Uppsala University

Supergravity and Kaluza-Klein dimensionalreductions

Author:Roberto Goranci

Supervisor:Giuseppe Dibitetto

May 13, 2015

Abstract

This is a 15 credit project in basic supergravity. We start with the supersym-metry algebra to formulate the relation between fermions and bosons. This projectalso contains Kaluza-Klein dimensional reduction on a circle. We then continuewith supergravity theory where we show that it is invariant under supersymmetrytransformations, it contains both D = 4, N = 1 and D = 11, N = 1 supergravitytheories. We also do a toroidal compactification of the eleven-dimensional super-gravity.

I

Contents1 Introduction 1

2 Supersymmetry algebra 32.1 Poincaré symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Representation of spinors . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Generators of SL(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Super-Poincaré algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Clifford algebra 73.1 The generating γ-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 γ-matrix manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Symmetries of γ-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Fierz rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 Majorana spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Differential geometry 114.1 Metric on manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Cartan formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Connections and covariant derivatives . . . . . . . . . . . . . . . . . . . 124.4 Einstein-Hilbert action in frame and curvature forms . . . . . . . . . . . 17

5 Kaluza-Klein theory 185.1 Kaluza-Klein reduction on S1 . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Free Rarita-Schwinger equation 216.1 Dimensional reduction to the massless Rarita-Schwinger field . . . . . . 22

7 The first and second order formulation of general relativity 237.1 Second order formalism for gravity and fermions . . . . . . . . . . . . . 237.2 First order formalism for gravity and fermions . . . . . . . . . . . . . . . 25

8 4D Supergravity 278.1 Supergravity in second order formalism . . . . . . . . . . . . . . . . . . 278.2 Supergravity in first order formalism . . . . . . . . . . . . . . . . . . . . 298.3 1.5 order formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.4 Local supersymmetry of N = 1, D = 4 supergravity . . . . . . . . . . . 30

9 Toroidal compactification 339.1 D=11 supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.2 11-Dimensional toroidal compactification . . . . . . . . . . . . . . . . . . 38

10 Conclusion 43

A Riemann tensor for Einstein-Hilbert action in D dimensions 45

B Equations of motion for Einstein-Hilbert action in D dimensions 47

C Covariant derivative of γν 48

D Variation of the spin connection 48

II

E Chern-Simons dimensional reduction 49

III

IV

1 IntroductionThe aim of this thesis is to study supergravity from its four-dimensional formulationto its eleven-dimensional formulation. We also consider compactifications of higherdimensional supergravity theories which results in extended supergravity theories infour-dimensions. Supergravity emerges from combining supersymmetry and generalrelativity.

General relativity is a theory that describes gravity on large scales. The predictionsof general relativity has been confirmed by experiments such as gravitational lensing andtime dilation just to mention a few. However general relativity fails to describe gravityon smaller scales where the energies are much higher than current energy scales testedat the LHC.

The Standard model took shape in the late 1960’s and early 1970’s and was con-firmed by the existence of quarks. The Standard model is a theory that describesthe electromagnetic the weak and strong nuclear interactions. The interactions betweenthese forces are described by the Lie group SU(3)⊗SU(2)⊗U(1) where SU(3) describesthe strong interactions and SU(2) ⊗ U(1) describes the electromagnetic and weak in-teractions. Each interaction would have its own force carrier. The Weinberg-Salammodel of electroweak interactions predicted four massless bosons, however a process ofsymmetry breaking gave rise to the mass of three of these particles, the W± and Z0.These particles are carriers of the weak force, the last particle that remains massless isthe photon which is the force carrier for the electromagnetic force.

We know that there are four forces in nature and the Standard model neglects grav-itational force because its coupling constant is much smaller compared to the couplingconstants of the three other forces. The Standard model does not have a symmetry thatrelates matter with the forces of nature. This led to the development of supersymmetryin the early 1970’s.

Supersymmetry is an extended Poincaré algebra that relates bosons with fermions.In other words since the fermions have a mass and all the force carriers are bosons,we now have a relation between matter and forces. However unbroken supersymmetryrequires an equal mass for the boson-fermions pairs and this is not something we haveobserved in experiments. So if supersymmetry exist in nature it must appear as abroken symmetry. Each particle in nature must have a superpartner with a differentspin. For example the gravitino which is a spin-3/2 particle has a superpartner called thegraviton with spin-2. Graviton in theory is the particle that describes the gravitationalforce. We now see why supersymmetry is an important tool in the development of atheory that describes the three forces in the Standard model and gravity force. Withsupersymmetry one could now develop a theory that unifies all the forces in nature. Onetheory in particular that describes gravity in the framework of supersymmetry is calledsupergravity.

Supergravity was developed in 1976 by D.Z Freedman, Sergio Ferrara and PeterVan Nieuwenhuizen. They found an invariant Lagrangian describing the gravitino fieldwhich coincides with the gravitational gauge symmetry. In 1978 E.Cremmer, B.Juliaand J.Scherk discovered the Lagrangian for eleven dimensional supergravity. The elevendimensional theory is a crucial discovery which is the largest structure for a consistent su-pergravity theory. Eleven dimensions is the largest structure where one has a consistenttheory containing a graviton, however there exist theories that contain higher dimen-sions which have interactions containing spin 5/2 particles. From the eleven-dimensionaltheory one can obtain lower-dimensional theories by doing dimensional reductions, onethen finds ten-dimensional theories known as Type IIA and Type IIB and these the-ories are related to superstring theories with the same name. Supergravity appears aslow-energy limit of superstring theory. The maximally extended supergravity theories

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start from the eleven-dimensional Lagrangian, one obtains maximally extended theo-ries by doing dimensional reductions, for example Type IIA is a maximally extendedsupergravity theory since it contains two supercharges in ten dimensions.

The outline of this thesis goes as follows. In section 2 we begin with constructingSupersymmetry algebra and its representations. In section 3 we discuss Clifford algebrawhich plays an important role in Supergravity because it contains spinors. In section4 we go trough some differential geometry, in particular the Cartan formalism anddifferential forms. In section 5 we consider the (D + 1) dimensional Einstein-Hilbertaction and perform a dimensional reduction on a circle this is known as the Kaluza-Kleinreduction. We obtain aD-dimensional Lagrangian that contains gravity, electromagneticfields and scalar fields.

In section 6 we go through the Rarita-Schwinger equation and discuss degree’s offreedom for the gravitino field. Section 7 contains first and second order formulationsof general relativity. In section 8 we consider the four-dimensional Supergravity theoryand show the invariance of its Lagrangian. Section 9 contains eleven-dimensional Su-pergravity theory where we construct a Lagrangian that is invariant. We also performdimensional reductions on the eleven-dimensional theory using Kaluza-Klein method toobtain lower dimensional theories.

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2 Supersymmetry algebraSupersymmetry is a symmetry that relates two classes of particles, fermions and bosons.Fermions have half-integer spin meanwhile bosons have integer spin. When one con-structs a supersymmetry the fermions obtains integer spin and the bosons obtain half-integer spin, each particles has its own superpartner. So for example a graviton hasspin-2 meanwhile its superpartner gravitino has spin-3/2.

The key references in this section will be [1], [2] and [3].

2.1 Poincaré symmetry

The Poincaré group ISO(3, 1) is a extension of the Lorentz group. The Poincaré groupis a 10-dimensional group which contains four translations, three rotations and threeboosts. The Poincaré group corresponds to basic symmetries of special relativity, it actson the spacetime coordinate xµ as

xµ → x′µ = Λµνxν + aµ . (2.1)

Lorentz transformation leaves the metric invariant

ΛT ηΛ = η

Λ is connected to the proper orthochronous group SO(3, 1) and the metric is given asη = (−1, 1, 1, 1). The generators of the Poincaré group are Mµν , P σ with the followingalgebras

[Pµ, P ν ] = 0 (2.2)[Mµν , P σ] = i(Pµηνσ − P νηµσ) (2.3)

[Mµν ,Mρσ] = i(Mµσηνρ +Mνρηµσ −Mµρηνσ −Mνσηµρ) (2.4)

where P is the generator of translations and M is the generator of Lorentz transforma-tions. The corresponding Lorentz group SO(3, 1) is homeomorphism to SU(2), wherethe following SU(2) generators correspond to Ji of rotations and Ki of Lorentz boostsdefined as

Ji = 12εijkMjk , Ki = M0i

and their linear combinations

Ai = 12

(Ji + iKi) , Bi = 12

(Ji − iKi) .

The linear combinations satisfy SU(2) commutation relations such as

[Ji, Jj ] = iεijkJk (2.5)[Ji,Kj ] = iεijkKk (2.6)

[Ki,Kj ] = −iεijkKk (2.7)[Ai, Aj ] = 0 (2.8)[Bi, Bj ] = 0 (2.9)[Ai, Bj ] = 0 . (2.10)

We can interpret J = A + B as physical spin. There is a homeomorphism SO(3, 1) ∼=SL(2,C) which gives rise to spinor representations. We write down a 2 × 2 hermitianmatrix that can be parametrized as

x =(x0 + x3 x1 − ix2x1 + ix2 x0 − x3

). (2.11)

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This will lead to a explicit description of the (12 , 0) and (0, 1

2) representation and this isthe central treatment for fermions in quantum field theory. 1

The determinant of equation (2.11) gives us x2 = −xµηµνxν which is negative of theMinkowski norm of the four vector xµ. This gives rise to a relation between the linearspace of the hermitian matrix and four-dimensional Minkowski space. The transforma-tion xµ → xµΛ under SO(3, 1) leaves the square of the four vector invariant, but whathappens to the determinant if we consider a linear mapping 2.

Let N be a matrix of SL(2,C) and consider the linear mapping

x→ NxN †

the four-vectors are linearly related i.e xµ → xµΛ and from the 2 × 2 matrix we canobtain the explicit form of the matrix Λ. The linear transformation gives us that thedeterminant is preserved, and the Minkowski norm is invariant. This means that thematrix Λ must be a Lorentz transformation. Since SO(3, 1) ∼= SU(2) ⊗ SU(2) is validthen the topology of SL(2,C) is connected due to the fact that Λ is a Lorentz trans-formation of SO(3, 1) and the relation SO(3, 1) ∼= SL(2,C). The spinor representationwill play a important role in the future chapters.

2.2 Representation of spinors

Let us consider the spinor of representations of SL(2,C). The corresponding spinors arecalled Weyl spinors. The (1

2 , 0) is given by the irreducible representation of the chirality2L ≡ (ψα, α = 1, 2) and the spinor representation (0, 1

2) is given by 2R = (χα, α = 1, 2).The first representation gives us the following Weyl spinor

ψ′α = Nα

βψβ, α, β = 1, 2 (2.12)

this is the left-handed Weyl spinor. The other representation is given by

χ′α = N∗

αβχβ, α, β = 1, 2 (2.13)

this is the right-hand Weyl spinor. These spinors are the representation of the basicrepresentation of SL(2,C) and the Lorentz group. From the previous chapter we cannow see the relation between the Lorentz group and SL(2,C). Introducing anotherspinor εαβ which will be a contravariant representation of SL(2,C) defined as

εαβ = εαβ =(

0 1−1 0

)= −εαβ = −εαβ (2.14)

hence we see that the spinor ε raises and lowers indices. Our representations of the Weylspinors then becomes

ψα = εαβψβ, χα = εαβχβ . (2.15)

When we have mixed indices of both SO(3, 1) and SL(2,C) the transformations of thecomponents xµ should look the same as we have seen in section 2.1. We saw that thedeterminant is invariant under transformation of SO(3, 1) or a transformation via thematrix x = xµσ

µ, hence

(xµσµ)αα → Nαβ(xνσν)βγN∗

αγ = Λµνxνσµ

1The reason why we can find spinor representations is that we have written our 2 × 2 matrix withPauli matrices such that x = xµσµ

2Since we know that we have a relation between Minkowski space and a linear space of the hermitianmatrix, it is valid to consider a linear mapping of the hermitian matrix

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the right transformation rule is then given by

(σµ)αα = Nαβ(σν)βγ(Λ−1)µνN∗

αγ

we can also introduce the matrices σµ defined as

(σν)αα = εαβεαα(σµ)ββ

2.3 Generators of SL(2,C)

We can define the tensors σµν , σµν as antisymmetric products of σ matrices

(σµν)αβ = i

4(σµσν − σν σµ)αβ (2.16)

(σµν)αβ = i

4(σµσν − σνσµ)αβ (2.17)

which satisfy the Lorentz algebra

[σµν , σλρ] = i(ηµρσνλ + ηνλσµρ − ηµλσνρ − ηνρσµλ) (2.18)

The finite Lorentz transformation is then represented as

ψα → exp(− i

2ωµνσ

µν)α

βψβ (2.19)

which is the left handed spinor, the right hand spinor is defined as

χα → exp(− i

2ωµν σ

µν)α

βχβ (2.20)

We can obtain some useful identities concerning σµ and σµν , defined as

σµν = 12iεµνρσσρσ (2.21)

σµν = −12iεµνρσ ¯σρσ (2.22)

these identities are known as the self duality and anti self duality. These identities areimportant when we discuss Fierz identities which rewrite bilinear of the product of twospinors as a linear combination of products of the bilinear of the individual spinors. Wewill use Fierz identities when we consider Clifford algebra.

2.4 Super-Poincaré algebra

In this section we will consider the Super-Poincaré group which is an extension of thePoincaré group but with a new generator known as the spinor supercharge QAα , where αis the spacetime spinor index and A = 1, . . . ,N labels the supercharges. A superalgebracontains two classes of elements , even and odd. Let us introduce the concept of Gradedalgebras.

Let Oa be a operator of the Lie algebra then

OaOb − (−1)ηaηbObOa = iCeabOe (2.23)

where ηa takes the value

ηa =

0 : Oa bosonic generator1 : Oa fermionic generator

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This structure relations include both commutators and anti commutators in the pattern[B,B] = B, [B,F ] = F and F, F = B. We know the commutation relations in thePoincaré algebra, we now need to find the commutation relations between the super-charge and the Poincaré generators.

The following commutation relations with the supercharge generator are defined as

[Qα,Mµν ] = (σµν)αβQβ (2.24)[Qα, Pµ] = 0 (2.25)Qα, Qβ = 0 (2.26)Qα, Qβ = 2(σµ)αβPµ (2.27)

When N = 1 then we have a simple SUSY, when N > 1 then we have a extendedSUSY. Let us consider N = 1 and the commutator (2.27). We consider the N = 1SUSY representation, in any on-shell supermultiplet 3 the number nB of bosons shouldbe equal to the number nF of fermions4.

Proof. Consider the fermion operator (−1)F = (−1)F defined via

(−)F |B⟩ = |B⟩, (−)F |F ⟩ = −|F ⟩

The new operator (−)F anticommutes with Qα, since

(−)FQα|F ⟩ = (−)F |B⟩ = |B⟩ = Qα|F ⟩ = −Qα(−)F |F ⟩ → (−)F , Qα = 0 (*)

Next, consider the trace

Tr

(−)F Qα, Qβ

= Tr

(−)FQαQβ + (−)F QβQα

= Tr−Qα(−)F Qβ +Qα(−)F Qβ

= 0

It can also be evaluated using Qα, Qβ = 2(σµ)αβPµ

Tr

(−)FQα, Qβ

= Tr

(−)F 2(σµ)αβPµ

= 2(σµ)αβpµ Tr(−)F = 0

where Pµ is replaced by the eigenvalues pµ for the specific state. The conclusion is

0 = Tr(−)F =∑bosons

⟨B|(−)F |B⟩+∑

fermions

⟨F |(−)F |F ⟩

=∑bosons

⟨B|B⟩ −∑

fermions

⟨F |F ⟩ = nB − nF

Each supermultiplet contains both fermion and boson states which are known asthe superpartners of each other. This proof also shows that there are equal number ofbosonic and fermionic degrees of freedom. This statement is only valid when the Super-Poincaré algebra holds. It is also valid when we have auxiliary fields that closes thealgebra off-shell i.e when off-shell degree’s of freedom disappear on-shell. If we considerthe massless representation the eigenvalues are pµ = (E, 0, 0, E), the algebra is

Qα, Qβ = 2(σµ)αβPµ = 4E(

1 00 0

)αβ

3a supermultiplet is a representation of a SUSY algebra4However there exist some examples where one also has off-shell supermultiplet that have equal

amount of bosonic and fermionic degree’s of freedom. Off-shell equality holds for some extended SUSYand higher dimensional theories.

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letting α take the value of 1 or 2 gives us the following commutation relations

Q2, Q2 = 0 (2.28)Q1, Q1 = 4E . (2.29)

We can define the creation- and annihilation operators a and a† as

a = Q1

2√E, a† = Q1

2√E

(2.30)

we can see that the annihilation operator is in the representation (0, 12) and has the

helicity λ = −1/2 and the creation operator is in the representation (12 , 0) and has the

helicity λ = 1/2. We can build the representation by using a vacuum state of minimumhelicity λ, lets call it |Ω⟩, if we let the annihilation operator act on this state we getzero. Letting the creation operator act on the vacuum state we obtain that the wholemultiplet consisting of

|Ω⟩ = |pµ, λ⟩, a†|Ω⟩ = |pµ, λ+ 1/2⟩

so for example we have that the graviton has helicity λ = 2 and the superpartnergravitino has the helicity λ = 3/2.

3 Clifford algebraIn section 1.2 we only considered Weyl spinors. We will now introduce Clifford algebrawhich uses gamma matrices. The Clifford algebra satisfies the anticommutation relation

γµγν + γνγµ = 2ηµν1 (3.1)

these matrices are the generating elements of the Clifford algebra. The key referencesof this section are [2] and [4].

