Large $N$ Invariants of Torus Links in Lens Spaces

Kushal Chakraborty and Suvankar Dutta

Indian Institute of Science Education and Research Bhopal

Bhopal bypass, Bhopal 462066, India

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

We compute the invariants for a class of knots and links in arbitrary
representations in $S^{3}/\mathbb{Z}_{p}$ in the large $k$ (level), large $N$
(rank) limit, keeping $N/(k+N)=\lambda$ fixed, in $U(N)$ and $Sp(N)$ Chern-
Simons theories. Using the relation between the saddle point description and
collective field theory, we first find that the invariants for the Hopf link
and unknot are given by the on shell collective field theory action. We next
show that the results of these two invariants can be used to compute the
invariants of other torus knots and links. We also discuss the large $N$ phase
structure of the Hopf link invariant and observe that the same may admit a
Douglas-Kazakov type phase transition depending on the choice of
representations and $\lambda$.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary : correlation functions in Chern-Simons theory
  3. 3 Knot invariants in $U(N)$ Chern-Simons theory
    1. 3.1 Large $N$ analysis of two point function : invariant for the Hopf link
    2. 3.2 Invariants for a class of torus knots in $S^{3}$
    3. 3.3 Large $N$ phases of two point correlators in $U(N)$ Chern-Simons theory
    4. 3.4 An example : explicit computation of two point function and study of phase structure for Wigner semicircle distributions
    5. 3.5 Calculation of partition function
  4. 4 $Sp(N)$ Chern-Simons theory and knot invariants
    1. 4.1 The partition function
    2. 4.2 Two point correlator and Hopf link invariants
  5. 5 Discussion
  6. A Framing dependence of partition function
  7. B Review of $U(N)_{k}$ Chern-Simons theory on Seifert Manifold
  8. C Explicit derivation of Hamilton-Jacobi equation
    1. C.1 $U(N)$ Chern-Simons theory
    2. C.2 $Sp(N)$ Chern-Simons theory
  9. D From eigenvalue density to Young tableau density
    1. D.1 $Sp(N)$

##  1 Introduction

Knot theory is an interesting field in mathematics. A knot is a smooth
embedding of a circle $\mathcal{S}^{1}$ to a three manifold . A link is a
collection of disjoint knot. An important question in knot theory is whether
two knots (or links in general) are equivalent by ambient isotopy (i.e. if one
can continuously deform one knot into the other without breaking it). _Knot
invariants_ 111Knot invariants in general mean invariants for both knots and
links. are used to distinguish between different in-equivalent knots

A variety of knot invariants in three dimensions can be obtained from the
correlation functions of Wilson loop (WL) operators in Chern-Simons (CS)
theory [1]. The CS theory is topological at classical as well as quantum
level. The only interesting observables in this theory are WLs along oriented
knots and links. Since WLs are independent of metric, their expectation values
are topological and hence they are bona fide candidates for the knot
invariants. Witten proved that these topologically invariant correlation
functions are precisely the generalised knot invariants in three manifolds
[1]. For $G\equiv SU(N)$, $G\equiv SU(2)$ and $G\equiv SO(N)$ these
correlation functions give the HOMFLY-PT polynomial, the Jones polynomial and
the Kauffman polynomials respectively when the representations associated with
the WLs are in fundamental representations. In general the correlation
functions of WLs in arbitrary representations provide a wide class of
topological invariants. These are known as _coloured polynomials_. From the CS
theory perspective such invariants are well defined and we also understand the
source of representations associated with the invariants. In mathematics such
colored invariants have been defined by Reshetikhin and Turaev [2] and used to
construct new quantum invariants of 3-manifolds.

Although CS theory can be solved exactly, the calculation of knot invariants
is difficult.

Figure 1: Hopf link.

There exists a large literature on the computation of generalised coloured
invariants for different links and knots. Some of them can be found in [3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In this paper we consider
$U(N)$ and $Sp(N)$ CS theories and compute the knot invariants in the
following limit

$\displaystyle\displaystyle N,k\rightarrow\infty\quad\text{keeping
$\lambda=\frac{N}{k+N}$ fixed}.$ (1.1)

Here, $N$ is the rank of the gauge group and $k$ is the level of the CS
theory. This limit is called _double scaling limit_. We use the saddle point
technique to calculate the invariant for the Hopf link, shown in fig.1 in lens
space $S^{3}/\mathbb{Z}_{p}$ for any representations $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$ associated with two unknots222The computation presented in
this paper could be extended to find the invariants for hyperbolic knots and
links. For example Borromean links (for 3 given representations
$\mathcal{R}_{1}$, $\mathcal{R}_{2}$ and $\mathcal{R}_{3}$). Therefore such
calculations might be helpful in understanding the volume conjecture [19, 20,
18, 21] and it’s generalised version.. We then show that with this result in
hand one can compute the invariants for the class of torus knots (2,$k$=odd)
and links (2,$k$=even) following the method developed in [3].

Since the representations $\mathcal{R}_{1/2}$ are integrable, in the large $N$
limit we describe these representations in terms of eigenvalue distributions
of unitary matrices. As a result, in the double scaling limit the problem can
be studied with the aid of a collective field theory333Similar analysis was
done by Gross and Matytsin [22, 23] in the context of 2d Yang-Mills theory.
[24]. We find that the Hopf link invariant satisfies the Hamilton-Jacobi
equation with $p\lambda$ playing the role of time and the Hamiltonian being
the free collective field theory Hamiltonian. Therefore, large $N$ (classical)
value of the Hopf link invariant, evaluated on the classical solutions, is
given by the on shell action of the free collective field theory. The
Hamilton’s equations are similar to the continuity and Navier-Stokes equations
of an incompressible fluid with negative pressure. Thus the study of Hopf link
invariant boils down to the study of time evolution of an one dimensional
incompressible fluid with the boundary conditions depending on
$\mathcal{R}_{1/2}$.

The knowledge of the Hopf link invariant should be enough to find the
invariants for the class of other torus knots (2,$k$=odd) and links
(2,$k$=even) using the generalised Alexander-Conway skein relations
recursively. However when we place representations other than fundamental it
becomes difficult. In [3] Ramadevi et.al. developed a technique to get the
invariants for links made of braids up to four strands. A class of invariants
for links and knots including torus knots can be obtained from the eigenvalues
of half-twist braid matrices. We use these relations and show that in the
double scaling limit the invariants for a wide class of knots and links can be
computed from the result of the Hopf link invariant. We also emphasise that
our results are valid for any arbitrary large $N$ representations associated
with the knots.

We also study the large $N$ phase structure of two point correlation functions
of CS theory. We show that in the double scaling limit the two point
correlation function (Hopf link invariant) admits a third order phase
transition depending on the choice of $\mathcal{R}_{1/2}$ and $\lambda$. Such
a phase transition is similar to the Douglas-Kazakov phase transition [25].

#### Main results :

The main results of this paper are following.

  * •

We show that in the $N$ limit how the computation of two point correlation
functions of CS theory for Hopf link boils down to computation of on shell
action of a free collective field theory. To be precise, the calculation of
Hopf link invariant turns out to be equivalent to the time evolution of an one
dimensional incompressible fluid with the boundary conditions depending on
$\mathcal{R}_{1/2}$.

  * •

The Hopf link invariant in the large $N$ limit is equal to the on-shell
partition function of a collective field theory. The result is given by eqn
(3.32). The functions $\sigma_{1/2}(\theta)$ contain information of the
representations $\mathcal{R}_{1/2}$ associated with the Hopf link.

  * •

We also show that in the large $N$ limit the invariants for torus knots (2,
$k$=odd) and links (2, $k$=even) can be computed from the Hopf link invariants
and are given by (3.42).

  * •

We study the large $N$ phase transition in two point correlation functions of
CS theory.

The plan of the paper is as follows. In section 2 we review the correlation
functions of WLs in CS theory in Seifert manifold in different framings.
Section 3 provides the detailed calculation of the Hopf link invariant and
other torus knot invariants in $U(N)$ CS theory. We also discuss the large $N$
phase structure of the Hopf link invariant in this section. We conclude the
section 3 with an example. Large $N$ phase structure of $Sp(N)$ CS theory and
correlations functions are discussed in section 4. We end the paper with a
discussion 5. In appendix A we discuss how to obtain the correlation functions
in different manifolds using surgery and their framing dependence. In other
appendices we elaborate various technical details used in the main text.

