is defined.

Remark 5.2.8. A quantum field theory with such a property is usually called topologicalquantum field theory (TQFT).


Given f ∈ C∞(T ∗Rn) and two points a, b ∈ R, we have

f(Q(b), P (b))− f(Q(a), P (a)) =

∫ b

a

(∂if(Q(t), P (t))

d

dtQi(t) + ∂if(Q(t), P (t))

d

dtPi(t)

)dt

= limr!∞

∫ +∞

−∞

(∂if(Q(t), P (t))

d

dtQi(t) + ∂if(Q(t), P (t))

d

dtPi(t)

)λr(t)dt

= limr!∞

Xf,r(S),

where (λr) is a sequence of smooth, compactly supported functions that converges almost

everywhere to the characteristic function of the interval [a, b] and Xf,r is the local vector field

Xf,r(t) = (−∂if(Q(t), P (t)), ∂if(Q(t), P (t)))λr(t).

Remark 5.2.9. Recall that M is a space of maps. So a vector field on the target T ∗Rn

generates a local vector field on M. Let Xf be the local vector field corresponding to the

Hamiltonian vector field Xf of f . Then Xf,r is given by multiplying Xf by λr.

If we restrict Xf,r to M(q, p), we can observe by Lemma 5.2.5 that it is divergence-free.Hence, we get

〈(f(Q(b), P (b))−f(Q(a), P (a)))O〉0(q, p) = limr!∞

⟨Xf,r(S)O

⟩0(q, p) = i~ lim

r!∞

⟨Xf,r(O)

⟩0(q, p),

which vanishes if O does not depend on (Q(t), P (t)) for t ∈ [a, b]. This means that we canmove the evaluation point at least as long we do not meet another evaluation point.

5.2.5. Associativity. Let us prove the associativity of the Moyal product by usingthe path integral description considering the expectation value of the observable Of,g,h;u,v,w.There will be three propagators appearing which correspond to the three different ways ofpairing u, v, w. Note that the function θ will not see the difference, since

θ(w − u) = θ(w − v) = θ(v − u) =1

2.

We group the propagators by considering first only those between u and v and only thereafterthe other ones. This will imply that

〈Of,g,h;u,v,w〉0 = (f ⋆ g) ⋆ h.

If we group the propagators such that we first consider those between v and w and then theothers, we get

〈Of,g,h;u,v,w〉0 = f ⋆ (g ⋆ h).

This proves associativity of the Moyal product.

Remark 5.2.10. We can also prove this result by using the independence of evaluation points.This then implies

limv!u+

〈Of,g,h;u,v,w〉0(q, p) = limv!w−

〈Of,g,h;u,v,w〉0(q, p).

The left hand side corresponds to evaluating first the expectation value of Of,g;u,v then placingthe result at u and computing the expectation value of O〈Of,g;u,v〉0,h;w. This will give the result

(f ⋆ g) ⋆ h. Repeating this computation on the right hand side, we get associativity.


5.2.6. The evolution operator. Let us consider the deformed action SH as in (5.2.2).We restrict ourselves to a time-independent Hamiltonian h and let it act from the instant ato the instant b with a < b. Then we consider the action SH with

(5.2.5) H(q, p, t) = h(q, p)χ[a,b](t),

where χ[a,b] denotes the characteristic function on the interval [a, b]. The aim is to computethe evolution operator

U(q, p, T ) :=

∫

M(q,p)exp (iSH/~)

∫

M(q,p)exp (iS/~)

,

where T = b− a. Observe that

U(q, p, T ) =

⟨exp

(i

~

∫H(Q(t), P (t), t)dt

)⟩

0

(q, p)

=

⟨exp

(i

~

∫ b

ah(Q(t), P (t), t)dt

)⟩

0

(q, p).

Now we can express the integral in terms of Riemann sums as∫ b

ah(Q(t), P (t), t)dt = lim

N!∞

T

N

∑

1≤r≤N

h(Q(a+ rT/N), P (a + rT/N)).

Hence, we get

U(q, p, T ) = limN!∞

⟨ ∏

1≤r≤N

exp

(i

~

T

Nh(Q(a+ rT/N), P (a+ rT/N))

)⟩

0

(q, p)

= limN!∞

⟨Oexp( i

~

TNh),...,exp( i

~

TNh);a+ T

N,a+2 T

N,...,a+T

⟩0(q, p)

= limN!∞

(exp

(i

~

T

Nh

))⋆N(q, p) = exp⋆

(i

~h

)(q, p).

Remark 5.2.11. Note that the previous result also includes negative powers of ~. However,everything is well-defined since each term in the power series expansion of exp⋆ is actually aLaurent series in ~.

5.2.7. Perturbative evaluation of integrals. Let M be an n-dimensional manifoldand S ∈ C∞(M). We want to understand how we can compute an integral of the formZ :=

∫M exp(iS/~)dnx. Consider first the case where S has a unique critical point x0 ∈ M

which is nondegnerate. Then we can write

S(x0 +√~x) = S(x0) +

~

2d2x0S(x) +Rx0(

√~x),

where Rx0 is a formal powers series given by the Taylor expansion of S starting with the

cubic term in√~x with x ∈ Tx0M. Let A be the Hessian of S at x0 with respect to the

Euclidean metric 〈 , 〉. In particular, we write

d2x0S(x) = 〈x,Ax〉.


Theorem 5.2.12 (Stationary phase expansion). The stationary phase expansion (or alsosaddle-point approximation) of the integral Z :=

∫M exp(iS/~)dnx is given by

Z = ~n2 exp(iS(x0)/~)

∫

Mexp

(i

2〈x,Ax〉

)∑

r≥0

1

r!Rrx0(

√~x)dnx

= (2π~)n2 exp(iS(x0)/~) exp

(iπ signA

4

)1√

|detA|∑

r≥0

1

r!

⟨Rrx0(

√~x)⟩0,

where 〈〉0 denotes the Gaussian expectation value with respect to the nondegenerate symmet-ric matrix A.

Expectation values as in Definition 5.1.1 are then given by

(5.2.6) 〈O〉 =

∑r≥0

1

r!

⟨O(x0 +

√~x)Rrx0(

√~x)⟩0∑

r≥0

1

r!

⟨Rrx0(

√~x)⟩0

.

Remark 5.2.13. Note that the denominator is of the form 1+O(~) and thus the expectationvalue can be computed as a formal power series in ~. One can consider the computation in agraphical way by associating a vertex of valence k to the term of degree k in Rx0 and a vertexof valence ℓ to the term of degree ℓ in O. By Wick’s theorem (Theorem 5.1.5) one has toconnect these vertices in pairs in all possible ways the half-edges emanating from each vertex.The result is a collection of Feynman diagrams with a weight associated to each of them.The expectation value is then a sum over all Feynman diagrams of their weights. A Feynmandiagram with vertices coming only from Rx0 , i.e. a Feynman diagram appearing only in thedenominator of (5.2.6), is called a vacuum diagram. A combinatorial fact of (5.2.6) is that anexpectation value is given by the sum over graphs which do not have a connected componentwhich is a vacuum diagram (see [Zin94; Pol05] for more details on Feynman diagrams).