3.1 The generating γ-matrices

Let is discuss the Clifford algebra associated with the Lorentz group in D dimensions.We can construct a Euclidean γ-matrices which satisfy (3.1) with Minkowski metric ηµν .The representation of the Clifford algebra can be written in terms of σ matrices whichare hermitian with square equal to 1 and cyclic

γ1 = σ1 ⊗ 1⊗ 1⊗ . . .γ2 = σ2 ⊗ 1⊗ 1⊗ . . .γ3 = σ3 ⊗ σ1 ⊗ 1⊗ . . .γ4 = σ3 ⊗ σ2 ⊗ 1⊗ . . .γ5 = σ3 ⊗ σ3 ⊗ σ1 ⊗ . . .. . . = . . .

there are two representations of this algebra mainly an even representation and anodd representation. Let us assume that D = 2m is even then the dimension of therepresentation is 2D/2, this means that we need m factors to construct the γ-matrices.For odd representations we have D = 2m + 1 which gives us the same dimension ofthe representation. This is due to the fact that we need γ2m+1 matrices to constructthe algebra but since we only keep the factor of m and delete a σ1 matrices whichdoes not increase the dimension. So the general construction of the algebra gives a

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representation of the dimension 2D/2. We can construct the Lorentzian γ-matrices bypicking any matrix from the Euclidean construction and multiplying it by i and label itγ0 for the time-like direction. The hermiticity property of the Lorentzian γ are definedas

㵆 = γ0γµγ0 . (3.2)

To preserve the Clifford algebra we introduce the definition of conjugacy

γ′µ = U−1γµU . (3.3)

We only consider hermitian representation in which (3.2) holds, then the matrix U hasto be unitary. Given two equivalent representations the transformation matrix U isunique.

3.2 γ-matrix manipulation

We need to define some γ-matrix manipulation in order to later use it for fermionspin calculations. Clifford algebra are needed to explore the physical properties of thsefields. These manipulations are valid in odd and even dimensions D. Consider the indexcontraction such as

γµνγν = (D − 1)γµ . (3.4)

In general we obtain

γµ1...µrν1...νsγνs...ν1 = (D − r)!(D − r − s)!

γµ1...µr . (3.5)

The general order reversal symmetry is defined as

γν1...νr = (−)r(r−1)/2γνr...ν1 (3.6)

the sign factor (−)r(r−1)/2 is negative for r = 2, 3 mod 4. Another useful property isthe contraction which is defined as

γµ1µ2γν1...νDεν1...νD = D(D − 1)εµ2µ1ν3...νDγν3...νD (3.7)

γ-matrix without index contraction in the simplest case is defined as

γµγν = γµν + ηµν . (3.8)

This follows directly from the definitions: the antisymmetric part of the product isdefined to be γµν and the symmetric part is ηµν . In general one writes the totallyantisymmetric Clifford matrix that contains all the indices and then add terms of possibleindex pairings. Another example is

γµνργστ = γµνρστ + 6γ[µν[τδ

ρ]σ] + 6γ[µδν [τδ

ρ]σ] (3.9)

3.3 Symmetries of γ-matrices

The Clifford algebra of 2m×2m matrices for both even and odd representations, one candistinguish the antisymmetric and the symmetric matrices with a symmetry propertycalled charge conjugation matrix. There exist a unitary matrix C such that each matrixCγA is either antisymmetric or symmetric. Symmetry only depends on the rank r ofthe matrix γA, so we can write

(Cγ(r))T = −trCγ(r), tr = ±1 (3.10)

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where γA is the basis of the Clifford algebra defined as

γA = 1, γµ, γµ1µ2 , . . . , γµ1...µD .

For rank r = 0 and 1 we obtain

CT = −t0C, γµT = t0t1CγµC−1 (3.11)

Given two possibilities in the representation for even dimensions we can construct theClifford algebra with Pauli matrices

C+ = σ1 ⊗ σ2 ⊗ σ1 ⊗ σ2 ⊗ . . . , t0t1 = 1 (3.12)C− = σ2 ⊗ σ1 ⊗ σ2 ⊗ σ1 ⊗ . . . , t0t1 = −1 (3.13)

for odd dimensions only one of the two can be used. To preserve the algebra the chargeconjugation matrix transforms as

C ′ = UTCU . (3.14)

The charge conjugation needs to be unitary C† = C−1 in any representation.

3.4 Fierz rearrangement

In this section we study the importance of the completeness of the Clifford algebra basisγA. Fierz rearrangement properties are frequently used in supergravity, these propertiesinvolve changing the pairing of spinors in product of spinor bilinears.

Let us derive the basic Fierz identity. Using spinor indices defined as

MαµMµβMγνM

νδ = δαβδγ

δ . (3.15)

We can expand the matrix M in the complete basis γA using our spinor indices weobtain

δαβδγ

δ = 12m

∑A

(mA)αδ(γA)γβ (3.16)

the coefficients are mA = 2−mδαδδγ

β(γA)βγ . Therefore we obtain the basic rearrange-ment lemma

δαβδγ

δ = 12m

∑A

(γA)αδ(γA)γβ . (3.17)

The Fierz rearrangement is valid for any set of four anticommuting spinor fields. Thebasic Fierz identity (3.17) gives us

(λ1λ2)(λ3λ4) = − 12m

∑A

(λ1γAλ4)(λ3γAλ2) (3.18)

Useful Weyl spinor identities, which involves manipulating σ matrices interacting withtwo spinors, the symmetry properties are gives as

ψσµχ = −χσµψ (3.19)ψσµσνχ = χσν σµψ (3.20)ψσµνχ = −χσµνψ . (3.21)

These manipulations will be useful later on.

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Let us define the Majorana conjugate of any spinor λ using its transpose and the chargeconjugation matrix

λ = λTC (3.22)

This equation is useful in SUSY and supergravity in which symmetry properties of γ-matrices and of spinor bilinears are important and these properties are determined byC. Using the definition of Majorana conjugate and equation (3.10) we obtain

λγµ1...µrχ = trχγµ1...µrλ (3.23)

the minus sign obtained by changing the Grassmann valued spinor components. Thesymmetry property is valid for Dirac spinors, but its main application is with Majoranaspinors. Therefore we call it Majorana flip relations.

3.5 Majorana spinors

We consider the reality constraint and write down the following equations

ψ = ψC = B−1ψ∗, i.e ψ∗ = Bψ (3.24)

the constraint is compatible with Lorentz symmetry, the equation (3.24) is the definingcondition for Majorana spinors. In order to have Majorana spinors we consider whenD = 4, using that CT = −t1C and equation (3.12). These equations and the conditionindeed satisfies the equation (3.24). There are representations of γ-matrices that are realand may be called really real representations. Here is a real representation for D = 4

γ0 =(

0 1−1 0

)= iσ2 ⊗ 1 (3.25)

γ1 =(

1 00 −1

)= σ3 ⊗ 1 (3.26)

γ2 =(

0 −iσ2iσ2 0

)= σ1 ⊗ σ1 (3.27)

γ3 =(

0 σ3σ3 0

)= σ1 ⊗ σ3 (3.28)

. (3.29)

The physics of Majorana spinors is the same, in any Clifford algebra the complex conju-gate can be replaced with the charge conjugation. For example the complex conjugateof χγµ1...µrψ where χ and ψ are Majorana, is computed in the following way

(χγµ1...µrψ)∗ = (χγµ1...µrψ)C = χ(γµ1...µr )Cψ = χγµ1...µrψ

where we have used ψC = ψ and χC = χ.

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4 Differential geometryIn this section we will discuss differential geometry, where we will formulate the Cartanformalism also known as vielbein, spin connection and p-forms mainly. In supergravityfermions couple to gravity. The key references in this section will be [4] and [5].

4.1 Metric on manifold

A metric or inner product on a real vector space V is a bilinear map from V ⊗V → R.Theinner product of two vectors u and v ∈ V must satisfy the following properties:

• bilinearity, (u, c1v1 + c2v2) = c1(u, vi) + c2(u, v2) and (c1v1 + c2v2, u) = c1(v1, u) +c2(v2, u)

• non-degeneracy, if (u, v) = 0 ∀v ∈ V then u = 0

• symmetry (u, v) = (v, u) .

The metric on a manifold is smooth assignment of an inner product map on eachTp(M) ⊗ Tp(M) → R 5. In local coordinates the metric is specified by a covariantrank-2 symmetric tensor field gµν and the inner product of two contravariant vectorsUµ(x) and V µ(x) is gµνUµ(x)V µ(x) which is a scalar field. We can summarize theproperties of a metric by the line element

ds2 = gµνdxµdxν . (4.1)

non-degeneracy means that det(gµν) = 0 so the inverse metric gµν exist as a rank-2symmetric contravariant tensor which satisfy

gµρgρν = gνρgρµ = δµν . (4.2)

We can use this relation to raise and lower indices e.g Vµ(x) = gµνVν(x) and ωµ(x) =

gµνων(x). In gravity the metric has the following signature −+ + + . . .+. The metrictensor gµν may be diagonalized by an orthogonal transformation (O−1)µa = Oaµ, and

gµν = OaµDabObν (4.3)

with positive eigenvalues λa in Dab = diag(−λ0, λ1, . . . , λD−1)

4.2 Cartan formalism

Let us define an important auxiliary quantity

eaµ(x) =√λa(x)Oaµ(x) . (4.4)

In four dimensions this quantity is known as the tetrad or vierbien. In general dimensionsit is called vielbein but when we discuss gravity we prefer the term frame field. We canwrite the metric as

gµν = eaµηabebν (4.5)

where ηab = diag(−1, 1, . . . , 1) is the metric of flat D-dimensional Minkowski space-time. Given a x-dependent matrix Λab(x) which leaves ηab invariant, which allows us toconstruct the equation (3.5) with a Lorentz transformation i.e

e′aµ (x) = Λ−1a

b(x)ebµ(x) . (4.6)5i.e the tangent bundle of a smooth manifold M assigns to each fixed point of M a vector space

called the tangent space and each tangent space can be equipped with an inner product.

11

all frame fields related by a local Lorentz transformation are viewed as equivalent. LocalLorentz transformations in curved spacetime differ from global Lorentz transformationif Minkowski space. We require that the frame field eaµ transform as a covariant vectorunder coordinate transformations

e′aµ(x′) = ∂xρ

∂x′µeaρ(x) . (4.7)

The vielbein eaµ has an inverse frame field eµa which satisfy eaµebµ = δab and eaµeν

a =δµν . The vielbein transforms under a local transformation, hence

eµa = gµνηabebν , eµagµνe

νb = ηab . (4.8)

This term shows that the inverse frame field can be used relate a general metric to theMinkowski metric. The second relation of (4.8) indicates that eµa form an orthonormalset of Tp(M). Since it is non-degeneracy the det eµa = 0 we have a basis of each tangentspace. Any contravariant vector field has a unique expansion in the new basis, i.eV µ = V aeµa with V a = V µeaµ. The V a are the frame components of the original vectorfield V µ. The vector fields transform under a Lorentz transformation, i.e V ′a = Λ−1a

bVb.

We can then use eµa and eaµ to transform vector and tensor fields back and forth betweena coordinate basis with indices µ, ν, . . . and a local Lorentz basis with indices a, b, . . . inthe metric ηab. We can do an exercise to show how all these transformations work:

UµVµ = gµνUµVν = eaµηabe

bνU

µV ν = ηabUaV b = UaVa

where the second term is obtain from Vµ = gµνVν , the third term we use equation (3.8)

and the last term we use the same relation as the second term. We have now constructeda new local invariance, we have enlarge the symmetry of GR to be general coordinatetransformations and Lorentz transformations. We need frame fields to describe fermionsin general relativity.

We can also use the frame field eaµ to define the new basis Λp(M) of differential forms.The local Lorentz basis of 1-form is

ea ≡ eaµdxµ . (4.9)

For 2-forms the basis consist of the wedge product ea ∧ eb and so on. Local frames areuseful when we consider fermions coupled to gravity, because spinors transform underLorentz transformations.

4.3 Connections and covariant derivatives

Contravariant derivative on a manifold is a rule to differentiate a tensor of type (p, q)producing a tensor (p, q + 1). We need to introduce the affine connection Γρµν . Onvector fields the covariant derivative is defined as

∇µV ρ = ∂µVρ + ΓρµνV ν (4.10)

∇νVν = ∂νVν − ΓρµνVµ . (4.11)

In supergravity we will use the frame field eaµ. Gravitational theories with fermions isantisymmetric in Lorentz indices ωab = ωabµ dx

µ. The components ωabµ are called the spinconnections because they describe the spinors on the manifold.

Given the 1-forms ea we examine the 2-form

dea = 12

(∂µeaν − ∂νeaµ)dxµ ∧ dxµ (4.12)

12

the antisymmetric components transforms as (0, 2) tensor under coordinate transforma-tion. Using the Lorentz transformation (4.6) we obtain the following equation

de′a = Λ−1a

bdeb + dΛ−1a

b ∧ eb (4.13)

the second term spoils the vector transformation so we need to add an extra term thatabsorbs the term involving dΛ−1a

b. We introduce the spin connection which is a 2-form,giving us

dea + ωab ∧ eb ≡ T a . (4.14)

If ωab is defined to transform under Lorentz transformation as

ω′ab = Λ−1a

cdΛcb + Λ−1acω

cdΛdb (4.15)

then T a transforms as a vector, T ′a = Λ−1abT

b. We know that any tensor transformsas a vector i.e

T a′µ′b′ν′ = Λa′

a∂xµ

′

∂xµΛb′b

∂xν

∂xν′ Taµbν

so when we do a Lorentz transformation in the frame field i.e e′aµ → Λ−1a

bebµ we

have to add an extra term when taking the covariant derivative of the transformation.Therefore when performing a covariant derivative on the frame field we obtain a (0, 2)tensor and a Lorentz vector which coincides with the connection transformation. The2-form T a is called the torsion 2-form of the connection and (4.14) is called the firstCartan structure equation.

Spinor fields Ψ(x) are crucial for supergravity theories, we describe the spinor fieldsthrough their local frame components. The local Lorentz transformation rule

ψ′ = exp(−1

4λabγab

)ψ (4.16)

determines the covariant derivative

Dµψ =(∂µ + 1

4ωµabγ

ab)ψ (4.17)

this will be very useful when we do calculations in supergravity.

Let us transform Lorentz covariant of the vector and tensor frame fields to the coordi-nate basis where they become covariant derivatives with respect to general coordinatetransformations. We can write out the spin connection in terms of the affine connectionΓρµν . The quantity ∇µV ν = eνaDµV

a is the transform to coordinate basis of a framefield and covariant vector field. We can show that this quantity can take the form ofequation (4.10) and (4.11)

∇µV ρ = eρaDµVa

= eρaDµ(eaνV ν)= ∂µV

ρ + eρa(∂µeaν + ωµabebν)V ν

(4.18)

where we have used that DµVν = ∂µV

ν + ωµabV

ν and eρaeaν = δρν . We can show that

∇µVν = eaνDµVa in the same way

∇µVν = eaνDµVa = eaνDµ(eρaVρ)= eaνDµe

ρaVρ + eaνe

ρaDµVρ

= ∂µVν + eρa(∂µeaν + ωµabebν)Vρ

(4.19)

13

These two examples can be written down in a general Lorentz tensor of type (p, q). Thetransformation to coordinate basis is given by

∇µT ρ1...ρpν1...νq

≡ eρ1a1 . . . e

ρpape

b1ν1 . . . e

bqνqDµT

a1...ap

b1...bq(4.20)

define the tensor of type (p, q + 1) with the properties of the covariant derivative. Theaffine connection relates to the spin connection by

Γρµν = eρa(∂µeaν + ωµabebν) (4.21)

we can rewrite this as∂µe

aν + ωµ

abebν − eaρΓρµν = 0 . (4.22)

This property is called vielbein postulate. When we considered a connection without spinconnection we had an restriction mainly the that the connection is metric compatiblei.e ∇gµν = 0. We have a similar restriction on the connection above,

∇µgνρ = ∂µgνρ − Γσµνgσρ − Γσµρgνσ = 0 (4.23)

the metric postulate means that the tensor is covariantly constant. The Γρµν is theChristoffel symbols defined as

Γρµν = 12gρσ(∂µgσν + ∂νgµσ − ∂σgµν) (4.24)

this is the torsion-free connection. When the torsion is present the affine connection isnot symmetric rather

Γρµν − Γρνµ = T ρµν (4.25)

Using equation (4.23) we can calculate the nonvanishing torsion, we start with

0 = ∇agbc +∇bgca −∇cgab (4.26)

using the metric compatibility property we obtain the following equation

∂agbc + ∂bgca − ∂cgab = Γadbgdc + Γadcgbd + Γbdagcd + Γbdcgda − Γcdagdb − Γcdbgad= 2Γ(a

db)gdc + 2Γ(a

dc)gbd + 2Γ(b

dc)gda

= 2Γ(adb)gdc + T[a

dc]gbd + T[b

db]gda

(4.27)

this connection is not unique. If the torsion vanishes we get back the affine connection.

