Topological Quantum Field Theories inTopological Recursion

Campbell WheelerPaul Norbury

The University of Melbourne

Outline

We aim to find an elementary construction of a class of 2-dimensional TopologicalQuantum Field Theories that arise out of Topological Recursion. We know that thereis some 2-dimensional Topological Quantum Field Theory contained in the outputof some Topological Recursion due to results in Cohomological Field Theory [6].We consider a graphical approach to both 2-dimensional Topological Quantum FieldTheory and Topological Recursion and use these to try and find this construction.

Topological Quantum Field Theories

Firstly we will discuss some of the physical motivation behind Topological QuantumField Theories then describe there axioms and how these can be interpreted usingcategory theory. Then we’ll see how they are motivated by the physics. Many ofthese ideas are attributed to M. Atiyah and E. Witten and the following papers andbook go into much more depth than we will here [1] [7]. We will however try to keepthe discussion simpler.

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Motivation

Not surprisingly the motivation behind the axioms of a Topological Quantum FieldTheory comes from Quantum Field Theory. To define a Quantum Field Theory onsome manifold we must specify some action, S(φ), which is a function of the fields,φ, on the manifold. This action completely determines the physics on this manifold.When we say physics we mean expectation values and correlation functions betweenphysical observables which is the output of a Quantum Field Theory. One waycalculations are carried out is through Feynman integrals. They are of the followingform

〈W 〉 =

∫WeiS(φ)dφ

where the right is an integral over all fields of the theory. In analogy with thepartition function of statistical mechanics we can define what is called the partitionfunction of our theory which is like the expectation value of 1. It is given by thefollowing

Z =

∫eiS(φ)dφ

Now for particularity nice actions, that are independent of the metric given to themanifold the Quantum Field Theory is on, the partition function Z will be an invari-ant of the manifold. An example of such an action is the Chern-Simons action [1] [4].These integrals aren’t well defined so we try to formulate axioms that will not rely onthese ill defined integrals but capture the information they hold. The direct link be-tween the axioms is complex and needs more discussion than we will go into. See [4]for more. The axioms given next are based on axioms given in [4] which are basedon axioms given in [1].

Axioms

A (n+1)-dimensional Topological Quantum Field Theory associates to every n-dimensional(smooth compact orientable) manifold, Σ, a vector space, Z(Σ), and to every (n+1)-dimensional (smooth compact orientable) manifold with boundary, M , to a linearfunctional, Z(M), with the following properties. (A visual aid is given below)

• (Vacuum) Z(∅) = C

• (Duality) Z(Σ∗) = Z(Σ)∗

• (Multiplicity) Z(Σ1 t Σ2) = Z(Σ1)⊗ Z(Σ2)

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These aren’t the only rules but they are the main ones and give a feel for how itworks. We wish to simplify the discussion so we don’t need to list an extensive listof axioms. It is important to note that gluing is achieved through contracting, thenatural map from the tensor product of a vector space and it’s dual to the field. Thiswill require us gluing boundary components with opposite orientation. See [1] [4] formore.

Functorality

The notation used above to denote the vector space, Z(Σ), associated to the n-dimensional (smooth compact orientable) manifold, Σ, is for a reason. The reasonis that if we list the way the association works with all the rules and we know alittle category theory we can see that this association is actually a functor from twocategories. It is a functor from the category of (n+1)-dimensional cobordisms tothe category of vector spaces (both with some extra structure). Noting this we candefine a (n+1)-dimensional Topological Quantum Field Theory as follows.

An (n+1)-dimensional Topological Quantum Field Theory is a functor, Z, from thecategory of (d+1)-dimensional cobordisms to vector spaces.

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This definition is compact but using the rules directly is often clearer and in whatfollows we won’t really touch on the fact that a Topological Qunatum Field Theory,Z, is a functor.

2-dimensional Topological Quantum Field Theories

We will build up to the statement of a classification theorem of 2-dimensional Topo-logical Quantum Field Theories. We will use results from the classification of surfacesto do this. We will then also describe some algebra structure every 2-dimensionalTopological Quantum Field Theory will have and use this to show that every 2-dimensional Topological Quantum Field Theory is equivalent to some FrobeniousAlgebra and visa versa.