##  2 Preliminary : correlation functions in Chern-Simons theory

In this section we review the correlation functions in CS theory on Seifert
manifolds in different framings. Experts may skip this section.

The topological nature of classical CS theory is preserved even at the quantum
level (correlation functions) but at the cost of a choice of _framing_.
Physical observables (correlations of WLs) are completely determined in terms
of topological data of the three manifold $\mathbb{M}$ up to a framing. In
order to understand the framing dependence in detail we first state an
important connection between CS theory and Wess-Zumino-Witten (WZW) model [1].
Quantisation of CS theory with gauge group $G$ and level $k$ in $\mathbb{M}$
with boundary $\Sigma$ ($=\partial\mathbb{M}$) renders a physical Hilbert
space $\mathcal{H}(\Sigma)$ which is isomorphic to space of conformal blocks
of WZW model with an affine Lie algebra $\mathbf{g}_{k}$. Using this
connection, correlations of WL operators in CS theory can be written in terms
of observables of the WZW model.

To show the framing dependence explicitly we consider the CS theory on a
Seifert manifold $\mathcal{M}_{(g,p)}$. A Seifert manifold is a non-trivial
circle bundle over genus $g$ Riemann surface $\Sigma_{g}$ with the first Chern
class $p$. Seifert manifolds for a generic $p$ can be obtained from
$\mathcal{M}_{(g,0)}$ (which is a product of genus $g$ Riemann surface and a
circle $\Sigma_{g}\times S^{1}$) by surgery. Different choices of surgeries
give different framings of $\mathcal{M}_{(g,p)}$. The $n$-point correlation
functions in CS theory are defined as

$\displaystyle\displaystyle\langle\mathcal{W}_{\mathcal{R}_{1},\cdots\mathcal{R}_{n}}^{\mathrm{K}_{1},\cdots,\mathrm{K}_{n}}\rangle$
$\displaystyle=\int[DA]e^{iS_{CS}}\prod_{a=1}^{n}\mathcal{W}_{\mathcal{R}_{a}}^{\mathrm{K}_{a}}(A)$
(2.1)

$\text{where}\
\mathcal{W}_{\mathcal{R}}^{\mathrm{K}}(A)=\text{Tr}_{\mathcal{R}}U_{\mathrm{K}}\
\text{with}\ U_{\mathrm{K}}=P\exp\oint_{\mathrm{K}}A$ and $S_{CS}$ is the
standard CS action with gauge field $A$. Denoting the $n$-point correlations
of WLs by $\mathcal{W}^{(g,p)}_{\mathcal{R}_{1}\cdots\mathcal{R}_{n}}[G,k]$
one can write them in the following form [26, 27, 28]

$\displaystyle\displaystyle\mathcal{W}^{(g,p)}_{\mathcal{R}_{1}\cdots\mathcal{R}_{n}}[G,k]$
$\displaystyle=\sum_{\mathcal{R}}\mathcal{K}^{(p)}_{\mathcal{R}_{1}\mathcal{R}}\mathcal{W}_{\mathcal{R}\mathcal{R}_{2}\cdots\mathcal{R}_{n}}(\Sigma_{g}\times
S^{1},G,k)$ (2.2)

where $\mathcal{K}^{(p)}$ depends on the framing. Note that being topological
$\mathcal{W}^{(g,p)}_{\mathcal{R}_{1}\cdots\mathcal{R}_{n}}[G,k]$ does not
explicitly depend on the geometry of the knots $\mathrm{K_{i}}$s. It depends
on the _linking number_ of the link made out of these knots.
$\mathcal{W}_{\mathcal{R}\mathcal{R}_{2}\cdots\mathcal{R}_{n}}(\Sigma_{g}\times
S^{1},G,k)$ is the $n$-point correlation function in $\Sigma_{g}\times S^{1}$,
given by

$\displaystyle\displaystyle\mathcal{W}_{\mathcal{R}\mathcal{R}_{2}\cdots\mathcal{R}_{n}}(\Sigma_{g}\times
S^{1},G,k)=\sum_{\mathcal{R}}\mathcal{S}_{0\mathcal{R}}^{2-n-2g}\prod_{a=1}^{n}\mathcal{S}_{\mathcal{R}\mathcal{R}_{a}}.$
(2.3)

See appendix A for a detailed discussion. Here
$\mathcal{S}_{\mathcal{R}\mathcal{R}^{\prime}}$ and
$\mathcal{T}_{\mathcal{R}\mathcal{R}^{\prime}}$ are the modular transform
matrices associated with the highest weight representations of the affine Lie
algebra $\mathbf{g}_{k}$ under inversion and translation of modular parameter
respectively (see [29] for details).

Figure 2: Links of unknots.

They satisfy

$\displaystyle\displaystyle\mathcal{S}^{2}=(\mathcal{S}\mathcal{T})^{3}=\mathbb{I}.$
(2.4)

The summation on the right hand side of (2.2) runs over integrable
representations of $\mathbf{g}_{k}$.
$\mathcal{W}^{(g,p)}_{\mathcal{R}_{1}\cdots\mathcal{R}_{n}}[G,k]$ in (2.2)
represents a topological invariant of links of $n$ unknots as shown444The
figure 2 describes the linking of the knots schematically. The exact diagrams
of such links in $\mathcal{M}_{(g,p)}$ is complicated. Mathematically the $p$
dependence in the correlation function comes (after surgery) through the
operator $\mathcal{K}^{p}$. in fig 2.

When $p=1$ and $g=0$ the Seifert manifold is a three sphere $S^{3}$. On
$S^{3}$, there exists a canonical choice of framing, given by
$\mathcal{K}^{(1)}=\mathcal{S}$. In canonical framing the correlation of $n$
WLs (2.2) is given by

$\displaystyle\displaystyle\mathcal{W}_{\mathcal{R}_{1}\cdots\mathcal{R}_{n}}[S^{3},G,k]=\mathcal{S}^{2-n}_{0\mathcal{R}_{1}}\prod_{a=2}^{n}\mathcal{S}_{\mathcal{R}_{1}\mathcal{R}_{a}}.$
(2.5)

The invariant for the link diagram in fig 2 in Seifert manifold can be
obtained from $\Sigma_{g}\times S^{1}$ by surgery with the choice
$\mathcal{K}^{(p)}=(TST)^{p}$. This particular choice is called Seifert
framing. In Seifert framing, therefore, the invariant is given by [26, 27, 28]

$\displaystyle\displaystyle\mathcal{W}^{(g,p)}_{\mathcal{R}_{1}\cdots\mathcal{R}_{n}}[G,k]=\sum_{\mathcal{R}}\mathcal{T}_{\mathcal{R}\mathcal{R}}^{-p}\mathcal{S}_{0\mathcal{R}}^{2-n-2g}\prod_{a=1}^{n}\mathcal{S}_{\mathcal{R}\mathcal{R}_{a}}.$
(2.6)

The main focus of this paper is to study the large $N$ structure of these knot
invariants.

##  3 Knot invariants in $U(N)$ Chern-Simons theory

In this section we consider the gauge group $G=U(N)$. For our purpose, we
define a modified WL operator $\widetilde{W}_{\mathcal{R}}$

$\widetilde{\mathcal{W}}_{\mathcal{R}}=\frac{\mathcal{W}_{\mathcal{R}}}{\mathcal{S}_{0\mathcal{R}}}.$
(3.1)

The $n$-point correlation functions of these modified WL operators are given
by,

$\displaystyle\displaystyle\widetilde{\mathcal{W}}^{(g,p)}_{\mathcal{R}_{1}\cdot\cdot\mathcal{R}_{n}}(N,k)=\sum_{\mathcal{R}}\mathcal{S}_{0\mathcal{R}}^{2-2g-n}T_{\mathcal{R}\mathcal{R}}^{-p}\prod_{a=1}^{n}\frac{\mathcal{S}_{\mathcal{R}\mathcal{R}_{a}}}{\mathcal{S}_{0\mathcal{R}_{a}}}.$
(3.2)

$\mathcal{S}$ and $\mathcal{T}$ for $u(N)_{k}$ are given by (B.2, B.5). For a
given representation $\mathcal{R}_{a}$ we define a set of $N$ variables
$\\{\theta^{(a)}_{1},\cdots\theta^{(a)}_{N}\\}$