We can now consider curvature, recall that we defined the curvature as

Rabcd = ∂cΓdab − ∂dΓcab + ΓcaeΓdeb − ΓdaeΓceb (4.28)

we also have another connection which will have curvature too

(∇c∇d −∇d∇c)V µ = ∂c∂dVµ + (∂cωdµσ)V σ + ωd

µσ∂cV

σ − Γced(∂eV µ + ωeµσV

σ)+ ωc

µλ(∂dV λ + ωd

λσV

σ)− (c↔ d)(4.29)

we would like this to be the spin curvature and a torsion term. We look at the termsinvolving V and no derivatives of V , and identify

Rcdµσ = ∂cωd

µσ − ∂dωcµσ + ωc

µλωd

λσ − ωdµλωcλσ (4.30)

14

as the curvature of the spin connection. The remaining terms are

(−Γced + Γdec)(∂eV µ + ωeµσV

σ) = −Tced∇eV µ (4.31)

Because of the antisymmetry in cd, it is also a (0, 2) tensor under general coordinatetransformation, called the curvature tensor. Thus we can define the curvature 2-form

Rµσ = 12Rµσabdx

a ∧ dxb (4.32)

using equation (4.30) we then see that the curvature 2-form is related to connection1-form by

dωµσ + ωµν ∧ ωνσ = Rµσ (4.33)

this is known as the second Cartan structure equation. We can derive the Bianchiidentities for the curvature tensor, but first we will introduce the basis of 1-forms

Eµ = eaµdxa. (4.34)

We can define the torsion 2-form

T λ = eaλ 1

2Tbacdx

b ∧ dxc . (4.35)

We can use these p-forms to obtain the Cartan first and second structure equations in amuch faster way. We consider the first structure equation where we will use the equation(4.34), rewriting the first structure equation and doing the calculation yields us

dEµ + ωµν ∧ Eν = d(eaµdxa) + ωaµνdx

a ∧ ebνdxb

= (∂beaµ)dxb ∧ dxa + ωaµνeb

νdxa ∧ dxb

= 12

(∂aebµ − ∂beaµ + ω µa νeb

ν − ω µb νea

ν)dxa ∧ dxb

= 12

(Γ ca be

µc − ω µ

a σeσb − Γ c

b aecµ + ω µ

b σeaσ + ω µ

a νebν − ω µ

b νeaν)dxa ∧ dxb

= 12T ca b ec

µdxa ∧ dxb = Tµ

this is Cartan’s first structure equation, where we have used ∂beaν = Γ cb aecν − ωbνσeaσ

and contracted the remaining terms which disappears when doing so. We can do thesame with the curvature

dωµσ + ωµν ∧ ωνσ = d(ωµσadxa) + ωµνadxa ∧ ωνσ

= dωµσadxa + ωµνadx

a ∧ ωνσbdxb

= 12

(∂bωµσa − ∂aωµσb + ωµνaω

νσb − ω

µνbω

νσa)dxa ∧ dxb

= 12Rµσabdx

a ∧ dxb = Rµσ

(4.36)

We can now find the Bianchi identities for Cartan’s first and second structure equation.Taking d of Cartan’s first structure equation gives us

dTµ = d(dEµ + ωµν ∧ Eν)= dωµν ∧ Eµ − ωµν ∧ dEν

= (Rµσ − ωµρ ∧ ωρσ) ∧ Eν − ωµν ∧ (T ν − ωνσ ∧ Eσ)= Rµσ ∧ Eν − ωµρ ∧ ωρσ ∧ Eν − ωµν ∧ T ν + ωµν ∧ ωνσ ∧ Eσ

= Rµσ ∧ Eν − ωµν ∧ Tµ

(4.37)

15

and taking the d of Cartan’s second structure equation gives us

dRλµ = d(dωλµ + ωλρ ∧ ωρσ)= dωλρ ∧ ωρσ − ωλρ ∧ dωρσ= (Rµσ − ωµρ ∧ ωρσ) ∧ ωρσ − ωλρ ∧ (Rρσ − ωρδ ∧ ωδσ)= Rµσ ∧ ωρσ − ωλρ ∧Rρσ .

(4.38)

Using the following relation Raµνρ = Rµνbaebρ, we obtain

R aµνρ +Rνρµa +R a

ρµν = −DµTνρa −DνTρµ

a −DρTµνa (4.39)

DµRab

νρ +DνRab

ρµ +DρRab

µν = 0 . (4.40)

The derivatives Dµ in the equation above are Lorentz covariant derivatives and containonly the spin connection. The first relation is the first Bianchi identity for the curvaturetensor, but we include a torsion tensor. The second relation is known as the Bianchiidentity for for the curvature. The commutator of Lorentz covariant derivatives leads toRicci identities. We define them as

[Dµ, Dν ]Φ = 12RµνabM

abΦ (4.41)

[Dµ, Dν ]V a = RµνabV

b (4.42)

[Dµ, Dν ]Ψ = 14Rµνabγ

abΨ (4.43)

where Ψ is the spinor fields, Φ is a field transforming in a representation of properLorentz group with the generator Mab.

16

4.4 Einstein-Hilbert action in frame and curvature forms

Let us show that the Einstein-Hilbert action for GR can be written as the integral of avolume form involving the frame and curvature.

1(D − 2)!

∫εabc1...cD−2e

c1 ∧ . . . ∧ ecD−2 ∧Rab

= 1(D − 2)!

∫εabc1...cD−2e

c1 ∧ . . . ∧ ecD−2 ∧ 12Rabcde

c ∧ ed

= 12(D − 2)!

∫εabc1...cD−2e

c1 ∧ . . . ∧ ecD−2 ∧ ec ∧ edRabcd

= −12(D − 2)!

∫εc1...cD−2abε

c1...cD−2cdRabcddV

= −22(D − 2)!

∫(D − 2)!δcdc1

abc1. . . δcD−2

cD−2RabcddV

=−∫δacδb

dRabcddV = −∫dDx

√−det gR

(4.44)

where we have used the equation (4.32) and volume form defined as

dV = dDx√−det g (4.45)

andea1 ∧ . . . ∧ eaq ∧ eb1 ∧ . . . ∧ ebp = −εa1...aqb1...bpdV . (4.46)

This form of the action reveals how the connection with torsion is realised in the physicalsetting of gravity coupled to fermions.

17

5 Kaluza-Klein theoryIn this section we will consider Kaluza-Klein theory which will be very useful when wedo dimensional reductions. In this chapter we will go through dimensional reduction ofEinstein-Hilbert action (EH) in D + 1 dimensions. The dimensional reduction of E-Haction will yield us a D-dimensional Einstein field equations, the Maxwell equations forelectromagnetic fields and an wave equation for scalar fields. The key references in thissection are [1] and [6]

5.1 Kaluza-Klein reduction on S1

Let us assume we start from Einstein gravity in (D + 1) dimensions described by theEinstein-Hilbert Lagrangian

L =√−gR (5.1)

where R is the Ricci scalar and g denotes the determinant of the metric tensor. Wenow want to reduce the action to D dimensions by doing so we need to compactify oneof the coordinates on a circle S1 of radius L. We can expand all the components ofD + 1-dimensional metric as a Fourier series

gMN =∑n

g(n)NM (x)einz/L (5.2)

If we do this then we obtain infinite numbers of fields in D dimensions by labelled theFourier mode number n. So if we consider a scalar field in 5 dimensions defined by theaction

S5D =∫d5x∂M ϕ∂M ϕ (5.3)

we set the extra dimensions x4 = z defining a circle with the radius L with z = z+ 2πL,the spacetime is now M4 × S1. Using the Fourier expansion where we expand aroundϕ, the coefficients describe infinite amount of D-dimensional scalar fields, which satisfythe equation of motion

2ϕn −n2

L2ϕn = 0 . (5.4)

This means that we have infinite amount of Klein-Gordon equations for massive D-dimensional fields and that each ϕn is a D-dimensional particle with mass m2

n = n2

L2 . Weassumed that the circle is very small, which tells us that the particles should be aroundthe Planck mass. However since the modes are way beyond intergalactic scales, we cannot observed these types of particles and we can thus neglect them.

Going back to the EH action, the Kaluza-Klein ansatz will be gMN (x, z) where themetric tensor will not depend on the z coordinate. The index M runs over (D + 1)values of the higher dimension which splits into D values of lower dimension, i.e takesthe value associated with the compactification. Let us define the (D + 1)-dimensionalmetric in terms of D-dimensional fields gµν , Aµ and ϕ

ds2D+1 = e2αϕds2

D + e2βϕ(dz +A)2 . (5.5)

This ansatz is independent of z thus we have express the components of the higherdimensional metric gMN which are given by the lower dimensional fields

gMN =(e2αϕgµν + e2βϕAµAν e2βϕAµ

0 e2βϕ

)(5.6)

18

To calculate the Ricci tensor for the Einstein-Hilbert Lagrangian we must choose avielbein basis defined as

ea = eαϕea, ez = eβϕ(dz +A) . (5.7)

We can now calculate the one-forms, where we set the torsion tensor to zero.

dea = αdϕeαϕ ∧ ea − eαϕdea

= α∂bϕe−αϕeb ∧ ea − ωab ∧ eb

(5.8)

and the other one-form can be calculated in the same way

dez = βdϕ ∧ eβϕ(dz +A) + eβϕdA

= βe−αϕ∂aϕea ∧ ez + 1

2e(β−2α)ϕFabe

a ∧ eb(5.9)

where we have used that F = dA defined as

F = 12Fabe

a ∧ eb = 12

(∂aAb − ∂bAa)ea ∧ eb . (5.10)

We can now obtain the spin connection by using Cartan’s first structure equation (3.14),

ωab = ωab + αe−αϕ(∂bϕea − ∂aϕeb)− 12F abe(β−2α)ϕez (5.11)

ωaz = −βe−αϕ∂aϕez − 12F abe

(β−2α)ϕeb . (5.12)

Now that we have found the spin connection we can obtain the two-form by using Car-tan’s second structure equation. Before we obtain the curvature, we want to choosethe values of α and β such that the dimensionally-reduced Lagrangian is defined asL =

√−gR. If we do not fix the constants our Lagrangian takes the form L =

e(β−(D−2)α)ϕ√−gR, we now see that we have to set β = −(D − 2)α to achieve theLagrangian we defined at the beginning. This is called the Einstein frame and it comesfrom our metric ansatz where we have Weyl rescaling e2αϕ and e2βϕ which results inmodified Ricci tensors. We also want to ensure that the scalar field ϕ acquires a kineticterm with a canonical normalisation 1

2(∂ϕ)2 in the Lagrangian. Therefor we choose theconstant to be

α2 = 12(D − 1)(D − 2)

, β = −(D − 2)α (5.13)

The Ricci tensor are

Rab = e−2αϕ(Rab −

12∂aϕ∂bϕ− αηab2ϕ

)− 1

2e−2DαϕFa

cFbc (5.14)

Raz = 12e(D−3)αϕ∇b

(e−2(D−1)αϕFab

)(5.15)

Rzz = (D − 2)αe−2αϕ2ϕ+ 14e−2DαϕF 2 . (5.16)

We can obtain the Ricci scalar R = ηABRAB = ηabRab + Rzz,

R = ηab(e−2αϕ

(Rab −

12∂aϕ∂bϕ− αηab2

)− 1

2e−2DαϕFa

cFbc

)+ (D − 2)αe−2αϕ2ϕ+ 1

4e−2DαϕF 2

= e−2αϕ(R− 1

2(∇ϕ)2 + (D − 3)α2ϕ

)− 1

2e−2DαϕF 2 .

(5.17)

19

We can now determine the determinant of the metric tensor,√−g = e(β+Dα)ϕ√−g = e2αϕ√−g (5.18)

the Einstein-Hilbert Lagrangian now takes the form

L =√−g

(R− 1

2(∇ϕ)2 − 1

4e−2(D−1)αϕF 2

). (5.19)

Doing the integral over the z coordinate now yields us the following action

SEH = 116πGD+1

N

∫dD+1√−gR

= 2πL16πGD+1

N

∫dDx√−g

(R− 1

2(∇ϕ)2 − 1

4e−2(D−1)αϕF 2

).

(5.20)

the scalar field ϕ is called a dilaton. We can work out the field equations for thisLagrangian that tells us more about the fields. If one were to set the scalar field tozero we would obtain the Einstein-Maxwell Lagrangian in D dimensions, but this is notallowed due to the equations of motion.

Rµν −12Rgµν = 1

2

(∂µϕ∂νϕ−

12

(∂ϕ)2gµν

)+ 1

2e−2(D−1)αϕ

(F 2µν −

14F 2gµν

)∇µ

(e−2(D−1)αϕFµν

)= 0

2ϕ = −12

(D − 1)αe−2(D−1)αϕF 2 .

(5.21)

the reason we have dropped the 2ϕ term in the Lagrangian is because it is a totalderivative in L which doesn’t contribute to the field equations. From the last equationin (5.21) we see that we cannot set ϕ = 0 because it involves a F 2 term on the right-handside. This means that the lower dimensional fields prevent the truncation of the scalar[6]. We neglected the massive modes in our Fourier expansion but assume we kept themassive modes, would we then be able to set the equation of motion of the massivefield to zero. When we do a dimensional reduction on a circle our fields must transformcovariantly under lower-dimensional U(1) gauge symmetry and under diffeomorphismof the lower-dimensional spacetime. Thus any mixture of massless modes with towers ofmassive modes will be avoided. Lower-dimensional solutions must be solutions to higher-dimensional theories which guarantees consistent truncation. So in other words whenthe Kaluza-Klein ansatz satisfies the equations of motion we have consistent truncation.

20

6 Free Rarita-Schwinger equationLet us now introduce a free spin-3/2 field. We only consider fields that don’t interactand we consider them separately. In particular we consider Ψµ(x) as a free field, thegauge transformation is

Ψµ(x)→ Ψµ(x) + ∂µεx . (6.1)

We will also assume that Ψµ and ε are complex spinors with 2[D/2] spinor components forspacetime dimension D. The gauge field Ψµ(x) should have anti-symmetric derivativeof the gauge potential ∂µΨν − ∂νΨµ which is invariant under gauge transformation. Wenow consider a action that is Lorentz invariant, first order in derivatives and invariantunder gauge transformations (6.1) and conjugate transformation of Ψµ. We also demandthat the action is hermitian so that the equations of motion for Ψµ is Dirac conjugateof Ψµ. We obtain the following action

S = −∫dDxΨµγ

µνρ∂νΨρ (6.2)

we see that the action contains a third rank of Clifford algebra γµνρ which satisfies allthese properties. Using the variational principle on the action we obtain the Euler-Lagrange equation given as

γµνρ∂νΨρ = 0 . (6.3)

Let us consider the on-shell degree’s of freedom for the Rarita-Schwinger field. We needto fix the gauge so we impose the constraint

γiΨi = 0 . (6.4)

We rewrite the components of (6.3) as ν → 0 and ν → i which gives us

γij∂iΨj = 0−γ0γij(∂0Ψj − ∂jΨ0) + γijk∂jΨk = 0 .

(6.5)

Using γij = γiγj − δij and the gauge condition we obtain

∂iΨi = 0 . (6.6)

Using the same gamma-matrix manipulation on the second equation in (6.5) and multi-plying it with γi as well as using the gauge condition, we obtain that the two first termsare zero due to Ψ0 = 0. The second equation then takes the form

γ0∂0Ψ0 + γj∂jΨi = γµ∂µΨi = 0 . (6.7)

We have now shown that the components of Ψi satisfy a Dirac equation. Hence theclassical degree’s of freedom after imposing the gauge condition and (6.6) is given by(D − 3)2[D/2].

The spinor field Ψ is a representation of SO(D − 2) which would result in on-shelldegree’s of freedom given by (D − 2)[D/2]−1. However when one subtracts the γ-traceof the vector-spinor representation, we end up with a irreducible representation thatcontains (D − 3)[D/2] components.

We can rewrite the equation (6.3) using the γ-matrix relation γµγµνρ = (D − 2)γνρ,

which imples that the equation γνρ∂νΨρ = 0 is zero in spacetime dimension D > 2. Wealso use that γµνρ = γµγνρ − 2ηµ[γρ]. Using this we obtain the following equation

γµ(∂µΨν − ∂νΨµ) = 0 . (6.8)

21

This equation is equivalent to the equation of motion above by applying γν and obtainγνρ∂µΨρ = 0. We can also apply ∂ρ to obtain the following equation of motion definedas

/∂(∂ρΨν − ∂νΨρ) = 0 . (6.9)

If we now take SUSY into account we know that the spin-3/2 particle should have asuperpartner. If the spin-3/2 particle from the Rarita-Schwinger equation represents agravitino then the superpartner should be a graviton. The graviton is expected to bemassless, but the gravitino is expected to have mass.

6.1 Dimensional reduction to the massless Rarita-Schwinger field

Let us apply dimensional reduction the Rarita-Schwinger field in D+ 1 dimensions withD = 2m. We assume that the field Ψµ(x, y) is periodic in y so that the Fourier seriesinvolves modes with half-integer k. This yields us only massive modes since k = 0 occurs.Let us derive the wave equation of a massive gravitino in D-dimensional Minkowskispace. We impose a gauge condition ΨDk(x) = 0 thus eliminating the field componentΨD(x, y). Writing out the wave equation using γD = γ∗ where µ = D and µ = D − 1:

γνρ∂νΨρk = 0 (6.10)

(γµνρ∂ν − ik

Lγ∗γ

µρ)Ψρk = 0 . (6.11)

We can obtain the equation of motion for a massive particle from (6.11) by applyingthe chiral transformation Ψ = e−iπγ∗/4Ψ′. We can rewrite the chiral transformation as aphase factor (cos(2β) + iγ∗ sin(2β))Ψ then we see that the only contribution is the sinusterm and the Fourier series mode eiky/LΨ gives rise to the massive modes since k = 0,therefore our equation of motion takes the form

(γµνρ∂ν −mγµρ)Ψρ = 0 (6.12)

where m = k/L which is the mass of the vector fields. One can further investigate theconstraints on the equation of motion by finding γµΨν = 0 and letting µ = 0 whichyields us the following constraint

(γij∂i −mγj)Ψj = 0 . (6.13)

Gathering the information about the constraints of the equation of motion gives usinformation about degrees of freedom.

γµΨµ = 0 (6.14)(γij∂i −mγj)Ψj = 0 (6.15)

(/∂ +m)Ψµ = 0 . (6.16)

The initial data are the values t = 0 of the Ψµ restricted by the fist two equations above.The complex scalar field Ψµ with D × 2D/2 degrees of freedom contains (D − 2)× 2D/2

independent classical degrees of freedom and thus 12(D − 2) × 2D/2 on-shell physical

states [4].

We can obtain a more general action defined as

S = −∫dDxΨµ(γµνρ∂ν −mγµρ −m′ηµρ)Ψρ (6.17)

thus from the dimensional reduction we obtained that the mass term for massive grav-itino should be mΨµγ

µνΨν and the extra term in the action is a Lorentz invariant termwhich has the dimension of mass.

22

7 The first and second order formulation of general relativityIn this chapter we will discuss the first and second order formulation of GR, which willbe important when we consider supergravity for N = 1. The second order formulationin which the metric tensor or the frame field describes gravity. If fermions are involvedwe must use the frame field.

In first order a.k.a Palatini formalism one starts with an action in which eaµ and ωµabare independent variables and the Euler-Lagrange equations are first order in derivatives.When we couple gravity to spinor fields the ωµab field equation contains terms bilinear inthe spinors and the solution is ωµab = ωµab(e)+Kµab with a contorsion tensor determinedas a bilinear expression in the spinor fields.

