Topological field theory approach to intermediate statistics

Random matrix models provide a phenomenological description of a vast variety of physical phenomena. Prominent examples include the eigenvalue statistics of quantum (chaotic) systems, which are conveniently characterized using the spectral form factor (SFF). Here, we calculate the SFF of unitary matrix ensembles of infinite order with the weight function satisfying the assumptions of Szeg\"o's limit theorem. We then consider a parameter-dependent critical ensemble which has intermediate statistics characteristic of ergodic-to-nonergodic transitions such as the Anderson localization transition. This same ensemble is the matrix model of U(N) Chern-Simons theory on a three-dimensional sphere, and the SFF of this ensemble is proportional to the HOMFLY invariant of (2n,2)-torus links with one component in the fundamental and one in the antifundamental representation. This is one of a large class of ensembles arising from topological field and string theories which exhibit intermediate statistics. Indeed, the absence of a local order parameter suggests that it is natural to characterize ergodic-to-nonergodic transitions using topological tools, such as we have done here.


Introduction
1.1 Random Matrix Theory in disordered and complex systems: brief overview The idea of Wigner [1] to describe complex physical systems treating its Hamiltonian matrix as random has found since then a wide variety of applications.One of the main interests and challenges of modern theoretical physics to which random matrix theory has been very successfully applied is the description of interacting many-particle systems subject to a certain degree of randomness.Physically, this randomness is often caused by a true physical disorder, originating for instance from irregularities in a crystal lattice or by the presence of impurities.One can also have auxiliary phenomenological randomness representing the fact that the interactions in the system are too complicated to be described in microscopic detail, which is the case, for instance, for heavy nuclei.Further, quantum noise induced when a system is in contact with an external bath is a source of a temporal randomness.Random matrix theory (RMT) allows one to deal with such problems on a phenomenological level.This theory cannot answer questions about the microscopic details of a system, but it focuses instead on universal relations and scaling properties of relevant quantities.Indeed, one of the main results of RMT is the existence of universality classes (see [2] for survey), in which the symmetry of the system determines the class and, consequently, the statistical properties of the energy spectrum.
RMT models disordered and/or complicated Hamiltonians as matrices with random elements distributed according to a certain probability.Certain general physical symmetries (like time-reversal symmetry) provide restrictions on how the matrix elements are correlated.This leads to a different classes of random matrices [3], see the classic book by Mehta [4] and a contemporary overview of RMT by Forrester [5].Here, we will consider ensembles of Hermitian or unitary matrices, in particular, their eigenvalue statistics.One of the questions RMT deals with is the statistics of the GUE, which is an ensemble of Hermitian random matrices H with Gaussian weight function.This entails that its eigenvalues are distributed according to a U pN q-invariant Gaussian probability distribution P pHq " expr´αTrV pHqs, where V pHq " H 2 and α is a real positive parameter.Other classes correspond to ensembles of real symmetric matrices, with the probability measure being invariant under orthogonal transformations, or self-dual Hermitian matrices with probability distribution invariant under symplectic transformations [4].Another notable generalization is the notion of circular Random matrix Ensemble (RME), where the eigenvalues are distributed across the complex unit circle instead of the real line.Although many properties are common to the Gaussian and circular ensembles, certain objects are easier to calculate in the circular case, which is why these ensembles are the focus of this paper.
For typical systems, which obey a so-called Eigenstate Thermalization Hypothesis (see [6], [7] for a recent review), almost every energy level contains "seeds" of thermal behavior (even for isolated systems) leading to the chaotic nature of the RMT statistics.Therefore, quantum states belonging to this type are called ergodic.The delocalized (in the context of disorder or chaotic phase is described by Wigner-Dyson statistics, P psq " s β e as 2 , where s is the difference between consecutive energy levels, β " 1, 2, 4 for the unitary, orthogonal and symplectic cases respectively and a is a constant.As the strength of randomness increases, one can encouter a transition to the situation where states of a system are going to be localized in some basis.This could be a basis of states relevant for the description of localization in real space (Anderson localization) or in the Hilbert space (many-body localization).Deep inside a localized phase, the behavior of the system is nonergodic and the RMT level's statistics follows a Poisson distribution, P psq " e ´s.This type of statistics is usually found in quantum integrable systems, where an infinite number of conserved charges significantly constrains the dynamics.

Intermediate statistics and corresponding RMT approaches
Quantum systems whose classical counterparts are somewhere in between ordered and chaotic have spectral statistics that exhibit a mixture of Wigner-Dyson and Poissonian features, which we will generally refer to as intermediate statistics.An important example of such a system is given by disordered electrons, where increasing the disorder strength leads to greater deviation from Wigner-Dyson universality.At the point of transition between extended and localized regimes the wave functions are multifractal [8], which entails that intersecting the wave function at various amplitudes gives a set of varying fractal dimensions depending on the amplitude.A natural question occurs: is it possible to unveil some universality, perhaps based on RMT, for the ergodic-to-nonergodic transition itself for a broad range of systems?Some works in the literature hint at this possibility.There were several proposals in this directions [9], [10], [11], [12], [13], [14], [15], [16].Since Anderson transition occurs in real space, the RME symmetry should be broken in some way: this is a general feature required for the RMT to describe the transition.One obvious class of RMT's should therefore has a manifestly broken symmetry.A notable example of these theories are the banded, non-invariant RMT's [8].The probability distribution P pHq " expp´ř i,j |H ij | 2 {A ij q is defined by the variance matrix A ij " r1 `pi ´jq 2 {B 2 s ´1, which is clearly non-invariant with respect to the unitary transformations of the form H Ñ U HU : .It was explicitly demonstrated that this ensembles describes an intermediate statistics and the multifractal wave functions [17].
However, one can also have intermediate statistics in ensembles where the symmetry is not explicitly broken, i.e. for which the measure is invariant with respect to the transformations from the corresponding group.We focus on these ensembles here.Generically speaking, one can classify ensembles according to the asymptotic behaviour of the confining potential.Let us consider a power-law asymptotic scaling, V phq " |h| α for |h| " 1 .If the exponent α satisfies α ą 1, we talk about steep confinement.When on the other hand α ă 1, we deal with a weakly confined Random Matrix Ensemble.A particular weakly confined RME may be obtained from the generic one by a limiting procedure.
Consider a potential of the form V α phq " γ ´1h ´2p|h| α ´1q 2 for large |h|.In the limit α Ñ 0 at fixed h, we find the following confining potential which shall be called log-Gaussian critical RME (or a log 2 -RME) [18].It was realized that several classes of invariant RMT's, such as (1) exhibit intermediate statistics in terms of eigenvalues and multifractal behavior in terms of statistics of its eigenfunctions.Remarkably, both the spectral statistics and eigenvector multifractality at the mobility edge were found to match the matrix ensemble prediction at the exact same value of q [19].This behavior is somehow reminiscent of the spontaneous symmetry breaking conjectured in [19], [20].
The intermediate RME exists in a 'circular' guise, i.e.where the matrices under consideration are unitary instead of Hermitian, so that the eigenvalues are on the complex unit circle.In this case the potential is given by which, upon exponentiating, is proportional to the third Jacobi theta function.More details are given in section 3.2.Again, due to the fact that certain expressions are more tractable in the circular case, we focus on this representation.