$\theta_{i}^{(a)}=\frac{2\pi}{N+K}\left(h_{i}^{(a)}-\frac{N-1}{2}\right)$
(3.3)

where $h_{i}^{(a)}=n_{i}^{(a)}+N-i$ are the hook numbers of the Young diagram
associated with $\mathcal{R}_{a}$ and $n_{i}^{(a)}$ is number of boxes in the
$i^{th}$ row. Since $\mathcal{R}_{a}$s are integrable representations of
$u(N)_{k}$, $n_{i}^{(a)}$s satisfy

$\displaystyle\displaystyle-\frac{k}{2}\leq n_{N}^{(a)}\leq\cdots\leq
n_{1}^{(a)}\leq\frac{k}{2}.$ (3.4)

It is easy to check that in the double scaling limit the variables
$\theta_{i}^{(a)}$ ranges from $-\pi$ to $\pi$. These $\theta^{(a)}_{i}$ can
be thought of as eigenvalues of unitary matrices. However, there is a
difference. The minimum gap between two eigenvalues is $2\pi/(k+N)$.
Therefore, in large $N$ limit an integrable representation can be described in
terms of a distribution of these eigenvalues, denoted by
$\sigma_{(a)}(\theta)$ and defined as

$\displaystyle\displaystyle\sigma_{a}(\theta)=\frac{1}{N}\sum_{i=1}^{N}\delta(\theta-\theta_{i}^{(a)}).$
(3.5)

In the double scaling limit (1.1), $\sigma_{a}(\theta)$ satisfies the
constraint

$\displaystyle\displaystyle\sigma_{a}(\theta)\leq\frac{1}{2\pi\lambda}$ (3.6)

which follows from the fact that two eigenvalues have a minimum separation
$2\pi/(k+N)$. Considering the above change of variables one can compute the
ratio of $\mathcal{S}_{\mathcal{R}\mathcal{R}_{a}}$ and
$\mathcal{S}_{0\mathcal{R}_{a}}$. It turns out to be

$\displaystyle\displaystyle\frac{\mathcal{S}_{\mathcal{R}\mathcal{R}_{a}}}{\mathcal{S}_{0\mathcal{R}_{a}}}=e^{-\frac{2i}{N}\sum_{i}(h_{i}-\frac{N-1}{2})\sum_{i}\theta^{(a)}_{i}}\chi_{R}(\theta^{(a)}_{i})$
(3.7)

where $h_{i}$s are the hook numbers associated with $\mathcal{R}$ and
$\chi_{\mathcal{R}}(\theta)$ is the character of the $U(N)$ in the
representation $\mathcal{R}$. Using the expression for $\mathcal{T}$ (B.5) the
$n$-point correlation function can be written as,

$\displaystyle\displaystyle\widetilde{\mathcal{W}}^{(g,p)}_{\mathcal{R}_{1}\cdot\cdot\mathcal{R}_{n}}(N,k)=\sum_{\mathcal{R}}\frac{q^{-\frac{p}{2}\mathcal{C}_{2}(\mathcal{R})}e^{-i\Theta\mathcal{C}_{1}(\mathcal{R})}}{\mathcal{S}_{0\mathcal{R}}^{2g+n-2}}\prod_{a=1}^{n}\chi_{\mathcal{R}}(\theta^{(a)}).$
(3.8)

where

$\displaystyle\displaystyle q=e^{2\pi i/(k+N)}$ (3.9)

and

$\displaystyle\displaystyle\Theta=2\sum_{a=1}^{n}\left(\frac{1}{N}\sum_{i=1}^{N}\theta^{(a)}_{i}\right).$
(3.10)

This correlation function is similar to the partition function of a _q_
-deformed Yang-Mills theory with a $\Theta$-term on a genus $g$ Riemann
surface with $n$ boundaries [22, 26]. The distributions of holonomies on those
boundaries are given by $\\{\theta^{(a)}_{i}\\}$. The only difference between
(3.8) and the partition function of a _q_ -deformed 2d Yang-Mills theory is
that the sum in (3.8) runs over integrable representations.

When $\mathcal{R}_{1}=\cdots=\mathcal{R}_{n}=0$, the correlation function
(3.8) gives the partition function of $U(N)$ CS theory of level $k$ on Seifert
manifold. The large $N$ phase structure of this theory for $g=0$ was studied
in [30]. The theory undergoes a third order phase transition in the double
scaling limit (1.1). We have reviewed the result in appendix B.

In this section we explicitly compute the two point correlation functions
($n=2$) in the _double scaling limit_ (1.1) and show that using the result of
two point correlation function one can compute the invariants for a class of
torus knots (2,k=odd) and links (2,k=even).

Two point correlators in $S^{3}/\mathbb{Z}_{p}$, which is a Seifert manifold
with $g=0$ for any $p$, is given by

$\displaystyle\displaystyle\mathcal{W}_{\mathcal{R}_{1},\mathcal{R}_{2}}[S^{3}/\mathbb{Z}_{p},N,k]=\sum_{\mathcal{R}}\mathcal{T}_{\mathcal{R}\mathcal{R}}^{-p}\mathcal{S}_{\mathcal{R}\mathcal{R}_{1}}\mathcal{S}_{\mathcal{R}\mathcal{R}_{2}}$
(3.11)

and represents a topological invariant for the Hopf link (fig 1). In the large
$N$ limit the two point function admits a genus expansion (perturbative part)
[31]

$\displaystyle\displaystyle\ln\left[\mathcal{W}_{\mathcal{R}_{1},\mathcal{R}_{2}}[S^{3}/\mathbb{Z}_{p},N,k]\right]=\sum_{h=0}^{\infty}N^{2-2h}\mathrm{W}^{(h)}_{\mathcal{R}_{1},\mathcal{R}_{2}}(\lambda).$
(3.12)

Our goal is to compute the leading contribution
$\mathrm{W}^{(0)}_{\mathcal{R}_{1},\mathcal{R}_{2}}(\lambda)$ for any
$\mathcal{R}_{1/2}$ using the saddle point technique.

The modified two point correlation function (3.1) is given by

$\begin{split}\widetilde{\mathcal{W}}_{\mathcal{R}_{1}\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},N,k)=\sum_{\mathcal{R}}q^{-\frac{p}{2}\mathcal{C}_{2}(\mathcal{R})}e^{-i\Theta\mathcal{C}_{1}(\mathcal{R})}\chi_{\mathcal{R}}(\theta^{(1)})\chi_{\mathcal{R}}(\theta^{(2)})\end{split}$
(3.13)

where $\theta^{(1/2)}$ are eigenvalues corresponding to the representation
$\mathcal{R}_{1/2}$. We consider the representations $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$ such that the eigenvalues are symmetrically distributed
about _zero_ and hence the $\Theta$ term drops out from (3.13). Expressing the
characters of $U(N)$ in terms of Schur polynomial we can explicitly write down
the two point correlation functions in the following form,

$\begin{split}\widetilde{\mathcal{W}}_{\mathcal{R}_{1}\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},N,k)=\sum_{\\{y_{i}\\}}\frac{e^{2\pi
ip\frac{N^{2}}{24}}}{2^{N(N-1)}}\frac{\det||e^{iNy_{j}\theta^{(1)}_{k}}||}{\prod_{i<j}^{N}\sin\left(\frac{\theta^{(1)}_{i}-\theta^{(1)}_{j}}{2}\right)}\frac{\det||e^{iNy_{j}\theta^{(2)}_{k}}||}{\prod_{i<j}^{N}\sin\left(\frac{\theta^{(2)}_{i}-\theta^{(2)}_{j}}{2}\right)}\
e^{-p\pi iN\lambda\sum_{j=1}^{N}y_{j}^{2}}\end{split}$ (3.14)

where

$\displaystyle\displaystyle y_{j}=\frac{1}{N}\left(h_{j}-\frac{N-1}{2}\right)$
(3.15)

is the shifted hook number for the representation $\mathcal{R}$.

In general it is difficult to find an exact expression for the two point
function. However, in the large $N$ limit we see that the right hand side of
(3.14) is dominated by a single representation for a specific class of
$\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ and it is indeed possible to find an
exact expression for
$\widetilde{\mathcal{W}}_{\mathcal{R}_{1}\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},N,k)$.