7.1 Second order formalism for gravity and fermions

We consider the massless Dirac field Ψ(x), the action is defined as

S = S2 + S1/2 =∫dDxe

( 12κ2 e

µaeνbRµν

ab(ω)− 12

Ψγµ∇µΨ + 12

Ψ←−∇µγµΨ). (7.1)

We can obtain the relationδS2(e, ω)δωµab

∣∣ω=ω(e) = 0 (7.2)

by using that the variation of the curvature is given by

δRµνab = Dµδων

ab −Dνδωµab (7.3)

thus making δRµνab ∝ Dµe

aν − Dνe

aµ. Using Cartan’s first structure equation we

have that the spin connection can be expressed as a unique torsion free spin connectiondefined as

ωµab(e) = 2eν[a∂[µeν]

b] − eν[aeb]σeµc∂νeσc . (7.4)

The covariant derivatives defined from previous chapters:

∇µΨ = DµΨ = (∂µ + 14ωµ

abγab)Ψ (7.5)

Ψ←−∇µ = Ψ←−Dµ = Ψ(←−∂ µ −

14ωµ

abγab) . (7.6)

Frame fields are used to transform frame vector indices to a coordinate basis, e.gγµ = eµaγ

a = gµνγν . Thus γµ transforms as a covariant vector under coordinate trans-formation. The covariant derivative of γµ is therefore

∇µγν = ∂µγν + 14ωµ

ab[γab, γν ]− Γρµνγρ . (7.7)

The spin connection appears with the commutator as required for a spinor. We canobtain the following result

∇µγν = γa(∂µeav + ωµabebν − Γρµνeaρ) = 0 (7.8)

which tells us that covariant derivatives commute with multiplication by γ-matrices.One obtains this by using the vielbein postulate (4.22). This relation holds for anyaffine connection the complete derivation can be seen in the appendix appendix C. Forexample the Dirac field ∇µ(γνΨ) = γν∇µΨ, and from the (7.1) one can find the equationof motion for the massless covariant Dirac equation

γµ∇µΨ = 0 . (7.9)

23

Let us now take the variation of the action defined in (7.1) with respect to eaµ, the goalis to find the Einstein field equations for a conserved symmetric fermionic stress tensorTµν . The variation of the action is

δS =∫dDxe

( 1κ2

(ebνRµνab −

12eaµR

)δeaµ

− 12

Ψγa←→∇ µΨδeaµ − 18

Ψγµ, γabΨδωµab) (7.10)

we have used (7.2), and we have dropped the terms that are satisfied by the equation ofmotion (7.9) in the fermionic part of the Lagrangian. Let us now consider the variationof the spin connection in the last part of the action above, when we integrate by partswe will obtain a stress tensor. The variation of the spin connection is defined as

ebρeaνδωµ

ab = D[µδaν]e

aρ −D[νδ

aρ]e

aµ + (D[ρδ

aµ]e

aν . (7.11)

The anti commutator is equal to a third rank Clifford matrix eµc γcab. Using the variationof the spin connection above we see that it is totally anti symmetric in µ, ν and ρ, thuswe are able to write the last term

−14

ΨγµνρΨeaνebρδωµab = −14

ΨγµνρΨebρ∇µδeνb . (7.12)

One obtains this relation by doing some γ-matrix manipulations using 12γ

a, γb andinserting this into the anti-commutation relation above, after doing some calculationswe obtain that this is 1

4Ψγ[aηb]µΨδωµab using that the gamma matrices are constantmatrices i.e γµ = eµaγ

a and using (7.11) one obtains the result in (7.12). Integratingby parts, and using the Dirac equation (7.9) and γ-matrix manipulations we obtain

−14eaρδ

aνΨ(γν

←→∇ ρ − γρ←→∇ ν)Ψ = 14

Ψ(γa←→∇ µ − γµ

←→∇ ρeρa)Ψδeaµ (7.13)

Thus we arrived at the final expression defined as

δS =∫dDx

( 1κ2

(ebνRµνab −

12eaµR

)δeaµ − 1

4Ψ(γa←→∇ µ − γµeρa

←→∇ ρ

)Ψδeaµ

).

(7.14)since δeaµ is arbitrary we can set it to zero thus we obtain the Einstein equation of theform

Rµν −12gµνR = κ2

4Ψ(γµ←→∇ ν + γν

←→∇ µ

)Ψ . (7.15)

The Einstein equations are consistent only if the matter stress tensor is covariantlyconserved Tµν = 0, and symmetric Tµν = Tνµ. This follows from the requirements thatthe matter action is invariant under general coordinate transformation and local Lorentztransformations.

Let us show that the stress tensor is Lorentz invariant, we consider the action

S =∫dDxL(e, ω(e), ψ) (7.16)

where ψ is the spinor field, thje independent fields transform as

δeaµ = −λabebµ, δΨ = −14λabγ

abΨ, δψµ = −14λabγ

abψµ . (7.17)

The ψµ field is the gravitino field that appears in the Rarita-Schwinger equation andλab is the infinitesimal Lorentz transformation. We assume that the action is invariantso the variation is given by

δS =∫dDx

(δS

δeaµδeaµ + δS

δψδψ

)= 0 (7.18)

24

From the first term we obtainT aµ = 1

e

δS

δaµ(7.19)

Thus if the equation of motion (7.9) is satisfied then the equation above becomes

δS =∫dDxeTµνeaνe

bµλab(x) = 0 (7.20)

since λab(x) is antisymmetric the stress tensor must be symmetric. Symmetry is guar-anteed only if the fermion equation of motion is satisfied.

7.2 First order formalism for gravity and fermions

The field equation for ωµab contains terms bilinear in the spinors giving a connection withtorsion ωµab = ωµab(g) +Kµab. We will use the action (7.1) and the covariant derivative(7.6) for the first order action. In first order formalism the frame field and the spinconnection ωµab are independent variables and that the curvature tensor Rµνab can beconstructed from this. The field equation for the spin connection gives a connectionwith the torsion

ωµab = ωµab(e) +Kµab(ϕ) (7.21)

The variation of the gravitational action is

δS2 = 12κ2

∫dDx eeµae

νb

(Dµδων

ab −Dνδωµab)

(7.22)

=∫dDx√−geµaeνbDµδων

ab . (7.23)

The Lorentz covariant derivatives can be integrated by parts. We have to include theaction of Dµ on each of the three factors in

√−geµaeνb. Using the relation ∂µ

√−g =√

−g Γρρµ(g), we obtain

δS2 = −∫dDx√−g

((Γρρµ(g)eµa +Dµe

µa)eνb + eµbDµe

νb

)δων

ab (7.24)

the firs two terms can be written as

(Γρρµ(g)− Γρρµ)eµa +∇µeµa . (7.25)

The last term is a total covariant derivative of eµa which vanishes as a consequence ofthe vielbein postulate. Using that Γρµν = Γρµν(g)−Kµν

ρ and the contorsion tensor

K[µ[νρ] = −12

(T[µν]ρ − T[νρ]µ + T[ρµ]ν) (7.26)

thus we can rewrite the equation (7.25) as

(Γρρµ(g)− Γρρµ)eµa = Taρρ (7.27)

The last term in the action can be written as

Dµeνb = −eρb(Dµe

cρ)eνc (7.28)

The integrand of the second term is then

−eµaeρbDµecρeνcδων

ab = −12eµae

ρb(Dµe

cρ −Dρe

cµ)eνcδωνab

= −12Tab

νδωνab

(7.29)

25

this follows from equation (4.12). Combining all the information above we arrive at thefinal expression of the variation of S2 defined as

δS2 = 12

∫dDx√−g (Tabν − Taρρeνb + Tbρ

ρeνa) δωνab . (7.30)

Let us look at the connection variation of the spinor action using equation (7.6), weobtain

δS1/2 = −14

∫dDx e ΨγνabΨδωνab . (7.31)

We can now find a equation for the torsion tensor, by considering δωνab to be arbitraryand thus we obtain

Tabν − Taρρeνb + Tbρ

ρeνa = 12κ2ΨγabνΨ . (7.32)

We have seen that the spin connection in first order formalism has a connection withthe torsion tensor i.e ωµab = ωµab(e) + Kµab(ϕ), thus we see that the spin connectionωµab is a function of the other fields. The contorsion tensor appears when we couplegravity to fermions

26

8 4D SupergravityWe have now arrived to supergravity theory, which is a locally invariant supersymmetrytheory. This means that the action is invariant for SUSY transformations where ε(x) isa function of the spacetime coordinates. A supergravity theory is nonlinear which is ainteracting field theory that contains gauge and gravity multiplets and other multipletsdescribing scalar fields. The gauge multiplet consist of a frame field eaµ(x) describingthe graviton and N of vector-spinor fields Ψi

µ(x) where i = 1, . . . ,N . In this section wewill focus on D = 4 with N = 1.

8.1 Supergravity in second order formalism

We will not start with the second order formalism of supergravity. We have the Einstein-Hilbert Lagrangian which should be interpret as the kinetic term for gravity and weshould add a kinetic term for the spin 3/2 field which is known as the Rarita-SchwingerLagrangian. The action should be of form S = S2 +S3/2, hence the supergravity actionis defined as

SSG = 12κ2

∫dDx

(e eaµebνRµνab − e ψµγµνρDνψρ

)(8.1)

where e stands for the determinant of eaµ, the gravitino covariant derivative is given by

Dνψρ = ∂νψρ + 14ωνabγ

abψρ . (8.2)

The supersymmetry transformations laws of the vielbein and the gravitino are given by

δeaµ = 12εγaψµ (8.3)

δψµ = Dµε(x) . (8.4)

The gravitino is the gauge field of local supersymmetry, and one can interpret (8.3) asa bosonic vielbein that transforms into its superpartner. We have to check that thetransformations satisfy the supersymmetry algebra. The commutator of two supersym-metry transformations should give rise to a suitable combination of local supersymmetrytransformations. Thus the commutator of two infinitesimal supersymmetry transforma-tions yields space-time dependent vector field εγµε. Consider the commutator of twosupersymmetry transformations on the vielbein is

[δε1 , δε2 ]eaµ = 2δ[ε1δε2]eaµ

= δε2(ε1γaψµ)− δε1(ε2γ

aψµ)= ε1γ

aDµε2 − ε2γaDµε1

= Dµ(ε2γaε1) .

(8.5)

We define the space-time vector field ξµ = ε2γaε1, which yields us

[δε1 , δε2 ] = ξνDµeνa + eν

a∂µξν (8.6)

We have used that Dµ is the covariant derivative which acts as a space-time derivativeon ξν . For now we set our torsion tensor to zero thus we have a relation from the vielbeinpostulate that allows us to write

D[µeaν] = 0 . (8.7)

Considering this property and the equation (8.6), we can write the covariant derivativeas

[δε1 , δε2 ]eµa = ξν∂νeµa + eν

a∂µξν + ξνων

abeµ

b (8.8)

27

the first and second term are infinitesimal action of a diffeomorpishm on the frame fieldand the last term as a infinitesimal internal Lorentz transformation δΛeµ

a with Λab =ξνων

ab. We have shown that the commutator of two supersymmetry transformations

can be written as[δε1 , δε2 ]eµa = δξ + δΛ (8.9)

which is a combination of local symmetries: diffeomorphisms and Lorentz transforma-tions. So the space-time dependent vector field ξµ is a element of the group of localdiffeomorphism on space-time. We have to treat the space-time metric as a dynamicalobject because it is coordinate invariant.

Let us go back to our action we want to show that the action (8.1) is invariant undersupersymmetry transformations. We start with the gravitational part of the action

δS2 = 12κ2

∫dDx e eaµebνRµνab (8.10)

we will follow the same procedure as we did in section (7.1), where we take the curvaturetensor to depend on the spin connection ω, hence Rµνab(ω). Thus the variation of thegravitational action under (8.3) is

δS2 = 12κ2

∫dDx

((2e(δeaµ)ebν + (δe)eaµebν)Rµνab + eeaµebνδRµνab

)= 1

2κ2

∫dDx e

(Rµν −

12gµνR

)(−εγµψµ) .

(8.11)

Due to symmetry Rµνab = Rνµba gives the factor of 2 in the first line, the secondterm vanishes because of the variation on the action where we set the variations on theboundary to zero.

Let us now consider the gravitino variation. We want to partial integrate so we varyδψµ and multiply with 2. We then obtain

δS3/2 = − 12κ2

∫dDx e ε

←−Dµγ

µνρDνψρ

= 1κ2

∫ddx e εγµνρDµDνψρ = 1

8κ2

∫dDx e εγµνρRµνabγ

abψρ

(8.12)

we integrated by parts and used (7.8) to obtain the second line and we replaced the ∇µby Dµ because of antisymmetry and no absence of torsion. The last expression can beobtain by using the Ricci identity (4.43). We have to evaluate the γµνργab. We contractthe Riemann tensor with the gamma matrices and using the relation (3.9), we obtain

Rµνabγµνργab = γµνρabRµνab + 6Rµν [ρ

bγµν]b + 6γ[µRµν

ρν]

= γµνρabRµνab + 2Rµνρbγµνb + 4Rµνµbγνρb

+ 4γµRµνρν + 2γρRµννµ .(8.13)

The first and second term vanishes from the Bianchi identity (4.39) with a zero torsionand the third term vanishes because of symmetry clash between the symmetric Riccitensor Rνb = Rµν

µb and the anti symmetry of γνρb. We have now obtain the variation

on the gravitino field, given as

δS3/2 = 12κ2

∫dDx e

(Rµν −

12gµνR

)(εγµψν) (8.14)

We see that δS2 + δS3/2 = 0 and thus confirms local supersymmetry to linear order inψµ for any spacetime dimension.

28

8.2 Supergravity in first order formalism

In the previous section we only considered linear order terms but it becomes morecomplicated when we go beyond linear order, due to complication in establishing localsupersymmetry for supergravity theory. Let us now consider the spin connection ωµabas an independent variable. Considering the action (8.1), we will use the connectionvariation (7.30) of S2. We are left with the spin variation on the spin-3/2 field, weobtain the following variation

δS3/2 = − 18κ2

∫dDx e

(ψµγ

µνργabψρ)δων

ab . (8.15)

The rank 3 terms from (3.9) vanish because of antisymmetry in the gravitino indicesµ, ρ, therefore we obtain

ψµγµνργabψρ = ψµ

(γµνρab + 6γ[µeν [be

ρ]a])ψρ . (8.16)

Using the equation (7.30) we can solve S2 + S3/2 = 0, we obtain the following solutionto the torsion

Tabν = 1

2ψaγ

νψb + 14ψµγ

µνρabψρ . (8.17)

The first formalism determines a connection with torsion in supergravity. The fifth rankterm in the torsion vanishes in D = 4, and we obtain the same torsion as in section (7.2).We will see that writing the spin connection as ω = ω(e)+K will leave the supergravityaction invariant under the transformation rules (8.3) and (8.4).

We can substitute the torsion above in the action (8.1) to obtain the second order actionof supergravity.

S = 12κ2

∫d4x e

(R(e)− ψµγµνρDνψρ + LSG,torsion

)(8.18)

where

LSG,torsion = − 116

((ψργµψν)(ψργµψν + 2ψργνψµ)− 4(ψµγ · ψ)(ψγ · ψ)

)(8.19)

8.3 1.5 order formalism

The second order action (8.18) is invariant under supersymmetry transformations, withthe connection

ωµab = ωµab(e) +Kµab , (8.20)

Kµνρ = −14

(ψµγρψν − ψνγµψρ + ψργνψµ) (8.21)

that includes the gravitino torsion. For higher order terms using first and second or-der formalism becomes very complicated to solve, because of the variations on differentorders in the gravitino field. In the first order the spin connection is an independentvariable this approach becomes complicated when we couple matter multiplets to su-pergravity. The simplest approach to supergravity is between the first and second ordercalled the 1.5 formalism, we can apply this formalism in any dimensions.

We are working in the second order formalism since there are only two independentfields ψµ and eµa. Let us consider a functional of three variables S[e, ω, ψ], we use thechain rule to calculate its variation in the second order formalism

δS[e, ω(e) +K,ψ] =∫dDx

(δS

δeδe+ δS

δωδ(ω(e) +K) + δS

δψδψ

). (8.22)

29

We can neglect all the δω variations, because in 1.5 order formalism we don’t need tocalculate the algebraic field equation δS/δω = 0.

We can summarize this

• Use the first order form of the action S[e, ω, ψ] and the transformation rules δeand δψ with an unspecified connection

• Ignore the connection variation and calculate

δS =∫dDx

(δS

δeδe+ δS

δψδψ

)(8.23)

8.4 Local supersymmetry of N = 1, D = 4 supergravity

We will now use the 1.5 order formalism to show that supergravity in D = 4 and N = 1is invariant under the transformations rules (8.3) and (8.20) and (8.21). To simplify thegravitino action we introduce the highest rank Clifford element γ∗ = iγ0γ1γ2γ3. We canexpress the third rank Clifford matrices as

γabc = −iεabcdγ∗γd, γµνρ = −iεµνρσγ∗γσ . (8.24)

The first relation holds for local frames and the second is needed for the coordinatebasis. We can now rewrite the gravitino action using these relations and we obtain

S3/2 = i

2κ2

∫d4xεµνρσψµγ∗γσDνψρ . (8.25)

The advantage of this form is that the frame field variation is needed only in γσ insteadof eγµνρ. Since we are working in 1.5 order formalism we can ignore the variation δω ofRµνρσ, so we consider the following terms

δS = δS2 + δS3/2,e + δS3/2,ψ + δS3/2,ψ (8.26)

where the first term is the variation of the gravity and the second term is the variationof frame field in S3/2, meanwhile the third and forth term are variations of ψ and ψ.The variation of the gravity was obtain from (8.11)

δS2 = 12κ2

∫dDx e

(Rµν(ω)− 1

2gµνR

)(−εγµψµ) . (8.27)

and the second term in (8.26) is obtained by the variation of the frame field which isobtained in the same fashion as the second order formalism

δS3/2,e = i

4κ2

∫d4xεµνρσ(εγaψσ)(ψγ∗γaDνψρ) . (8.28)

The variation of the S3/2,ψ is given by

δS3/2,ψ = i

2κ2

∫d4xεµνρσψµγ∗γσDνDρε

= i

16κ2

∫d4xψµε

µνρσγ∗γσγabRνρab(ω)ε

(8.29)

the last line is obtained from the equation (4.43), and we were allowed to shift thederivative Dν due to partial integration. Let us now write down the variation of S3/2,ψ,where we will use the relation (3.23) with t3 = 1. We obtain

δS3/2,ψ = i

2κ2

∫d4xεµνρσψρ

←−Dνγ∗γσDµε (8.30)

30

we can now use the covariant derivative given in previous chapters

ψρ←−Dν = ∂νψρ −

14ψρωνabγ

ab (8.31)

if we then partial integrate we can use the same property as above which gives us theright to shift derivatives and we now obtain the following variation

δS3/2,ψ = −i2κ2

∫d4xεµνρσψργ∗ ((Dνγσ)Dµε+ γσDνDµε)

= −i2κ2

∫d4xεµνρσψργ∗

(12Tνσ

aγaDµε−18γσγ

abRµνab(ω)ε).