5.2.8. Infinite dimensions. Let us look at the situation of computing (5.2.6) in thecase when M is infinite-dimensional. We want to recall the following definition.

Definition 5.2.14 (Local function). A function on a space of fields on some manifold M islocal if it is the integral on M of a function that depends at each point on finite jets of fieldsat that point.

Remark 5.2.15. If the action is a local function, A will be a differential operator. For ourpurpose, we will need its Green function (the propagator) and in the Gaussian expectationvalues of ORr and Rr we will use (5.1.3). Instead of summing over indices we will haveto integrate over the Cartesian products of the source manifold. The normal ordering willexclude all the graphs with edges having the same endpoints (these edges are called tadpolesor short loops) and thus integration is restricted to the configuration space of the sourcemanifold. However, this is usually not enough to make the integrals converge as the Greenfunctions are very singular when the arguments approach each other. The general procedureto get rid of divergencies is called renormalization. Anyway, for our case (i.e. the case of aTQFT), the configuration-space integrals associated to Feynman diagrams without tadpolesdo indeed converge.


Remark 5.2.16. If the action S has more critical points and all of them are nondegenerate,the asymptotic expansion is obtained by computing the stationary phase expansion (Theorem5.2.12) around each critical point and then summing all these terms together. The expressionfor expectation value is then no longer given by (5.2.6). It can happen that one of thecritical points dominates the others so that one can forget them. In physical theories thisis usually done by regarding exp(iS/~) as the analytic continuation of the exponential ofminus a poisitive-definite function, called Euclidean action, so that the dominating criticalpoint is the absolute minimum. There are still examples where there is no way to do it as allcritical points appear as saddle points. In such a case, one can consider another option whichconsists of selecting one particular critical point (called the sector) and expanding only aroundthis point. In this case, (5.2.6) is again the correct expression for the infinite-dimensionalextension.

As already mentioned, it can often happen that the critical points are degenerate. A simplecase is given when critical points are parametrized by a finite-dimensional manifold Mcrit.Then, using Fubini’s theorem for the finite-dimensional case, we can rewrite the integralsimilarly as in (5.2.3) as ∫

M(· · · ) =

∫

x0∈Mcrit

µ(x0)

∫

M(x0)(· · · )

where∫M(x0)

denotes the asymptotic expansion of the integral in the complement to Tx0Mcrit

of a formal neighborhood of x0, while µ denotes a measure on Mcrit (which is determined inthe finite-dimensional case and has to be chosen in the infinite-dimensional case as part ofthe definition). If the Hessian is constant on Mcrit, we can express the expectation value as

〈O〉 =

∫

x0∈Mcrit

µ(x0)∑

r≥0

1

r!

⟨O(x0 +

√~x)Rrx0(

√~x)⟩0∫

x0∈Mcrit

µ(x0)∑

r≥0

1

r!

⟨Rrx0(

√~x)⟩0

,

where 〈〉0(x0) denotes the Gaussian expectation value which is computed by expandingaround x0 orthogonally to Tx0Mcrit. A possible choice for the measure µ is the delta measurepeaked at some point x0. In this case, as we already did before, the expectation value willbe denoted by 〈〉(x0).

5.2.9. A simple generalization. Note that we have only considered the perturbativeexpansion with formal expansion parameter ~. It might happen that there is another smallexpansion parameter, or some coefficient in S which is much smaller than ~ or in our setting isan element of ~2R[[~]]. In such a case, we can define the Gaussian part by using the quadratic~-independent term of S/~ and consider all the other terms as the perturbation R. It canalso happen that the perturbation term R contains quadratic and linear terms as well. Inparticular, we can have an action function of the form

(5.2.7) S(y, z) = 〈y,Bz〉+ f(y, z),

where B is a nondegenerate matrix and f is a function quadratic in z. If we work aroundthe critical point y = z = 0, we can rescale z by ~ to get

S(y, ~z) = ~〈y,Bz〉 + ~f(y, z)


and consider f as the perturbation to the Gaussian theory defined by B.

5.2.9.1. Quantum mechanics. Recall that the topological action (5.2.1) defines the Moyalproduct in terms of the expectation value of the observable (5.2.4), which is independent ofthe evaluation points. Consider a Hamiltonian H as in (5.2.5) and define the expectation

f ⋆a,b;u,vg(q, p) :=

∫

M(q,p)exp(iSH)/~)Of,g;u,v

∫

M(q,p)exp(iSH/~)

=

⟨exp

(i~

∫ b

ahdt

)Of,g;u,v

⟩

0

(q, p)

⟨exp

(i~

∫ b

ahdt

)⟩

0

(q, p)

which will no longer be independent to u and v (a < u < v < b) nor will it define anassociative product. By the computations before, we get

f ⋆a,b;u,vg =exp⋆

(i~(u− a)h

)⋆ f ⋆ exp⋆

(i~(v − u)h

)⋆ g ⋆ exp⋆

(i~(b− v)h

)

exp⋆(i~(b− a)h

)

Let us consider the perturbative computation for the Hamiltonian

h(q, p) =1

2Gij(q)pipj ,

at p = 0 and for f and g only depending on q. By setting Q(t) = q+Q(t) and P (t) = p+P (t),we get

SH = ~

∫ +∞

−∞P (t)

d

dtQ(t)dt+

~2

2

∫ b

aGij(q + Q(t))Pi(t)Pj(t)dt.

Hence, f ⋆a,b;u,vg(q, 0) is the ratio of the expectation value of exp(i~2

∫ ba G

ij(q + Q)PiPjdt)Of,g;u,v

and the expectation value of exp(i~2

∫ ba G

ij(q + Q)PiPjdt). The propagator pairs the P ’s to

Q’s, so it is better to consider oriented graphs. The Feynman diagrams will be orientedgraphs with

(1) a vertex at u with no outgoing arrows,(2) a vertex at v with no outgoing arrows,(3) vertices between a and b with exactly two outgoing arrows.

Moreover, graphs containing tadpoles or vacuum subgraphs will not be allowed. See Figures5.2.2, 5.2.3 and 5.2.4 for examples of allowed and not allowed Feynman diagrams. Note thatto a vertex of first type with r incoming arrows we associate an r-th derivative of f , to avertex of second type with r incoming arrows we associate an r-th derivative of g and to avertex of third type with r incoming arrows we associate an r-th derivative of G. The orderin ~ is given by the number of vertices of the third type. So, at order zero, we have f placedat u and g placed at v. This yields the pointwise product of f and g. At order 1, we havethe graphs of Figure 5.2.2. Hence, we get

f ⋆a,b;u,vg(q, 0) = f(q)g(q)− i~

4(b− a+ 2u− 2v)Gij(q)∂if(g)∂jg(q)

− i~

4(b− a)Gij(q)[∂i∂jf(q)g(q) + f(q)∂i∂jg(q)] +O(~2).

5.3. SYMMETRIES AND THE BRST FORMALISM 139

Exercise 5.2.17. Compute some of the further orders.

a u

v

b a u v b a u v b

Figure 5.2.2. Feynman diagrams of order one.

a u

v

b a u v b a u v b

Figure 5.2.3. Feynman diagrams of order two.

a u

v

b a u v b a u v b

Figure 5.2.4. Feynman diagrams which are not allowed: tadpoles, vacuumsubdiagrams and both.

