Mauricio ROMO

AbstractThe author reviews some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G.The author focuses on the case of G=SL(N,C)and M being a knot complement:M=S3K.The main result presented in this note is the cluster partition function,a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral for G=SL(N,C).He also reviews various applications and open questions regarding the cluster partition function and some of its relation with string theory.

Keywords Chern-Simons theory,Knots,Cluster algebras

1 Introduction

In the recent years there has been a growing interest in the study of Chern-Simons(CS for short)theory with complex gauge group G,in particular when G=SL(N,C).One of the main reasons is its appearance in the so-called 3d-3d correspondence(see[1]for a review).This is a correspondence between supersymmetric 3d gauge theories and CS theory(which is a topological field theory)with complex gauge group.The correspondence arises from wrapping a certain class of 6-dimensional extended objects in M-theory,called M5-branes,on a 3-manifold M.This can be considered as part of a much broader line of research on the context of N=2 supersymmetric quantum field theories.We refer the interested reader to the comprehensive review[2].

The main goal of this note is not to review the 3d-3d correspondence but to summarize some properties and challenges present in CS theory with complex gauge group and,along with that,we will present in more detail a recently developed computational tool:The cluster partition function(see[3]).Formally,the partition function of CS theory with complex gauge group G is given in the form of a path integral over G-connections A:

The main question is then,how one can make sense of ZG(M).When G is a compact Lie group,many techniques for computing ZG(M)and hence to give a definition of(1.1),have been developed since the pioneering work(see[4])connecting ZG(M)with invariants of 3-manifolds,but for noncompact and/or complex G much less is understood.Foundational work on CS theory with complex gauge group can be found in[5],and subsequent development in the lines that we will focus on here,in[6–7].One way to approach the problem in the case∂M 6= ∅,is to interpret ZG(M)as a wavefunction.Geometric quantization of the restricted classical phase space of the theory1For the case of G compact this was carried out in[4,8–9].associates a Hilbert space H∂Mto ∂M.The CS partition function ZG(M)is then interpreted as a wavefunction in the following sense:

where Π represents a choice of polarization and X,the ‘position variables’(so,hX,Π |Ψi is a function of X which are mutually commuting variables).The topology of M(and possibly some extra data inherent to M)is what determines which vector|Ψi should be chosen.When M is a knot complement M=S3K,we have a toroidal boundary:and the restricted classical phase space corresponds to flat G-bundles on T2and comes naturally equipped with a symplectic structure.

One of the most e ff ective and well studied tools for computing partition functions ZG(M)=hX,Π |Ψi for the case of M being a cusped 3-manifold and G=SL(N,C)have been stateintegral models.We will give a brief summary on development of these models in Section 3.The model we will present in detail,the cluster partition function,shares many properties with state-integral models and is expected to be equivalent in some cases.

The cluster partition function provides a way to define the CS path integral.In other words,it gives a prescription to compute a functionwhich should be interpreted as the wavefunction ZG(M).Our focus will be on the case of gauge group G=SL(N,C)over a 3-manifold M which corresponds to a(hyperbolic)knot complement on S3.Moreover,we will actually see thatis more well suited for the case that M can be obtained from a mapping torus construction.The cluster partition function was originally proposed in[10]based on ideas of[11],however important modifications were done in[3]to get it to the form we will present it here.The functionwe obtain is not immediately obvious that is well defined as a nonperturbative invariant but we can propose perturbative topological invariants of M starting fromWe will look at this problem in more detail in Subsection 5.1.

This note is organized as follows:In Section 2 we will start reviewing classical aspects of the CS path integral for gauge group G=SL(N,C)focusing on comparison with the case of compact gauge group,boundary conditions for the case of M=S3K and some interpretations and results of the perturbative expansion of ZG(M).In Section 3 we will review the canonical quantization of the boundary phase space for CS on M=S3K and the subsequent interpretation of ZG(M)as a wavefunction.In Section 4 we will come to the central theme of this note,we will review the derivation ofand explain its relation with ZG(M).For this we will start by reviewing the Fock-Goncharov construction of coordinates for the space of flat connections on a punctured Riemann surface Σ and its quantization which defines a Hilbert space HΣ.Then,we will defineas the trace over HΣof an operator.We will see that our definition requires M to be a mapping torus but its relation with cluster algebras(hence,its name)allows us to actually makes sense ofas function of a general quiver Q plus a sequence of mutations and permutations m acting on it.This will be an important point for Section 5 where we will review the applications ofand its general formMany of these applications are currently under study.Finally in the appendix we collect results about the quantum dilogarithm,a special function that plays a central role in the definition of