Connection to topological field and string theories
The intermediate RME described above was also found in a completely different context, namely, as a matrix model of U pN q Chern-Simons theory S 3 [21].Chern-Simons is a topological theory, indeed, Witten famously showed that its Wilson line expectation values are given by knot-and link invariants [22].We suspect that it is not a coincidence that the matrix model of a topological theory has intermediate statistics characteristics of ergodic-to-nonergodic transitions.Indeed, the absence of a natural local order parameter in ergodic-to-nonergodic transitions suggest that it is natural to use topological tools for its characterization.
There is, in fact, a relation between strongly Anderson-localized systems and noninteracting topological states [23].One of the most notable features of topological states of matter is the existence of propagating edge states, which are robust with respect to the application of arbitrarily strong perturbations at the boundary that break translational symmetry (e.g.disorder).The existence of extended, gapless degrees of freedom in strongly random fermionic systems is unusual, because of the phenomenon of Anderson localization.Thus, the degrees of freedom at the boundary of topological insulators (superconductors) must be of a very special kind, in that they entirely evade the phenomenon of Anderson localization.The problem of classifying all noninteracting topological insulators in d spatial bulk dimensions is equivalent to a classification problem of Anderson localization at the pd ´1q-dimensional boundary.Therefore a 10-fold classification scheme of noninteracting topological insulators [24] is equivalent to the Altland-Zirnbauer classification of (noninteracting) Anderson insulators [25].This correspondence however does not describe transition from ergodic to nonergodic phases.This begs the question: can the nonergodic phases and ergodic-to-nonergodic phase transitions be generally related to certain interacting topological states of matter?
Indeed, U pN q Chern-Simons theory is such an interacting topological system which describes ergodicto-nonergodic transitions.We conjecture that it is representative of a broader correspondence, and that the relevant tools are available in the topological part of the string theory.This provides a potential new bridge (apart from the AdS/CMT duality) between string theory and quantum many-body theory, from which a fruitful exchange of ideas can arise.This is the main motivation of our paper.
To further substantiate this claim, we note that close inspection of matrix model potentials which appeared in the context of topological strings (see e.g.[26], [27], [28]) shows that many, if not all, of them belong to the class of weak confinement potentials.As described above, weak confinement is a sigature of intermediate statistics.On the other hand it appears that many, if not all, of the known intermediate invariant RMT's that appeared in the condensed matter literature and which show a multifractal spectrum can be described by some of the variants of topological string theory.
In the simplest case of the Chern-Simons matrix model, the connection to string theory arises from the finding by Witten [29] that a CS U pN q theory on S 3 describes open topological strings on the co-tangent space T ˚S3 , in the presence of N D-branes wrapping S 3 .Later, Gopakumar and Vafa [30], [31] found that these models correspond to closed topological strings on other spaces, called conifolds.This correspondence was named geometrical transition between a so-called A and B models and is one of the manifestations of the gauge-gravity duality (see [32] for an extensive review).In the N Ñ 8 limit, whcih we focus on here, U pN q Chern-Simons theory on S 3 undergoes a so-called crystal melting transition [33], which is related to topological strings on certain Calabi-Yau manifolds [34].We conjecture that matrix models with a similar origin in topological string theory, such as those of U pN q Chern-Simons theories on general lens spaces or or ABJM theory, also exhibit intermediate statistics.

Summary of main results
To clarify the connection between intermediate RME and topological string theory, we set out to calculate the asymptotic Spectral Form Factor (SFF, defined explicitly in the next section) for the Chern-Simons matrix model.The SFF is one of the central objects in RMT, it has clear features which differentiate between ergodic and nonergodic behaviors.While our original motivation was the intermediate Chern-Simons matrix model mentioned above, the techniques we apply have far broader applicability.In particular, they can be applied to any matrix model with unitary matrices of infinite order and weight function satisfying the assumptions of Szegö's limit theorem [35].For this reason, we treat both the general and the specific cases, so that certain sections may be skipped depending on the interests of the reader.
• Spectral Form Factor To be more specific, we calculate the SFF for these matrix models by expressing it as a sum over weighted unitary integrals with the insertion of the Schur polynomials, which take the form of certain Toeplitz minors [36], [37], [38], [39].We assume we can write the weight function as f pzq " Epx; zqEpx; z ´1q or f pzq " Hpx; zqHpx; z ´1q, where Epx; zq pHpx; zqq is the generating function of elementary (homogeneous) symmetric polynomials.We find that the SFF is then given by where p n pxq are power sum polynomials.SFF's are typically characterized by what has been termed a dip-ramp-plateau shape, see e.g.[40], [41], [42], [43].We find that the dip arises from the disconnected SFF, i.e. xtrU n y 2 " p n pxq 2 .The factor n which saturates at n{N " 1 gives the ramp and plateau; this contribution arises from the connected SFF.

• Trace identities
From the calculation of the SFF, it is easy to show that, for m, n P Z Further, for a partition λ satisfying λ 1 `λt 1 ´1 ă n for some n P Z `.Then, Consider sintead the case where λ instead satisfies λ 1 `λt 1 ´1 ă n, and define m :" λ 1 `λt 1 ´1´n.Then, if m ď λ 1 ´λ2 and m ď λ t 1 ´λt 2 , (5) still holds.

• Dualities
It is easy to see that, upon replacing Epx; zq by Hpx; zq, we find exactly the same SFF.Indeed, for any set of variables x for which pp n pxqq 2 gives the same value for all n, f pzq " Epx; zqEpx; z ´1q and f " Hpx; zqHpx; z ´1q gives the same SFF.We suspect that this is an example of a larger class of dualities between various intermediate RME's.
• Application to Chern-Simons RME We apply these results to the matrix model with weight function given by the third Jacobi theta function, f pzq " This is the matrix model described above, which was introduced in [13] as a phenomenological model of intermediate statistics, and in [44] as [44] as a matrix model of U pN q Chern-Simons theory on S 3 .In the latter context, the SFF is given by a topological invariant, specifically, the HOMFLY invariant [45], of p2n, 2q-torus links with one component in the fundamental and the other in the antifundamental representation.As far as the authors are aware, these invariants have heretofore not appeared in the literature.As for all matrix models considered here, the SFF is given by a linear ramp which saturates at a plateau, plus a disconnected contribution.
Since the SFF corresponds to a p2n, 2q-torus link, it follows that the disconnected contribution is the product of two pn, 1q-torus knots.Calculating the invariant of an pn, 1q-torus knots for general N , we find that it is given by the q n -deformation of N , which simplifies even further upon implementing the limit N Ñ 8. We thus find the following expression for the SFF We plot this below for q " 0.9 k , k " 1, . . ., 9, where we add lines at x `pq x `q´x ´2q ´1 for continuous x as a guide to the eye.The trace identities in (4) and ( 5) of course apply to the Chern-Simons matrix model as well, where the latter entails that one can 'unlink' an pn, 1q-torus knot in the fundamental representation and an unknot in representation λ.

◆ k=1
Figure 1: The SFF given in (106) plotted for n " 1, 2, . . ., 19, with q " 0, 9 k , k " 1 . . ., 9. The continuous lines are added to guide the eye.For q farther from 0, the disconnected contribution becomes larger, so that the dip is more pronounced and the SFF displays greater deviations from a simple linear ramp.There are dashed lines which indicate kn = constant, in particular kn " 1, . . ., 9.
From (93), it follows that lines with kn = constant lie at 45 degrees for any SFF calculated here, i.e. any SFF given by (89).Note that these SFF's saturate at a plateau at n{N " 1, which is, of course, not indicated in this plot.