Classification of Surfaces

Theorem - Classification of surfaces with boundary A (smooth connectedcompact orientable) surface with boundary is completely determined by its numberof connected boundary components and its genus (i.e. the number of handles). [6]

This means we will write a given surface with boundary as Mg,n where g representsthe genus and n represents the number of connected boundary components.

Boundary Components Note also that the each connected boundary componentis isomorphic to S1, the circle. This means that from the axiom of multiplicity we canget the vector space of any boundary component from the vector space associated tothe circle. We do this by applying the axiom which will give us some tensor productof the vector space Z(S1).

Constructing Surfaces It is important to note that every connected surface (be-sides the sphere, two holed sphere/cylinder and the torus) can be built out of gluingthree holes spheres (pairs of pants in the language of [6] or trinions in the languageof [4]). This is easy to see after we have the classification theorem and a pen andpaper. From this we can see that every surface can be built using one holed spheres,two holed spheres and there holed spheres.

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Classification

From the construction of surfaces and the possible boundary components we cansee that the following association completely determines a 2-dimensional TopologicalQuantum Field Theory.

| Z(S1) 1 η c |

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Every vector space and surface can be gotten by taking disjoint unions or by pastingone holed spheres and three holed spheres. We use the reversed orientation of thecylinder to paste things together. Note that in this way the cylinder acts like a semi-metric (i.e. can be negative) as we can use it as a non-degenerate inner product.Pasting and disjoint union are governed by the axioms so once these are specifiedeverything is determined.

Algebra Structure

We can view the three holed sphere as giving us some kind of multiplication when wecontract (glue) it to a cylinder. The following is a visual proof that this multiplicationis associative.

In fact this multiplication is unital, commutative and associative. Also note thatwith the cylinder we have a non-degenerate inner product that satisfies the following

〈a, b · c〉 = 〈a · b, c〉

the proof is again given visually below.

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This collection of facts describes a Frobenious algebra and in fact every Frobeniousalgebra is determined by the same four things we listed above. A vector space, com-mutative and associative multiplication, a unit and a non-degenerate inner productthat satisfies the above statement.

Semi-Simple

If a Frobenious algebra has a basis {e1, ..., en} that satisfies the relation below wecall this Frobenious algebra semi-simple.

ei · ej = δijei

Note δij is the Kronecker delta. From this definition one can see that we have

1 =n∑i

ei

Calculations are then very trivial if we are given such as basis. In fact given such abasis the following numbers will determine our Frobenious algebra.

{〈1, e1〉, ..., 〈1, en〉}

Using some properties of the Frobenious algebra this is the same as the following.

{〈e1, e1〉, ..., 〈en, en〉}

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These numbers are then what we need to describe a semi-simple 2-dimensional Topo-logical Quantum Field Theories. This is interesting as if we have a semi simple Topo-logical Quantum Field Theory there is a list of numbers that completely determinethe theory.

Problem

Finding the idempotent basis is not necessarily easy. To find the basis in terms ofa general basis we need to solve multiple quadratic equations in multiple variables.This becomes computationally difficult when we have large numbers of variables andequations. An example of the kind equations needed to be solved is given here

α2 + 3β2 + 2γ2 = α 2αβ + 4βγ = β γ2 + 2αγ + 3β2 = γ

(This example came from the center of the group algebra generated by S3 and is onlya subset of the list of equations needed to be solved to find the basis) There are afew solutions to these equations that are interesting when we consider the full set ofequations. They are given here.

α = β = γ =1

6α = −β = γ =

1

6α =

2

3, β = 0 and γ =

−1

3

They will actually be the coefficients needed to switch the standard basis for thecenter of the group algebra to the idempotent basis.

Graphical Approach

We are going to describe a graphical method that can be used to compute variouslinear functionals from the four things that classify our surface. This method isreminiscent of that used when discussing Feynman Diagrams.

Pair of Pants Decomposition to a Graph

As mentioned in the classification of surfaces every surface can be constructed outof pairs of pants, that is three holed spheres. Instead of considering a possibleconstruction (note that in general there will be many possible ways to construct agiven surface out of pairs of pants) we will now consider a possible decompositiongiven some surface. Once given this decomposition we will associate to this a trivalentgraph (each internal vertex having three edges going into it). The way we will do

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this is by letting each pair pants be a vertex and each boundary component or gluedboundary be represented by an edge. This process is illustrated here. Note that thenumber of edges and vertices (3g− 3 +n and 2g− 2 +n respectively) is independentof our decomposition.