2 Chern-Simons Theory with Complex Gauge Group

In this section we will review some basic facts about Chern-Simons theory with complex gauge group G2Even though in this section we will make statements that hold for general G,in the rest of this note we will set G=SL(N,C)..Consider a compact 3-manifold M,possibly with boundaries and a complex Lie group G.Fix a principal G-bundle EG→M and consider a connection A:

It is important to remark that G includes large gauge transformations3These are topologically nontrivial gauge transformations under which shifts by πZ..For k∈Z and s∈C,then eiSCSis gauge invariant(see[5]).Define

where the boundary conditions should be imposed.If we restrict to s∈R,then ZG(M)is an oscillatory integral.In such a case,.However,other values of s ∈ C are also of interest.For instance,ZG(M)can also define a unitary theory for s∈iR(see[5]).Making sense of ZG(M)when s 6∈R is possible via analytic continuation(see[7]).We summarize the main cases treated in[7].

Compact gauge groupConsider CS theory with gauge group a compact Lie group H.The construction is analogous as before,just changing G by H,so the path integral for the connection A∈Conn(EH)is given by an integral over YH=Conn(EH)/H:

analytic continuation to k∈C is done in two steps(see[5,7]).First change the domain of integration to bewhere the subindex C stands for complexification andare the topologically trivial gauge transformations g:M→HC.Then define ZH(M)as the integration over a middle dimensional contoursuch thatgoes to 0 along every asymptotic direction of C.The contour C can be constructed by downward Morse flow usingas a Morse function.It is important to remark that C is not unique but one can impose extra conditions such as ZH(M)must coincide with the integration overwhen k∈Z and conditions on the behavior for k→∞to constraint the choices of C.Another important point is that,because of the way C is chosen,it is not invariant under large gauge transformations.Indeed if one naively just change the domain of integration in(2.9)tothen(2.9)will vanish if k 6∈Z(see[7]).

Compact gauge group with Wilson loopConsider now inserting a Wilson loop along a knot K⊂M in(2.9).A Wilson loop is specified by K and an irreducible representation R of H.Then

We can write the holonomy operatoras an integral over a quantum mechanical system,whose physical Hilbert space is identified with R,coupled to A(see[7,12]):

where U is the space of maps u:S1→H/TH,4Here THdenotes the maximal torus of H.α is the highest weight of R,seen as an element of THand Θα=Tr(αg−1dg)is a 1-form in H/TH.By writing the Wilson loop this way,we can write ZH(K)as an integral over YH×U.Then,analytic continuation on k follows the same steps as the case without Wilson loop but replacing the integration domain by a middle dimensional contouris the space of maps

Noncompact gauge groupHere we will comment on the case of analytic continuation in s of(2.8).Renaming the connectionand takingas independent of A already makes(2.8)non-invariant under large gauge transformations,as it can be seen from its explicit form:

The functionals CS[A]/4π andare defined modulo 2π but independent shifts will not leave(2.12)invariant.So,integration over Y×Y is already ill defined.Only if both shifts are equal,(2.12)is invariant.The solution proposed in[7]is to define the integral over,the smallest covering of Y×Y on which(2.12)is well defined,i.e.,we quotient out only by the gauge transformations that leave(2.12)invariant.Then,the analytic continuation of(2.8)can be defined by integration over a middle dimensional contourwhich makes the partition function convergent.Let us remark on the relation with the case of compact gauge group(see[7]).Denote H⊂G the unique compact subgroup satisfying HC=G.Denote the analytic continuation of CS partition function with group H and level k∈C as

where Jαdenotes a valid integration cycle.Since,and(2.12)factorizes in an A and andependent piece,C can be factorized ascycles in the basis of cycles under whichare convergent.5Here h∨denotes the dual Coxeter number of H and the reason for this shift is explained in[7].Then we can write