Outline of the paper
The sections are organized as follows.In section 2, we set up the general framework of random matrix ensembles and introduce important objects including the SFF.In section 3, we treat U pN q Chern-Simons theory on S 3 and its expression as a matrix model, after which we consider the expression of knot and link invariants as matrix integrals.In section 4, we review the computation of such matrix integrals using their expression in terms of Toeplitz minors.These Toeplitz minors, in turn, are given by symmetric polynomials in terms of variables determined by the weight function.We then express the assumptions of Szegö's theorem as requirements on these symmetric polynomials, in particular the power sum polynomials.Further, we find in this section that, although the expression in this work are generally valid for N Ñ 8, in certain cases they are valid for finite N as well.
In section 5, we set out to compute the SFF using the techniques outlined in the previous sections.
Using fundamental relations in the theory of symmetric polynomials, we derive the results for general weight function outlined in the previous subsection.The specific case of the SFF of the Chern-Simons matrix model is worked out in section 5.2.We then consider the broader implications of these calculations in the concluding remarks.In the appendices, the reader can find more details about q-deformations and symmetric polynomials, with special attention given to Schur polynomials.

Random matrix theory
We will consider random matrix ensembles, which have partition functions in the form of a matrix integral, ż dM P pM q .
Here, P pM q is the probability density function associated to M .Consider first the case where the matrices M are Hermitian, so that they can be diagonalized by a unitary transformation.Integrating over U pN q leads to an eigenvalue expression of the form [4] Z where C N is some multiplicative constant and wpxq is called the weight function.Choosing where α is some positive numerical constant, leads to the familiar Gaussian unitary ensemble (GUE) with weight function wpxq " expp´αx 2 q.This ensemble is characterized by fully extended eigenvectors and strong eigenvalue repulsion, which we will collectively refer to as Wigner-Dyson statistics.It was conjectured in the 1980's [46], [47], [48] that the eigenvalues of quantum systems whose classical counterpart is chaotic exhibit Wigner-Dyson statistics (after an unfolding procedure, which is to say, a rescaling of the energies such that the average inter-energy spacing is equal to unity).This conjecture has been so extensively corroborated that Wigner-Dyson statistics are nowadays seen almost as a definition of quantum chaos.
We will also consider ensembles where the matrices of the ensemble are themselves unitary, first introduced as so-called circular ensembles by Dyson [3].Being unitary, the eigenvalues of these matrices are distributed across the complex unit circle.Such unitary ensembles have a partition function of the form where we denote the matrices under consideration by U .For wpx i q=constant, (11) reduces to Dyson's circular unitary ensemble (CUE). in the limit N Ñ 8, the CUE and GUE exhibit the same bulk statistics after unfolding, i.e. the CUE also described systems whose classical counterpart is chaotic [4], [49].
While the Wigner-Dyson ensembles described above provide excellent phenomenological descriptions of quantum chaotic systems, they, naturally, fail to describe systems with intermediate spectral statistics.
An example of such a system consists of disordered electrons at the mobility edge of the Anderson localization transition [50], [8].Muttalib and collaborators introduced a family of random matrix ensembles [13] depending on some parameter 0 ď q ď 1.This matrix ensemble appears in two guises, analogous to GUE and CUE.In case the matrices under consideration are Hermitian, the weight function is of the following "log-squared" form In the expression above, we define q ": e ´gs , where g s is the string coupling constant in the manifestation of Chern-Simons theory as a topological string theory on the cotangent space.The domain of wpxq in ( 12) is the positive real line.In case the matrices we consider are themselves unitary, the weight function is given by wpφq " Θ 3 pe iφ ; qq " That is, the weight function is given by Jacobi's third theta function, which is defined on the complex unit circle.

Density of states and spectral form factor
An important object in random matrix theory is the density of states, given by where we used the fact that The density of states, averaged over the matrix ensemble, gives the probability of finding an eigenvalue at φ. From these level densities, we can construct the n-point density correlation functions for n " 2, . . .and various related quantities.An important example thereof which is often used to characterize the eigenvalue statistics of various ensembles is the spectral form factor (SFF), which is the Fourier transform of the density-density correlation function [4].Starting from the density-density correlation function, The SFF is then given by the expansion coefficients of e inpθ´φq , n P Z `, rescaled by a factor N , [4], [49], The choice of normalization is made so that the CUE SFF saturates at unity.For future convenience, we also define the connected part of the SFF For the CUE and GUE, the SFF is characterized by a linear ramp which saturates at n " N .For intermediate statistics, Kpnq displays deviations from this behavior which will be detailed below.
3 Chern-Simons matrix model and knot/link invariants

Knot operator formalism
We review the construction of Chern-Simons partition functions and knot invariants using Heegaard splitting [22] and knot operators [51].Heegaard splitting provides a way to calculate the Chern-Simons partition functions of certain three-manifolds, which we denote by M .We construct M by taking two separate three-manifolds M 1 and M 2 which share a common boundary Σ, i.e.BM 1 » Σ » BM 2 .M is then constructed by acting on the common boundary Σ with some homeomorphism f and then gluing M 1 and M 2 together, which we write as In this construction, we take the boundaries of M 1 and M 2 to have opposite orientation, so that M is a closed manifold.Writing the Hilbert space of Σ as HpΣq and its conjugate as H ˚pΣq, performing the path integral over M 1 gives a state |Ψ M1 y P HpΣq, whereas performing the path integral over M 2 to find a state xΨ M2 | in the conjugate Hilbert space H ˚pΣq due to the fact that the boundaries of M 1 and M 2 have opposite orientation.The homeomorphism f induces a map U f on HpΣq whose action we denote by The partition function is then given by In a seminal paper [22], Witten found that HpΣq is the the space of conformal blocks of the corresponding Wess-Zumino-Novikov-Witten (WZNW) model on Σ at level k.In case there are no marked points on Σ where Wilson lines are cut, i.e. if all Wilson lines can be embedded on Σ, HpΣq is given by the characters of the WZNW model on Σ.We will be considering only the latter case.
A relatively simple example of a Heegard splitting is given by the division of S 3 into two three-balls that share a boundary Σ " S 2 .The only knot that can be embedded on S 2 is the unknot, which is the trivial example of an unknotted circle.We therefore do not consider this example any further.Let us instead consider the case where M 1 and M 2 are given by solid tori S 1 ˆD2 which share a boundary torus BM 1 " S 1 ˆS1 " BM 2 .The manifolds which can be constructed via such a Heegaard spltting on a torus are known as lens spaces [52].The simplest example of a lens space is found by taking f to be the identity map.In this case, we glue the two copies of D 2 along their boundaries to form S 2 , so that the resulting space is given by S 2 ˆS1 .We normalize the Chern-Simons partition function for S 2 ˆS1 to unity.Let us consider an example where we act on T 2 with a nontrivial homeomorphism.
The group of homeomorphisms of T 2 is given by SLp2; Zq, which consists of matrices of the form SLp2; Zq is generated by the modular S and T -transformations.Representing the 1-cycles of the torus by basis vectors ˜1 0 ¸and ˜0 1 ¸, the S and T -transformations can be written as That is, S interchanges the 1-cycles and reverses the orientation of the torus, while T cuts open the torus along a 1-cycles to form a cylinder, twists one end of the cylinder by 2π, and glues the two ends of the cylinder back together.Consider the case where we glue two solid tori M 1,2 along their boundaries after acting with an S-transformation.Since S-transformations exchange the 1-cycles on the torus, the contractible cycle of M 1 is glued to the non-contractible cycle of M 2 and vice versa.We thus find a closed three-manifold with no non-contractible cycles which, due to the Poincaré conjecture, is homeomorphic to S 3 .
The construction of torus knots is analogous to the construction of lens spaces in the sense that, if we insert a Wilson line corresponding to an unknot on the boundary torus, we can act with arbitrary SLp2; Zq transformation on the torus which turns the unknot into a non-trivial torus knot.Let us denote the torus knot operators, to be defined more precisely below, by W pp,qq λ , where λ labels the irreducible representation of the Wilson line and p and q are integers which count the winding of the knot around non-contractible and contractible cycle of the torus, respectively.Note that p and q are coprime for torus knots, whereas for p and q not coprime we would get a torus link, which is a generalization of a torus knot with more than one component (i.e. more than one knotted piece of string).The number of components of a torus link equals the greatest common divisor of p and q.
From the definition of the S and T -transformations, it is clear that they act on torus knot as follows S ´1W pp,qq S " W pq,´pq , T ´1W pp,qq T " W pp,q`pq .
For example, if we insert an unknot around the non-contractible cycle of the torus and act n times with the T -transformation, we get a knot which still winds around the non-contractible cycle once but which now also winds around the contractible cycle n times.Note that this is topologically still an unknot; the additional winding around the contractible cycle only gives rise to a multiplicative framing factor.Similar knots will play an important role in the comparison with random matrix theory, to be outlined below.
It is easy to see that modular transformations map the set of torus knots into itself, as these transformations do not change the number of components.Indeed, for any pair of coprime integers pp, qq, one can easily see that pp, q `pq are also coprime, so that the number of components is unchanged under modular transformations.Further, due to Bézout's lemma [53], there is an SLp2; Zq-transformation corresponding to any pair of coprime integers, so that we can construct any torus knot by acting on an unknot with an SLp2; Zq-transformation.The explicit form for the knot operators mentioned above was found by Labastida, Llatas, and Ramallo [51], using the relation to WZNW-models previously found by Witten [22].Let us summarize the salient points of the knot operator formalism.As mentioned above, HpΣq is given by the conformal blocks of the corresponding WZNW-model on Σ with group G at level k.In the case of Σ " T 2 without marked points, which we will be considering henceforth, HpΣq consists of the characters of integrable representations of the corresponding WZNW-model.We denote the set of fundamental weights by tv i u and Weyl vector by ρ " where y is the dual Coxeter number of G, which equals N for G " U pN q and N ´1 for G " SU pN q.
Remember that an irrep with highest weight Λ " ř i Λ i v i corresponds to a Young tableau where the length of the i th row is given by where I equals N in the case of U pN q and N ´1 in the case of SU pN q.See appendix 8.2 or e.g.section 13.3.2 of [54] for more background information on partitions and their role in representation theory.From now on we will take G " U pN q so that y " N .We will denote ket states corresponding to p by |py, which can be chosen in such a way that they form an orthonormal basis.
The only further ingredient we need are the explicit expressions for the Hilbert space operators induced by the modular transformations.We simply state these here, further details may be found in [51] T pp 1 " δ p,p 1 e 2πiphp´c{24q , In the above expressions, W is the Weyl group, pwq is the signature of Weyl reflection w, c is the central charge of the WZNW-model, and h p is the conformal weigth of the primary field corresponding to p, which is given by