5.3. Symmetries and the BRST formalism

If the action S has symmetries encoded in a free action of a Lie group G (gauge theory withgauge group G), the critical points will never be nondegenerate since they always will appearin G-orbits. This is also the case for the Poisson sigma model for linear Poisson structuresand hence we need to understand how to deal with these type of theories.


5.3.1. The main construction: Faddeev–Popov ghost method. LetM be a finite-dimensional manifold together with a measure µ. Let G be a compact Lie group which isendowed with an invariant measure (Haar measure) with a measure-preserving free actionon M. Moreover, we assume that M/G is a manifold. Then an invariant function f is thepullback of a function f ∈ M/G and

I :=1

Vol(G)

∫

Mfµ =

∫

M/Gfµ,

where µ is the measure induced by µ on the quotient. If we consider a section of a principalbundle π : M!M/G, we can rewrite I as an integral on the image of this section. Assumethat this integral is locally given by the zero set of a function F : M′

! g where M′ ⊂ M issuch that π(M′) = M/G and g denotes the Lie algebra of G. In the physics literature thecondition F = 0 is called gauge fixing and F is called gauge-fixing function. For x ∈ M′,let A(x) be the differential dF (x) restricted to the vertical tangent space at x. This can beidentified with the Lie algebra g and thus we can regard A(x) as an endomorphism of g anddenote by J its determinant. In the physics literature J is called Faddeev–Popov determinant.Then we get

I =

∫

M′

fδ0(F )Jµ,

where δ0 denotes the delta function at 0 ∈ g. Let us now rewrite I such that it is suitablefor the stationary phase expansion formula. We need to write δ0(F ) and J in exponentialform. Therefore, we use the Fourier transform of the delta function and the Grassmann (odd)integration as in 5.1.3. Denote by 〈 , 〉 the canonical pairing between g∗ and g. Then wehave

I = C

∫f(x)µ(x) exp

(i

~〈λ, F (x)〉

)ω(λ) exp

(i

~〈c, A(x)c〉

),

where the integral is over x ∈ M′, λ ∈ g∗, c ∈ Πg, c ∈ Πg∗ and where C is a constantdepending on ~ and on the choice of the top form ω ∈ ∧top

g∗. Note that we have added theprefactors i

~for later purposes.

Remark 5.3.1. In the physics literature c is called ghost and c is called antighost. λ is usuallycalled Lagrange multiplier.

Note that if we choose a basis ci of g∗, then∧ g∗ can be identified with the algebra generatedby the cis with the relations

cicj = −cjci, ∀i, j.The generators ci ∈ g∗ are often called ghost variables. Similarly, one can introduce antighostvariables ci ∈ g.

Remark 5.3.2. The determinant of A can also be obtained in terms of an ordinary Gaussianintegral over g×g∗. However, then one needs to put A−1 into the exponential. In the infinite-dimensional case we may use the techniques of Feynman diagrams only when dealing withlocal functions. If the action of the group is local, A will be a differential operator, so 〈c, Ac〉will be a local function, in contrary to the quadratic function defined in terms of the Greenfunction A−1.


The expectation value of an invariant function g with respect to a given invariant action Scan be written as

(5.3.1) 〈g〉 =

∫

Mexp(iS/~)gµ

∫

Mexp(iS/~)µ

=

∫

Mexp(iSF /~)gµ

∫

Mexp(iS/~)µ

=: 〈g〉F ,

where

M := M′ ×Πg× g∗ ×Πg∗,

SF := S + 〈λ, F 〉 − 〈c, A, c〉,and µ := µω. The function SF is usually called the gauge-fixed action.

5.3.2. The BRST formalism. Note that, by construction, the right hand side of(5.3.1) is independent of F . Moreover, the assumptions we had are quite restrictive, namely,we want to have a measure-preserving action on M by a compact Lie group G and we haveto assume that the principal bundle M!M/G is trivial. On the other hand, to define

(5.3.2) 〈g〉F =

∫

Mexp(iSF /~)gµ

∫

Mexp(iS/~)µ

we only need the infinitesimal action

X : g! X(M)

γ 7! Xγ

of a Lie algebra g on M. In this case A is simply given by

Aγ = LXγF, γ ∈ g.

Moreover, we want to relax the condition that F−1(0) defines a section an rather require thatA(x) should be nondegenerate for all x ∈ M′. Of course we also require that the integralsare well-defined and that the denominator of (5.3.1) does not vanish.Let F ⊂ C∞(M′, g) be the space of allowed gauge-fixing functions. As in this case 〈g〉F iswell-defined, it makes sense to consider it at more general instances. Observe also that if theLie group is not compact, an invariant function is not a test function. It is then understoodthat it is replaced by a test function that in a neighborhood of the zeros of F coincideswith the given function. To avoid cumbersome notation, we will never explicitly change thefunction, so we will write e.g.

∫Rδ0(x)dx = 1, without mentioning that the constant function

1 is replaced by a test function which is one in a neighborhood of zero. However, since thegauge-fixing function F is arbitrarily chosen, we want the conditions on 〈g〉F to be locallyindependent of F .

Definition 5.3.3 (Gauge-fixing independence). A locally constant function on F is calledgauge-fixing independent.


Theorem 5.3.4. Let X : g ! M be an infinitesimal action of the Lie algebra g on themanifold M. If S and g are invariant functions and

(5.3.3) divXγ +Tr adγ = 0, ∀γ ∈ g,

then 〈g〉F is gauge-fixing independent.

The condition in Theorem 5.3.4 basically tells us that the divergence of X is a constant func-tion. In particular, for the case when the Lie algebra is unimodular (i.e. Tr adγ = 0 for all γ),the condition says that the infinitesimal action must be measure-preserving. The discussionsbefore are covered by Theorem 5.3.4 since the Lie algebra of a compact Lie group is indeedunimodular.

5.3.2.1. The BRST operator and the proof of Theorem 5.3.4. The infinitesimal action of gon M gives C∞(M) the structure of a g-module. Therefore, we can consider the Lie algebracomplex

∧g∗⊗C∞(M). Note that the Lie algebra differential δ is in particular a derivation

on∧

g∗ ⊗ C∞(M) and moreover defines a vector field on M × Πg. Recall that a vectorfield on the superspace Πg is an element of

∧g ⊗ g. If g is a Lie algebra, the commutator

can be regarded as an element of∧2

g∗ ⊗ g and as such it is a vector field on Πg. Vectorfields on M×Πg can then be identified with elements of

∧g∗⊗ g⊗C∞(M)⊕∧ g∗⊗X(M).

The commutator tensor the constant function 1 is an element of∧2

g∗ ⊗ g ⊗ C∞(M) whilethe infinitesimal action of g on M is an element of g∗ ⊗ X(M). As such they define vectorfields on M× Πg and δ is their sum. If we choose a basis ei of g and denote by fkij thecorresponding structure constants, the algebra of functions on Πg can be identified with thegraded commutative algebra with odd generators ci (ghost variables) and

δck = −1

2fkijc

icj .

On functions f ∈ C∞(M), we get that δ acts by

δf = ciLXeif.