Noncompact gauge group with Wilson loopNow consider a finite dimensional representation R of H defined as before.Then we can lift this to a holomorphic or anti-holomorphic representation of G=HC.We choose a holomorphic lift.The partition function:

is already divergent,so we have no way to canonically define a contour of integration for it.The solution suggested in[7]is to define this path integral as

where ZH(K,t− h∨)is the analytic continuation of ZHwith the insertion of a Wilson loop,defined as before,with just a shift of the level.So,in this case the partition ZG(K)completely factorizes.At the perturbative level,inthis is equivalent to the statement that the expectation value of Wilson loops in a holomorphic representation obtained by complexifying a real one does not contain any new information.This is stated in Exercise 6.32 in[13]and also remarked in[6]and[7].

2.1 Boundary conditions for G=SL(N,C)and M=S3K and classical phase space

From now on we will focus on the case of G=SL(N,C)and M being a knot complement in S3:6Given a knot K embedded on S3,set N(K)a tubular neighborhood of K then M=S3K is the compact manifold given by S3minus the interior of N(K).

Then M has a nonempty toroidal boundary:

therefore,we distinguish two cycles,γmand γl,in ∂M.The first cycle is called meridian and is given by the cycle that is contractible in N(K).γlis called longitudinal and is given by the transversal cycle to γmin ∂M,intersecting it at a single point.

Boundary conditions for A are given by specifying the conjugacy class of its holonomy along a cycle aγm+bγl(a,b∈ Z).For us,it will be better to consider specifying the holonomy along γl.For example if we fix the holonomy around γlto a generic element of SL(N,C)(i.e.,an element such that all its eigenvaluesare distinct),we get

where the∼relations means up to conjugation by SL(N,C).The classical phase space is given by extremizing the CS actionand imposing the boundary conditions.The extrema of SCSis given by flat connections(F=dA+A∧A):

modulo gauge transformations.Flat connections modulo gauge transformations are in one-toone correspondence with representations of π1(M)into the gauge group:

where SL(N,C)acts on Hom(π1(M),SL(N,C))by conjugation.The space Mflat(M),also called SL(N,C)-character variety of M,should be described as a Lagrangian submanifold7One way to motivate this,is that a semiclassical solution can be characterized by a Lagrangian submanifold in the context of geometric quantization(see[14]).LMof Mflat(∂M)(see[15]),the moduli space of flat connections in ∂M.LMis Lagrangian with respect to the Weil-Petersson symplectic form on Mflat(Σ),given by

We should make a few remarks before closing this subsection.First,note that the equations describing LMare holomorphic.This is because Mflat(∂M)admits a hyper-K¨ahler structure(see[17])and LMis a holomorphic subvariety.For the case N=2,there is only one equation A(l,m)and it is known to correspond to the classical A-polynomial(see[18])of the knot(see[6]).

2.2 Perturbative results for Chern-Simons theory with complex gauge group

Perturbation theory of CS with compact gauge group has been studied for many years,citing all the existent literature will be overwhelming,so,instead we point out the interested reader to the review[19].For the case of complex gauge group,perturbative aspects of CS theory has been analyzed in[6],[16]and[20–21]for the case G=SL(2,C).One of the main interests in this case is because of the connection between the perturbative expansion with the volume conjecture[22–24]and the generalized volume conjecture[6]8For a review see[25]..Define the constants

Recall that for the knot complement we are interested in fixing the boundary condition byfixing the holonomy of A around one of the peripheral cycles.Suppose that we fix Holγl(A)as a boundary condition,then the eigenvalues of Holγm(A)are fixed via(2.24).There is expected to be a finite number of solutionslabeled by a finite set of pointsTherefore,perturbatively,we expect that the partition function takes the factorized form:

As a final remark,as we mentioned before,this expansion for G=SL(2,C),9This is the statement of the generalized volume conjecture in[6].coincides with the expansion of the SU(2)colored Jones polynomialaround n→∞,k→∞while keeping u=fi xed(the parameter u is associated with the holonomy eigenvalues that we fix as boundary conditions(see[25]).This expansion in~can also be obtained by topological recursion applied to the ‘character variety of K’(see[26–27]),this is,the Lagrangian LMdefined in the previous subsection.