Chern-Simons matrix model
Let us consider how the matrix model description of Chern-Simons theory arises.As explained above, S 3 can be constructed via a Heegaard splitting along a torus on which we act with an S-transformation.
We thus find that the Chern-Simons partition function on S 3 is given by ZpS 3 q " x0|S|0y " S 00 .
We plug in the expression for S 00 from equation (27) and use Weyl's denominator formula, where α are the positive roots of U pN q.Expressing the roots of U pN q in terms of Dynkin coordinates x i , we find Lastly, we define a new set of variables y i :" e N gs`xi , in which the partition function is given by [44] ZpS 3 q " e ´p7N 3 gs{12`N 2 gs{2´N gs{24q N !
Alternatively, we can use the following expression where we repeat the definition of the third Jacobi Theta function This gives where we added another summation over the Weyl group in the first equality and (33) in the second.
Lastly, the Weyl group W is isomorphic to the symmetric group S N so that |W | " N !. Using the Weyl denominator formula again leads to [34][55].
Note that ( 36) and ( 32) correspond precisely to the matrix ensemble introduced by [13], given in (13) and (12), respectively.Note also that, using the Jacobi triple product formula, Θ 3 can be written as a specialization of Epx; zq, the generating function of the elementary symmetric polynomials.We can also replace Epx; zq by Hpx; zq at the cost of transposing all representations involved in the calculation, this amounts to replacing Θ 3 by 1 Θ3 .Since the SFF for n ď N is invariant under transposition of all representations (see e.g. ( 78)), the calculation of the SFF done below is also valid for the case where the weight function of of the form 1 Θ3 .Indeed, the above argument applies to any specialization i.e. to any choice of variables x i .We will therefore replace Epx; zq by Hpx; zq in the computations below to avoid having to transpose all representations.

Computing torus knot and link invariants in the Chern-Simons matrix model
Now that we have looked at the Chern-Simons partition function, we consider knot and link invariants and their computation in the Chern-Simons matrix model.First, we consider the multiplication properties of knot operators.If we take W K λ to be a knot operator corresponding to a knot K in representation λ, we can write The coefficients N ν λµ in (37) are the fusion coefficients of the WZNW-model.When both k and N are much larger than any of the representations under consideration, N R R1R2 are given by Littlewood-Richardson coefficients.This allows us to construct the invariants of torus links.We label a torus link by P, Q P Z, where the number of components is given by S " gcdpP, Qq and the representations are labelled by j P t1, . . ., Su.These links are given by [51], [56], [57] S ź j"1 where N µ λ1,...,λ S are generalized Littlewood-Richardson coefficients appearing in the product of representations λ 1 b ¨¨¨b λ S .
We now outline the computation of torus knot and link invariants using the matrix model for U pN q Chern-Simons on S 3 .The simplest knot, the unknot, is given by the ensemble average of the matrix trace in the corresponding representation [58].That is, If we diagonalize a matrix U to give diagpd 1 , d 2 , . . ., d N q, it is well known that where s λ pxq is the Schur polynomial in variables x i corresponding to representation λ.The reader can consult appendix 8.3 or the book by Macdonald [59] or Stanley [60] for further information on Schur polynomials.In the remainder of this work, we will often write traces without specified representations, in which case the trace is taken in the fundamental representation.
In general, we can assign an orientation to a knot or component of a link, which corresponds to a continuous non-zero tangent vector along K.When we project a knot or link into the plane, we can assign a sign `or ´to each crossing, as in figure 3. We denote by λ the representation conjugate to λ.We then have [21] tr λ U ´1 " tr λ U .
in the language of knot theory, taking tr λ U to tr λ U ´1 corresponds to inverting the orientation of the component carrying representation λ.Of course, for the unknot, this does not matter, as reverting the orientation can be compensated by a parity transformation.The same is true for the Hopf link, as overcrossings can be freely changed into undercrossings.To convince oneself of this point, one can assign an orientation to both components of the Hopf link in figure 2, and rotate one component along an axis parallel to the projection plane whilst keeping the other component fixed.For more complicated knots or links, such as the p4, 2q-torus link on the right hand side of figure 2, overcrossings can no longer be turned into undercrossings and inverting the orientation of one component will generally lead to a different expectation value.Let us consider more complicated objects involving integer powers of U .
Generally, any product of traces of any GLpN, Cq matrix U , S α " ptrU q α1 ptrU 2 q α2 . . .ptrU s q αs , α i P N , can be expanded in characters of GLpN, Cq, denoted by χ λ pU q, with characters of the symmetric group S l as expansion coefficients, where l " ř i α i [61].If U P U pN q, the characters are given by Schur polynomials, see appendix 8.3 for more background.We then have where ř R is a sum over all Young tableaux with total number of boxes equal to l, and χ R pCp kqq is the character of the symmetric group S l in representation R evaluated at the conjugacy class of S l given by cycle lengths α 1 , α 2 , . . ., α k .Despite its concise notation, (77) is generally rather difficult to compute due to the sum over partitions of l.However, in certain cases the above expression can be calculated.Taking U P U pN q with eigenvalues d i and choosing α 1 " n and α i " 0 for i ‰ 1, we find [61] trU n " ÿ where we used the fact that characters of the symmetric group satisfy In words, (44) states that trU n is given by the sum over hook-shaped irreps with n boxes, which appear with alternating signs.One may recognize from (44) that this is the expression of the n th power sum polynomial in terms of Schur polynomials.For n " 4, one can express (44) in terms of Young diagrams as follows.
-+ -One can show [58], [62], [63] that xtrU n y gives the invariant of an pn, 1q-torus knot [64], which differs from any pn, mq-torus knot only by a framing factor.Equation ( 44) gives an expansion of of trU n in terms of Schur polynomials.Explicit expressions can be found in section 5.2.As noted above, an pn, 1q-torus knot is topologically equivalent to an unknot and differs only due to framing [58].
However, terms of the form xtrU n trU ´ny, such as appear in the SFF, give p2n, 2q-torus links, which are not topologically trivial for any n P Zzt0u.