(8.32)

The torsion term comes from when we add the Christoffel symbols and use antisymmetryin νσ, this relation comes from (4.25). We see that the Riemann term R(ω) is equal inboth variations on the gravitino fields thus we can add them and we obtain the followingvariation

δS3/2,ψ + δS3/2,ψ = −i2κ2

∫d4xεµνρσψργ∗

(12Tνσ

aγaDµε−14γσγabRµν

ab(ω)ε)

(8.33)

Using the following γ-matrix manipulation

γσγab = γσab + 2eσ[aγb] = iedσεabcdγ∗γc + 2eσ[aγb] (8.34)

where we use the relation from (8.24), using this relation and plugging it back intothe action (8.33) and keeping in mind that ψµγcε = −εγcψµ. We consider the partsseparated, let us start with the first part and we see that we have a contraction of twoLevi-Civita symbols, using εµ1...µnν1...νpε

µ1...µnρ1...ρp = −p!n!δρ1...ρpν1...νp . The contraction of

two Levi-Civita with the Riemann tensor R(ω) becomes

εµνρσεabcdedσRνρ

ab = −2e(eµae

νbeρc + eµb e

νceρa + eµc e

νaeρb

)Rνρ

ab(ω)

= 4e(Rc

µ(ω)− 12eµcR(ω)

).

(8.35)

Using this and plugging it back in to the action (8.27), with the majorana spinor con-dition above we see that these terms cancel out. We see that we have obtained thesame result as in the second order formalism of supergravity. Which tells us that wehave a linear local supersymmetry. The second term in (8.34) to the integrand in (8.32)involves the factor

εµνρσRνρσb(ω) = −εµνρσDνTρσb (8.36)

where we have used the Bianchi identity (4.39), where the derivative Dν is a Lorentzcovariant derivative and contain only the spin connection acting on the index b. We areleft with

δS2 + δS3/2,ψ + δS3/2,ψ = −i4κ2

∫d4xεµνρσψµγ∗γa (TρσaDνε+ (DνTρσ

a)ε) (8.37)

We now have the variation of the frame field for the S3/2 action,

δS3/2,e = i

4κ2

∫d4xεµνρσ(εγaψσ)(ψγ∗γaDνψρ) . (8.38)

We need to reorder the spinors using Fierz rearrangement, using

(γµ)αβ(γµ)γδ = 12m

∑A

vA(ΓA)αβ(ΓA)γβ (8.39)

31

the expansion coefficient vA = (−)rA(D−2rA) where rA is the tensor rank of the Cliffordbasis element ΓA, thus for even dimensions D = 2m we have that 0 ≤ rA ≤ D. Applyingthis to the integrand above we obtain

(εγaψσ)(ψγ∗γaDνψρ) = −14∑A

(−)rA(4− 2rA)(εΓAγ∗Dνψρ)(ψµΓAψσ)

= 12

(εγaγ∗Dνψρ)(ψµγaψσ)

= (εγ∗γaDνψρ)Tµνa .

(8.40)

Due to antisymmetry in µσ of the action S3/2,e with the contraction of the Levi-Civitasymbols, allows us to write the last line using the D = 4 torsion tensor defined as (8.17).We insert this into (8.38) and reorder the (ε . . . Dψ) bilinear and exchange the indicesµρ to obtain

δS3/2,e = −i4κ2

∫d4xεµνρσTρσ

aψµ←−Dνγ∗γaε . (8.41)

We have now expressed all the variations in terms of the torsion tensor and adding themtogether we obtain

δS = −i4κ2

∫d4xεµνρσ

(Tρσ

a(ψµγ∗γaDνε+ ψµ←−Dνγ∗γbε) +DνTρσ

bψµγ∗γbε)

= −i4κ2

∫d4xεµνρσ∂ν

(Tρσ

bψµγ∗γbε)

= 0 .(8.42)

We have shown that the action is invariant up to a total derivative. We also see that thespin connection among the three terms in the second line cancel, thus we have showedthat the N = 1, D = 4 supergravity theory is locally supersymmetric.

32

9 Toroidal compactificationWe have seen Kaluza-Klein reductions on a circle, in this chapter we will consider thetoroidal compactification of the eleven-dimensional supergravity Lagrangian. We willreduce the 11D supergravity Lagrangian down to 4D. Performing dimensional reductionswill yield us e.g Type IIA and Type IIB theories which are low-energy theories ofsuperstring theory with the same name. The main references in this chapter will be [6],[10] and [11] .

9.1 D=11 supergravity

We will now consider supergravity in 11 dimensions, before we construct the Lagrangianand the transformation rules we must first investigate the field content for D = 11supergravity.

The field content of D = 11 theory, consists of elfbein eaµ, a Majorana spin 3/2 ψµand a antisymmetric gauge tensor with three indices Aµνρ, this was first formulated by[8]. To obtain the set of fields we simply calculate the number of states. The formulasfor counting the states are defined as

eaµ = d(d− 3)2

, ψµ = (d− 3)2(d/2)

2, Aµ1,...µp =

(d− 2p

). (9.1)

We see that we obtain 44 degrees of freedom for the vierbein and we have 84 degrees offreedom. This comes from the fact that we need equal amounts of boson and fermionstates.

We know that doing a dimensional reduction on T 7 generates the D = 4, N = 8supergravity theory, which contains one graviton gµν , seven vectors gµi and gij contains28 scalars. The Majorana field decomposes into 8 spin 3/2 fields, and 56 1/2 fields.The N = 8 theory contains 28 vectors and the missing 21 are supplied by the 3-formcomponent Aµij . The form components Aijk contain 35 scalars and the componentsAµνi give additional 7 scalars. This comes from the fact that antisymmetric tensorfields in four dimensions are equivalent to scalars or vector fields by a symmetry knownas duality. Thus we have 70 scalars, combining all the fields we end up with the samedegrees of freedom as D = 11 N = 1 supergravity theory.

We can now begin to construct the 11-dimensional Lagrangian and its transformationrules. We begin with the 3-form potential Aµνρ this field must be coordinate invariantand local Lorentz invariant, it must also transform under gauge transformation involvinga parameter θνρ. The theory involves a gauge invariant 4-form field strength Fµνρσ. Theequations are

δAµνρ = ∂µθνρ + ∂νθρµ + ∂ρθµν (9.2)Fµνρσ = ∂µAνρσ − ∂νAρσµ + ∂ρAσµν − ∂σAµνρ (9.3)

∂[τFµνρσ] = 0 (9.4)

the last equations follows from the Bianchi identity, this follows from the fact that lettinga covariant derivative act on F = dA gives us zero when written in differential form.

We will also use Majorana flip relation, we have the same properties as in fourdimensions

χγµ1...µrλ = trλγµ1...µrχ, t0 = t3 = 1, t1 = t2 = −1, tr+4 = tr . (9.5)

We know that the 11-dimensional supergravitry theory must have graviton and gravitinoterms like we had in the D = 4 theory, we also have to include the 3-form potential

33

Aµνρ plus several terms which we need to find by evaluating the gravitino field and thepotential. We begin with the action defined as

S = 12κ2

∫d11x e

(eaµebνRµνab − ψµγµνρDνψρ −

124FµνρσFµνρσ + . . .

)(9.6)

where . . . are the terms we need to find. Let us evaluate the gravitino and 3-form poten-tial now using the second order formalism where we have a torsion-free spin connectionωµab(e). The transformation rules are given by

δeaµ = 12εγaψµ (9.7)

δψµ = Dµε+ (aγαβγδµ + bγβγδδαµ)Fαβγδε (9.8)

δAµνρ = −13cε(γµνψρ + γνρψµ + γρµψν) . (9.9)

We have expressed the transformations rules that contain numerical constants a, b, c,which we have to determine. The action for a free theory where we don’t have anyinteractions, is given by

S0 = 12

∫d11x

(−ψµγµνρ∂νψρ −

124FµνρσFµνρσ

)(9.10)

Let us now take the variation of this action, which is needed to determine the numericalconstants. We vary the action w.r.t ψµ and Aµνρ

δS0 =∫d11x ε

((aγαβγδµ − bγβγδδαµ)Fαβγδγµνρ∂νψρ −

16cγνρψσ∂F

µνρσ)

=∫d11x ε

((−aγαβγδµ + bγβγδδαµ)∂νFαβγδγµνρψρ −

16cγνρψσ∂µFµνρσ

) (9.11)

where we have used

δψµ = ε(aγαβγδµ − bγβγδδαµ)Fαβγδγµνρ . (9.12)

The factor of 2 disappears because we vary the fields twice hence we can combine themand only vary one field, the variation of the field strength tensor was obtained by using(9.9). We now see that we need to reduce the products of the γ matrices, to sums overrank 6, rank 4 and rank 2 elements of ΓA of the Clifford algebra. We start with the firstterm in (9.11)

γαβγδµγµνρFαβγδ = (D − 6)γαβγδνρFαβγδ + 8(D − 5)γαβγ[νF ρ]

αβγ

− 12(D − 4)γαβFαβνρ(9.13)

we can obtain this by using γ-matrix manipulations. In the first term we see that µ canrun over all D values except αβγδ and νρ which gives us (D − 6) factor in front of thefirst term. We can use the same logic for the other terms. In the second term we seethat ν can run over all D values except the five indices. We can also choose four indicesfrom the first factor and two indices from the second term which gives us the factor 8.We also have one contraction from the δ-index thus combining all this we obtain thesecond term. The third term contains a contraction from the γ-index and the same logicapplied we obtain the last term.

Let us now take the second term

γβγδγµνρFµβγδ = −γνραβγδFαβγδ − 6γαβγ[νF ρ]αβγ + 6γαβFανρ . (9.14)

34

This relation holds directly from the definition of the Clifford algebra (3.1), we have seenthis relation in (3.8) and (3.9). We first write the totally antisymmetric Clifford matrixthat contains all the indices and then add possible pairings. We also use the same logicas above hence the constants in front of the second and third term.

We can now use these relations above in the variational action (9.11) to solve thenumerical constants. By doing so we see that the rank 6 elements disappear due to theBianchi identity (9.4) we are left with the rank 4 elements and the rank 2 elements ofthe Clifford algebra. We can summarize this as

8a(D − 5) + 6b = 0 (9.15)

12a(D − 4) + 6b = 16c (9.16)

we solve for D = 11 and we obtain that a = c216 and b = −8a. The transformation rules

are then given as

δψµ = ∂µε+ c

216

(γαβγδµ − 8γβγδδαµ

)Fαβγδε (9.17)

δAµνρ = −cεγ[µνψρ] . (9.18)

We now have to solve the c constant, we do so by examining the commutator of twoSUSY transformations and require that they agree with the local supergravity. Let uswrite down the SUSY transformation for the gauge potential, defined as

[δ1, δ2]Aµνρ = − 1216

c2ε2γ[µν(γαβγδρ] − 8γβγδδαρ]

)ε1Fαβγδ − (1↔ 2) . (9.19)

We know that each contraction in these terms contains a pair of indices so the contri-butions to the Clifford algebra will be of rank 1, 3, 5 , 7. We can do the same γ-matrixmanipulations as above, but we can simplify this by observing that the parameters ε1and ε2 are antisymmetric and we only obtain contributions from the rank 1 and rank5 terms. We also see that the first term does not have any rank 1 terms because wewould need to contract three pairs of indices to obtain a non-vanishing rank 1 term,and the antisymmetrization do not allows this. We can see this by doing the γ-matrixmanipulation where the contraction is between the antisymmetric indices hence why weare not allowed to do this. We can see that the rank 3 terms vanish when we considerthe antisymmetry of the parameters ε1 and ε2, using the relations in (3.11) and a veryuseful table in [2]

d( mod 8) S A ε η

0 0,3 2,1 -1 +10,1 2,3 -1 -1

1 0,1 2,3 -1 -12 1,0 3,2 -1 -1

1,2 3,0 +1 +13 1,2 0,3 +1 +1

Table 1: The S and A stands for symmetric and antisymmetri, for a detailed descriptionof the table we refer to [2]

note that ε = ±t0 and η = ±t0t1. We can see that when we do the calculation for rank3 elements they all cancel with each other.

The second term is already given in (3.9). We know from the definition that theSUSY algebra contain a spacetime translation which involves a rank 1 bilinear ε1γ

σε2

35

from the second term in (9.19). We then obtain the commutator

[δ1, δ2]Aµνρ = −49c2ε1γ

σε2Fσµνρ (9.20)

SUSY requires that the rank 5 elements vanish in the term (9.19) which they do.Let us interpret the SUSY commutation relation by looking at the field strength

defined in (9.3). The first term ∂σAµνρ is the space-time translation from the SUSYalgebra, the remaining terms just add up to the 3-form potential. The gauge field isproportional gauge transformation θµν = −ε1γ

σε2Aµνν . These gauge transformationscan be found when one considers the super Yang-Mills theory and also in local super-symmetry for supergravity theories.

We can now find the c constant, from above we see that c2 = 9/8. We choose thepositive root and obtain c = 3/2

√2.

We can find the supercurrent in (9.11) by considering the spinor parameter ε de-pending on xµ, taking the variation of the action and performing partial integration.We find that the variation contains a term that is proportional to Dνε whose coefficientsare the supercurrent

J ν =√

296

(γαβγδνρFαβγδ + 12γαβFαβνρ)ψρ . (9.21)

This is a new term to the 11-dimensional supergravity. We can now extend the resultsfrom the free system to a interacting system. We consider a general frame field eaµ(x)and consider the general ε(x), we then have the transformation rules

δeaµ = 12εγaψµ (9.22)

δψµ = Dµε+√

296

(γαβγδνρFαβγδ + 12γαβFαβνρ)ψρ (9.23)

δAµνρ = −3√

24εγ[µνψρ] (9.24)

and the action

S = 12κ2

∫d11x e

(eaµebνRµνab − ψµγµµρDνψρ −

124FµνρσFµνρσ

−√

296ψν(γαβγδνρFαβγδ + 12γαβFαβνρ)ψρ + . . .

).

(9.25)

We still need to figure out the rest of the Lagrangian. We have seen that the first andsecond term cancel in previous section and that εFαβγδψρ also cancel.

We are now left with the variation of the field strength εF 2ψ and the variation ofψFψ term. Writing the variational terms explicitly,

δLFF = 148

(4εγµψν − 1

2gµν εγ · ψ

)Fµ

ρστFνρστ

δLψFψ = 196× 144

ε(γα

′β′γ′δ′ν + 8γβ′γ′δ′

δα′

ν

)Fα′β′γ′δ′

× (γαβγδνρFαβγδ + 12γαβFαβν)ψρ .

(9.26)

We have now used the 1.5 order formalism and taking the linear parts in the δS, theterms of the form εψ in the variation vanish as it did in D = 4 supergravity. Theproducts of γ-matrices contain rank 9, 7, ,5 ,3, 1 terms. For a detailed description onhow to solve these γ-matrices we refer to [9]. The rank 1 terms in (9.26) cancel between

36

each other and several rank 3, 5 and 7 terms cancel withing the term ψFψ. We havethe rank 9 terms, which we can obtain from the γ-matrix products

γα′β′γ′δ′

νγαβγδνρ = (D − 9)γα′β′γ′δ′αβγδ + . . . = 2γα′β′γ′δ′αβγδρ + . . . (9.27)

γβ′γ′δ′

γαβγδα′ρ = −γα′β′γ′δ′αβγδρ + . . . (9.28)

where the dots are the lower ranked terms which we ignore. Combining these results weare left with

δLFF + δLψFψ = − 116× 144

εγα′β′γ′δ′αβγδρFα′β′γ′δ′Fαβγδ (9.29)

this is the term that survives from the variational terms. In order to make the Lagrangianinvariant the terms must cancel, hence we need Clifford algebra in odd spacetime di-mensions D = 2m+ 1. For D = 11 the generating matrices are 32× 32, using

γ±µ = (γ0, γ1, . . . , γ(2m−1), γ2m) = ±γ∗ (9.30)

we obtain that the rank 9 Clifford element is related to rank 2 by this equation. Thusthe Clifford algebra is given as

γα′β′γ′δ′αβγδρ = − 1

2eϵα

′β′γ′δ′αβγδρµνγνµ . (9.31)

Rewriting the equation (9.29) we get

e(δLFF + δLψFψ

)= 1

32× 144ϵα

′β′γ′δ′αβγδρµν εγνµψρFα′β′γ′δ′Fαβγδ

= 43√

2× 32× 144ϵα

′β′γ′δ′αβγδρµν(δAµνρ)Fα′β′γ′δ′Fαβγδ

(9.32)

We are left with the following variation, called the Chern-Simons term for the 3-formpotential

SC−S =∫A3 ∧ F4 ∧ F4 (9.33)

there is a U(1) gauge symmetry where the gravitino is charged and for which the 3-formpotential is the gauge field. We also note that F4 = dA3 is the U(1) field strength4-form. The Chern-Simons term is gauge invariant up to a total derivative. The actiontransform as ∫

A3 ∧ F4 ∧ F4 =∫A3 ∧ F4 ∧ F4 +

∫dΛ2 ∧ F4 ∧ F4

=∫A3 ∧ F4 ∧ F4 +

∫d(Λ2 ∧ F4 ∧ F4)

(9.34)

the last term vanishes due to the Bianchi identity dF4 = 0, and we have used the gaugetransformation A3 → A3 + dΛ2. This variation does not produce any further variationssince there are no frame fields in the expression. A typical property of the Chern-Simonsaction always guarantees gauge invariance.