###  3.1 Large $N$ analysis of two point function : invariant for the Hopf
link

We follow the work of Gross and Matytsin [22, 23] to do the large $N$ analysis
of (3.14). In the large $N$ limit the integrable representation on the right
hand side of (3.13) can be denoted by a density function $\rho(y)$,

$\displaystyle\displaystyle\rho(y)=-\frac{dx}{dy(x)},\quad\text{where}\quad
y(x)=\frac{y_{i}}{N},\ x=\frac{i}{N}.$ (3.16)

Since $i$ runs from 1 to $N$, $x\in[0,1]$. Also from the discreteness of
$y_{i}$s (eqn. 3.15) it follows that

$\displaystyle\displaystyle\rho(y)\leq 1.$ (3.17)

Following [30], after a wick rotation in complex $p$ plane $(p\to-ip)$ we
introduce555Since the correlation function is an analytic function of $p$ one
can Wick rotate the final answer back to the original $p$.

$\displaystyle\displaystyle\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}$
$\displaystyle=$ $\displaystyle 2^{N(N-1)}e^{-\pi
pN^{2}/12}\widetilde{\mathcal{W}}_{\mathcal{R}_{1}\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},N,k)$
(3.18)

and a new variable $A$ in place of $p\lambda$

$\displaystyle\displaystyle A=2\pi p\lambda.$ (3.19)

One can show from (3.14) that
$\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}$ satisfies

$\begin{split}2N\frac{\partial\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}}{\partial
A}=\frac{1}{D[\theta^{(a)}]}\sum_{k=1}^{N}\frac{\partial^{2}}{\partial\theta_{k}^{(a)^{2}}}[D[\theta^{(a)}]\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}]\end{split}$
(3.20)

where

$D[\theta^{(a)}]=\prod_{i<j}^{N}\sin\left(\frac{\theta^{(a)}_{i}-\theta^{(a)}_{j}}{2}\right)\quad\text{for
$a=1,2$}.$ (3.21)

Assuming $\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}$ is
dominated by a single representation in the large $N$ limit, we choose

$\displaystyle\displaystyle\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}=e^{N^{2}F_{N}}\quad\text{and
$\lim_{N\to\infty}F_{N}=F$}$ (3.22)

and after a little algebra we find that $F$ satisfies (see appendix C for
detailed calculation)

$\begin{split}2\frac{\partial F}{\partial
A}=\int\sigma_{a}(\theta)\Big{[}\Big{(}\frac{\partial}{\partial\theta}\frac{\delta
F}{\delta\sigma_{a}(\theta)}\Big{)}+2U(\theta)\frac{\partial}{\partial\theta}\frac{\delta
F}{\delta\sigma_{a}(\theta)}+U(\theta)^{2}-\frac{\pi^{2}}{3}\sigma_{a}(\theta)^{2}\Big{]}.\end{split}$
(3.23)

Considering $F$ to be of the following form666Here the cut-integral
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int d\theta^{\prime}$ represents integration over
$\theta^{\prime}$ over the valid range except the point $\theta$. Such
integrals sometimes are denoted by $P\int$.

$F[\sigma_{1}(\theta),\sigma_{2}(\theta)|A]=S[\sigma_{1}(\theta),\sigma_{2}(\theta)|A]-\frac{1}{2}\sum_{a=1}^{2}\int\sigma_{a}(\theta)d\theta\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\sigma_{a}(\theta^{\prime})\log[\sin\Big{(}\frac{\theta-\theta^{\prime}}{2}\Big{)}]d\theta^{\prime}$
(3.24)

one can recast the equation (3.23) as

${}\begin{split}\frac{\partial S}{\partial
A}=\frac{1}{2}\int\sigma_{a}(\theta)\Big{[}\Big{(}\frac{\partial}{\partial\theta}\frac{\delta
S}{\delta\sigma_{a}(\theta)}\Big{)}^{2}-\frac{\pi^{2}}{3}\sigma_{a}(\theta)^{2}\Big{]}d\theta.\end{split}$
(3.25)

As discussed in [22] this equation can be thought of as the Hamilton–Jacobi
equation with Hamiltonian

$\begin{split}H[\sigma,\Pi]=\frac{1}{2}\int\sigma(t,\theta)\Big{[}\Big{(}\frac{\partial\Pi(t,\theta)}{\partial\theta}\Big{)}^{2}-\frac{\pi^{2}}{3}\sigma(t,\theta)^{2}\Big{]}d\theta\end{split}$
(3.26)

of a $(1+1)$ dimensional field theory with field $\sigma(t,\theta)$ and
conjugate momentum $\Pi(t,\theta)=\frac{\delta S}{\delta\sigma(t,\theta)}$.
The Hamilton’s equations of motion are given by,

$\begin{split}\frac{\partial\sigma}{\partial t}+\frac{\partial(\sigma
v)}{\partial\theta}&=0\\\ \frac{\partial v}{\partial t}+v\frac{\partial
v}{\partial\theta}-\pi^{2}\sigma\frac{\partial\sigma}{\partial\theta}&=0\end{split}$
(3.27)

where

$\displaystyle\displaystyle
v(t,\theta)=\frac{\partial\Pi(t,\theta)}{\partial\theta}.$ (3.28)

These equations are similar to the continuity and Navier-Stokes equations of
an one dimensional fluid moving on a circle with density $\sigma(t,\theta)$,
velocity $v(t,\theta)$ and a negative pressure. Finding a solution of the
equation (3.25) for $S(\sigma_{1},\sigma_{2})$ is equivalent to solve these
fluid equations with the boundary conditions

$\begin{split}\sigma(t=0,\theta)&=\sigma_{1}(\theta)\\\
\sigma(t=A,\theta)&=\sigma_{2}(\theta).\end{split}$ (3.29)

Suppose $(\bar{\sigma}(t,\theta),\bar{v}(t,\theta))$ is a solution of fluid
equations with the desired boundary conditions. The quantity
$S(\sigma_{1},\sigma_{2},A)$ evaluated on this solution is therefore given by

$\displaystyle\displaystyle\bar{S}(\sigma_{1},\sigma_{2},A)=\frac{1}{2}\int
H[\bar{\sigma}(0,\theta),\bar{v}(0,\theta)]dA+\text{const}.$ (3.30)

Using the series of definitions (3.24), (3.22) and (3.18) one can write the
knot invariant for Hopf link in $S^{3}/\mathbb{Z}_{p}$ as

$\displaystyle\displaystyle\begin{split}\mathcal{W}_{\mathcal{R}_{1}\mathcal{R}_{2}}=\mathcal{S}_{0\mathcal{R}_{1}}\mathcal{S}_{0\mathcal{R}_{2}}\widetilde{\mathcal{W}}_{\mathcal{R}_{1}\mathcal{R}_{2}}=\exp\left[N^{2}\left(\bar{S}(\sigma_{1},\sigma_{2},A)+\frac{\pi
p}{12}-\ln 2\right)\right].\end{split}$ (3.31)

Hence from (3.12) we find

$\displaystyle\displaystyle\mathrm{W}^{(0)}_{\mathcal{R}_{1}\mathcal{R}_{2}}=\bar{S}(\sigma_{1},\sigma_{2},A)+\frac{\pi
p}{12}-\ln 2.$ (3.32)

Thus we see that the invariant for the Hopf link with any arbitrary large
representations of $u(N)_{k}$ is given by the on shell free collective field
theory action with the boundary conditions (3.29). Further, since the fluid
equations (3.27) are dispersionless KdV equation (Burger equation), the whole
exercise to find the Hopf link invariant boils down to solving a
dispersionless KdV equation with a set of boundary conditions. According to
our notation if $\mathcal{R}_{2}$ (or $\mathcal{R}_{1}$) is zero, then the
corresponding two point function
$\mathcal{W}_{\mathcal{R}_{1}\mathcal{R}_{2}}$ becomes one point function and
gives the invariant for unknot. $\mathcal{R}_{2}=0$ corresponds to a
$\sigma_{2}$ given by (3.60).

Our next goal is to find out the invariants for other class of torus knots
with the result of the Hopf link invariant in our disposal.

###  3.2 Invariants for a class of torus knots in $S^{3}$

In this section we discuss how knot invariants for other torus knots in
$S^{3}$ (a Seifert manifold with $g=0$, $p=1$) can be computed in the large
$N$ limit from the result of the Hopf link.

Torus knots are special kinds of knots which can be put on the surface of a
torus. A formalism of knot operators was developed in [7] to compute the
invariants of torus knots. In [3] the authors developed a different method to
obtain the invariants of links made from braids of up to four strands. It was
shown in [3] that for special types of links as shown in figure 3(a) and 3(b),
the knot invariants can be written in terms of the eigenvalues of the half-
twist matrix.

(a) For any values of $m$ such braiding gives $m$ links of two unknots in
representations $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ with the same
orientations.

(b) For any values of $m$ such braiding gives $2m$ crossings of two unknots in
representations $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ with opposite
orientations.