Matrix integrals and Toeplitz minors
We review the computation of the unitary group integral over Schur polynomials using a method outlined in [38] and [39], which in turn draw from results derived by Bump and Diaconis [36], Tracy and Widom [37], among others.We start from an absolutely integrable function on the unit circle in We will specifically be considering the case where d k " d ´k, so that f pe iθ q is real-valued.We further require that f pe iθ q satisfies the assumptions of Szegö's theorem.That is, we write f pe iθ q as and demand that From the Fourier coefficients of f , we construct a Toeplitz matrix (i.e. a matrix that is constant along its diagonals) of infinite order, T pf q " pd j´k q j,kě1 .
We denote by T N pf q the N by N principal submatrix of T pf q, i.e. matrix obtained from T pf q by taking its first N rows and columns.We will see that various matrix integrals with weight function f can be expressed as minors of T N pf q, that is, as determinants of matrices obtained from T N pf q by removing a (necessarily equal) number of rows and columns.For a unitary matrix U with eigenvalues, we write e iθ1 , e iθ2 , . . ., e iθ N , We employ Weyl's integral formula [65] to express the integral of f pU q over U pN q with respect to the de Haar measure as where the angles satisfy 0 ď θ k ă 2π.The expression for the Vandermonde determinant in (134) allows us to use an identity due to Andreiéf, sometimes referred to as Heine or Gram identity [66].
Take g j and h j , j P t1, 2, . . ., N u, to be two sequences of integrable functions on some measure space with measure µ, then Choosing g j pe ´iθ q " e ipN ´jqθ " h j pe iθ q and dµpe iθ q " f pe iθ q dθ 2π , we find where d k are again the Fourier coefficients of f , Now let λ " pλ 1 , . . ., λ m q and µ " pµ 1 , . . ., µ n q be partitions of |λ| " ř pλq i λ i and |µ| " ř pµq j µ j , respectively.Here, λ i , µ j P Z `and p.q is the length of the partition.Ordering as λ i ě λ i`1 and similarly for µ j , these partitions label Young tableaux in the standard way.One then obtains a Toeplitz minor T λ,µ N pf q via the following procedure: • We start from T N `κpf q, where κ " maxtλ 1 , µ 1 u • If λ 1 ´µ1 ą 0, we remove the first λ 1 ´µ1 colums from T N `κpf q, otherwise we remove µ 1 ´λ1 rows.
• We then keep the first row and remove the next λ 1 ´λ2 rows, after which we again keep the first row and remove the next λ 2 ´λ3 rows and so on and so forth.
• We repeat the third step where we replace λ i by µ i and where we remove columns instead of rows Note that the second step ensures that the resulting matrix T λ,µ N pf q is N by N .We write s λ pU q " s λ pe iθ1 , e iθ2 , . . .q, where s λ are Schur polynomials, which we review in appendix 8.3.The determinant of T λ,µ N pf q can then be expressed as [36], [67] D λ,µ N pf q :" det T λ,µ N pf q " ż U pN q s λ pU ´1s qs µ pU q f pU qdU One can recognize the pattern of striking rows and columns involved in the construction of T λ,µ N pf q, as the index j is shifted to j ´λj and k to k ´µk .One can easily verify that, for two functions of the form ape iθ q " ÿ kď0 a k e ikθ , bpe iθ q " the associated Toeplitz matrix satisfies Let us therefore write f pe iθ q as follows f pe iθ q " Hpx; e iθ qHpy; e ´iθ q , where Hpx; zq is the generating function of the homogeneous symmetric polynomials h k given in ( 125) and where we assume that h k pxq and h k pyq are square-summable, i.e.
Gessel [68] showed that, for f as in (58), where one should note that the sum runs over all partitions ν with at most N rows.Here, we only consider the case where y " x P R, but the expressions here easily generalize to x ‰ y and x, y P C, subject to the assumptions of Szegö's theorem.Equation ( 60) can then be generalized as [38], [39] ż s λ pU ´1qs µ pU q f pU qdU " In the above expressions, we can replace Hpx; zq by Epx; zq if we simultaneously transpose all partitions.
Let us therefore consider the Jacobi triple product expansion p1 `qk´1{2 e iθ qp1 `qk´1{2 e ´iθ q " pq; qq 8 Epx; e iθ qEpx; e ´iθ q , where we define x " pq 1{2 , q 3{2 , . . .q in the last line and where Θ 3 is the Jacobi theta function.Choosing f pe iθ q " a pq; qq 8 Epx; e iθ qEpx; e ´iθ q and taking into account the notation established in ( 50), then f pU q is the weight function of the Chern-Simons partition function.This example is treated extensively in [39], more details and proofs can be found there.Using (53) with d k " q k 2 {2 , we see that the partition function is given by which is a well-known result true for general N .

Infinite N
Let us now take the limit N Ñ 8. From ( 61) and the fact that [Chapter I.
where the sums run over all partitions, we have [38], [39] W λµ :" Taking (65) with µ " H, we see that calculating the matrix integral of a single trace in some representation ( 55) is given by the following procedure.The Schur polynomial s λ pU q in (65) has the eigenvalues as its variables.The evaluation of the integral amounts to replacing the eigenvalues of the Schur polynomials by the variables x i in f pzq " Epx; zqEpx; z ´1q or f pzq " Hpx; zqHpx; z ´1q.Taking f pe iθ q equal to Θ 3 pe iθ q in (62), we have the following expression for the Hopf link expectation value where one should note that the representations are transposed due to the fact that Θ 3 pe iθ q is expressed in terms of Epx; zq rather than Hpx; zq.

Let us now consider what the assumptions of Szego's theorem imply for a function of the form f pzq "
Epx; zqEpx; z ´1q or f pzq " Hpx; zqHpx; z ´1q.Let us consider first the case f pzq " Epx; zqEpx; z ´1q.We repeat the top line of (125), so that Therefore, and ( 48) is written as where we ignore an irrelevant factor 2. We see that Epx; zq by Hpx; zq, we have, so that the assumptions of Szegö's theorem are given by (71) as well.