We have now obtained all the transformations rules. We can thus write the 11-dimensional Lagrangian as

S = 12κ2

∫d11x e

(eaµebνRµνab − ψµγµµρDν

(ω + ω

2

)ψρ −

124FµνρσFµνρσ

−√

2192

ψν(γαβγδνρ + 12γαβgγνgδρ)ψρ(Fαβγδ + Fαβγδ)

− 2√

2(144)2 ϵ

α′β′γ′δ′αβγδρµνFα′β′γ′δ′FαβγδAµνρ

).

(9.35)

37

We replaced the spin-connection field by ω with (ω + ω)/2 in the covariant derivativeof the gravitino kinetic term and we also replaced the field-strength with (Fαβγδ +Fαβγδ)/2, these substitutions ensure that the field equations are supercovariant. Thehatted connection and field strength are given by

ωµab = ωµab(e) +Kµab , (9.36)

ωµab = ωµab(e)−14

(ψµγbψa − ψaγµψb + ψbγaψµ) , (9.37)

Kµab = −14

(ψµγbψa − ψaγµψb + ψbγaψµ) + 18ψνγ

νρµabψρ , (9.38)

Fµνρσ = 4∂[µAνρσ] + 32√

2ψ[µγνρψσ] . (9.39)

The action is invariant under the transformations defined in (9.22− 9.24)

9.2 11-Dimensional toroidal compactification

Before we start with the dimensional reduction on the eleven-dimensional supergravityLagrangian we first have to investigate the reductions of antisymmetric tensor fieldstrengths. We will consider the reduction from (D + 1) to D dimensions, we assumethat we have an n-index field strength in higher dimension which we denote Fn. Letus rewrite the field strength in terms of the gauge potential An−1 so that Fn = dAn−1.Thus after reduction we obtain two dimensional reduced potentials, namely (n−1) whichlies in the D-dimensional spacetime and the (n−2) which lies is in the S1 direction. Wecan express this as

An−1(x, z) = An−1(x) +An−2(x) ∧ dz (9.40)

the field strength can be expressed as

Fn = Fn + Fn−1 ∧ (dz +A1) (9.41)

where A1 is the metric reduction. Thus we obtain that the D-dimensional field strengthare given by

Fn = dAn−1 − dAn−2 ∧ A1, Fn−1 = dAn−2 (9.42)

we can then conclude that the field strengths after dimensional reduction will meandifferent things. For each reduction we do we obtain one field strength of same rank asthe one we reduce from and the other field strength has (n− 1) rank.

The m-th reduction for the gauge-potential can now be written as

An−1 = An−1 +A(n−2)a ∧ dza + 12!A(n−3)ab ∧ dza ∧ dzb

=m,n∑p=0

1p!A(n−1−p)a1...ap

∧ dza1 ∧ . . . ∧ dzap

(9.43)

and the field strength can be written as

Fn =m,n∑p=0

1p!F(n−p)a1...ap

∧ dza1 ∧ . . . ∧ dzap . (9.44)

We have now established the dimensional reduction of field strengths and gauge-potentials.Let us now begin with finding the D-dimension Lagrangian for the toroidal compact-

ifiation, we consider the (D + 1)-dimensional Lagrangian of the form where we followthe notations used in [11]

L = eR− 12e(∂ϕ)2 − 1

2n!eeaϕF 2

n (9.45)

38

doing a dimensional reduction on this Lagrangian gives us the following Lagrangian

L = eR− 12e(∂ϕ)2 − 1

4ee−2(D−1)αϕF2

− 12n!

ee−2(n−1)αϕ+aϕF 2n −

12(n− 1)!

ee2(D−n)αϕ+aϕF 2n−1

(9.46)

where we have the 2-form defined as F = dA, so for each time we reduce the Lagrangianwe obtain a new two-form field strength from the metric, and one dilaton scalar so therewill be (D − 11) dilaton scalars. The dilaton factor of the field strength in (9.45) takesthe general form of eaD+1·ϕD+1 where ϕD+1 = (ϕ1, ϕ2, . . . , ϕ10−D). We can see that thereis no dilaton factor in the eleven-dimensional supergravity Lagrangian so we set a11 = 0,we can obtain the lower dimensional vectors a by the following recursion formula

aD =(aD+1, x

√2

(D − 1)(D − 2)

)with x =

−(n− 1), for Fn → Fn(D − n), for Fn → Fn−1−(D − 1), for F

.

(9.47)So for each reduction we obtain n-form field strength from the n-form in (D+ 1) dimen-sion, an (n − 1)-form from the n-form field strength and the 2-form from the metric.We note that the 2-form appears in the D dimensional Lagrangian for the first timesince there is no dilaton vectors in the eleven-dimensional supergravity theory we setaD+1 = 0, from this formula we can obtain all the factors for the dilaton vectors thatappear when we do a dimensional reduction. One can easily calculate the prefactorsusing this formula.

Let us denote the dilaton vector for the 4-form as FMNPQ, the 3-forms as FMNPi andthe 2-forms as FMNij 1-forms as FMijk by a, ai, aij , aijk where i labels the internal (11-D) indices in D dimensions. We also obtain 2-forms FMN i and 1-forms F jMi with i < jwhich comes from the dimensional reduction of the vielbein. We denote the vielbeindilaton scalars with bi and bij , we summarize this with this table:

FMNPQ vielbein4-form: a = −g3-forms: ai = fi − g2-forms: aij = fi + fj − g bi = −fi1-forms: aijk = fi + fj + fk − g bij = −fi + fj

Table 2: Dilaton vector

where the vectors g and f have (11−D) components in D dimensions and are givenby

g = 3(s1, s2, . . . , s11−D) (9.48)

fi = (0, 0 . . . , 0︸ ︷︷ ︸i−1

, (10− 1)si, si+1, . . . , s11−D) (9.49)

where si =√

2/((10− i)(9− i)). We can now obtain the Lagrangian for the toroidalcompactification of the eleven-dimensional supergravity theory.

We write down the compact form of the elevel-dimensional Lagrangian, given as

L = eR− 148eF 2

4 + 16F4 ∧ F4 ∧ A (9.50)

the Chern-Simons term needs to be treated separately, which we will come back to. Wecan apply the rules above to the kinetic term where we consider a modification to the

39

equation (9.44), where we now include the dilaton vectors and order them as followingi < j < k, which follows directly from the Table 2. The field strength dimensionalreduction equation takes the general form

1n!F 2n =

n−1∑p=0

∑a1<...<ap

1(n− p)!

(F(n−p)a1...ap)2e(− n−1

3 g+fa1 +...+fap )ϕ (9.51)

since we have a 4-form in our Lagrangian the dimensional reduction of the kinetic termusing the equation above and use the dilaton scalar notations in the table

124F 2

4 = 124ea·ϕF 2

4 + 16∑i

eai·ϕ(F i3)2 + 12∑i<j

eaij ·ϕ(F ij2 )2 +∑i<j<k

eaijk·ϕ(F ijk1 )2 (9.52)

and the 2-forms coming from the vielbein reduction are

124F 2

4 = 12∑i

ebi·ϕ(F i2)2 +∑i<j

ebij ·ϕ(F ij1 )2 . (9.53)

The complete Lagrangian for the toroidal compactification of the eleven-dimensionalsupergravity theory is gives as

L = eR− 12e(∂ϕ)2 − 1

48eea·ϕF 2

4 −112∑i

eai·ϕ(F i3)2 − 14∑i<j

eaij ·ϕ(F ij2 )2

− 12∑i<j<k

eaijk·ϕ(F ijk1 )2 − 14∑i

ebi·ϕ(F i2)2 − 12∑i<j

ebij ·ϕ(F ij1 )2 + LFFA .(9.54)

We have now written out the toroidal compactified Lagrangian, let us now consider theChern-Simons term. We begin with writing out the 3-form gauge potential amd the4-form field strengths associated with the Chern-Simons term

F4 = F4 + F i3 ∧ dzi −12F ij2 ∧ dz

i ∧ dzj − 16F ijk1 ∧ dzi ∧ dzj ∧ dzk (9.55)

F4 = F4 + F l3 ∧ dzl −12F lm2 ∧ dzl ∧ dzm − 1

6F lmn1 ∧ dzl ∧ dzm ∧ dzn (9.56)

A3 = A3 +Ap2 ∧ dzp − 1

2Apq1 ∧ dz

p ∧ dzq − 16Apqr0 ∧ dzp ∧ dzq ∧ dzr . (9.57)

Note that we have F4∧F4∧A3, thus giving us 64 terms in total. The complete expressionis written out in the appendix. We also note that the amount of indices the terms havewill tell us what dimension the Lagrangian will have after the compactification on atorus. Before we go on let us define a important property of differential forms, namely

d(ωp ∧ ωq) = dωp ∧ ωq + (−1)pωp ∧ dωq (9.58)

When we do our calculations we set the right-hand side to zero this is due to havinga eleven-dimensional theory. When we let a differential operator act on a form weincrease its degree by one, which would result in a having p-forms of higher degree thanour theory where our Lagrangian is written out as an 11-form. From a more technicalpoint of view we see that letting a differential operator act on F4 ∧ F4 ∧ A3 will resultin having Bianchi identities dF4 = 0, hence why we set the right-hand side to zero.

Let us begin with the simplest case where we have terms with 1 index, we write outthe Lagrangian as

Ln=1FFA = 1

6

∫ (F4 ∧ F4 ∧Ai2 ∧ dzi + F4 ∧ F i3 ∧ dzi ∧A3 + F i3 ∧ dzi ∧ F4 ∧A3

). (9.59)

40

The next step is to partial integrate the terms. Let us consider the second term wherewe now perform a partial integration and use the differential form property above∫ (

F4 ∧ F i3 ∧ dzi ∧A3)

=∫ (

dAi2 ∧ F4 ∧A3)∧ dzi

=∫ (

Ai2 ∧ d(F4 ∧A3))∧ dzi

=∫ (

Ai2 ∧ (dF4 ∧A3 + F4 ∧ dA3))∧ dzi

=∫ (

Ai2 ∧ F4 ∧ F4)∧ dzi

=∫ (

F4 ∧ F4 ∧A2)εi

(9.60)

where we have used the differential form property in the first line, in the third line wesee that the first term we obtain is zero due to the Bianchi identity. We are now leftwith something that has form as the first term. Applying the same procedure to the lastterm should give us the same terms as above, one could easily see from the similaritiesof the terms 6. We can now gather all the terms and obtain the following Lagrangian

Ln=1FFA = 1

6

∫ (F4 ∧ F4 ∧A2 + F4 ∧ F4 ∧A2 + F4 ∧ F4 ∧A2

)εi

= 16

∫3(F4 ∧ F4 ∧A2)εi = 1

2

∫(F4 ∧ F4 ∧A2)εi .

(9.61)

We have now found the Chern-Simons term for D=10 Lagrangian. Note that we nowhave a Lagrangian of 10-form, we can thus conclude that each time we do a dimensionalreduction we integrate over the coordinate in the toroidal direction which results inhaving one dimension less. We refer to the appendix for the rest of the dimensional re-ductions. Let us write down all the Lagrangian one obtains after dimensional reduction:

D = 10 : 12F4 ∧ F4 ∧A2 ,

D = 9 :(1

4F4 ∧ F4 ∧Aij1 −

12F i3 ∧ F

j3 ∧A3

)εij ,

D = 8 :( 1

12F4 ∧ F4 ∧Aijk0 − 1

6F i3 ∧ F

j3 ∧A

k2 + 1

2F i3 ∧ F

jk2 ∧A3

)εijk ,

D = 7 :(1

6F4 ∧ F i3 ∧A

jkl0 −

14F i3 ∧ F

j3 ∧A

kl2 + 1

8F ij2 ∧ F

kl2 ∧A3

)εijkl ,

D = 6 :( 1

12F4 ∧ F ij2 ∧A

klm − 112F i3 ∧ F

j3 ∧A

klm0 + 1

8F ij2 ∧ F

kl2 ∧Am2

)εijklm ,

D = 5 :( 1

12F i3 ∧ F

jk2 ∧A

lmn0 + 1

48F ij2 ∧ F

kl2 ∧Amn1 − 1

72F ijk1 ∧ F lmn1 ∧A3

)εijklmn ,

D = 4 :( 1

48F ij2 ∧ F

kl2 ∧A

mnp0 − 1

72F ijk1 ∧ F lmn1 ∧Ap2

)εijklmnp ,

D = 3 : − 1144

F ijk1 ∧ F lmn1 ∧Apq1 εijklmnpq ,

D = 2 : − 11296

F ijk1 ∧ F lmn1 ∧Apqr0 εijklmnpqr .

These are all the reduced Lagrangian we can obtain from the D = 11 Chern-Simonsterm.

6One could choose to partial integrate the first term and obtain something similar to F4 ∧ F i3 ∧ A3

but we chose to partial integrate the other two terms to obtain the maximally ranked field-strengths inour Lagrangian.

41

These reductions are from the unmodified field strengths, i.e the field strength re-duction (9.55 − 9.57) are not expressed in the vielbein basis. However if one were toconsider dualisations as [10] one would need modified field strengths which are expressedin vielbein basis. Let us re express the toroidal coordinates dzi in terms of

hi = dzi +Ai1 +Aij0 dzj . (9.62)

We obtain this from the dimensional reduction of the metric ansatz

ds211 = e

13 g·ϕds2

D +∑i

e2γi·ϕ(hi)2 (9.63)

where γi = 16 g −

12 fi. We have considered the field strengths in the Chern-Simons

term to be associated with the gauge potentials e.g F4 = dA3 and so one, in general onewould also obtain non-linear Kaluza-Klein modifications. If we express the field strengthreductions in terms of (9.62), we obtain the following modified field strengths

F4 = F − γijF i3 ∧ Aj1 −

12γikγjlF ij2 ∧ A

k1 ∧ Al1 + 1

6γilγjmγknF ijk1 ∧ Al1 ∧ Am1 ∧ An1

F i3 = γijF j3 − γijγklF jk2 ∧ A

l1 −

12γijγkmγlnF jkl1 ∧ Am1 ∧ An1

F ij2 = γkiγljF kl2 − γkiγljγmnF klm1 ∧ An1F ijk1 = γliγmjγnkF klm1

(9.64)

these modified field strengths appear in the kinetic term of the Lagrangian (9.54). Themodified field strengths that come from the vielbein are given by

F i2 = F i2 − γjkFij1 ∧ A

k1

F ij1 = γkjF ik(9.65)

Notice that we do not have any 0-form potentials, this is because Aij0 are defined onlyfor i < j, in other words Aij0 = 0 for i ≥ j. This can be seen when we to dimensionalreduction from the Kaluza-Klein vielbein ansatz by taking the i’th step of the reductionprocess. We obtain the following relation

dzi = γil(hl −Al1) (9.66)

one then sees that the 0-form potentials are eliminated7, notice that γij = 0 when i > jand γij = 1 for i = j.

If one were to consider brane’s we would need to have these modified Chern-Simonsterms in the kinetic term.

For example in [11] the authors attempt to find p-brane solutions in maximal su-pergravity, where they find the equations of motion of the supergravity Lagrangian andimpose additional constraints on the Chern-Simons modifications of the field strengths.These constraints gives rise to which field strengths can be dualised in certain dimen-sions.

7A detailed mathematical description can be found in [11]

42

10 ConclusionIn this thesis we have seen how Supergravity theories are invariant under the local SUSYtransformations where the spinor parameters ε(x) are arbitrary functions of spacetimecoordinates. Showing that the Lagrangians are invariant tells us that Supergravity isconsistent as a classical theory of graviton and gravitino.

In section 5 we saw how one obtained a theory containing gravity, electromagneticfields and scalar fields from a (D + 1)-dimensional Einstein-Hilbert action by usingKaluza-Klein theory. This method turns out to be very important when consideringSupergravity theories. When we performed a dimensional reductions of the eleven-dimensional Supergravity theory on a 7-torus, we obtained the degrees of freedom ofD = 4, N = 8 theory. The theory contains one graviton gµν , seven vectors gµi and28 scalars gij . The Majorana field is decomposed into eight gravitinos and 56 spin-1/2fields this is obtained from calculating the states for the Rarita-Schwinger field. Weknow from Supersymmetry that a theory should contain equal amount of bosonic andfermionic states on-shell degrees of freedom, so we are missing bosons to account forthe fermions. These bosons are obtained from the 3-form gauge potential Aµij whichsupplies 21 scalars. The form components Aijk contain 35 scalars and the componentsAµνi give additional seven scalars. This completes the field content of the D = 4, N = 8theory.

We also saw that we obtained ten-dimensional Supergravity theories, Type IIA andType IIB which are low-energy limits of superstring theories of the same name.

Type IIB and and SO(6)-gauged D = 5 supergravities have important applications inAdS/CFT-correspondence. The modified field strengths appear in the dimensional re-duction of the Chern-Simons term in the eleven-dimensional supergravity. The modifiedfield strengths are needed to guarantee gauge invariance in the kinetic term. When oneconsiders external sources such as brane worlds one needs the modified field strengths.With the modified field strengths one can consider dualisations of higher dimensionaltheories to lower dimensional theories. The eleven-dimensional Supergravity theory com-bined with M2 and M5-brane solutions is the basis for M-theory, which is the theorythat contains all the various string theories.

AcknowledgmentsI would like to thank Giuseppe for his guidance and the useful discussions during ourmeetings. I would also like to thank Johan Asplund and Daniel Neiss for taking theirtime to go through some of the calculations.