3 Quantization of Classical Phase Space

In this section we will consider and review the problem of quantizing the classical phase space of CS theory with G=SL(N,C).This can be done,for example,using the Hamiltonian formalism(canonical quantization)(see[4]).Put the theory on M=R×Σ where Σ is a Riemann surface possibly with boundaries/punctures.

On R×Σ we fix the natural gauge A0=0,where x0is the R direction on R×Σ.Then,the CS action(2.3)takes the form:

moreover,from(3.1)we can derive the Poisson brackets

this provides the classical phase space with a symplectic structure.Consider the case at hand,so the boundary is a torus:

then,the symplectic form induced from the Poisson brackets can be explicitly computed in terms of the holonomies around the cycles γm,γl:

by choosing a real polarization we can quantize this phase space and the holomorphic part of the partition function(with fixed boundary condition)should satisfy

as was shown in[6]and in[28],after choosing a real polarization oferatorcorresponds to the quantization of the A-polynomial of the knot K(see[18]),however,the quantization of A(l,m)and,more generally,of the system Ai(l,m)10A recursive method to find the quantization of the system of polynomial equations Ai(l,m)(i=1,···,N−1)is proposed in[16].is a complicated problem in general and this is not a practical way to compute Zα’s.The computation of CS partition function can be made by the use of the so called state integral models.The basic idea behind state-integral models is to start from a triangulationof M and assign a wave function to each∆i,which can be then glued together via process called symplectic gluing.For G=SL(2,C)and level(k=1,s),a state-integral model for shaped triangulations was developed in[29–31]and its perturbative invariants further studied in[16].An alternative state-integral model for G=SL(2,C)and level(k=1,s)was proposed in[28]and its perturbative invariants further analyzed in[20].Nonperturbative aspects of the previous cases were recently studied in[32].The case of level(k=0,s)was developed in[33].For the case of G=SL(N,C)and level(k=1,s),a perturbative model was developed in[15].Very recently,the case of arbitrary level(k,s)was studied in[34–35].In the next section we will focus on a particular model which is called the cluster partition function.

4 Cluster Partition Function for Mapping Cylinder/Torus

We came to the main section of this note.Here we will review the construction of the cluster partition function developed in[3].We will motivate it by its relation to ZG(M)for G=SL(N,C)at level(k=1,s)and M=S3K,when M is a mapping torus.However in[3]the case(k=0,s)was also analyzed and more importantly,as we will see,the cluster partition function can be constructed starting from any cluster algebra associated to a quiver.

4.1 Fock-Goncharov coordinates and quantization of

The classical phase space we are interested in is the moduli space of SL(N,C)- flat connections on a punctured Riemann surface Σ modulo gauge transformations with prescribed holonomy around the punctures.Suppose that Σ has h boundaries anddenotes the generators of π1(Σ)corresponding to boundary curves.Also,consider a setof h conjugacy classes ρaof SL(N,C).Then,define

The space of framed PGL(N,C) flat connections on Σ,which we denote by XN(Σ),is a subspaceFrom now on we will work with XN(Σ),but the final answer is expected to lift to a function inA similar situation occurs in[34].To define XN(Σ)we need to consider FN,the space of complete flags in CN,then here PGL(N,C)acts on FNby left multiplication.The space XN(Σ)is a ramified cover of Mflat(Σ)and it has been extensively studied in[36].One of the results in[36]is the existence of a birational morphism:

The coordinates on XN(Σ1,1)constructed in[36]can be encoded on a quiver diagram,that can be drawn on top of an ideal triangulation of Σ (this is a triangulation of Σ where the vertices corresponds to punctures),with N0:=N2−1 nodes and adjacency matrix Qi,j11For a recent concise review of the construction of this space and the quiver diagram see the Appendix A of[37].(see Figure 1 for an example).

Figure 1 On the left is an ideal triangulation of Σ1,1and the right is the FG quiver for N=3.