Colourings of Graphs

Once we have graph we can start to determine our linear functional. The way we dothis is by colouring the external vertices with the vectors we wish to input into ourfunctional. A colouring is basically just some labelling. So we label the end of anedge by some vector. We can only consider basis elements as the input as anythingelse will be some linear combination of these and the linear functionality means wecan calculate that using the linearity. Once we have an input we wish to compute, wecolour the ends of each edge (i.e. each edges has two colours) by the basis vectors andconsider all possible colourings. We associate a weight to each colouring and thensum over all possible colourings. Say we consider a 2-dimensional Frobenious algebrawith basis {e0, e1}. The following is an example of a colouring for the TopologicalQuantum Field Theory described below in Benefits.

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Weight of a Colouring

The weight associated to a colouring comes from associating a weight to each edgeand vertex then multiplying these together. These weights come from the reciprocalof the cylinder of the two coloured vectors of an edge (the cylinder = edge = η) andthe value of the pair of pants of the three coloured vectors of a vertex (the pair ofpants = vertex = c) respectively. Note that one needs to check that when we sumover all colourings we get the the same result no matter what decomposition we use.This is of course true.

Benefits

For a graph with 6 internal edges we already have 212 possible colourings for a twodimensional vector space so this doesn’t seem to help in any calculation. Howeverif we choose a nice basis most of the weights may go to zero and then we can ruleout all but few colourings which greatly simplifies the kind of calculation we’d haveto do otherwise which would involve large strings of tensor products. An especiallynice example is with the idempotent basis which gives either zero or one possiblenon-zero colouring for each input. For a slightly more complex example let {e0, e1}be a basis for Z(S1). Consider the following outputs for the cylinder and the pair ofpants.

η(ei, ej) =

{1 if i 6= j0 if i = j

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c(ei, ej, ek) =

{1 if i+ j + k is odd0 if i+ j + k is even

This is an example that arises in Cohomological Field Theories. Notice only colour-ings with opposite vectors on each edge will be non-zero. In fact with this examplethere will be either 0 or 2g non-zero colourings and each colouring will have weigh 1in this way we can actually determine the following result.

Z(Mg,n)(ei1 , ..., ein) =

{2g if g + n−

∑nj=1 ij is odd

0 if g + n−∑n

j=1 ij is even

The idea is that either we have no colourings or that each handle gives us two possiblecolourings. A picture is given here to give an intuitive idea of the argument one woulduse to prove this.

Topological Recursion

We will give a brief introduction to Topological Recursion and how we can describeit through a similar graphical approach to that used in the topological quantum fieldtheory. We will talk about some key differences and similarities.

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Plane Curves and Symplectic Invariants

A plane curve is a subset of C2 defined by a polynomial equation in the two variables.The set is given by the expression {(x, y) ∈ C2 | p(x, y) = 0} for some polynomial, pin x and y. This set will be some surface or a one dimensional complex manifoldimmersed in C2. This will not necessarily be an embedding and we may have selfintersection (e.g. the Klein Bottle can be immersed in R3 but can’t be embedded).Eynard and Orantin developed a sequence of meromorphic differentials ωgn to recur-sively define symplectic invariants F g = ωg0 on a genus zero immersed plane curvewhere the branch points in the coordinate x are simple (have only one twist). Theseinvariants are invariant under automorphisms of C2 that preserve the symplecticform dx ∧ dy. See the introduction in [5] and also [2] and [3].

The Recursion

The recursion that they defined has the following base and kernel which are definedin terms of the immersion of the plane curve, (x, y). Note this kernel will need onlybe defined close to the branch points of x in it’s second variable and that when z isnear a branch z denotes the unique point such that x(z) = x(z).

ω01(z1) = y(z1)dx(z1) ω0

2(z1, z2) = B(z1, z2) =dz1dz2

(z1 − z2)2

K(z1, z) =

∫ zzB(z1, z

′)

2(y(z)− y(z))dx(z)

From these kernels the recursion is defined as follows.

ωgn+1(z1, zS) =∑a∈A

Resz=a

K(z1, z)(ωg−1n+2(z, z, zS) +

∑ItJ=Sg1+g2=g

ωg1|I|+1(z, zI)ωg2|J |+1(z, zJ)

)

where S = {2, ..., n+ 1}, I 6= ∅ and J 6= ∅ and A is the set of branch points.