The vector field δ can be extended to M by adding it to the vector field on g∗⊗Πg∗. Moreover,using the dual basis ei of g∗, the algebra of functions can be identified with the gradedcommutative algebra with odd generators ci and even generators λi. There we define

(5.3.4) δci = λi, δλi = 0, ∀i.

More abstractly, observe that a polynomial vector field on g∗ × Πg∗ is an element of∧

g ⊗Sym(g) ⊗ (g∗ ⊕ g∗). The identity operator can then be regarded as an element of g ⊗ g∗.Using the inclusion map

i : g⊗ g∗ !∧

g⊗ Sym(g)⊗ (g∗ ⊕ g∗),

a⊗ b 7! 1⊗ a⊗ (b⊕ 0),

we can regard it as a vector field on g∗×Πg∗. Moreover, note that this vector field correspondsto the de Rham differential on Πg∗.


Lemma 5.3.5. The map δ is a differential (i.e. an odd derivation that squares to zero) on∧g∗ ⊗ C∞(M) ⊗∧ g⊗ Sym(g). It has degree one with respect to the grading

deg(α⊗ f ⊗ β ⊗ γ) := deg(α) − deg(β).

In fact, we can rewrite Lemma 5.3.5 by saying that δ is a cohomological vector field on M. Inthe physics literature, δ is called the BRST operator. A function on M can also be considered

as a function on M. By definition of δ, a function f is invariant if and only δf = 0.

Exercise 5.3.6. Show thatdiv δ = divXc +Tradc.

Note that gauge-fixing function F : M ! g can be regarded as an element of C∞(M)] ⊗ g.Using the inclusion

C∞(M)⊗ g ! C∞(M)⊗∧

g !∧

g∗ ⊗ C∞(M)⊗∧

g⊗ Sym(g),

we can associate to F a function ΨF on M. With the same notations as above, we have

ΨF = ciFi.

The odd function ΨF is called gauge-fixing fermion.

Exercise 5.3.7. Show thatSF = S + δΨF .

Assume now that the action S is invariant, i.e. δS = 0.

Lemma 5.3.8. Let g be a function on M. If δg = 0 and δ is divergence-free, then

IF :=

∫

Mexp(iSF /~)gµ,

is gauge-fixing independent.

Proof. Let (Ft) be a family of gauge-fixing functions. Then

d

dtIFt =

i

~

∫

Mδ

(d

dtΨFt

)exp(iSF /~)gµ

=i

~

∫

Mδ

(d

dtΨFt exp(iSF /~)g

)µ =

i

~

∫

Mdiv δ exp(iSF /~)gµ = 0.

Proof of Theorem 5.3.4. Using Lemma 5.3.8 together with Exercise 5.3.6 we imme-diately get the proof.

An particular case of a δ-closed function is a δ-exact function. These functions are irrelevantas for computing expectation values. In fact,∫

Mexp(iSF /~)δhµ =

∫

Mδ (exp(iSF /~)h) µ = 0

if S is invariant and δ is divergence-free. We can now extend Theorem 5.3.4 to the followingtheorem.


Theorem 5.3.9. Let X : g ! M be an infinitesimal action of the Lie algebra g on themanifold M. If the action S is invariant and the BRST operator δ is divergence-free (i.e.(5.3.3) holds), then

(1) 〈g〉F is gauge-fixing independent for all g ∈ ker δ,(2) 〈g〉F for all g ∈ im δ.

Hence, the expectation value defines a linear function on the δ-cohomology.

Remark 5.3.10 (Ward identities). Note that point (2) produces identities relating expecta-tion values of different quantities. Such identities as called Ward identities and usually havenontrivial content.

Example 5.3.11 (Translations). Let M = g = R. Moreover, let g act by infinitesimaltranslations. Denoting by x the coordinate on M, we have δx = c and δc = 0. Assume Sand g are constant. Then

〈g〉F =

exp(iS/~)

∫δ0(F (x))F

′(x)

exp(iS/~)

∫δ0(F (x))F

′(x)

= g

if the denominator does not vanish. Clearly, 〈g〉F is gauge-fixing independent. Similarly, onecan treat rotation-invariant functions on M = S1. A section here is just a point. We take M′

to be a neighborhood of this point. After identifying M′ with R, we can proceed as abovewith F being any function with a single nondegenerate zero corresponding to the image ofthe section.

Example 5.3.12 (Plane rotations). Let M = R2 \ 0 and g = so(2) acting by infinitesimalrotations. Denoting by x and y the coordinates on R2, we have

δx = yc, δy = −xc, δc = 0.

Let M′ = x > 0. A possible choice for F is the function F (x, y) = y. Then

SF (x, y, c, λ, c) = S(x, y) + λy + cxc,

where S is the given rotation-invariant action. Then we get

〈g〉F =

∫ +∞

0exp(iS(x, 0)/~)g(x, 0)xdx

∫ +∞

0exp(iS(x, 0)/~)xdx

,

which is equal to the expected expression∫

Mexp(iS(x, y)/~)g(x, y)dxdy∫

Mexp(iS(x, y)/~)dxdy

for a rotation-invariant function g.


5.3.3. Infinite dimensions. In the infinite-dimensional setting, we want to consider(5.3.2) whenever it makes sense as a perturbative expansion. In this case M is an infinite-dimensional manifold, g an infinite-dimensional Lie algebra acting freely on M and S some

local action function on M. The space M and the BRST operator δ will be exactly defined

as above and δ will still be a cohomological vector field on M. A gauge-fixing function Fwill be allowed whenever the corresponding A is nondegenerate and the critical point of theaction at a zero of F is also nondegenerate. Then, for suitable functions g, we will be able todefine 〈g〉F as a perturbative expansion using Feynman diagrams. Note that an observablein this setting is a δ-cohomology class g for which 〈g〉F is well-defined. In fact, Theorem5.3.9 will hold whenever div δ = 0. Note that, however, the divergence of δ is not defined apriori and has to be understood in terms of expectation values. The usual way to proceed isto assume Theorem 5.3.9 to hold and to derive from it properties of the expectation values.Once they are properly defined in terms of Feynman diagrams, one can check whether theidentities hold and call any deviation an anomaly.Note also that if our aims are of mathematical nature, Theorem 5.3.9 provides a source for alot of interesting conjectures (which, fortunately, in most cases turn out to be true).

5.3.3.1. The trivial Poisson sigma model on the plane. Consider a 2-dimensional general-ization of Section 5.2 which is also the basis for the study of the Poisson sigma model. Let ξand η be 0-form and 1-form on the plane respectively. We assume that they vanish at infinitysufficiently fast (e.g., as Schwarz functions) so that we may define the action

(5.3.5) S :=

∫

Ση dξ,

with Σ = R2. Note that we have dropped the wedge product to avoid cumbersome notationand we will stick to this convention for the rest of the discussion. The space of fields is givenby M = Ω0

0(R2)⊕ Ω1

0(R2). On M we have an action of the abelian Lie algebra g = Ω0

0(R2),

given by the monomorphism i d, where

gd−! Ω1

0(R2)

i−!M.

The action function S is clearly invariant. The BRST differential δ is given on coordinatesby

δξ = 0, δη = dc.