Finite N
Although the expressions given above were derived for N Ñ 8, some of them can in fact be generalized to finite N in case the number of distinct non-zero variables x j is smaller than N .From equations ( 60), (61), and ( 64), we see that, for finite N and f pzq " Hpx; zqHpx; z ´1q, ş s λ pU qs µ pU ´1q f pU qdU ş f pU qdU " empty set, as in point 2, we recover the disconnected part of the SFF, which is given by the square of n´1 ÿ r"0 p´1q r s pn´r,1 r q pxq " xtrU n y . (82) The remaining terms, coming from point 3, is given by the square of n´1 ÿ r"0 ÿ ν‰H ν‰pn´r,1 r q p´1q r s pn´r,1 r q{ν pxq (83) At first sight, this may seem like a rather rather complicated expression.Let us factor the expression in (83) into two separate sums over r and s and consider one such sum for a fixed choice of ν " p1q.
Remembering that s pn´r,1 r q{p1q " h n´r´1 e r and using equation (2.6') of [59], we find for a single such sum, Taking n " 4, the above identity can be expressed in terms of Young diagrams as follows. - The identity ř n´1 r"0 p´1q r h n´r´1 e r " 0 can be seen from Hpx; tqEpx; ´tq " 1, see equation (125).Equation (84) can then be found by checking every order of t in Hpx; tqEpx; ´tq.One can see from these considerations that any term corresponding to a single choice of ν in (83) is equal to zero.The contribution for general ν " pa, 1 b q Ă pn ´r, 1 r q with ν ‰ H and ν ‰ pn ´r, 1 r q is given by n´a ÿ r"b p´1q r h n´r´a e r´b " n´b´a ÿ r"0 p´1q r h n´b´a´r e r " 0 . ( The SFF is then given by That is, the SFF is given by a linear ramp and plateau, which arises from the connected expectation values, plus the disconnected component given by xtrU n y 2 .In terms of the so-called dip-ramp-plateau classification, we see that the dip arises fully from the disconnected expectation values, whereas the ramp and plateau arise fully from the connected part.
We now compute the disconnected SFF.We have the following expression for Schur polynomials of hook-shaped reps, see I.3 Example 9 of [59], This gives xtrU n y " n´1 ÿ r"0 p´1q r pn ´rqh n´r pxqe r pxq " p n pxq " The functions p n pxq are the power-sum polynomials, mentioned in appendix 8.2.The fact that we get power-sum polynomials should not be surprising due to the statements below equations ( 44) and (65).Namely, trU n is the n th power sum polynomial in the eigenvalues of U , and the evaluation of the matrix integral of a single matrix trace amounts to replacing the eigenvalues by the variables x i .This allows one to easily find, or, indeed, generate SFF's of various weight functions which can be expressed as f pzq " Epx; zqEpx; z ´1q or Hpx; zqHpx; z ´1q.Namely, for any choice of variables x " px 1 , x 2 , . . .q, we have Below equation ( 70), we show that the assumptions of Szegö's theorem require so that the disconnected part of the SFF goes to zero.Hence, we see that the plateau of the SFF is exact, that is Let us now give some basic expressions for the SFF itself.From the form of (86), we can give an expression for the behaviour of the SFF upon rescaling x i .The linear ramp remains unaffected by rescaling as it is independent of choice of variables x i .Further, since xtrU n y " ř n´1 r"0 p´1q r s pn´r,1 r q pxq is a sum of polynomials of degree n in x i , we have upon rescaling as x j Þ Ñ Ax j , where A is some number, Further, we take x j Þ Ñ px j q k with k P Z `, we have, writing This can naturally be generalized to k P R if we take the label n of p n pxq to be a general real number.
We plot an example of an SFF in figure 1.In the figure, we indicate lines with kn "constant, which lie at 45 degrees.Although this SFF was computed for a specific choice of weight function, it follows from equation (93) that lines of constant kn always lie at 45 degrees.The linear ramp then corresponds to kn Ñ 8.
For a finite number of variables, the calculation of the SFF from (89) is rather straightforward.In case we have a very large number of non-zero variables, p n pxq is generally rather hard to calculate, except for certain known examples.Let us take x k " 1{pk `1q 2 .Using the well-known product expansion of the hyperbolic sine as sinhpπtq " πt ś kě1 ´1 `t2 k 2 ¯, we have Further, we have, where ζpsq is the Riemann zeta function.The SFF for weight function ( 94) is therefore given by

General trace identities
We now consider some expectation values of trU n with some more general objects.For example we can conclude from the arguments leading to (86) that the connected part of xtrU n trU ´ky, for k, n P Z `, is given by In particular, let us take k ă n.In that case, any ν P pk ´s, 1 s q for all s P t0, . . ., k ´1u necessarily satisfies |ν| ď k ă n, so that pn ´r, 1 r q{ν ‰ H for any partition pn ´r, 1 r q, r P t0, . . ., n ´1u.Using (85), the result is again zero.Note that equation (97) can easily be found for the CUE case from bosonization [70][71].More generally, let us consider expectation values of the form Since fixing any ν Ď pn ´r, 1 r q in (98) with ν ‰ pn ´r, 1 r q gives zero upon summing over r, we only get a nonzero answer for terms for which ν " pn ´1, 1 r q Ď λ.That is, where the boundaries on the sum arise from the fact that we only sum over those representations pn ´r, 1 r q which satisfy pn ´r, 1 r q Ď λ.Equation (99) greatly simplifies certain calculations.For example, consider pn ´r, 1 r q Ę λ @ r P t0, . . ., n ´1u.Another way to write this is that λ 1 `λt 1 ´1 ă n.We then have, Let us represent λ in Frobenius notation as λ " pa 1 , . . ., a k |b 1 , . . ., b k q with a i and b j non-negative integers satisfying a 1 ą ¨¨¨ą a k and b 1 ą ¨¨¨ą b k .In this case, a 1 `b1 `1 gives the number of boxes in the upper left hook of λ, or, equivalently, the hook-length of the top left box in λ, labelled by x " p1, 1q in the notation of appendix 8.3.In this notation, (100) states that xtrU ´ntr λ U y c " 0 if a 1 `b1 `1 ă n.For the specific case of the Chern-Simons matrix model this identity has an interesting interpretation which we comment on in section 5.2.
We can find similar identities for certain representations λ with a 1 `b1 `1 ą n.Define m :" a 1 `b1 `1´n and consider λ " pa|bq " pa 1 , . . ., a k |b 1 , . . .b k q satisfying m ď a 1 ´a2 ´1 and m ď b 1 ´b2 ´1, or, equivalently, m ď λ 1 ´λ2 and m ď λ t 1 ´λt 2 , respectively.Let us take µ " pa 2 , . . ., a k |b 2 , . . ., b k q, which is constructed from λ by removing the first row and column.For any rep pn ´r, 1 r q satisfying pn ´r, 1 r q Ď λ, we then have λ{pn ´r, 1 r q " pa 1 `1, 1 b1 q{pn ´r, 1 r q ˆµ " pa 1 `1 ´n `rq ˆp1 b1´r q ˆµ . (101) That is, λ{pn ´r, 1 r q factorizes as the skew partition of two hook shapes times the partition obtained from λ by deleting the top-left hook.In terms of Young diagrams, an example is given by the following.