Figure 3: Braiding of knots in $S^{3}$

The half-twist matrix $\mathcal{B}(\mathcal{R}_{1},\mathcal{R}_{2})$
introduces right-handed half-twists in parallelly oriented strands carrying
representations $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ whereas,
$\hat{\mathcal{B}}(\mathcal{R}_{1},\bar{\mathcal{R}}_{2})$ introduces right-
handed half-twists in oppositely oriented strands carrying representations
$\mathcal{R}_{1}$ and $\bar{\mathcal{R}}_{2}$. The dimensions of $\mathcal{B}$
and $\hat{\mathcal{B}}$ depend on the number of irreducible representations in
the product $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ ($\bar{\mathcal{R}}_{2}$).
Denoting the link invariants in fig 3(a) and 3(b) by
$\mathcal{V}[\mathcal{L}_{2m}(\mathcal{R}_{1},\mathcal{R}_{2})]$ and
$\mathcal{V}[\hat{\mathcal{L}}_{2m}(\mathcal{R}_{1},\bar{\mathcal{R}}_{2})]$
respectively it was shown in [3] that they are given by

$\displaystyle\displaystyle\mathcal{V}[\mathcal{L}_{2m}(\mathcal{R}_{1},\mathcal{R}_{2})]=\sum_{\mathcal{R}}\mathrm{dim}_{q}\mathcal{R}\
(\lambda^{+}_{\mathcal{R}}(\mathcal{R}_{1},\mathcal{R}_{2}))^{2m}$ (3.33)

and

$\displaystyle\displaystyle\mathcal{V}[\hat{\mathcal{L}}_{2m}(\mathcal{R}_{1},\bar{\mathcal{R}}_{2})]=\sum_{\mathcal{R}}\mathrm{dim}_{q}\mathcal{R}\
(\lambda^{-}_{\mathcal{R}}(\mathcal{R}_{1},\bar{\mathcal{R}}_{2}))^{2m}$
(3.34)

where $\lambda^{\pm}_{\mathcal{R}}(\mathcal{R}_{1},\mathcal{R}_{2})$ are the
eigenvalues of $\mathcal{B}(\mathcal{R}_{1},\mathcal{R}_{2})$ and
$\hat{\mathcal{B}}(\mathcal{R}_{1},\bar{\mathcal{R}}_{2})$ respectively. They
are given by

$\displaystyle\displaystyle\begin{split}\lambda^{+}_{\mathcal{R}}(\mathcal{R}_{1},\mathcal{R}_{2})&=(-1)^{\epsilon_{\mathcal{R}}}q^{\mathcal{C}_{2}(\mathcal{R}_{1})+\mathcal{C}_{2}(\mathcal{R}_{2})+\frac{|\mathcal{C}_{2}(\mathcal{R}_{1})-\mathcal{C}_{2}(\mathcal{R}_{2})|}{2}-\frac{\mathcal{C}_{2}(\mathcal{R})}{2}}\\\
\lambda^{-}_{\mathcal{R}}(\mathcal{R}_{1},\bar{\mathcal{R}}_{2})&=(-1)^{\epsilon_{\mathcal{R}}}q^{-\frac{|\mathcal{C}_{2}(\mathcal{R}_{1})-\mathcal{C}_{2}(\mathcal{R}_{2})|}{2}+\frac{\mathcal{C}_{2}(\mathcal{R})}{2}}\end{split}$
(3.35)

where $q$ is given by (3.9). The sum on the right hand side runs over distinct
irreducible representations of
$\mathcal{R}_{1}\otimes\mathcal{R}_{2}(\bar{\mathcal{R}}_{2})$. The factor
$(-1)^{\epsilon_{\mathcal{R}}}$ depends on the symmetric or anti-symmetric
properties of $\mathcal{R}$ in the product of
$\mathcal{R}_{1}\otimes\mathcal{R}_{2}$. For $m=1$,
$\mathcal{V}[\mathcal{L}_{2m}(\mathcal{R}_{1},\mathcal{R}_{2})]$ and
$\mathcal{V}[\hat{\mathcal{L}}_{2m}(\mathcal{R}_{1},\mathcal{R}_{2})]$ give
the invariants for the Hopf link in $S^{3}$ with the same and opposite
orientations respectively.

Denoting

$\displaystyle\displaystyle\mathcal{G}(\mathcal{R}_{1},\mathcal{R}_{2})=q^{\mathcal{C}_{2}(\mathcal{R}_{1})+\mathcal{C}_{2}(\mathcal{R}_{2})+\frac{|\mathcal{C}_{2}(\mathcal{R}_{1})-\mathcal{C}_{2}(\mathcal{R}_{2})|}{2}}$
(3.36)

the knot invariant
$\mathcal{V}[\mathcal{L}_{2m}(\mathcal{R}_{1},\mathcal{R}_{2})]$ can be
written as

$\displaystyle\displaystyle\mathcal{V}[\mathcal{L}_{2m}(\mathcal{R}_{1},\mathcal{R}_{2})]=\mathcal{G}(m,\mathcal{R}_{1},\mathcal{R}_{2})\sum_{\mathcal{R}}\mathrm{dim}_{q}\mathcal{R}\
q^{-m\mathcal{C}_{2}(\mathcal{R})}.$ (3.37)

In the double scaling limit (1.1), using the change of variables (3.3) and
denoting the large representations $\mathcal{R}_{1/2}$ by $\sigma_{1/2}$
respectively the invariants (3.36) can be written as,

$\displaystyle\displaystyle\mathcal{V}[\mathcal{L}_{2m}(\mathcal{R}_{1},\mathcal{R}_{2})]=\hat{\mathcal{G}}(m,\lambda,\sigma_{1},\sigma_{2})\mathcal{F}(m,\lambda)$
(3.38)

where

$\displaystyle\hat{\mathcal{G}}(m,\lambda,\sigma_{1},\sigma_{2})$
$\displaystyle=\exp\Bigg{[}\frac{imN^{2}}{2\pi\lambda}\left(\int
d\theta\theta^{2}\left(\sigma_{1}+\sigma_{2}\right)+\frac{1}{2}\left|\int
d\theta\theta^{2}(\sigma_{1}-\sigma_{2})\right|\right)-\frac{i\pi m\lambda
N^{2}}{3}\Bigg{]}$

and

$\displaystyle\displaystyle\mathcal{F}(m,\lambda)=\int\left[\mathcal{D}\theta(x)\right]\exp\left[\frac{N^{2}}{2}\int
dx\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
dy\log\sin\left|\frac{\theta(x)-\theta(y)}{2}\right|-\frac{imN^{2}}{2\pi\lambda}\int
dx\ \theta(x)^{2}\right].\qquad$ (3.40)

The function $\mathcal{F}(m,\lambda)$ is difficult to calculate as the
functional integration over $\theta(x)$ does not run over all possible
configurations (since $\mathcal{R}$ in (3.33) runs over irreducible
representations of
$\mathcal{R}_{1}\otimes\mathcal{R}_{2}(\bar{\mathcal{R}}_{2})$ only). However
in the large $N$ limit one can find the invariant for any $m$ knowing the
value of the same for $m=1$. As we have mentioned that for $m=1$,
$\mathcal{V}[\mathcal{L}_{2m}(\mathcal{R}_{1},\mathcal{R}_{2})]=\mathcal{W}_{\mathcal{R}_{1}\mathcal{R}_{2}}(\lambda)$,
the invariant for the Hopf link evaluated in the last section777Up to an
analytic continuation in $\lambda:\lambda\rightarrow i\lambda$. This is due to
the fact that in deriving $\mathcal{W}_{\mathcal{R}_{1}\mathcal{R}_{2}}$ we
did an analytic continuation in $p$.. Now from the expression of the function
$\mathcal{F}(m,\lambda)$ we see that

$\displaystyle\displaystyle\mathcal{F}(m,\lambda)=\mathcal{F}(1,\lambda/m)$
(3.41)

since the space of functional integration over $\theta(x)$ remains unchanged
(for given $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$) as we vary $m$. Therefore
we find that in the double scaling limit the link invariants for the class of
links shown in figure 3(a) are given by

$\displaystyle\displaystyle\mathcal{V}[\mathcal{L}_{2m}(\mathcal{R}_{1},\mathcal{R}_{2})]=\left(\frac{\hat{\mathcal{G}}(m,\lambda,\sigma_{1},\sigma_{2})}{\hat{\mathcal{G}}(1,\lambda/m,\sigma_{1},\sigma_{2})}\right)\mathcal{W}_{\mathcal{R}_{1}\mathcal{R}_{2}}(\lambda/m).$
(3.42)

Figure 4: For any odd values of $m$, such braiding gives a knot in
representations $\mathcal{R}_{1}$ with $m$ crossings.