In order to define a gauge-fixing function, we want to choose a Riemannian metric on R2.Denote by ∗ the Hodge star operator induced by the Riemannian metric. Then, using theHodge star, we can define the pairing

(5.3.6) 〈α, β〉∗ :=

∫

R2

(∗α)β, α ∈ Ωj(R2), β ∈ Ωk(R2), j, k = 0, 1, 2.

Let d∗ := ∗d∗ be the formal adjoint of the de Rham differential and choose the gauge-fixingfunction F (ξ, η) = d∗η. Note that different metrics will give different gauge-fixing functions.one can show that the corresponding operator A is then given by the Laplacian on 0-formswhich is invertible for the given conditions at infinity. There is also a unique critical point,i.e. a solution to dξ = dη = 0, satisfying the gauge-fixing condition d∗η = 0, in particular,ξ = η = 0. Hence, this gauge-fixing is indeed allowed.


Using integration, we can identify g∗ with Ω2(R2). The corresponding gauge-fixing fermionis

ΨF =

∫

R2

c d∗η.

This gives the gauge-fixed action

(5.3.7) SF =

∫

R2

(η dξ + λd∗η − cd∗dc

).

Using the pairing as in (5.3.6), we can rewrite the gauge-fixed action as

SF =1

2〈φ,dφ〉∗ − 〈∗c,∆c〉∗,

where ∆ := d∗d + dd∗ denotes the Hodge Laplacian,

φ :=

ξηλ

∈ Ω0

0(R2)⊕Ω1

0(R2)⊗Ω2

0(R2)

and

d :=

0 ∗d 0∗d 0 d∗0 d∗ 0

.

The propagators between φ and c or c are clearly zero. Hence, we get

〈∗c(z)c(w)〉0 = −i~G0(w, z),

where G0 denotes the Green function of the Hodge Laplacian acting on functions. Now inorder to get the propagator between two fields φ, we need to invert the symmetric operatord. For this, we first compute its square

d2 =

∆ 0 00 ∆ 00 0 ∆

.

Then, we can observe that d−1 = dd−2 and hence

d−1 =

0 ∗d∆−1 0∗d∆−1 0 d ∗∆−1

0 d ∗∆−1 0

.

In particular, we have

〈ξ(z)η(w)〉0 = i~ ∗z dzG1(z, w) = i~ ∗w dwG0(w, z),

where G1 denotes the Green function of the Hodge Laplacian acting on 1-forms. Let usintroduce the superfields

ξ := ξ − d∗c,

η := c+ η.(5.3.8)


Then we can define a superpropagator

(5.3.9) i~θ(z, w) := 〈ξ(z)η〉0 = 〈ξ(z)η(w)〉0 − 〈d∗c(z)c(w)〉0= i~(∗zdz + ∗wdw)G0(w, z) ∈ Ω1(C2(R

2)),

where C2(R2) denotes the configuration space of two points in R2 as in Remark 4.3.7.

Lemma 5.3.13. If we choose the Euclidean metric, then

θ =dφE2π

,

where d denotes the differential on C2(R2) and φE(z, w) the Euclidean angle between a fixed

reference line and the line passing through z and w.

Proof. The Green function for the Euclidean Laplacian in two dimensions is given by

G0(z, w) =1

2πlog(|z − w|),

where | | denotes the Euclidean norm. In complex coordinates we have

G0(z, w) =1

4πlog((z − w)(z − w)).

Then

dzG0(z, w) =1

4π

(dz

z − w+

dz

z − w

).

The Euclidean Hodge star operator in complex coordinates gives ∗dz = idz and ∗dz = idz.Hence, we have

∗zdzG0(z, w) =1

4πi

(dw

w − z− dw

w − z

).

Summing everything up, we get

θ =1

4πi

(dz − dw

z − w− dz − dw

z − w

)=

1

4πid log

(z − w

z − w

).

One the other hand, we have z −w = |w − z| exp(iφ), which gives

φ =1

2ilog

(z − w

z − w

),

and hence we get the claim.

Remark 5.3.14. The cohomology class of θ is in fact the generator of H1(C2(R2),Z). It is

not difficult to see that other choices of metric will still give the same cohomolgy class. Onecan easily note that ∗wθ(z, w) is the Green function of the operator P := ∗d∆−1∗ which is aparametrix for the de Rham differential on forms that vanish at infinity, in particular,

dP + Pd = id.

The convolution relating P to θ can be written as

Pα = −π2(θπ∗1α), α ∈ Ωj0(R2), j = 0, 1, 2,


with π1 and π2 being the projections to R2. Then

dPα− Pdα = −(π2)∗(dθπ∗1α) + π∂∗ (θ)α,

where π∂∗ (θ)(w) denotes the integral of θ along a limiting small circle around w. Since P is aparametrix and α is arbitrary, we can see that in general θ is closed and integrates to 1 alongthe generators of H1(C2(R

2),Z).

5.3.3.2. Expectation values. Note that any function of ξ is BRST-invariant, i.e. we canconsider the evaluation of ξ at some point u. A function of

∫γ η is also invariant for any

closed curve γ. Hence, the expectation value⟨ξ(u)

∫

γη

⟩

0

=: i~Wγ(u), u 6∈ im γ,

is independent of the gauge-fixing. Since we also have

i~Wγ(u) =

⟨ξ(u)

∫

γη

⟩

0

,

we can immediately see that Wγ is in fact the winding number of γ around u. This numberis in fact invariant under deformations of γ or displacements of u, which indicates that thetheory is topological. For example, let us deform γ to γ′. Denoting by σ a 2-chain whoseboundary is γ − γ′, we get

Wγ(u)−Wγ′(u) =

⟨ξ(u)

∫

σdη

⟩

0

.

Moreover, introduce the sequence of divergence-free vector fields Xr(ξ, η) = λr⊕0, where (λr)is a sequence of functions that converges almost everywhere to the characteristic function ofthe image of σ. Then we get

Wγ(u)−Wγ′(u) = limr!∞

〈ξ(u)Xr(S)〉0 = i~ limr!∞

〈Xr(ξ(u))〉0 = 0,

under the assumption that u does not belong to σ.

5.3.3.3. The trivial Poisson sigma model on the upper half-plane. Consider now the action(5.3.5) where Σ = H2. As a boundary condition, we impose that the 1-form η vanishes whenrestricted to the boundary ∂H2 = R × 0. The Lie algebra acting on the space of fields isgiven by 0-forms on H2 vanishing on ∂H2. The BRST complex can then be defined exactlyas before and we can choose the same gauge-fixing function. We define the superpropagatoras in (5.3.9) with the difference that we will denote it by ϑ instead of θ in order to avoid anyconfusion.

Lemma 5.3.15. If we choose the Euclidean metric, then

ϑ =dφh2π

,

where d denotes the differential on C2(H2) and φh(z, w) denotes the angle between the vertical

line through w and the geodesic joining w to z in the hyperbolic Poincare metric (recall Figure4.3.4).


Proof. The Green function GH2

0 of the Laplacian on H2 is the restriction to H2 of theGreen function of the Laplacian on R2 plus a harmonic function such that the sum satisfies

the boundary conditions. In complex coordinates, we need GH2

0 (w, z) = 0 whenever w is real.This can be done by setting

GH2

0 (w, z) = G0(w, z) −G0(w, z).