/ =
Since pa 1 `1, 1 b1 q{pn ´r, 1 r q is a product of a row and a column, we can again use (84) to find p´1q r s λ{pn´r,1 r q " p´1q n´a1´1 s µ m ÿ k"0 p´1q k h m´k e k " 0 . (102)

The SFF of the Chern-Simons matrix model
As noted before, the SFF of the Chern-Simons matrix model corresponds to a p2n, 2q-torus link with one component in the fundamental and the other in the antifundamental representation.Whereas expressions for link invariants of the form xtrU n1 trU n2 . . .y with n i ě 2 have appeared in the literature [69], [72], [73], expressions with powers of mixed signature, to the best of the authors' knowledge, have not.The expressions presented in the previous section allow us to calculate precisely those objects.
In particular, the SFF, is again given by (86).We can easily calculate the non-trivial part of the SFF, xtrU n y 2 , for |q| ă 1, by using the expression in terms of power-sum polynomials.However, it is instructive to see how this arises from the functional form of this object as a function of q.Let us apply (140) to the hook-shaped representation pa, 1 b q, which gives the following expression for general N s pa,1 b q px i " q i´1 q " q 1 2 bpb`1q rN `a ´1s! rN ´b ´1s!ra ´1s!rbs!ra `bs .
Note that, although we take N Ñ 8 here, we first calculate these objects for general N .This is particularly useful in knot theory, as the expression for general N may allows one to distinguish various knots and links which may have the same invariant when one ignores the dependence on N .
We now use ( 77) and ( 66) to calculate xtrU n y, which, for the lowest values of n, is given by xtrU y " q 1{2 1 ´qN 1 ´q " q 1{2 rN s q , @ trU 2 D " qp1 ´q2N q 1 ´q2 " qrN s q 2 , @ trU 3 D " q 3{2 p1 ´q3N q where one should note that the variables in (66) are given by x j " q j´1{2 .One can see a simple pattern emerge in (66).Indeed, using (77) and taking into account the comments made below (65), we see that xtrU n y " p n px j " q j´1{2 q " q n{2 N ÿ j"1 q npj´1q " q n{2 1 ´qnN 1 ´qn " q n{2 rN s q n .(105) That is, the asymptotic pn, 1q-torus knot invariant is given by the q n -deformation of N times a factor q n{2 .As far as the authors are aware, this statement has heretofore not appeared in the literature.
As mentioned above, as well as in appendix 8.3, the limit N Ñ 8 simplifies these expressions even further.This is due to the fact that q nN " 0 for n ě 1, so that quantum dimensions for reps with finite column lengths depend only on the hook lengths.Upon this simplification, the final expression for the SFF is then given by The SFF is plotted in 1 for n " 1, . . ., 20, with q " 0, 9 k , k " 1, . . ., 9.

General identities for the Chern-Simons matrix model
The identities we derived in 5.1.1 apply to the Chern-Simons matrix model as well, in which case they have an interpretation in terms of knot and link invariants.For example, take (100), which says that, for λ satisfying pn ´1, 1 r q Ę pn ´r, 1 r q @r P t1, . . ., n ´1u, In terms of knot and link invariants, the above expression entails that expectation value of the product of an pn, 1q-torus link with an unknot in representation λ with opposite orientation equals the product of their expectation values.
Another trace identity derived in 5.1.1 is equation (107).This equation expresses the fact that a particle in the (anti)fundamental rep winding n times around a article in rep some λ will give a vanishing connected expectation value if the λ 1 ´λt 1 ´1 ´n ď λ 1 ´λ2 and λ 1 ´λt 1 ´1 ´n ď λ t 1 ´λt 2 .Further, it is worth emphasizing that, using (99), one can calculate pretty much any object of the form as all the objects appearing on the right hand side of the above expression are skew Schur polynomials with variables x i " q i´1{2 , to which we can apply the q-hook length formula in equation (140).

Overview and Conclusions
Here, we put forward a conjecture that many (if not all) examples of invariant RMT's which exhibit intermediate statistics are given by matrix models of topological field or string theories.We explicitly support this conjecture by the example of the matrix model introduced by Muttalib, which is the matrix model of U pN q Chern-Simons model on S 3 .The latter model is directly related to A and B topological string models via Gopakumar-Vafa duality.
To calculate the SFF of this model, we consider general infinite order unitary matrix models with weight functions satisfying the assumptions of Szegö's theorem.We find that the SFF's for these models have a surprisingly concise form, with the connected SFF giving rise to the linear ramp and plateau, while the disconnected part gives rise to a dip.Moreover, from the assumptions of Szegö's theorem, it follows that the dip had to go to zero, so that the plateau is exact.Further, we derive certain identities on expectation values of products of traces, as well as the behavior of the SFF under certain changes of the weight function.
We then apply these general results to the matrix model for U pN q Chern-Simons theory for S 3 , studied by Muttalib and collaborators for its intermediate statistics.The SFF of this model is a topological (knot) invariant.In particular, it is given by the HOMFLY invariant p2n, 2q-torus links with one component in the fundamental and the other in the antifundamental representation which, to the best of the authors knowledge, did not appear in the literature before.It displays the hallmark characteristics of intermediate statistics, with a dip that becomes more pronounced as we move further away from the CUE limit, q Ñ 0. Due to the form of the SFF, one can identify various matrix models which have the same SFF, an immediate example of which is given by replacing Epx; zq by Hpx; zq.
Indeed, the present work provides the tools to shed more light on the connections between topological field theories and intermediate statistics; we believe that the matrix models which arise in topological string theory are natural tools for describing ergodic-to-nonergodic phase transitions.This paper provides a first example of what we suspect to be a broader connection.

Acknowledgements
We would like to thank Wouter Buijsman, Oleksandr Gamayun, Alex Garkun, and Miguel Tierz for 8 Appendices

q-Numbers
We review some basic facts and useful relations involving q-numbers, which are so-called q-deformations of more familiar (generally complex) numbers.We will only be considering q-deformation of positive integers here, which are defined as rns q " p1 `q `¨¨¨`q n´1 q " 1 ´qn Other definitions of rns q , such as q ´n{2 ´qn{2 q ´1{2 ´q1{2 , also appear in the literature.Their common feature is that lim qÑ1 ´rns q " n . (110) Note that, for k, m, n P Z `satisfying m n " k, we have rms q rns q " rks q n , rr ¨ms rr ¨ns " rks q nr (111) for example, r8s q r2s q " 1 `q `¨¨¨`q 7 1 `q " 1 `q2 `q4 `q6 " r4s q 2 .(112) We will write rns q as rns henceforth and only specify the deformation parameter in case it is different from q. q-Factorials and q-binomials are defined as follows.For n, k P Z rN s! " p1 `qqp1 `q `q2 q . . .p1 `q `¨¨¨`q N ´1q , We then introduce the q-Pochhammer symbols, which is defined as pa; qq k " p1 ´aqp1 ´aqq . . .p1 ´aq k´1 q ." p1 ´qN qp1 ´qN´1 q . . .p1 ´qN´r`1 q p1 ´qqp1 ´q2 q . . .p1 ´qk q . (117) We see from this expression that, for q ă 1, we have q-Pochhammer symbols can be generalized as follows These are rather versatile objects.For example, Jacobi's third theta function can be expressed through the Jacobi triple product as Note that that the definition in (120) has q n 2 {2 rather than q n 2 as expansion coefficients, following the convention of e.g.[34].This is the origin of the differences with the expressions appearing e.g. in [69], which are related to the expressions here by taking q Ñ q 2 .