References[1] F.Quevedo, S.Krippendorf, O.Schlotter Cambrigde lectures on supersym-

metry and extra dimensions arXiv:1011.1491

[2] A. Van Proeyen Tools for supersymmetry arXiv:hep-th/9910030v6

[3] P.Binétruy Supersymmetry Oxford Graduate texts

[4] D.Z.Freedman, A. Van Proeyen Supergravity Cambridge university press(2012)

[5] M.Perry Applications of differential geometry to physics (2009)

43

[6] C.N Pope Kaluza-Klein theory people.physics.tamu.edu/pope/

[7] H. Samtleben Introduction to Supergravity http://www.itp.uni-hannover.de/saalburg/Lectures/samtleben.pdf

[8] E. Cremmer, B. Julia, J. Scherk Supergravity theory in 11 dimensionsPhys. Lett. B76 1978

[9] S. Naito, K. Osada, T. Fukui Fierz identities and invariance of 11-dimensional supergravity action Phys. Rev D 1986

[10] E. Cremmer, B.Julia, H.Lü, C.N Pope Dualisation of Dualities arXiv:hep-th/9710119v2

[11] H.Lü, C.N Pope p-brane Solitons in Maximal Supergravities arXiv:hep-th/951201v1

44

A Riemann tensor for Einstein-Hilbert action in D dimensionsThe derivatives of the ON-Basis are defined as

dEi = βe−βϕ∂jEj ∧ Ei − ωij ∧ Ej (A.1)

dE5 = αe−βϕ∂jϕEj ∧ E5 + 1

2e(α−2β)ϕFijE

i ∧ Ej . (A.2)

The spin connection can be obtain by using Cartan’s first structure equation and weobtain

ω5i = α∂iϕe

−βϕE5 + 12e(α−2β)ϕFijE

j (A.3)

ωij = ωij − βe−βϕ(∂iϕEj − ∂jϕEi)−12e(α−2β)ϕF ijE

5 (A.4)

Cartan’s second structure equation is defined as

Rµσ = dωµσ + ωµν ∧ ωνσ (A.5)

Let us now find the Riemann tensor R5i, we start with the first component dωµσ

dω5i = d(α∂iϕe−2βϕE5 + 1

2e(α−2β)ϕFijE

j)

= α∂j∂iϕe−2βϕEj ∧ E5 − αβ∂iϕe−2βϕdϕE5 + α∂iϕe

−βϕdE5

+ 12

(dFije(α−2β)ϕEj + (α− 2β)Fijdϕe(α−2β)ϕEj + e(α−2β)ϕFijdEj

= α∂j∂iϕe−2βϕEj ∧ E5 − αβ∂iϕe−2βϕ∂jϕE

j ∧ E5

+ α∂iϕe−βϕ(αe−βϕ∂jϕE

j ∧ E5 + 12e(α−2β)ϕFijE

i ∧ Ej)

+ 12∂kϕFije

(α−3β)ϕEk ∧ Ej + 12

(α− 2β)Fij∂kϕe(α−3β)ϕEk ∧ Ej

+ 12e(α−2β)ϕFij(βe−βϕ∂kE

k ∧ Ej − ωjk ∧ Ek)

= α∂j∂iϕe−2βϕEj ∧ E5 − αβ∂iϕe−2βϕ∂jϕE

j ∧ E5

+ α2∂iϕ∂jϕe−2βϕEj ∧ E5 + 1

2α∂iϕFije

(α−3β)ϕEi ∧ Ej

+ 12∂kFije

(α−3β)ϕEk ∧ Ej + 12

(α− 2β)Fij∂kϕe(α−3β)ϕEk ∧ Ej

+ 12β∂kϕFije

(α−3β)ϕEk ∧ Ej − e(α−2β)ϕFijωjk ∧ Ek

Now we consider the second part of the equation above and we obtain

ω5j ∧ ωj i = α∂jϕe

−βϕE5 ∧ ωj i − αβ∂jϕ∂jϕe−2βϕE5 ∧ Ej

+ αβ∂jϕ∂iϕe−2βϕE5 ∧ Ej + 1

2e(α−2β)ϕFkjE

k ∧ ωj i

+ 12βe(α−3β)ϕ∂kϕFijE

k ∧ Ej − 14e2(α−2β)ϕF kjF

kiE

j ∧ E5

+ 12βηi[jFk]l∂

lϕe(α−3β)ϕEk ∧ Ej

45

combining all the equations now we obtain the following Riemann tensor

Rzi = α∂j∂iϕe−2βϕEj ∧ E5 − αβ∂iϕe−βϕ∂jϕE

j ∧ E5 + α2∂iϕ∂jϕe−2βϕEj ∧ E5

+ 12α∂iϕFije

(α−3β)ϕEi ∧ Ej + 12βηi[jFk]l∂

lϕe(α−3β)ϕEk ∧ Ej

+ 12∂kFije

(α−3β)ϕEk ∧ Ej + 12

(α− 2β)Fij∂kϕe(α−3β)ϕEk ∧ Ej

+ 12β∂kϕFije

(α−3β)ϕEk ∧ Ej − e(α−2β)ϕFijωjk ∧ Ek

+ α∂jϕe−βϕE5 ∧ ωj i − αβ∂jϕ∂jϕe−2βϕE5 ∧ Ei

− αβ∂jϕ∂iϕe−2βϕE5 ∧ Ej + 12e(α−2β)ϕFkjE

k ∧ ωj i

+ 12βe(α−3β)ϕ∂kϕFijE

k ∧ Ej − 14e2(α−2β)ϕF j iF

jiE

j ∧ E5

simplify this by gather all the wedge products under same brackets we obtain

R5i = e−2βϕ

(α(α− 2β)∂iϕ∂jϕ+ α∂j∂iϕ+ ηijαβ∂kϕ∂

kϕ− 14e2(α−2β)ϕFkjF

ki

)Ej ∧ E5

+ e(α−3β)ϕ(1

2(α− β)∂iϕFkj −

12

(α− β)∂[kϕFj]i −12∂[kFj]i + 1

2βηi[jFk]l∂

lϕ

)Ek ∧ Ej

− 12e(α−3β)ϕ

(Fijω

jk ∧ Ek + Fjkω

ji ∧ Ek

)+ α∂jϕe

−βϕE5 ∧ ωj i

We have another curvature tensor to calculate namely Rij we follow the same procedureas above

Rij = dωij + ωiν ∧ ωνjWe start with the first component which yields us

dωij =βe−2βϕ∂i∂kϕEk ∧ Ej − βe−2βϕ∂kj∂kϕE

k ∧ Ei + β2e−2βϕ∂i∂kϕEk ∧ Ej

− βe−βϕ∂iϕωjk ∧ Ek − β2e−2βϕ∂jϕ∂jϕEj ∧ Ei + βe−βϕ∂jϕω

ij ∧ Ej

− β2e−2βϕ∂iϕ∂kϕEk ∧ Ej + β2e−2βϕ∂jϕ∂kϕE

k ∧ Ei + 12∂kϕF

ije

(α−2β)ϕEk ∧ E5

− 12

(α− 2β)∂kF ije(α−3β)ϕEk ∧ E5 + 12e(α−3β)ϕα∂jϕF

ijE

j ∧ E5 − 14F kjFkje

2(α−2β)ϕEi ∧ Ej

the second termωiν ∧ ωνj = ωi5 ∧ ω5

j + ωik ∧ ωkjWhich yields us the following expression without going through the entire calculation

ωiν ∧ ωνj =12α∂iϕFjke

(α−3β)ϕE5 ∧ Ek + 12Fijα∂jϕe

(α−3β)ϕEj ∧ E5

+ 14FijFjke

2(α−2β)ϕEj ∧ Ek + ωik ∧ ωkj − β∂kϕe−βϕωik ∧ Ej

+ β∂jϕe−βϕωik ∧ Ek + β2∂iϕ∂kϕe−2βϕEk ∧ Ej − β2∂iϕ∂jϕe

−2βϕEk ∧ Ek

− β2∂kϕ∂kϕe−2βϕEi ∧ Ej + β2∂kϕ∂jϕe

−2βϕEi ∧ Ek

+ 12β∂iϕF kje

(α−3β)ϕEk ∧ E5 − 12β∂kϕF

kje

(α−3β)ϕEi ∧ E5

− 12F ike

(α−2β)ϕE5 ∧ ωkj + 12β∂kϕF ike

(α−3β)ϕE5 ∧ Ej −12β∂jϕF

ikE

5 ∧ Ek

46

Gathering all the terms we obtain the following Riemann tensor

Rij =rij + Ez ∧ Eke(α−3β)ϕ(

(α− β)F ij∂kϕ+ 12∂kF

ij −

12

(α− β)(∂iϕFjk − F ik∂jϕ)

+ 12β(∂nϕF

njδik + F il∂

lϕηjk))

+ Ek ∧ Ele−2βϕ(β(∂j∂[kϕδ

il] − ∂[k∂

iϕηl]j)

− 14e2(α−2β)ϕ

(F ijFkl + F i[k|Fj|l]

)+ β2

(∂iϕ∂[kϕηl]j − ∂pϕ∂pϕδi[kηl]j − ∂jϕ∂[kϕδ

il])

+ βe−βϕ(∂kϕE

i ∧ ωkj − ∂kϕωik ∧ Ej)− 1

2βe(α−2β)ϕ +

(F kjω

ik ∧ E5 − F ikωkj ∧ E5

)B Equations of motion for Einstein-Hilbert action in D dimen-

sionsThe derivation for the equations of motion for Einstein-Hilbert action in D dimensionswhere the action is given as

SEH = κ−1∫dDx√−g

(R− 1

2(∇ϕ)2 − 1

4e−2(D−1)αϕF 2

)(B.1)

we vary this action with respect to the metric gµν

δSEH = κ−1∫dDxδ

(√−gR− 1

2√−g(∇ϕ)2 − 1

4√−ge−2(D−1)αϕF 2

)= κ−1

∫dDx

(√−gδRµνgµν +

√−gRµνδgµν +Rδ

√−g

− 12√−g(∂µδϕ∂νϕ+ ∂µϕ∂

νδϕ)− 12√−g(∇ϕ)2δgµν

− 12

(∂ϕ)2δ√−g − 1

4√−ggµνe−2(D−1)αϕ(δFµνFµν + FµνδF

µν)

− 14√−gF 2

µνe−2(D−1)αϕδgµν + 1

4F 2e−2(D−1)αϕδ

√−g

+ 12√−g(D − 1)αδϕe−2(D−1)αϕF 2

)we set ∂µδϕ∂νϕ = ∂µϕ∂

νδϕ and δFµνFµν = FµνδF

µν and all the boundary terms areset to zero. Using the relation

δ√−g = −1

2√−ggµνδgµν (B.2)

and integrating by parts we obtain

47

δSEH = κ−1∫dDx

(√−gRµνδgµν −

12√−gRgµνδgµν −

√−g∂µδϕ∂νϕ

− 12√−g∂µϕ∂νϕδgµν + 1

4√−g(∂ϕ)2gµνδg

µν

− 12√−ggµνFµνδ∇µAνe−2(D−1)αϕ − 1

4√−gF 2

µνe−2(D−1)αϕδgµν

− 18√−ggµνF 2e−2(D−1)αϕδgµν + 1

2√−g(D − 1)αδϕe−2(D−1)αϕF 2

)= κ−1

∫dDx√−g(δgµν

(Rµν −

12Rgµν −

12

(∂µϕ∂νϕ+ 12

(∂ϕ)2gµν

− 14e−2(D−1)αϕ(F 2

µν −14F 2µνgµν)

)− δAν∇µ

(Fµνe

−2(D−1)αϕ)

− δϕ(∂µ∂νϕ− 1

2(D − 1)αe−2(D−1)αϕF 2

).

The equations of motion can be read of as

Rµν −12Rgµν = 1

2

(∂µϕ∂νϕ−

12

(∂ϕ)2gµν

)+ 1

2e−2(D−1)αϕ

(F 2µν −

14F 2gµν

)∇µ

(e−2(D−1)αϕFµν

)= 0

2ϕ = −12

(D − 1)αe−2(D−1)αϕF 2

(B.3)

C Covariant derivative of γν

∇µγν = ∂µγν + 14ωµ

ab[γab, γν ]− Γρµνγρ

= ∂µγν + 14ωµ

ab[γ[aγb], γν ]− Γρµνγρ

= ∂µγν + 18ωµ

ab([γaγb, γν ]− (a↔ b))− Γρµνγρ

= ∂µγν + 18ωµ

ab(γaγbγν − γνγaγb)− (a↔ b))− Γρµνγρ

= ∂µγν + 18ωµ

ab(γaγbγν − γaγνγb − 2ηaνγb − (a↔ b))− Γρµνγρ

= ∂µγν + 18ωµ

ab(γaγbγν − γaγbγν − 2ηaνγb + 2ηbνγa − (a↔ b))− Γρµνγρ

= ∂µγν + 18ωµ

ab(2γaηbν − 2γbηaν − (a↔ b))− Γρµνγρ

= ∂µγν + ωµabγ[aηb]ν − Γρµνγρ

= γa(∂µeaν + ωµabebν − Γρµνeaρ) = 0

(C.1)

D Variation of the spin connectionWe are interested in finding the variation of the spin connection which will be usefulwhen doing calculations in the second order formalism of gravity and fermions. We startby varying the first Cartan structure equation

dδea + ωab ∧ δeb + δωab ∧ eb = 0

48

Let us multiply with ηacecρ and do a cyclic permutation of µ, ν and ρ with the sign

+−+

ηac(D[µδ

aν]e

cρ + δω[µ

abeν]ecρ

)−ηac

(D[νδ

aρ]e

cµ + δω[ν

abeρ]ecµ

)ηac

(D[ρδ

aµ]e

cν + δω[ρ

abeµ]ecν

)= 0

writing out the explicit terms of the spin connection we obtain

ηac

(D[µδ

aν]e

cρ −

12δωµ

abeνbecρ + 1

2δωµ

abeµbecρ

)−ηac

(D[νδ

aρ]e

cµ −

12δων

abeρbecµ + 1

2δωρ

abeνbecµ

)ηac

(D[ρδ

aµ]e

cν −

12δωρ

abeµbecν + 1

2δωµ

abeρbecν

)= 0

we see that certain terms vanish and we have contracted the metric so all the c arereplaced with a and we obtain

ebρeaνδωµ

ab = D[µδaν]e

aρ −D[νδ

aρ]e

aµ +D[ρδ

aµ]e

aν (D.1)

E Chern-Simons dimensional reduction

F4 ∧ F4 ∧ A3 =(F4 + F i3 ∧ dzi −

12F ij2 ∧ dz

i ∧ dzj − 16F ijk1 ∧ dzi ∧ dzj ∧ dzk

)∧(F4 + F l3 ∧ dzl −

12F lm2 ∧ dzl ∧ dzm − 1

6F lmn1 ∧ dzl ∧ dzm ∧ dzn

)∧(A3 +Ai2 ∧ dzi −

12Aij1 ∧ dz

i ∧ dzj − 16Aijk0 ∧ dzi ∧ dzj ∧ dzk

)(E.1)

In order to obtain the dimensional reductions from eleven-dimensional supergravity tofour-dimensional supergravity, we need to expand the expression above and we obtaina very messy expression. In order to avoid having a expression that is to long we willbreak it into 4 parts, for the simplicity.

F4 ∧ F4 ∧ A3 = F4 ∧ F4 ∧A3 + F4 ∧ F4 ∧Ai2 ∧ dzi −12F4 ∧ F4 ∧Aij1 ∧ dz

i ∧ dzj

− 16F4 ∧ F4 ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk + F4 ∧ F l3 ∧ dzl ∧A3

+ F4 ∧ F l3 ∧ dzl ∧Ai2 ∧ dzi −12F4 ∧ F l3 ∧ dzl ∧A

ij1 ∧ dz

i ∧ dzj

− 16F4 ∧ F l3 ∧ dzl ∧A

ijk0 ∧ dzi ∧ dzj ∧ dzk − 1

2F4 ∧ F lm2 ∧ dzl ∧ dzm ∧A3

− 12F4 ∧ F lm2 ∧ dzl ∧ dzm ∧Ai2 ∧ dzi + 1

4F4 ∧ F lm2 ∧ dzl ∧ dzm ∧Aij1 ∧ dz

i ∧ dzj

+ 112F4 ∧ F lm2 ∧ dzl ∧ dzm ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk

− 16F4 ∧ F lmn1 ∧ dzl ∧ dzm ∧ dzn ∧A3

− 16F4 ∧ F lmn1 ∧ dzl ∧ dzm ∧ dzn ∧Ai2 ∧ dzi

+ 112F4 ∧ F lmn1 ∧ dzl ∧ dzm ∧ dzn ∧Aij1 ∧ dz

i ∧ dzj

+ 136F4 ∧ F lmn1 ∧ dzl ∧ dzm ∧ dzn ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk

49

this is the expression for the first extensions that contains 16 terms, we see that weshould in total have 64 terms in total. The next extensions are expressed as

F4 ∧ F4 ∧A3 = F i3 ∧ dzi ∧ F4 ∧A3 + F i3 ∧ dzi ∧ F4 ∧Ai2 ∧ dzi −12F i3 ∧ dzi ∧ F4 ∧Aij1 ∧ dz

i ∧ dzj

− 16F i3 ∧ dzi ∧ F4 ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk + F i3 ∧ dzi ∧ F l3 ∧ dzl ∧A3

+ F i3 ∧ dzi ∧ F l3 ∧ dzl ∧Ai2 ∧ dzi −12F i3 ∧ dzi ∧ F l3 ∧ dzl ∧A

ij1 ∧ dz

i ∧ dzj

− 16F i3 ∧ dzi ∧ F l3 ∧ dzl ∧A

ijk0 ∧ dzi ∧ dzj ∧ dzk − 1

2F i3 ∧ dzi ∧ F lm2 ∧ dzl ∧ dzm ∧A3

− 12F i3 ∧ dzi ∧ F lm2 ∧ dzl ∧ dzm ∧Ai2 ∧ dzi + 1

4F i3 ∧ dzi ∧ F lm2 ∧ dzl ∧ dzm ∧Aij1 ∧ dz

i ∧ dzj

+ 112F i3 ∧ dzi ∧ F lm2 ∧ dzl ∧ dzm ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk

− 16F i3 ∧ dzi ∧ F lmn1 ∧ dzl ∧ dzm ∧ dzn ∧A3 −

16F i3 ∧ dzi ∧ F lmn1 ∧ dzl ∧ dzm ∧ dzn ∧Ai2 ∧ dzi

+ 112F i3 ∧ dzi ∧ F lmn1 ∧ dzl ∧ dzm ∧ dzn ∧Aij1 ∧ dz

i ∧ dzj

+ 136F i3 ∧ dzi ∧ F lmn1 ∧ dzl ∧ dzm ∧ dzn ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk

F4 ∧ F4 ∧A3 = 12F ij2 ∧ dz

i ∧ dzj ∧ F4 ∧A3 −12F ij2 ∧ dz

i ∧ dzj ∧ F4 ∧Ai2 ∧ dzi

+ 14F ij2 ∧ dz

i ∧ dzj ∧ F4 ∧Aij1 ∧ dzi ∧ dzj

+ 112F ij2 ∧ dz

i ∧ dzj ∧ F4 ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk − 12F ij2 ∧ dz

i ∧ dzj ∧ F l3 ∧ dzl ∧A3

− 12F ij2 ∧ dz

i ∧ dzj ∧ F l3 ∧ dzl ∧Ai2 ∧ dzi + 14F ij2 ∧ dz

i ∧ dzj ∧ F l3 ∧ dzl ∧Aij1 ∧ dz

i ∧ dzj

+ 112F ij2 ∧ dz

i ∧ dzj ∧ F l3 ∧ dzl ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk

+ 14F ij2 ∧ dz

i ∧ dzj ∧ F lm3 ∧ dzl ∧ dzm ∧A3 + 14F ij2 ∧ dz

i ∧ dzj ∧ F lm3 ∧ dzl ∧ dzm ∧Ai2 ∧ dzi

− 18F ij2 ∧ dz

i ∧ dzj ∧ F lm3 ∧ dzl ∧ dzm ∧Aij1 ∧ dzi ∧ dzj

− 124F ij2 ∧ dz

i ∧ dzj ∧ F lm3 ∧ dzl ∧ dzm ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk

− 112F ij1 ∧ dz

i ∧ dzj ∧ F lmn ∧ dzl ∧ dzm ∧ dzn ∧A3

− 112F ij1 ∧ dz

i ∧ dzj ∧ F lmn ∧ dzl ∧ dzm ∧ dzn ∧Ai2 ∧ dzi

− 124F ij1 ∧ dz

i ∧ dzj ∧ F lmn ∧ dzl ∧ dzm ∧ dzn ∧Aij1 ∧ dzi ∧ dzj

− 172F ij1 ∧ dz

i ∧ dzj ∧ F lmn ∧ dzl ∧ dzm ∧ dzn ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk

50

F4 ∧ F4 ∧A3 = −16F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F4 ∧A3 −

16F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F4 ∧Ai2 ∧ dzi

+ 112F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F4 ∧Aij1 ∧ dz

i ∧ dzj

+ 136F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F4 ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk

− 16F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F l3 ∧ dzl ∧A3 −

16F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F l3 ∧ dzl ∧Ai2 ∧ dzi

+ 112F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F l3 ∧ dzl ∧A

ij1 ∧ dz

i ∧ dzj

+ 136F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F l3 ∧ dzl ∧A

ijk0 ∧ dzi ∧ dzj ∧ dzk

+ 112F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F lm3 ∧ dzl ∧ dzm ∧A3

+ 112F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F lm3 ∧ dzl ∧ dzm ∧Ai2 ∧ dzi

− 124F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F lm3 ∧ dzl ∧ dzm ∧Aij1 ∧ dz

i ∧ dzj

− 172F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F lm3 ∧ dzl ∧ dzm ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk

+ 136F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F lmn3 ∧ dzl ∧ dzm ∧ dzn ∧A3

+ 136F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F lmn3 ∧ dzl ∧ dzm ∧ dzn ∧Ai2 ∧ dzi

− 172F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F lmn3 ∧ dzl ∧ dzm ∧ dzn ∧Aij1 ∧ dz

i ∧ dzj

− 1216

F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F lmn3 ∧ dzl ∧ dzm ∧ dzn ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk .

Continue with terms with 2 indices

Ln=2FFA =

∫ (− 1

2F4 ∧ F4 ∧Aij1 ∧ dz

i ∧ dzj + F4 ∧ F3 ∧ dzi ∧Aj2 ∧ dzj

− 12F4 ∧ F ij2 ∧ dz

i ∧ dzj ∧A3 + F i3 ∧ dzi ∧ F4 ∧Aj2 ∧ dzj

+ F i3 ∧ dzi ∧ Fj3 ∧ dz

j ∧A3 −12F ij2 ∧ dz

i ∧ dzj ∧ F4 ∧A3

) (E.2)

rearranging the terms using the property of changing sign each time we run a p-formthrough another q-form, we obtain

Ln=2FFA =

∫ (− 1

2F4 ∧ F4 ∧Aij1 ∧ dz

i ∧ dzj − F4 ∧ F i3 ∧Aj2 ∧ dz

i ∧ dzj

− 12F4 ∧ F ij2 ∧A3 ∧ dzi ∧ dzj − F4 ∧ F i3 ∧A

j2 ∧ dz

i ∧ dzj

− F i3 ∧ Fj3 ∧A3 ∧ dzi ∧ dzj + 1

2F ij2 ∧ F4 ∧A3 ∧ dzi ∧ dzj

) (E.3)

Note that the factors in front of the various terms can help us see what type of term weshould obtain after integration. Consider the first term we see that we have 4 ∧ 4 ∧ 1i.e we have four “leg” in the first and second field strength and one “leg” in the gaugepotential. So we should perform partial integration on the terms with the same factor

51

in front but different setup of legs in each term 8.

−F4 ∧ F i3 ∧Aj2 ∧ dz

i ∧ dzj = −dA3 ∧Ai2 ∧ Fj3 ∧A

i2 ∧ dzi ∧ dzj

= −A3 ∧ d(F i3 ∧Aj2) ∧ dzi ∧ dzj

= −A3 ∧ F i3 ∧ Fj3 ∧ dz

i ∧ dzj

= −F i3 ∧ Fj3 ∧A3 ∧ dzi ∧ dzj

we see that after partial integration we obtain a term that has the same form as thefifth term in the integral, doing a partial integration on the term that has a 1/2 factorin front gives us

−12F4 ∧ F ij2 ∧A3 ∧ dzi ∧ dzj = −1

2F4 ∧ dAij1 ∧A3 ∧ dzi ∧ dzj

= +12Aij1 ∧ d(F4 ∧A3) ∧ dzi ∧ dzj

= −12Aij1 ∧ F4 ∧ F4 ∧ dzi ∧ dzj

= −12F4 ∧ F4 ∧Aij1 ∧ dz

i ∧ dzj .

We have now performed partial integration on two terms with different setup of legs,the same procedure applies to the other terms and we obtain

Ln=2FFA = 1

6

∫ (− 3

2F4 ∧ F4 ∧Aij1 − 3F i3 ∧ F

j3 ∧A3

)dzi ∧ dzj

=∫ (−1

4F4 ∧ F4 ∧Aij1 −

12F i3 ∧ F

j3 ∧A3

)εij

(E.4)

thus we have found the D = 9 C-S term. We can continue with this procedure andobtain dimensional reductions all the way down to D = 2. Let us continue with 3indices which should yield us a D = 8 C-S term. Gathering all the terms now with 3indices gives us the following Lagrangian

Ln=3FFA = 1

6

∫ (− 1

6F4 ∧ F4 ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk − 1

2F4 ∧ F i3 ∧ dzi ∧A

jk1 ∧ dz

j ∧ dzk

− 12F4 ∧ F ij3 ∧ dz

i ∧ dzj ∧Ak2 ∧ dzk −16F4 ∧ F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧A3

− 12F i3 ∧ dzi ∧ F4 ∧Aij1 ∧ dz

i ∧ dzj + F i3 ∧ dzi ∧ F i3 ∧ dzj ∧Ak2 ∧ dzk

− 12F i3 ∧ dzi ∧ F

jk2 ∧ dz

j ∧ dzk ∧A3 −12F ij2 ∧ dz

i ∧ dzj ∧ F4 ∧Ak2 ∧ dzk

− 12F ij2 ∧ dz

i ∧ dzj ∧ F k3 ∧ dzk ∧A3 −16F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F4 ∧A3

)(E.5)

We start by noticing that we have three terms with 1/6-term in front, these terms shouldyield us the same expression. Let us assume we want something of the form like the

8Hope this is understandable, if we think in this way we can save some time on the calculations

52

first term i.e 4∧ 4∧ 0∧ 1∧ 1∧ 1. The second term with the same factor is expressed as

−16F4 ∧ F ijk1 ∧ dzj ∧ dzj ∧ dzk ∧A3 = 1

6F4 ∧ F ijk1 ∧A3 ∧ dzi ∧ dzj ∧ dzk

= 16F4 ∧ dAijk0 ∧A3 ∧ dzi ∧ dzj ∧ dzk

= −16Aijk0 ∧ d(F4 ∧A3) ∧ dzi ∧ dzj ∧ dzk

= −16Aijk0 ∧ F4 ∧ F4 ∧ dzi ∧ dzi ∧ dzk

= −16F4 ∧ F4 ∧Aijk0 ∧ dzi ∧ dzj ∧ dzk

we see that this term has the same form as the first term mainly 4∧ 4∧ 0∧ 1∧ 1∧ 1, wealso obtain the same expression for the last term with the factor 1/6 one can also showthis but we see that the term resembles the term above. Let us now look on the termswith the factors 1/2 and see what expression we obtain after partial integration

−12F4 ∧ F i3 ∧ dzi ∧A

jk1 ∧ dz

j ∧ dzk = 12F4 ∧ F i3 ∧A

jk1 ∧ dz

i ∧ dzj ∧ dzk

= 12dA3 ∧ F i3 ∧A

jk1 ∧ dz

i ∧ dzj ∧ dzk

= 12A3 ∧ d(F i3 ∧A

jk1 ) ∧ dzi ∧ dzj ∧ dzk

= 12A3 ∧ F i3 ∧ F

jk2 ∧ dz

i ∧ dzj ∧ dzk

= 12F i3 ∧ F

jk2 ∧A3 ∧ dzi ∧ dzj ∧ dzk .

we can see that we obtain the same expression as the other 1/2 factor terms. Gatheringall of them together we obtain the following integral

Ln=3FFA = 1

6

∫ (−1

2F4 ∧ F4 ∧Aijk0 − F i3 ∧ F

j3 ∧A

k2 + 3F i3 ∧ F

jk2 ∧A3

)εijk

=∫ (− 1

12F4 ∧ F4 ∧Aijk0 − 1

6F i3 ∧ F

j3 ∧A

k2 + 1

2F i3 ∧ F

jk2 ∧A3

)εijk .

(E.6)

We thus see that we have 8-forms which yields us a 8-dimensional Chern-Simons term.

53

Let us continue now with terms that have four index. The following integral becomes

Ln=4FFA =

∫ (− 1

6F4 ∧ F i3 ∧ dzi ∧A

jkl0 ∧ dz

j ∧ dzk ∧ dzl

+ 14F4 ∧ F ij2 ∧ dz

i ∧ dzj ∧Akl1 ∧ dzk ∧ dzl

− 16F4 ∧ F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧Al2 ∧ dzl

− 16F i3 ∧ dzi ∧ F4 ∧Ajkl0 ∧ dz

j ∧ dzk ∧ dzl

− 12F i3 ∧ dzi ∧ F

j3 ∧ dz

j ∧Akl1 ∧ dzl ∧ dzk

− 12F i3 ∧ dzi ∧ F

jk2 ∧ dz

j ∧ dzk ∧Al2 ∧ dzl

− 16F i3 ∧ dzi ∧ F

jkl1 ∧ dzj ∧ dzk ∧ dzl ∧A3

+ 14F ij2 ∧ dz

i ∧ dzj ∧ F4 ∧Akl1 ∧ dzk ∧ dzl

− 12F ij2 ∧ dz

i ∧ dzj ∧ F k3 ∧ dzk ∧Al2 ∧ dzl

+ 14F ij2 ∧ dz

i ∧ dzj ∧ F kl2 ∧ dzk ∧ dzl ∧A3

− 16F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F4 ∧Al2 ∧ dzl

− 16F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F l3 ∧ dzl ∧A3

).

(E.7)

We see that the terms with the factor 1/6 should resemble each other and so on. Likealways we choose a term and perform a partial integration.

16F4 ∧ F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧Al2 ∧ dzl = −1

6F4 ∧ F ijk1 ∧Al2 ∧ dzi ∧ dzj ∧ dzk ∧ dzl

= 16dAijk0 ∧ F4 ∧Al2 ∧ dzi ∧ dzj ∧ dzk ∧ dzl

= 16Aijk0 ∧ d(F4 ∧Al2) ∧ dzi ∧ dzj ∧ dzk ∧ dzl

= 16Aijk0 ∧ F4 ∧ F l3 ∧ dzi ∧ dzj ∧ dzk ∧ dzl

= 16F4 ∧ F i3 ∧A

jkl0 ∧ dz

i ∧ dzj ∧ dzk ∧ dzl .

We thus see that this configuration becomes the first term in our integral above, afterpartial integration.

54

We continue with the 1/2 factor terms and see what we obtain after partial integra-tion

−12F i3 ∧ dzi ∧ F

jk2 ∧ dz

j ∧ dzk ∧Al2 ∧ dzl = −12F i3 ∧ F

jk2 ∧A

l2 ∧ dzi ∧ dzj ∧ dzk ∧ dzl

= 12dAij1 ∧ F

k3 ∧Al2 ∧ dzi ∧ dzj ∧ dzk ∧ dzl

= 12Aij1 ∧ d(F k3 ∧Al2) ∧ dzi ∧ dzj ∧ dzk ∧ dzl

= −12Aij1 ∧ F

k3 ∧ F l3 ∧ dzi ∧ dzj ∧ dzk ∧ dzl

= −12F i3 ∧ F

j3 ∧A

kl1 ∧ dzi ∧ dzj ∧ dzk ∧ dzl .

The terms left now are the terms with the factor 1/4, doing a partial integration on forexample the term

14F4 ∧ F jk2 ∧ dz

i ∧ dzj ∧Akl ∧ dzk ∧ dzl = 14dA3 ∧ F ij2 ∧A

kl1 ∧ dzi ∧ dzj ∧ dzk ∧ dzl

= −14A3 ∧ d(F ij2 ∧A

kl1 ) ∧ dzi ∧ dzj ∧ dzk ∧ dzl

= 14A3 ∧ F ij2 ∧ F

kl2 ∧ dzi ∧ dzj ∧ dzk ∧ dzl

= 14F ij2 ∧ F

kl2 ∧A3 ∧ dzi ∧ dzj ∧ dzk ∧ dzl .

We see that this term is the same as the other 1/4 factor terms, we can now add themtogether and obtain the following integral

Ln=4FFA =

∫ (16F4 ∧ F i3 ∧A

jkl0 −

14F i3 ∧ F

j3 ∧A

kl2 + 1

8F ij2 ∧ F

kl2 ∧A3

)εijkl (E.8)

55

Let us continue with 5 indices now, collecting all the term with 5 indices gives usthe following integral

Ln=5FFA = 1

6

∫ (+ 1

12F4 ∧ F ij2 ∧ dz

i ∧ dzj ∧Aklm0 ∧ dzk ∧ dzl ∧ dzm

+ 112F4 ∧ F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧Alm1 ∧ dzl ∧ dzm

− 16F i3 ∧ dzi ∧ F

j3 ∧ dz

j ∧Aklm0 ∧ dzk ∧ dzl ∧ dzm

+ 14F i3 ∧ dzi ∧ F

jk2 ∧ dz

j ∧ dzk ∧Alm1 ∧ dzl ∧ dzm

− 16F i3 ∧ dzi ∧ F

jkl1 ∧ dzj ∧ dzk ∧ dzl ∧Am2 ∧ dzm

+ 112F ij2 ∧ dz

i ∧ dzj ∧ F4 ∧Aklm0 ∧ dzk ∧ dzl ∧ dzm

+ 14F ij2 ∧ dz

i ∧ dzj ∧ F kl2 ∧ dzk ∧ dzl ∧Am2 ∧ dzm

− 112F ij2 ∧ dz

i ∧ dzj ∧ F klm1 ∧ dzk ∧ dzl ∧ dzm ∧A3

+ 112F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F4 ∧Alm1 ∧ dzl ∧ dzm

− 16F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F l3 ∧ dzl ∧Am2 ∧ dzm

+ 112F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F lm2 ∧ dzl ∧ dzm ∧A3

)Let us calculate the partial integration for one of the 1/12 terms and see what we obtain

112F4 ∧ F ijk1 ∧Alm1 ∧ dzi ∧ dzj ∧ dzk ∧ dzl ∧ dzm = − 1

12dAijk0 ∧ F4 ∧Alm1 ∧ dzijklm

= 112Aijk0 ∧ d(F4 ∧Alm1 ) ∧ dzijklm

= 112Aijk0 ∧ F4 ∧ F lm2 ∧ dzijklm

= 112F4 ∧ F ij2 ∧A

klm ∧ dzijklm

we introduced a short-hand notation to save space i.e, dzi∧dzj∧dzk∧dzl∧dzm = dzijklm.We continue with the factor 1/6 term and obtain the following

−16F ijk1 ∧ dzi ∧ dzj ∧ dzk ∧ F l3 ∧ dzl ∧Am2 ∧ dzm = 1

6F ijk1 ∧ F l3 ∧Am2 ∧ dzijklm

= 16dAijk0 ∧ F l3 ∧Am2 ∧ dzijklm

= −16Aijk0 ∧ d(F l3 ∧Am2 ) ∧ dzijklm

= −16Aijk0 ∧ F l3 ∧ Fm3 ∧ dzijklm

= −16F i3 ∧ F

j3 ∧A

klm0 ∧ dzijklm

56

and finally the last term gives us

14F i3 ∧ dzi ∧ F

jk2 ∧ dz

j ∧ dzk ∧Alm1 ∧ dzl ∧ dzm = 14F i3 ∧ F

jk2 ∧A

lm1 ∧ dzijklm

= 14dAi2 ∧ F

jk2 ∧A

lm1 ∧ dzijklm

= 14Ai2 ∧ d(F jk2 ∧A

lm1 ) ∧ dzjiklm

= 14Ai2 ∧ F

jk2 ∧ F

lm2 ∧ dzijklm

= 14F ij2 ∧ F

kl2 ∧Am2 ∧ dzijklm

we have now done all the partial integrations we need, collecting the results gives us

Ln=5FFA = 1

6

∫ (12F4 ∧ F ij2 ∧A

klm − 12F i3 ∧ F

j3 ∧A

klm0 + 3

4F ij2 ∧ F

kl2 ∧Am2

)εijklm

=∫ ( 1

12F4 ∧ F ij2 ∧A

klm − 112F i3 ∧ F

j3 ∧A

klm0 + 1

8F ij2 ∧ F

kl2 ∧Am2

)εijklm

(E.9)

57