Then we getϑ(z, w) = θ(z, w)− θ(z, w).

Since the hyperbolic angle is given by

φh(z, w) =1

2ilog

((z − w)(z − w)

(z − w)(z − w)

),

we conclude the claim.

Remark 5.3.16. Note that ϑ is the generator of H1(C2(H2),H2 × ∂H2;Z).

5.3.3.4. Generalizations. We can also consider a collection of n 0-forms ξi and n 1-formsηi with i = 1, . . . , n. Then we want to look at the action∫

Σηi dξ

i.

We can also think of ξ and η as forms taking values in Rn. The Lie algebra g of symmetrieswill then consist of the direct sum of n copies of the previous one; in other words it will be theabelian Lie algebra of Rn-valued 0-forms. Let ci for i = 1, . . . , n denote the generators of thealgebra of functions of Πg. Then we can define the BRST operator through δξi and δηi = ci.If we choose the gauge-fixing function to be Fi(ξ, η) = d∗ηi, everything remains the same asbefore. In particular, we can again introduce superfields ξi := ξi− d∗ci and ηi := ci+ ηi andcompute the superpropagator

(5.3.10)⟨ξi(z)ηj(w)

⟩0:=

i~θ(z, w)δij , on R2

i~ϑ(z, w)δij , on H2

Another generalization can be done by dropping the assumption that the 0-form field vanishesat infinity. More precisely, we denote by Xi a collection of maps to Rn with no conditionson the boundary or at infinity and consider the action

S =

∫

Σηi dX

i,

where Σ is either R2 or H2 and the ηis are 1-forms vanishing on the boundary and at infinity.Critical points are then pairs of constant maps together with closed 1-forms. They will be alsodegenerate modulo the action of the abelian Lie algebra of 0-forms. However, the degeneracywill be of a very simple type as it is parametrized by the finite-dimensional manifold Rn. Infact, it is enough to choose a measure on Rn and impose Fubini’s theorem. We will choose adelta-measure peaked at a point x ∈ Rn and require X to map the point ∞ to x. Note thatif we write Xi = xi + ξi, the ξi vanish at infinity and everything is reduced to the previouscase.A final generalization is to replace Rn by a manifold M . We think of Xi as a local coordinateexpression of a map X : Σ!M . For the action to be covariant, we need to assume that ηi(u)


for u ∈ Σ is the local coordinate expression of a 1-form on Σ taking values in the cotangentspace of M at X(u). In particular, we assume η ∈ Γ(Σ, T ∗Σ⊗X∗T ∗M). The space of fieldsM can then be identified with the space of vector bundle maps TΣ ! T ∗M and the actioncan be invariantly written as

S =

∫

Σ〈η,dX〉,

where 〈 , 〉 denotes the canonical pairing between the tangent and cotangent bundle of Mand dX denotes the differential of the map X regarded as a section of T ∗Σ⊗X∗TM . if wealso require X to map the point ∞ to a given point x ∈ M , we can expand around criticalpoints by setting X = x+ ξ with ξ : Σ! TxM and by regarding η as a 1-form taking valuesin T ∗

xM . Choosing local coordinates, we can identify TxM with Rn, where n = dimM , andreduce everything to the previous case.We also allow Σ to be any 2-manifold. The previous discussion will change drastically if Σ isnot simply connected, as the space of solutions modulo symmetries, with X = x, will now beparametrized by H1(Σ, T ∗

xM) and one has to choose a measure on this vector space as well.

5.4. The Poisson sigma model

Next we want to look at deformations of the trivial Poisson sigma model as discussed before.Here, we will describe how the action of the Poisson sigma model is expressed in order toderive Kontsevich’s star product out of it.

5.4.1. Formulation of the model. We want to formulate a deformation of the actionfunctional without introducing extra structure on Σ (which is either R2 or H2). Therefore,the terms we are allowed to add must be 2-forms on Σ given in terms of the fields Xi andηi. In particular, they need to be linear combinations of terms αij(X)ηiηj , β

ij(X)ηi dx

i

and γij(X)dXidXj. Note that we are not considering a term of the form φi(X)dηi, sinceintegration by parts reduces it to a term of the second type. The second and third terms canbe absorbed by a redefinition of η adding to it terms linear in η and dX. Hence, modulo fieldredefinitions, the most general deformation of the action has the form

S =

∫

Σ

(ηi dX

i +1

2ǫαij(X)ηiηj

)+O(ǫ2),

where ǫ is the deformation parameter (here typically ǫ = i~2 ) and α

ij is assumed to be skew-symmetric.

Remark 5.4.1. We want to show that it makes sense to only consider those deformations inwhich the αij are the components of a Poisson bivector field and that the BRST formalismis only available if the Poisson structure is affine.

Recall that the BRST operator before acted by δXi = 0, δηi = dci and δci = 0, with c ∈ Πgand g = Ω0

0(Σ,Rn). We would like to deform the trivial δ such that δS = O(ǫ2) for the new

S. Note that we will only consider the restriction of δ to M×Πg as its restriction to g∗×Πgneeds no deformation.

5.4. THE POISSON SIGMA MODEL 151

Lemma 5.4.2. Modulo field redefinitions, there is a unique BRST operator deforming thetrivial one such that δS = O(ǫ2) and δ2 = O(ǫ2) + R with R vanishing at critical points. Itacts by

δXi = −ǫαij(X)cj +O(ǫ2),

δηi = dci + ǫ∂iαjk(X)ηjck +O(ǫ2),

δci = −1

2ǫ∂iα

jk(X)cjck +O(ǫ2).

Moreover, R vanishes on the whole M×Πg if α is at most linear.

Proof. Recall that δ applied to X or η must be linear in the ghost variables c. Hence,the most general deformation of δ (without adding any extra structure on Σ) is of the form

δXi = ǫvij(X)cj +O(ǫ2),

δηi = dci + ǫ(ajki (X)ηjck + bji (X)dcj + dkij(X)dXjck

)+O(ǫ2),

for some functions vij , ajki , bji and d

kij on Rn. Thus, we get

δS = ǫ

∫

Σ

((ajki (X)ηjck + bji (X)dcj + dkij(X)dXjck)dX

i

+ ηi(dXr∂rv

ij(X)cj + vij(X)dcj) + αij(X)dciηj

)+O(ǫ2).

As the identity δS = O(ǫ2) must hold for any η, we get the following two equations

ajki (X)ckdXi + dXr∂rv

jk(X)ck + vjk(X)dck + αkj(X)dck = 0,(5.4.1)∫

Σ

(bji (X)dcj + dkij(X)dXjck

)dXi = 0.(5.4.2)

In particular, if we choose X to be the constant map (with value x), we deduce from (5.4.1)that

vjk(x)dck + αkj(x)dck = 0,

and since this has to hold for any c, we have

αjk = vjk.

Plugging into (5.4.2), we get

ajki (X)ckdXi − dXi∂iα

jk(X)ck = 0,

and since this has to hold for all c and X, we get

ajki = ∂iαjk.

Using integration by parts, (5.4.2) gives∫

Σ

(− dXr∂rb

ji (X)cj + dkij(X)dXjck

)dXi = 0,

and since this has to hold for all X and c, we finally get

dkij = ∂jbki .