We will denote these coordinates byThis construction have some remarkable features,one is that it equips XN(Σ1,1)with a Poisson structure12The Poisson structure in the yi=exp(Yi)coordinates is encoded in the matrixand on the other hand,the action of MCG(Σ)on these variables is given by cluster transformations generated by sequences of mutations of the quiver diagram(for a review of cluster algebras see[38]).A mutation at a node k of Q13In the following we will use Q to denote the quiver and the adjacency matrix of the quiver as well.corresponds to a change in the quiver’s adjacency matrix given by

Because of the symplectic structure onthere exists a geometric quantization of this space,the y variables are promoted to noncommutative ones:

the yis satisfy commutations relations consistent with the symplectic form,forming a noncommutative algebra that we denote AQ:

here we defined the logarithmic variables Yiand q is the quantization parameter.They satisfy the relations:

where the parametrization~=2πib2will prove useful in the subsequent sections.The quantization of the cluster transformations is given by

where eb(z)is the quantum dilogarithm function(see Appendix A of[37]for its definition and relevant properties),then we can construct an explicit operator implementing the mutations

4.2 Cluster partition function at level(k,s)=(1,s)

In this subsection we will present one of the main results of[3]:The cluster partition function.We will consider the case of CS level(k=1,s)14In[3]the case(k=0,s)is also analyzed,but here we will only focus on k=1..Let us first reparametrize the CS level in the following way:

where we consider b∈C,|b|=1,however analytic continuation to other values of b such as b∈R is possible.The reason of this parametrization of s comes from physics and the 3d-3d correspondence(see[39–41]),where the parameter b has a very concrete interpretation in terms of the geometry of the M5-branes.In terms of(4.16)the parameters in(2.25)become

therefore we can define the parameters q,as

the mapping torus Mϕis obtained by identifying both ends of Iϕ:

The quiver Q has N2−1 vertices and a very simple form(see[36])and can be written as a tessellation of the ideal triangulation of Σ1,1.See Section 4.5 of[3]for examples and applications of this,in the present context.The Poisson structure of the FG coordinates coincides with the holomorphic half of the Poisson structure that one can derive from Hamiltonian formalism(see[15]),putting CS theory on.Indeed,given the set ofwe can define the coordinates

After quantization,the operators yiact on a Hilbert space that we will denote by HQ.Before describing HQmore explicitly,we define a function,in terms of a chain of tfpermutations and mutations,which we denote simply by m16In our convention we read the products from left to right.:

and denote the quiver at ‘time’t by Q(t)(and we denote Q the original quiver):

Then,the function we want to define is

Now,we return to the problem of describing HQ.Finding a real polarization in terms of the Yioperators is not easy in general,so we resort on a very practical reparametrization.For this we need to double the amount of operators and introduce the Hermitian operators(ui,pi)satisfying the usual Heisenberg algebra with the parameter:

then,by simply writing

the Yis defined by this way automatically satisfy(4.10),but we are now working with the variables(ui,pi)which can be identified as usual position and momentum operators.Therefore we can choose a position representation,so HQis spanned by the basis|ui or equivalently span HQby a momentum basis|pi:

They also should satisfy the completeness relations17Here the measures are defined as du:

with this Hilbert space at hand.We can now write(4.28)in the following way(of course we could have chosen a position basis and the form would have been the same):

Using the properties of p and u operators,we can bring(4.33)to the form of an integral(see Appendix B of[3]for a detailed derivation of this result):

The total number of integration variables in(4.35)is 2tf−1.In(4.35)u(t)and p(t)denote just integration variables.We have left the integration contour undefined,in principle is the real line,but we may require deforming it for analytic continuation.For the perturbative expansion we will analyze in the following section,we do not need to specify a contour and in general,the question of admissible contours is an open question.The function(4.35)can be essentially thought of as the partition function on the mapping cylinder.To get the partition function on the mapping torus,we need to ‘wrap it up’.To be more precise,we need to be more specific on the properties of the sequences of mutations and permutations that can be identified with actions of.We denote=m the operator corresponding to an elementThen the quiver,under the action ofreturns to the same as the original one:but the yis do not have to18The operators acting on the yivariables are required to satisfy the relations of the mapping class group.For example for,suppose that we have a sequence of mutations and permutations that we associate to S∈SL(2,Z),then(see[36]).Then,under the action ofwe can identify the Hilbert spacesand so the trace ofmakes sense.This is the partition function we actually want to associate with the mapping torus:

but we need to make this more precise.By writing bϕ as a sequence m we can construct(4.35)associated to,then taking the trace seems straightforward:Identifyand then integrate over p(0).If we do such a thing,we will get a divergent result in general.The reason for this is that the integrand will be independent of some directions in the p(0)variables.This is expected and was already noted for example in[16,26]in the context of state integrals models.These directions have actually a very important meaning.At the level of the algebra AQwhat is happening is that its center is not trivial,some combinations of the Yivariables belong to the center of AQ,i.e.,they have vanishing commutator with allThere is a simple way to characterize these elements,which we denote by

When we are considering Mϕcorresponding to a knot complement,the geometric interpretation of Lαis that they should correspond to the longitudinal holonomy of the knot(see[3,43])19Note that,by definition.This means that we are working on the polarization where the longitudinal holonomy eigenvalues are taken as position variables.So,our partition function should be interpreted as

here,we have written the inserted delta functionsformula

Before closing this section we remark on a few important properties of(4.39).One can in principle writecorresponding to a different choice of polarization ofWe will not analyze this problem further here,but for instance,in[3]there is a more detailed study of the case when the chosen polarization corresponds to using the meridian holonomy eigenvaluesas position variables,therefore corresponding to a partition functionThe partition function(4.39),as constructed,using FG coordinates,is expected,and is shown,in various examples in[3],to be equivalent to the state integral model partition function studied in[15,28].However,this is only valid when we construct(4.39)starting from the quivers defined in[36].The construction presented here carries on for general quivers and sequences of mutations.This fact and some other properties of(4.39)makes it very well suited for applications which are not clear how do they carry on for the state integral models of[15,28].This will be the subject of the next section.

5 Applications of Cluster Partition Function

In this section we will give an overview of current and potential applications of the cluster partition function in(4.39).As a general remark,we emphasize that(4.39)can in principle be defined for any quiver Q and a given sequence of mutations of Q which lefts the yivariables associated to Q,invariant(and,for defining the expectation value(4.33)and(4.35)this last condition is not even necessary).So,we expect to have a much more broad range of applicability than just 3-manifolds Mϕ.

5.1 Perturbative invariants

We want to claim that when Mϕis a hyperbolic knot complement,then we can get perturbative invariants of knots21However,in general,we expect that we can do the same for any 3-manifold that can be obtained from a mapping torus construction..In order to study the perturbative invariants defined by(4.39)we proceed as in[16].First we determine the critical points that we will use to expand around.Introduce the following shortcut notations:

where p(L):=p(0)and recall that

so,the critical points of V(x;L)(i.e.,x such that∂xV=0)will be determined by the following equations:

since this change of variables plus renaming doesn’t introduce any extra factors(the determinant of the Jacobian of this transformation is 1),we obtain

where V(x;L)in(5.8)is understood as the potential after the change of variables we just defined.Now we can proceed with the perturbative expansion.For this we writewhereis a small perturbation and then use(6.8)to write

With this information,from(5.9)we can read the perturbative expansion ofand indeed,one can see that it takes the form(2.27)where the index α in(2.27)is labelling the choice of critical point x(c).The situation is very analogous to that from[20],so we can use their results.We first read the lower~degree terms in(2.27).Given a Laurent series f(~)on~,denote coe ff[f(~),~a]the coefficient of~ain f(~).Then,the first terms in(5.9)read

The terms Snwill be extracted from a sum of connected graphs.Consider GΓa connected graph with vertices of valences k ≥ 1.Then,we associate a weight to GΓ:To each k-vertex we associate a factorand a label tiand to each internal line connecting two vertices with labels tiand tja factorThen we can define the weight associated to GΓ:

Given a connected graph GΓ,it is easy to see that WΓ(GΓ)is of order~−V+Eor higher,where E is the number of internal lines and V is the number of vertices with valence k ≥ 3 in GΓ.After some computation one can show that E=V+L+V1+V2−1,where L is the number of loops and V1,V2are the number of 1-and 2-vertices respectively.Then,if we define

then

This concludes our analysis of perturbative invariants defined fromWe point out that in[45],was associated with topological invariants of knots.We expectto provide with novel invariants of knots or cluster variables in general.