In the similar way it is also possible to compute the link invariants
$\mathcal{V}[\hat{\mathcal{L}}_{2m}(\mathcal{R}_{1},\bar{\mathcal{R}}_{2})]$
from $\mathcal{W}_{\mathcal{R}_{1}\bar{\mathcal{R}}_{2}}(\lambda)$.

When $\mathcal{R}_{2}=\mathcal{R}_{1}$, one can construct a different braiding
as shown in fig (4). For even $m$ this is same as braiding in fig (3(a)) with
$\mathcal{R}_{2}=\mathcal{R}_{1}$. However for odd values of $m$ such braiding
gives a single knot. For example, when $m=3$ we get trefoil in $S^{3}$. $m=1$
gives an unknot. The invariant for this knot is given by

$\displaystyle\displaystyle\mathcal{V}[\mathcal{L}_{m}(\mathcal{R}_{1},\mathcal{R}_{1})]=\sum_{\mathcal{R}}\mathrm{dim}_{q}\mathcal{R}\
(\lambda^{+}_{\mathcal{R}}(\mathcal{R}_{1},\mathcal{R}_{1}))^{m}\quad.$ (3.43)

Following the same argument,
$\mathcal{V}[\mathcal{L}_{m}(\mathcal{R}_{1},\mathcal{R}_{1})]$ for any odd
$m$ can be evaluated from
$\mathcal{V}[\mathcal{L}_{1}(\mathcal{R}_{1},\mathcal{R}_{1})]$ in the double
scaling limit. The knot invariant
$\mathcal{V}[\mathcal{L}_{1}(\mathcal{R}_{1},\mathcal{R}_{1})]$ is given by
(3.32) with $\sigma_{2}$ representing a trivial representation (3.60).

###  3.3 Large $N$ phases of two point correlators in $U(N)$ Chern-Simons
theory

Large $N$ phase transition in $U(N)$ CS theory has been discussed in [30].
Similar phase transition is also observed in the correlation functions. In
this section we give qualitative arguments for such phase transitions.

It can be shown that if the fluid equations (3.27) admit a solution such that
there exists a _time_ $0\leq t^{*}\leq A$ when the velocity of the fluid is
zero, i.e.

$\displaystyle\displaystyle v(t^{*},\theta)=0$ (3.44)

then the fluid velocity and density satisfy an identity

$i\pi\sigma^{*}[\theta-(t-t^{*})(v(t,\theta)+i\pi\sigma(t,\theta))]=v(t,\theta)+i\pi\sigma(t,\theta))$
(3.45)

where

$\sigma^{*}(\theta)=\sigma(t^{*},\theta).$ (3.46)

The maximum value of the density is minimum at $t=t^{*}$. For such solution
one can show that the two point function (3.14) is dominated by an integrable
representation $\rho(y)$ given by (see appendix D for details)

$\pi\rho(-\pi\sigma^{*}(\theta))=\theta$ (3.47)

i.e. inverse of $\sigma^{*}(\theta)$. If the fluid equations do not admit any
such solution then there exists no real saddle points. Needless to mention the
existence of a real saddle point depends on the choice of $\sigma_{1/2}$ and
$\lambda$.

If the fluid equations admit the existence of a real saddle point for a given
$\sigma_{1/2}$ then depending on the value of $\lambda$ the system may undergo
a phase transition. In order to discuss this phase transition qualitatively we
assume that the initial and final densities $\sigma_{1}(\theta)$ and
$\sigma_{2}(\theta)$ are even functions of $\theta$ and they have gaps (i.e.
vanishes for $|\theta|$ greater than some $|\theta_{0}|\leq\pi$). The fluid
density $\sigma(t,\theta)$ starting from the configuration
$\sigma_{1}(\theta)$ at $t=0$ spreads out (i.e. the gap starts decreasing).
The absolute value of velocity of the fluid also decreases with time. At some
intermediate time $t^{*}$ when the velocity of the fluid is zero the density
has a maximum spread. After that fluid velocity starts increasing (in the
opposite direction) and the density starts contracting and reaches the final
configuration $\sigma_{2}(\theta)$ at $t=A$. The system will observe no phase
transition if the maximum spread of $\sigma(t,\theta)$ at $t=t^{*}$ is less
than $\pi$ or at max touches $\pi$. Otherwise the system will undergo a phase
transition. The Young diagram density $\rho(y)$ and have an upper cap
$\rho(y)\leq 1$. In addition the variable $y$ ranges between $\pm 1/2\lambda$.
Since the two functions $\sigma^{*}(\theta)$ and $\rho(y)$ are functional
inverse of each other, $\sigma^{*}$ having a gap means $\rho<1$. If
$\sigma^{*}$ is gap-less then $\rho$ develops a cap. Depending on the initial
conditions and $\lambda$ we can have four possibilities.

  * •

If $\sigma^{*}(\theta)$ has gap but no cap then $\rho(y)$ has a no cap but a
gap.

  * •

If $\sigma^{*}(\theta)$ is gap-less with no cap then the dominant Young
diagram $\rho(y)$ will have a cap but it is gap-less.

  * •

If $\sigma^{*}(\theta)$ has a cap with a gap then $\rho(y)$ has no gap but no
cap.

  * •

If $\sigma^{*}(\theta)$ has a cap and no-gap then $\rho(y)$ has a cap and no-
gap.

The last two cases are special in CS theory (unlike 2d YM theory [22]) as all
the representations are integrable representations. In the next section we
elaborate this qualitative discussion with an example.

###  3.4 An example : explicit computation of two point function and study of
phase structure for Wigner semicircle distributions

Wigner semicircle distribution is a probability distribution on real line
between $\pm R$

$\displaystyle\displaystyle f_{R}(x)=\frac{2}{\pi
R^{2}}\Re\left(\sqrt{R^{2}-x^{2}}\right).$ (3.48)

Such distributions appear in different contexts in physics and mathematics.
The eigenvalue density of Gaussian hermitian random matrix theory is given by
the semicircle distribution. The transition distribution of the limit shape of
asymptotic Young diagrams studied by Vershik-Kerov and Logan-Shepp [32, 33] is
given by semicircle distribution. In this section we consider the large $N$
distribution $\sigma_{1/2}(\theta)$ corresponding to $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$ are given by the semicircle distributions. The reason behind
this choice is that the fluid equations can be solved exactly. Our goal is to
explicitly calculate the knot invariant for the Hopf link for such
representations in the large $N$ limit and study the phase transition of two
point function.

We take the following semicircular distributions for the Young diagram density
corresponds to $\mathcal{R}_{1/2}$

$\displaystyle\displaystyle\rho_{\mathcal{R}_{1}/\mathcal{R}_{2}}=\frac{1}{\pi}\sqrt{L_{1/2}-\frac{L_{1/2}^{2}y^{2}}{4}}.$
(3.49)

Since $\mathcal{R}_{1/2}$ are integrable representations $L_{1/2}$ satisfies

$\displaystyle\displaystyle L_{1},L_{2}\leq\pi^{2}\quad\text{and}\quad
L_{1},L_{2}\geq 16\lambda^{2}.$ (3.50)

Suppose $u(y,t)$ denotes Young diagram distributions that interpolates between
$\rho_{\mathcal{R}_{1}}$ and $\rho_{\mathcal{R}_{2}}$ from $t=0$ to $t=A$. We
consider

$u(y,t)=\frac{1}{\pi}\sqrt{\mu(t)-\frac{\mu(t)^{2}y^{2}}{4}}$ (3.51)

such that

$\displaystyle\displaystyle\mu(0)=L_{1}\quad\text{and}\quad\mu(A)=L_{2}.$
(3.52)

Considering the relation $\theta=2\pi\lambda y$, the eigenvalue distribution
$\sigma(\theta,t)$ corresponding to $u(y,t)$ is given by

$\sigma(\theta,t)=\frac{1}{2\pi\lambda}u\left(\frac{\theta}{2\pi\lambda},t\right)=\frac{1}{2\pi^{2}\lambda}\sqrt{\mu(t)-\frac{\mu(t)^{2}\theta^{2}}{16\pi^{2}\lambda^{2}}}.$
(3.53)

For the choice of fluid density (3.53) the fluid velocity can be solved
exactly from the continuity equation and is given by

$\displaystyle\displaystyle
v(t,\theta)=-\theta\left(\frac{\dot{\mu}(t)}{2\mu(t)}\right)$ (3.54)

where $\dot{\mu}(t)$ is derivative of $\mu(t)$ with respect to $t$. Plugging
the ansatz for $\sigma(t,\theta)$ and the solution for $v(t,\theta)$ in the
Navier-Stokes equation we find a differential equation for $\mu(t)$

$\displaystyle\displaystyle\ddot{\mu}(t)-\frac{3\dot{\mu}(t)^{2}}{2\mu(t)}-\frac{1}{32\pi^{4}\lambda^{4}}\mu(t)^{3}=0.$
(3.55)

This is a second order non-linear ordinary differential equation and has a
simple solution up to two integration constants. These two constants can be
fixed from the boundary conditions (3.52).