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Diagrammatic Representation

This expression looks a little messy but there is a nice way to think about it thatmakes it much clearer. We view the ωgn as connected surfaces with n boundary pointsand genus g. Then we can picture the recursion as follows.

• View K as a pair of pants

• Consider all ways to glue a disjoint union of connected surfaces to two of theholes in the pair of pants to get n boundary components and genus g.

• Weight each possible gluing by the product of the ωgn’s associated to eachconnected surface being glued.

• Sum over all gluings considered and then sum over the residue at every branchpoint.

The process is given visually here.

Graphical Representation

We can represent these diagrams graphically like we did for the 2-dimensional Topo-logical Quantum Field Theories but there is much more structure that needs to beadded. Representing them graphically we can actually ignore the fact we are using arecursion and go straight to the expression we’re interested in. Doing this we don’tjust consider one pair of pants decomposition we consider every possible decomposi-tion. There is even more structure on top of that. A full description of this processcan be found in section 4.5 of [2].

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Research

A large portion of this project is actually trying to find the Topological QuantumField Theory in this Topological Recursion. We have came to some kind of resultbut this still needs more work. We’ll briefly highlight where we expect to see theTopological Quantum Field Theory. We’ll also briefly describe some ideas of howthe recursion could lead to a Topological Quantum Field Theory.

Where is the Topological Quantum Field Theory?

Here are some examples of the ωgn for the plane curve y2−yx+1 = 0 with x(z) = z+ 1z

and y(z) = z. Note z = 1z.

ω03(z1, z2, z3) =

(1

2

1

(1− z1)2(1− z2)2(1− z1)3− 1

2

1

(1 + z1)2(1 + z2)2(1 + z1)3

)dz1dz2dz3

ω11(z1) =

(1

16

1

(1− z1)4− 1

16

1

(1− z1)3− 1

32

1

(1 + z1)2− 1

16

1

(1 + z1)4+

1

16

1

(1 + z1)3+

1

32

1

(1 + z1)2

)dz1

We expect to see a Topological Quantum Field Theory contained in the coefficientof the highest order pole. We think of the poles as representing the input of variousbasis vectors. The dimension of our vector space will in fact be the number of poles.This is an interesting problem as there are many different coefficients in font of thesepoles including intersection numbers on the moduli space of curves which are quitecomplex mathematical objects.

A closer look at K

It is interesting to look at the behaviour of K when we consider close to the branchpoints of the second variable. We can write K as follows.

K(z1, z) =1

2

z − zy(z)− y(z)

1

(z − z1)(z − z1)dz1dx(z)

If we consider z → a where a ∈ A and assume dydz

(a) 6= 0 then we have the following.

K(z1, z)→1

2

(limz→a

z − zy(z)− y(z)

)1

(a− z1)2dz1dx(z)

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It seems that the limit in this formula may play an important role in this prob-lem. It seems similar to a derivative of y this may possibly be important. Takingthis limiting case of K makes it seem very similar to the B (note the second lastterm). So maybe this contains the information of the edge as well as the vertexwhich may be interesting. The other aspect that may be important is the terms ofthe form dz. This will add another derivative into the terms of the recursion (dz

dzto

be exact) so this will play an interesting role. The aim of this direction is try and seepossible coefficients that can be pulled out of the recursion to get back a TopologicalQuantum Field Theory.

Acknowledgements

I would like to thank AMSI and the University of Melbourne for giving me theopportunity to take part in this research project. I’d also really like to thank PaulNorbury for the countless number of conversations and for advice. This has been areally enjoyable experience and I think that this is largely due to the person I dida lot of the hands on work with. So lastly I’d like to thank Anupama Pilbrow forworking on the project with me and for the many valued conversations.

References

[1] M. Atiyah. The geometry and physics of knots. Cambridge University Press,1990.

[2] N. Orantin B. Eynard. Invarients of algebraic curves and topological expansion,2007.

[3] N. Orantin B. Eynard. Topological recursion in enumerative geometry and ran-dom matrices. Journal of Physics A: Mathematical and Theoretical, 42, 2009.

[4] R. Lawrence. An introduction to topological field theory, 1994.

[5] P. Norbury. Counting lattice points in the moduli space of curves, 2008.

[6] Paul Norbury. Various conversations, 2014/2015.

[7] E. Witten. Quantum field theory and the jones polynomial. Comunications inMathematical Physics, 1989.

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