Hence, we have shown that

δXi = −ǫαij(X)cj +O(ǫ2),

δηi = dci + ǫ(∂iα

jk(X)ηjck + d(bji (X)cj))+O(ǫ2),

which, after redefinition ci 7! ci − ǫbji (X)cj + O(ǫ2), gives the first two equations in Lemma5.4.2. For the last equation, we recall that the BRST operator on c must be quadratic in c,so its general form is

δci =1

2ǫf jki (X)cjck +O(ǫ2).

To determine the structure functions f jki , we can compute δ2. Note that δ2Xi = δ2ci = O(ǫ2).On the other hand,

δ2ηi = ǫ

(1

2d(f jki (X)cjci

)+ ∂iα

jk(X)dcjck

)+O(ǫ2)

= ǫ

((f jki (X) + ∂iα

jk(X))dcjck +

1

2dXr∂rf

jki (X)cjck

)+O(ǫ2).

At a critical point (where dXi = O(ǫ2)) the third summand of the last equation vanishes.Thus, δ2 = O(ǫ2) at critical points implies that

f jki = −∂iαjk,which proves the last equation in Lemma 5.4.2, Note also that

δ2ηi = −1

2ǫdxr∂r∂iα

jk(X)cjck,

which is zero (not only at critical points) whenever α is at most linear.

Next, we want to extend deformations beyond the first order in ǫ. Even without knowing thefollowing terms, we can already state the following Lemma.

Lemma 5.4.3. δ2 = O(ǫ3) at critical points only if α is Poisson.

Proof. Note that we have

δ2Xi = −ǫ2(αrk(X)ck∂rα

ij(X)cj +1

2αij(X)∂iα

rj(X)crcj

)+O(ǫ3).

Since this has to hold for all c, we get the Jacobi identity for α.

Remark 5.4.4. It is in fact possible to prove, under the assumption that α is Poisson, thatthis deformation is not only infinitesimal. In particular, we have the following theorem.

Theorem 5.4.5. Given a Poisson bivector field α, the odd vector field

δXi = −ǫαij(X)cj ,

δηi = dci + ǫ∂iαjk(X)ηjck,

δci = −1

2ǫ∂iα

jk(X)cjck,

5.5. DEFORMATION QUANTIZATION FOR AFFINE POISSON STRUCTURES 153

is cohomological for α at most linear or at critical points for all ǫ. Moreover,

S :=

∫

Σ

(ηi dX

i +1

2ǫαij(X)ηiηj

)

is δ-closed for all ǫ.


Remark 5.4.7. The geometrical meaning of Theorem 5.4.5 is that there is a distributionof vector fields on M under which the action is invariant. In general, this distribution isinvolutive only on the submanifold of critical points of S. It is involutive on the whole ofM whenever α is at most linear and in this case it can be regarded as the free, infinitesimalaction of a Lie algebra. The action S can be generalized to the case when one wants toconsider a Poisson manifold (M,α) instead of Rn. For this, one regards X as a map Σ!Mand, for a given map X, η is taken to be a section of T ∗Σ ⊗X∗T ∗M . If 〈 , 〉 denotes thecanonical pairing between the tangent and cotangent bundle of M and by α♯ the bundle mapT ∗M ! TM induced by the Poisson bivector field α (see also Section 3.2.1), we can write

S =

∫

Σ

(〈η,dX〉 + 1

2ǫ〈η, α♯η〉

).

5.4.2. Observables. For the case when Σ = H2, c has to vanish on the boundary. Thisimplies that δX(u) = 0 for u ∈ ∂Σ. Hence, we get

Of1,...,fk;u1,...,uk := f1(X(u1)) · · · fk(X(uk)),

f1, . . . , fk ∈ C∞(Rn), u1, . . . , uk ∈ ∂Σ ∼= R, u1 < · · · < uk,

are observables, i.e. δ-closed functions. In [CF00] it was shown that with the gauge-fixingd∗η = 0 for the Euclidean metric on Σ, one has

〈Of1,...,fk;u1,...,uk〉(x) = f1 ⋆ · · · ⋆ fk(x),where 〈〉(x) denotes the expectation value for X(∞) = x (and expanding only around thetrivial critical solution X = x, η = 0) while ⋆ denotes Kontsevich’s star product for the givenPoisson structure. In the next section will derive this result for the case when α is at mostlinear so that the BRST formalism is available.

5.5. Deformation quantization for affine Poisson structures

An affine Poisson structure on Rn is given by a Poisson bivector field α which is at mostlinear. The linear part α gives the dual of Rn a Lie algebra structure while the constant partis a 2-cocycle in the Lie algebra cohomology with trivial coefficients. Let us denote this Liealgebra by h. The fields of the Poisson sigma model for an affine Poisson structure are thena map X : Σ ! h∗ and a 1-form η on Σ with values in h. Since h is a Lie algebra, we canregard η as a connection 1-form on the trivial principal bundle P over Σ (with gauge groupany Lie group whose Lie algebra is h). For definiteness, we will fix Σ to be H2 and we will


require η to vanish at infinity and on the boundary. The action is then given by

S =

∫

Σ

(ηi dX

i +1

2αij(X)ηiηj

),

where

αij(x) = χij + xkf ijk

is the given affine Poisson structure on h∗. Using integration by parts, we can also rewrite itas

S =

∫

Σ

(〈X,Fη〉+

1

2χ(η, η)

),

where 〈 , 〉 denotes the canonical pairing between h and h∗ while

(Fη)i = dηi +1

2f jki ηjηk

is the curvature 2-form of the connection 1-form η. Note that there is a Lie algebra g, whichas a vector space consists of functions Σ ! h vanishing at infinity and on the boundary,that acts on the space of fields M and leaves the action invariant. The BRST operator onM × Πg has the form as in Theorem 5.4.5 with ǫ = 1. Geometrically, we can regard g asthe Lie algebra of infinitesimal gauge transformations of the principal bundle P ; the field ηactually transforms as a connection 1-form, while X transforms as a section of the coadjointbundle in case χ = 0. For χ 6= 0, we can regard X⊕1 as a section of the coadjoint bundle for

the Lie algebra h ∼= h⊕R obtained by central extension of h through χ. The BRST operatoron Πg∗ × g∗ has the usual form (5.3.4).

5.5.1. Gauge-fixing and Feynman diagrams. Choose a metric on Σ and define thegauge-fixing function F (X, η) = d∗η. The gauge-fixing fermion is given by

ΨF =

∫

Σ〈c,d∗η〉

and the gauge-fixed action is given by

SF =

∫

Σ

(ηi dX

i +1

2αij(X)ηiηj + λid∗ηi − ckd∗

(dck + ∂kα

ij(X)ηicj

)).

Fix the value of X at infinity to be given by x. We write X = x+ ξ, where the field ξ has tovanish at infinity. We can observe that SF has the same form as (5.2.7) (with y the collectionof ξ, c, λ and z the collection of η, c). Hence, we can write SF = S0 + S1 with

S0 =

∫

Σ

(ηi dX

i + λid∗ηi − ckd∗dck

),

S1 =

∫

Σ

(1

2αij(x+ ξ)ηiηj − ckd∗

(∂kα

ij(x+ ξ)ηicj

)),

5.5. DEFORMATION QUANTIZATION FOR AFFINE POISSON STRUCTURES 155

and regard S1 as a perturbation of S0. if we consider superfields ξ and η as in (5.3.8), wecan write

S1 =

∫

Σ

1

2αij(x+ ξ)ηiηj ,

where integration on Σ is understood to select the 2-form component.