First we consider a special case $L_{1}=L_{2}=L$. Solving equation (3.55) for
symmetric boundary conditions we find

$\begin{split}\mu(t)=\frac{4L\pi^{2}(8\pi^{2}\lambda^{4}+\sqrt{A^{2}L^{2}\lambda^{4}+64\pi^{4}\lambda^{8}})}{L^{2}(A-t)t+4\pi^{2}(8\pi^{2}\lambda^{4}+\sqrt{A^{2}L^{2}\lambda^{4}+64\pi^{4}\lambda^{8}})}.\end{split}$
(3.56)

Since $v(t,\theta)$ is proportional to $\dot{\mu}(t)$ it is easy to see that
the velocity is proportional to $A-2t$ and hence for $L_{1}=L_{2}=L$ the fluid
equations always admits a solution such that $v(t,\theta)=0$ at $t=A/2$. The
density $\sigma(t,\theta)$ coincide with $\sigma_{1}(\theta)$ at $t=0$ and
then starts spreading out as $t$ increases. The spreading is maximum at
$t=A/2$. Finally the density again starts contracting and finally becomes
$\sigma_{2}$ at $t=A$. See figure 5.

Figure 5: Plot of $\sigma(\theta,t)$ as a function of $\theta$ at $t=0$
(solid), $t=t^{*}=A/2$ (dashed) and any arbitrary $0<t<t^{*}$ (dotted).

If the maximum spread of $\sigma(t,\theta)$ at $t=A/2$ is less than $2\pi$
then $\mathcal{W}_{\mathcal{R}_{1}\mathcal{R}_{2}}$ do not observe any phase
transition. For

$\displaystyle\displaystyle
L\geq\frac{16\pi^{2}\lambda^{2}}{\pi^{2}-4p^{2}\lambda^{2}}$ (3.57)

$\sigma(A/2,\theta)$ always has a gap. Since there exists a point $t^{*}=A/2$
when velocity is zero, there exists a real dominant Young diagram which
maximises $\mathcal{W}_{\mathcal{R}_{1}\mathcal{R}_{2}}$. It is given by the
(3.47) and has the form

$\rho(y)=\frac{\sqrt{(8\pi^{2}\lambda^{2}+\sqrt{A^{2}L^{2}+64\pi^{4}\lambda^{4}})(4L-16\pi^{2}\lambda^{2}y^{2})-A^{2}L^{2}y^{2}}}{2L\pi}.$
(3.58)

Since $\sigma^{*}(\theta)$ has gap, the dominant Young diagram does not
saturate the upper bound. However at
$L=\frac{16\pi^{2}\lambda^{2}}{\pi^{2}-4p^{2}\lambda^{2}}$ the gap in
$\sigma^{*}(\theta)$ vanishes and the corresponding dominant Young diagram
touches the upper cap. Therefore
$\mathcal{W}_{\mathcal{R}_{1}\mathcal{R}_{2}}$ undergoes a Douglas-Kazakov
type phase transition at this critical value of $L$.

The value of the knot invariant can be obtained by calculating the Hamiltonian
on this solution and integrating over $A$. After an explicit calculation we
obtain the following result (substituting $A=2\pi p\lambda$)

$\begin{split}\mathrm{W}^{(0)}_{LL}(\lambda)&=-\frac{2\pi\lambda}{pL}+\frac{1}{2}\sqrt{1+\frac{16\pi^{2}\lambda^{2}}{p^{2}L^{2}}}+\frac{\pi
p}{12}-\ln 2-\frac{1}{2}\sinh^{-1}(\frac{pL}{4\pi\lambda}).\end{split}$ (3.59)

This is the HOMFLY-PT polynomial in the double scaling limit for Hopf link
where two representations associated with two unknots are same and given by
Wigner semicircle distributions. In general HOMFLY-PT is a polynomial of two
variables $q=e^{2\pi i/(k+N)}$ and $s=q^{N}$. However in the double scaling
limit the variable $q=1$ and $s=e^{2\pi i\lambda}$, hence
$\mathrm{W}^{(0)}_{LL}(\lambda)$ is a function of $\lambda$ only.

Surprisingly, the final expression for the action $S(\sigma_{1},\sigma_{2}|A)$
with $\sigma_{1/2}$, given by the same semicircle distribution of size $L$, is
similar to the limit shape of asymptotic Young diagrams [32, 33, 34] up to an
analytic continuation in $p$. In particular

$\frac{4}{\pi}\left(S(u)+1\right)_{u\rightarrow
iu}=\hat{\Omega}(u),\quad\text{where}\quad u=\frac{pL}{2\pi\lambda}$

and $\hat{\Omega}(u)$ is the limit shape. This is nothing but an observation,
we think.

One can also choose two different representation by considering the boundary
conditions $\mu(0)=L_{1}$ and $\mu(A)=L_{2}$. The expression for $\mu(t)$ can
also be computed exactly. From these expression we see that the existence of
dominant Young diagram depends on the values of $L_{1},\ L_{2}$ and $\lambda$.
For a given choice of the set $(L_{1},L_{2},\lambda)$ if the fluid velocity
never reaches zero then the there exists no real dominant Young diagram to
maximise (3.13). However one can calculate $S(\sigma_{1},\sigma_{2}|A)$ from
(3.25) and hence $\mathrm{W}^{0}_{L_{1}L_{2}}$. In this case
$\mathrm{W}^{0}_{L_{1}L_{2}}$ does not observe any phase transition. We see
that to get a real dominant Young diagram for $L_{1}>L_{2}$ the parameters
must satisfy
$A\geq\frac{8\sqrt{L_{1}-L_{2}}\pi^{2}\lambda^{2}}{L_{2}\sqrt{L_{1}}}$ and for
$L_{2}\geq L_{1}$ the relation is given by
$A\geq\frac{8\sqrt{L_{2}-L_{1}}\pi^{2}\lambda^{2}}{L_{1}\sqrt{L_{2}}}$. In
this case there exists a critical relation between $L_{1},L_{2}$ and $A$ which
determines whether the two point function $\mathcal{W}_{L_{1}L_{2}}$ will
undergo a phase transition or not. However such condition does not have any
handy expression but can be found numerically. One can also compute the knot
polynomial in this case.

###  3.5 Calculation of partition function

From (2.6) we see that we get back the partition function of CS theory on
Seifert manifold when $\mathcal{R}_{a}=0\ \forall a$ (no box in the Young
diagram). The phase structure of this theory in the aforementioned double
scaling limit was discussed in [30] by directly solving the saddle point
equation. As a consistency check, in this subsection we reproduce the same
result from the solution of fluid equations.

When $\mathcal{R}_{a}=0$ the corresponding $\sigma_{a}(\theta)$ is given by

$\displaystyle\sigma_{a}(\theta)=\begin{cases}\frac{1}{2\pi\lambda}&\text{for}\quad-\pi\lambda\leq\theta\leq\pi\lambda\\\
\\\ 0&\text{for}\quad\text{otherwise}.\end{cases}$ (3.60)

We expect that the solutions of the fluid equations (3.27) with the boundary
conditions that $\sigma(t,\theta)$ merges with (3.60) at $t=0$ and $t=A$ will
admit an intermediate time $t^{*}=A/2$ (follows from symmetry) when fluid
velocity is zero and the functional inverse of the fluid density at $t^{*}$
will give the dominant Young diagram representation obtained in [30].