Symmetric polynomials
We review here some basic aspects of symmetric polynomials in the variables x " tx 1 , x 2 , . . .u.The elementary symmetric polynomials are then defined as e k pxq " Some examples include Closely related are the complete homogeneous symmetric polynomials, defined as which contains all monomials of degree j.Note the difference in the summation bounds between (121) and (122).Some examples of these include Another example is the set of power-sum symmetric polynomials, Note that if a matrix U has x i as its eigenvalues, traces of moments of U are given by power-sum symmetric polynomials, that is, Defining z " e iθ as in (46), we have the following relations between the above polynomials [59] Epx; zq " Consider the example where x i " q i´1 , so that (see [59] I.2 examples 3 and 4) Similarly, so that Here e k is only defined for k ď N .From (118), we see that, for q ă 1 and N Ñ 8,

Schur polynomials
A somewhat less straightforward type of symmetric polynomial is the Schur polynomial, which reduces to some of the above examples in certain cases.Schur polynomials play an important role as characters of irreducible representations, often referred to as irreps, of general linear groups and subgroups thereof.
Irreps can be conveniently classified by partitions, and we use these terms interchangably in this work.
We will denote partitions as λ " tλ 1 , λ 2 , . . ., λ u, which are sequences of non-negative integers ordered as λ 1 ě λ 2 ě . . . .Typically, partitions are taken to have a finite number of elements, that is, only a finite number of λ i are non-zero, but we will impose no such restriction.The weight of a partition (not to be confused with the highest weight of the corresponding irrep) is given by the sum of its terms |λ| " ř i λ i and its length pλq is the largest value of i for which λ i ‰ 0. A semistandard Young tableau (SSYT) corresponding to λ is then given by a by positive integers T i,j satisfying 1 ď i ď pλq and 1 ď j ď λ i .These integers are required to increase weakly along every row and increase strongly along every column, i.e.T i,j ě T i,j`1 and T i,j ą T i`1,j for all i, j.Label by α i the number of times that the number i appears in the SSYT.We then define The Schur polynomial s λ pxq is given by [60].
where the sum runs over all SSYT's corresponding to λ i.e. all possible ways to inscribe the diagram corresponding to λ with positive integers that weakly increase along rows and strictly along columns.
We give an example of an SSYT corresponding to a Young diagram λ " p3, 2q.From (133) one can see that the contribution of the SSYT below would be given by x 2 1 x 2 x 2 3 .

3 2 3
We can see from the above definition that s p1 n q " e n , s pnq " h n , i.e. the Schur polynomial of a column or row of n boxes are given by a degree n elementary or homogeneous symmetric polynomial, respectively.Schur polynomials have a natural generalization to so-called skew Schur polynomials.In this case we have two diagrams λ and µ such that µ Ď λ i.e.
µ i ď λ i , @ i.We denote by λ{µ the complement of µ in the diagram corresponding to λ. Define a semistandard skew Young tableau corresponding to λ{µ similar to the above, namely, as an array of positive integers T ij satisfying 1 ď i ď pλq and µ i ď j ď λ i weakly increasing along rows and strictly increasing along columns.We then define the skew Schur polynomial corresponding to λ{µ as where the sum again runs over all SSYT's corresponding to λ{µ.Note that if µ is the empty partition, i.e. µ i " 0, @i, we have s λ{µ " s λ , and if λ " µ, s µ{µ " 1.Let us consider λ " p3, 2q and µ " p1q.
Below, we give an SSYT corresponding to the skew partition λ{µ, which would contribute x 2 1 x 2 x 3 to the skew Schur polynomial.Skew Schur polynomials can also be expressed in determinantal form.Using a matrix of the form M " px pN ´Kq j q N j,k"1 , we have the following expression for the Vandermonde determinant detpx pN ´Kq j q N j,k"1 " ź 1ďjăkďN px j ´xk q . (134) We then have s λ pU q " s λ px j q " The (skew) Schur polynomials can be expressed in terms of elementary symmetric polynomials e k pxq or complete homogeneous symmetric polynomials h k pxq via the following determinantal expressions, known as the Jacobi-Trudi identities s pµ{λq " detph µj ´λk ´j`k q pλq j,k"1 " detpe µ t j ´λt k ´j`k q λ1 j,k"1 " D λ,µ N pHpx; zqq , s pµ{λq t " detpe µj ´λk ´j`k q pλq j,k"1 " detph µ t j ´λt k ´j`k q λ1 j,k"1 " D λ,µ N pEpx; zqq where the partition λ t is obtained from λ by transposing the corresponding Young Other useful identities for our purposes are the following, which can be found in Chapter I.5 of [59], s λ{µ px 1 , . . ., x n q " 0 unless 0 ď λ t i ´µt i ď n for all i ě 1 . (138) Note that an example of (138) is given by the fact that e k px 1 , . . ., x N q " 0 for k ą N .We consider some Schur polynomials which are treated in I.3 examples 1-4 of [59].Schur polynomials with all variables equal to 1 give the hook-length formula for the dimension of the representation, that is s λ p1, . . ., 1q " where cpxq " j ´i for x " pi, jq P λ is the content of x P λ, hpi, jq " λ i `λt j ´i ´j `1 is its hook-length, and npλq " ř i pi ´1qλ i .If, instead, we choose variables as x i " q i´1 , we get the following q-deformation of the dimension of λ s λ px i " q i´1 q " q npλq ź xPλ rN `cpxqs rhpxqs ": q npλq dim q pλq .(140) The quantity dim q pλq is known as the quantum dimension, or q-dimension.It is given by the hook length formula (139) where numbers are replaced by q-numbers.If we consider knots and links as consisting of the world lines of anyons carrying some representations λ, . . ., dim q pλq gives the dimension of the Hilbert space of λ [74].Note that (111) implies that irreps with the same dimension can have different quantum dimensions.This is why Chern-Simons theory with 0 ă q ă 1 can distinguish between certain (un)knots which give identical results in the limit q Ñ 1 or q Ñ 0.
In fact, the above expression simplifies even further.In particular, one can easily see that, for |q| ă 1 and N Ñ 8, quantum dimensions for reps with finite column lengths depend only on the hook lengths.This is because q N ´k " 0 for k finite, so that, for λ such that cpxq is finite for all x P λ, In fact, since the Jacobi triple product expansion is only valid in case 0 ă |q| ă 1 and we take N Ñ 8 here, we see that the numerical values of the Schur polynomials considered here only depend on the hook-lengths of their components.Of course, one can still use the full functional form involving terms of the form q N to in the context of knot theory, as these functional forms can allow one to distinguish between knots or links which have the same hook-lengths.For example, the quantum dimensions of p2q and p1 2 q are different when considering their full, functional form.On the other hand these quantum dimensions are identical when we take into account the fact that q N " 0. Lastly, one should note that, since the hook-lengths are invariant under transposition, the quantum dimensions involved are invariant under transposition as well.

Figure 2 :
Figure 2: Two examples of p2n, 2q-torus links.The Hopf link, on the left, is the p2, 2q-torus link.On the right, we have the p4, 2q-torus link.

Figure 3 :
Figure 3: After projecting a knot or link in the plane, crossings are given a sign in the way indicated above.

as ř 8 k" 1
|p k pxq| k diverges otherwise.If we take x j to be real-valued, as we do in various explicit examples considered here, equation (71) requires that |x j | ă 1.The right requirement in (70) is strictly weaker than the left, so it does give rise to any additional restrictions.In the above expressions, if we replace valuable discussions and comments, and Vladimir Kravtsov for inspiring this study and useful discus-sions.This work is part of the DeltaITP consortium, a program of the Netherlands Organization for Scientific Research (NWO) funded by the Dutch Ministry of Education, Culture and Science (OCW).

s λ pxqs λ pyq " 8 źs λ pxqs λ t pyq " 8 źi" 1 ,j" 1 1
tableau and H refers to the empty partition i.e. the trivial representation.The objects on the right hand side of (136) are explained in section 4. Schur polynomials satisfy various useful identities, including the so-called Cauchy identity and its dual ÿ λ ´xi y j .
[51]vacuum state, that is, the state without any Wilson line inserted, is given by |ρy ": |0y .If we act with a knot operator corresponding to an unknot in representation corresponding to Λ, the result is[51] |ρy " |ρ `Λy " |py .