Remark 5.5.1. This shows that as long as the considered observables can be written asfunctions of the superfields and expectation values are computed in terms of the superprop-agators (5.3.10). If we denote the superpropagator graphically as an arrow from η to ξ, theperturbation S1 is represented by the two vertices as in Figure 5.5.1 with the bivalent vertex

corresponding to αij(x) = χij + xkf ijk and the trivial vertex corresponding to the structureconstants.

Figure 5.5.1. The two vertices

Let us now consider the observable Of1,...,fk;u1,...,uk as in Section (5.4.2). As the evaluationpoint, i.e., integration along a 0-cycle, is understood to select the 0-form component of adifferential form, we can write

Of1,...,fk;u1,...,uk := f1(x+ ξ(u1)) · · · fk(x+ ξ(uk)),

f1, . . . , fk ∈ C∞(Rn), u1, . . . , uk ∈ ∂Σ ∼= R, u1 < · · · < uk.

Computing then the expectation value 〈Of1,...,fk;u1,...,uk〉(x), we only need the superpropaga-tor. The Feynman diagrams then have three kind of vertices:

(1) bivalent vertices in the upper half-plane corresponding to αij(x),

(2) trivalent vertices in the upper half-plane corresponding to f ijk ,(3) ℓ-valent vertices with ℓ ≥ 0 with only incoming arrows at one of the boundary points

ui corresponding to the ℓ-th derivative of fi.

Recall that the normal ordering excludes all graphs containing a tadpole (i.e. an edge startingand ending at the same vertex). The combinatorics prevents automatically vacuum subgraphs(see Figure 5.5.2 for examples).For the case when k = 2 and considering the gauge-fixing d∗η = 0 with respect to theEuclidean metric on H2, i.e. with the superpropagator determined by the 1-form ϑ as inLemma 5.3.15, we get

(5.5.1) 〈Of,g;0,1〉(x) = f ⋆ g(x),

where ⋆ denotes Kontsevich’s star product for the given affine Poisson structure.

Exercise 5.5.2. Prove (5.5.1).


Figure 5.5.2. Example of an allowed graph and a non-allowed graph.

5.5.2. Independence of the evaluation point. We want to give a formal proof of theindependence of the expectation values of Of1,...,fk;u1,...,uk from the points u1, . . . , uk. Notefirst that

f(X(v)) − f(X(u)) =

∫ v

udXi∂if(X) =

∫ v

u

(dXi + d∗λi

)∂if(X)− δΦ,

where

Φ :=

∫ v

ud∗c∂if(X).

Let (ωr) be a sequence of 1-forms on Σ vanishing on the boundary and at infinity thatconverges to the measure concentrated on the interval (u, v) ∈ Σ. Denoting by (a, b), withb ≥ 0, the coordinates on Σ = H2, a possible choice for this sequence is

ωr(a, b) = rb exp(−rb2/2)χr(a)da,

where (χr) is a sequence of smooth, compactly supported functions converging almost every-where to the characteristic function of the interval (u, v). Let Yf,r be the local vector field

on M corresponding to the infinitesimal displacement of ηi by ωr∂if(X). Then

f(X(v))− f(X(u)) = limr!∞

Yf,r(SF )− δΦ.

If O is a BRST-observable depending on the fields outside the closed interval [u, v], we get

〈(f(X(v)) − f(X(u)))O〉 = i~ limr!∞

〈Yf,r(O)〉 − 〈δ(ΦO)〉 = 0.

5.5.3. Associativity. The independence of the evaluation points gives us

limv!u+

〈Of,g,h;u,v,w〉0(x) = limv!w−

〈Of,g,h;u,v,w〉0(x).

The left hand side corresponds intuitively to evaluating first the expectation value of Of,g;u,v,then replacing the result at u and finally computing the expectation value of O〈Of,g,;u,v〉0,h;w.

The result is then (f ⋆ g) ⋆ h. Repeating the computation on the right hand side, we getf ⋆ (g ⋆ h). This rather formal argument explains why one should expect the star productdefined by the Poisson sigma model to be associative.

5.6. THE GENERAL CONSTRUCTION 157

5.6. The general construction

We want to also state the theorem for general Poisson structures on Rd. However, we willnot provide a proof of the general construction here since one needs the more general gaugeformalism provided by Batalin and Vilkovisky.

5.6.1. The theorem of Cattaneo–Felder.

Theorem 5.6.1 (Cattaneo–Felder[CF00]). Consider the Poisson manifold (Rd, π) with Pois-son structure π together with the corresponding Poisson sigma model on a disk D. The fieldsare then given by a map X : D ! Rd and a 1-form η ∈ Γ(D,T ∗D ⊗X∗T ∗Rd) with boundarycondition η

∣∣S1 = 0. Moreover, let 0, 1 and ∞ be cyclically ordered points on ∂D = S1, i.e. if

we start at 0 and move counterclockwise on S1 we will first meet 1 and then ∞ (see Figure5.6.1), and impose the condition η(∞) = 0. Then Kontsevich’s star product between twofunctions f, g ∈ C∞(Rd) evaluated at the point x = X(∞) ∈ Rd is given by

f ⋆ g(x) =

∫

X(∞)=xf(X(1))g(X(0)) exp(iS(X, η)/~)D [X]D [η],

where the right hand side should be understood by perturbative expansion in terms of Feynmandiagrams.

Remark 5.6.2. The global field theoretic approach was given in [CMW20] using cutting andgluing methods for manifolds with boundary by introducing a formal global action as moregenerally developed in [CMW19]. Moreover, also the case for manifolds with corners was cov-ered. The main gauge formalism that was used is known as the BV-BFV formalism [CMR14;CMR17; CM20], which can bee seen as a extension of the Batalin–Vilkovisky formalism formanifolds with boundary. A first approach to a global formulation of the Poisson sigmamodel on closed manifolds using the Batalin–Vilkovisky formalism was given in [BCM12].

0 1

∞

D

Figure 5.6.1. Cyclically ordered points on ∂D = S1

5.6.2. Other similar constructions. The way of using quantum field theory to obtaincertain mathematical constructions has been proven to be a very interesting and deep methodrelating seemingly purely mathematical constructions to physics. Other examples includeWitten’s famous construction to obtain the 4-manifold invariants described by Donaldson[Don83] using a certain deformation of the supersymmetric Yang–Mills action and a specialtype of observables [Wit88]. Computing the expectation value of this observable with respect


to the theory formulated by this special action, one can recover the Donaldson polynomialsof the given 4-manifold. Another important and surprising result of Witten [Wit89] was thatthe expectation value of a Wilson loop observable representing a given knot with respect tothe Chern–Simons action [CS74; AS91; AS94] will give the Jones polynomial [Jon85] of theknot, which is an important knot invariant. These results have lead to many mathematicalconjectures, deeper insights in physics and created a whole new perspective towards theinterplay between mathematics and quantum field theory.

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