5.2 Exploring other branches of M fl at

As previously mentioned,the partition functionconstructed by using the quiver and coordinates specified in[36]is expected to correspond to a subspace ofThis is re flected in the fact that the partition functionis not the most general solution to(3.6)whenare operators obtained by quantization of the polynomial equations defining the classical solutions to the CS equations of motion.For the case of knot complements and N=2 there is only one equation A(l,m)=0 given by the A-polynomial of the knot(see[18]).All the A-polynomials have a factor(l−1)where l is the eigenvalue of the longitudinal holonomy.Even though is not yet proven rigourously,there is overwhelming evidence that for these cases the state integral model of[28](and,hence the cluster partition function with FG quiver),as further analyzed in[20]is only missing the branch given by l=1,also known as the‘abelian branch’,i.e.,all these partition functions are annihilated by a quantization ofFrom a physics perspective,it is expected that one can construct a partition function which gets contributions from all branches(see[1,39,46–47]).A proposal for a theory,in the case of N=2,which can contain all the possible branches was analyzed in[47].

However,even at the classical level,analyzing the space of flat SL(N,C)-connections for N>2 is a considerably more difficult task and was carried on only in some specific cases(see for instance[48–49]).Therefore,a quantization is not,in general,available.

For N>2 and higher,the problem of missing branches is even more severe.The abelian branch is still missing but also there new missing branches and they grow as N grows(see for example section 7.3 in[15]).There is no systematic analysis on a precise characterization of these missing branches,but physics gives us a guide:For knot complements,the longitudinal holonomies ρ are labelled by an embedding of SU(2)→ SU(N)(see[50]),and this corresponds to partitions of N of the form(we pick an ordering na≥na+1):

Note that we are using ρ to denote a holonomy as well as a partition of N.In other words we are fixing ρ inThese different choices of ρ are known in the physics literature as‘codimension 2 defects’.For the FG quiver case,this corresponds to the choice ρ =[1,···,1].For N=2,the only option is ρ=[1,1],but for N>2 more options arise and at the time this note is written.There is no known systematic way to compute partition functions of these other cases in general.One of the main claims in[3]is that the branches corresponding to different choices of partitions of N can be implemented by changing the quiver Q in[3],some quivers are worked out for some families of ρ’s,taking as inspiration the work of[51].Some very nontrivial consistency checks are preformed,but it is still an open question:How to determine the quiver corresponding to a given ρ in a systematic way?Given ρ there are some zeroth order conditions such that one can ask for a quiver Qρto be associated to it,these are

where lα=1,2,···denotes the number of times which α appears in ρ =[n1,···,ns],|Qρ|is the number of vertices of the quiver and

Therefore we expect that for a quiverdepends on parameters Lαwith α =1,···,|Ker(Qρ)|.Formulas for more general Riemann surfaces Σ are given in[3].We should remark that by this method,it is still unclear:How to obtain a partition function capturing the abelian branch?Another open problem is:Given a proposed quiver Qρwith ρ 6=[1,···,1],how to find sequences of mutations implementing the action of MCG(Σ)on the corresponding yicoordinates?This is done in some examples in[3],but a general,systematic method is lacking.

5.3 Wilson loop insertions

The insertion of Wilson loops along knots for CS theories with compact gauge group H,reviewed brie fly in Section 2,is a well established and an active subject of research in physics and mathematics,starting with the pioneering work(see[4])where the relation between expectation values of Wilson loops along knots K embedded in S3is related to the(colored)Jones polynomial of K.A Wilson loops is determined by the isotopy class of the knot K and a unitary representation R of H.More precisely,the Wilson loop operator is given by the trace on R of the holonomy of the connection along K:

where P stands for the path ordering operator.The un-normalized23We will work here with un-normalized expectation values,i.e.,we will not divide them by the partition function.expectation value of WR(K)is then computed by

Alternatively one can consider CS theory on the knot complement S3K and fix boundary conditions in terms of the holonomy of A along a non-trivial cycle of the boundary torus.There is a one-to-one correspondence between boundary conditions and expectation values of Wilson loops for the case of compact gauge group(see for example[9]).

In contrast,for the case of complex gauge group G,the correspondence between boundary conditions and Wilson loop expectation values is not completely well established.For instance,when G is noncompact,infinite dimensional representations exist that don’t come from a lift of a unitary representation of a real form of G.For a discussion about this in the case of G=SL(2,C)see[6]and[25].Motivated by physics considerations we would like to consider the problem of computing CS partition function,for gauge group G,on a knot complementin the presence of a Wilson loop along a nontrivial cycleFormally,this corresponds to computing

where R is the holomorphic lift of a finite dimensional representation of H⊂G,where H is a maximal compact subgroup of G.In the cases we consider that our knot complement comes from a mapping torus construction:and consider its natural lift to π1(M).When working in the FG coordinates associated to G=SL(N,C)and,there is a prescription to compute the holonomy Holγ(A),up to an SL(N,C)gauge transformation which acts by conjugation on Holγ(A)(see[36]).Therefore,we can compute the classical value of TrR(Holγ(A))in terms of the yicoordinates.It takes the following form:

The function WR(γ)can,in principle,be promoted to an operator in terms of yiand the quantization parameter q.For this class of Wilson loops this problem was addressed in[37]and the answer takes the form:

which can be reduced using the same techniques that we applied to the case without the insertion of WR(γ).Here we present the case of just one exponential expeasily obtained by linear extension:

for the details on the derivation of this result we refer the reader to[3].Notice that we can carry on a perturbative analysis in a similar fashion as in the previous section.First note that(5.27)can be written as

This is exactly the same(5.8)with the addition of the termwhich is defined by

Therefore,the critical points x(c)are the same as in the previous analysis without the insertion of the Wilson loop.Then,we can write the partition function around x(c),as in(5.9):

In order to write the modified invariants,which we denoteit is convenient to perform the shift

In particular,we denote the shift ofby

then,the perturbative analysis can be carried on in a completely analogous way as before,but with modified vertices:

Note that the number of terms infor a fixed degree in~,is finite.

A couple of remarks are in order.Consistency of this result in some nontrivial cases,was analyzed in[3]and it was found that it agrees with physics predictions.There are still various natural generalizations of this result that one can try to address.As we mentioned,the construction of WR(γ)relied on the fact that there exists an explicit expression in terms of the FG coordinates.However,for more general quivers associated with other regions ofsuch a result is not available.Evidently,for a general quiver,one can write an operator of the formin terms of the corresponding cluster coordinates,but its interpretation is not clear.Another possible generalization is changing γ by a more general cycle in π1(M).Explicit expressions for holonomies of cycles in π1(M)exists(see[15])but their quantization is not known,so,is not clear how to promote them to operators.Ultimately,the goal will be to take γ being another knot,linked to K,but it seems there are many more fundamental problems to solve before we can take that step.

6 Definition and Some Properties of Quantum Dilogarithm Function

The quantum dilogarithm function eb(z)was originally defined in[52–53]and for further properties as well as connection with other special functions we refer to some papers relevant in our context:[16,54–55].By defining the symbol

we can define eb(z)as quotient of infinite products:eb(z)is a meromorphic function for all values of b such thatwith zeroes and poles at

and its most relevant properties for us are,the periodicity:

and its Fourier transform:

Finally,let us remark on the asymptotic expansion of(see[16])

here Bnare the Bernoulli numbers and we are using the convention.The polylogarithm function Lis(x)is defined by

and the function Lis(x)for s<0 is given by

where S(n,k)are Stirling numbers of the second kind.

AcknowledgementsThe author would like to thank D.Gang,N.Kim and M.Yamazaki for collaboration in the works[3,56]in which this note is based.He also thanks M.Gabella and S.Lee for useful discussions.