In order to check our expectation we use the dominant Young diagram found in
[30]

$\displaystyle\displaystyle\rho(y)=\frac{p}{\pi}\tanh^{-1}\left[\sqrt{1-\frac{e^{-\frac{2\pi\lambda}{p}}}{\cosh(\pi\lambda
y)}}\right],$ (3.61)

for $0\leq\lambda\leq\frac{p}{\pi}\log[\cosh{(\pi/p)}]$. Inverse of this
function gives $\sigma^{*}(\theta)$. We now use the identity (3.45) to find
$v(t,\theta)$ and $\sigma(t,\theta)$. Since both velocity and density are real
functions of $t$ and $\theta$, we solve the real and imaginary parts of this
equation and find that $\sigma(t,\theta)$ matches with (3.60) at $t=0$ and
$t=A$. For $\lambda>\frac{p}{\pi}\log[\cosh{(\pi/p)}]$ the dominant Young
diagram distribution has a cap. It is difficult to invert that distribution to
find $\sigma^{*}(\theta)$. However, one can numerically check that other phase
also solves the fluid equations.

##  4 $Sp(N)$ Chern-Simons theory and knot invariants

A symplectic group $Sp(n,F)$ is a group of $2n\times 2n$ dimensional
symplectic matrices over a field $F$ under matrix multiplication. A $2n\times
2n$ dimensional matrix $A$ is a symplectic matrix if it satisfies the relation

$\displaystyle\displaystyle A^{T}\Omega_{2n}A=\Omega_{2n}$ (4.1)

where $\Omega_{2n}$ is a $2n\times 2n$ dimensional skew-symmetric matrix :
$\Omega_{2n}^{T}+\Omega_{2n}=0$. A standard choice of $\Omega_{2n}$ is

$\displaystyle\displaystyle\Omega_{2n}=\begin{pmatrix}0&\mathbb{I}_{n}\\\
-\mathbb{I}_{n}&0\end{pmatrix}.$ (4.2)

Irreducible representations of $Sp(n,F)$ are characterised by Young diagrams
with maximum $n$ number of rows.

CS theory for $Sp(N)$ gauge group is well studied. The large $N$ limit of
these theories and their connections with dual string theories were studied in
[35]. In this section we shall discuss the phase structure of the theory in
the aforementioned double scaling limit. Then we show that the analysis, given
in section 3.1, can be extended for $Sp(N)$ gauge group to obtain the
invariant for the Hopf link in terms of the collective field theory action.
The analysis is little different than that of a $U(N)$ theory since the
modular transformation matrices $\mathcal{S}$ and $\mathcal{T}$ for
$sp(N)_{k}$ affine algebra have different forms.

###  4.1 The partition function

The partition function of CS theory with gauge group $Sp(N)$ and level $k$ on
Seifert manifold is given by

$\mathcal{Z}[S^{3}/\mathbb{Z}_{p},Sp(N),k]=\sum_{\mathcal{R}}\mathcal{T}_{\mathcal{R}\mathcal{R}}^{-p}\mathcal{S}_{0\mathcal{R}}^{2}$
(4.3)

here $\mathcal{S}$ and $\mathcal{T}$ for $sp(N)_{k}$ affine lie algebra are
given by

$\begin{split}\mathcal{S}_{\mathcal{R}\mathcal{R}^{\prime}}=(-i)^{\frac{N(N-1)}{2}}\Big{(}\frac{2}{N+k+1}\Big{)}^{\frac{N}{2}}\det\left|\sin\Big{(}\frac{\pi
f_{i}(R)f_{j}(R^{\prime})}{N+k+1}\Big{)}\right|_{i,j=1}^{N}\end{split}$ (4.4)

where

$f_{i}(\mathcal{R})=n_{i}(\mathcal{R})-i+N+1\equiv h_{i}(\mathcal{R})$ (4.5)

and

$\mathcal{T}_{\mathcal{R}\mathcal{R}^{\prime}}=\exp\left[{-\frac{i\pi
N(N+1)}{12}}+{\frac{i\pi}{2(N+k+1)}\sum_{i=1}^{N}h_{i}(\mathcal{R})^{2}}\right]\delta_{\mathcal{R},\mathcal{R}^{\prime}}.$
(4.6)

We define new variables $\theta_{i}$

$\theta_{i}(\mathcal{R})=\frac{\pi
h_{i}(\mathcal{R}^{\prime})}{N+K+1}=\pi\lambda\frac{h_{i}(\mathcal{R})}{N},\quad\text{where}\
\lambda=\frac{N}{N+k+1}.$ (4.7)

The double scaling limit is given by as before : $N\rightarrow\infty,\
k\rightarrow\infty$ keeping $\lambda$ fixed. For an integrable representation
$\mathcal{R}$ we have $0\leq h_{i}(\mathcal{R})\leq N+k+1$, hence the new
variables $\theta_{i}(\mathcal{R})$ satisfies

$\displaystyle\displaystyle 0\leq\theta_{i}(\mathcal{R})\leq\pi.$ (4.8)

Although $\theta_{i}\geq 0$ for a $sp(N)$ representations, we introduce a
distribution function $\sigma^{sp}(\theta)$ which defines a symmetric
distribution of eigenvalues between $-\pi$ and $\pi$.

$\sigma^{sp}(\theta)=\frac{1}{2N}\sum_{i=1}^{N}\delta(\theta-\theta_{i})+\frac{1}{2N}\sum_{i=1}^{N}\delta(\theta+\theta_{i}).$
(4.9)

Introducing mirror images of the eigenvalues

$\theta_{-i}=-\theta_{i}$ (4.10)

$\sigma^{sp}(\theta)$ can be written as

$\displaystyle\displaystyle\sigma^{sp}(\theta)=\frac{1}{2N}\sum_{{i=-N}\atop{i\neq
0}}^{N}\delta(\theta-\theta_{i}).$ (4.11)

Hence $\sigma^{sp}(\theta)$ is a distribution of $2N$ eigenvalues : $N$
$\theta_{i}$s and their mirror images $\theta_{-i}$s. We should remember the
relation (4.10) while taking derivative with respect to $\theta_{i}$. We also
note that $\sigma^{sp}(\theta)$ has upper-cap given by

$\displaystyle\displaystyle\sigma^{sp}(\theta)\leq\frac{1}{2\pi\lambda}$
(4.12)

similar to the $U(N)$ case (3.6).

The partition function in the continuum limit is therefore given by

$\begin{split}\mathcal{Z}[S^{3}/\mathbb{Z}_{p},Sp(N),k]=\int[d\theta]\exp\left[{-\frac{N^{2}}{\lambda^{2}}S_{eff}[\sigma^{sp}]}\right]\end{split}$
(4.13)

where

$\begin{split}S_{eff}[\sigma^{sp}]=&-\lambda^{2}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\sigma^{sp}(\theta)\sigma^{sp}(\theta^{\prime})\log\Big{[}4\sin^{2}\Big{(}\frac{\theta-\theta^{\prime}}{2}\Big{)}\Big{]}d\theta
d\theta^{\prime}\\\
&+\frac{2p\lambda}{\pi}\int_{-\pi}^{\pi}\sigma^{sp}(\theta)\Big{(}\frac{\theta^{2}}{4}-\frac{\pi^{2}}{12}\Big{)}+\frac{2p\pi\lambda(1-\lambda)}{12}.\end{split}$
(4.14)

The saddle point equation is given by

$\displaystyle\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{-\pi}^{\pi}\sigma^{sp}(\theta^{\prime})\cot\Big{(}\frac{\theta-\theta^{\prime}}{2}\Big{)}d\theta^{\prime}=\frac{p}{2\pi\lambda}\theta.$
(4.15)

Thus we see that the saddle point equation for $Sp(N)$ CS theory on
$S^{3}/\mathbb{Z}_{p}$ is the same as that of $U(N)$ CS theory (B.10). Hence
in the large $N$ limit the phase structure of these two theories are
identical. Therefore, in the double scaling limit, $Sp(N)$ CS theory on
$S^{3}/\mathbb{Z}_{p}$ admits a third order phase transition at
$\lambda=p{\log(\cosh\frac{\pi}{p})}/{\pi}$.

###  4.2 Two point correlator and Hopf link invariants

Though the structure of the modular $\mathcal{S}$ and $\mathcal{T}$ matrices
for $sp(N)_{k}$ affine algebra is different than that for $u(N)_{k}$, the two
point correlator for the Hopf link admits a description in the language of
incompressible fluid and hence can be written in terms of the on shell action
of a free collective field theory. The calculation follows the similar line as
what we did for $U(N)$ theory but the intermediate steps are different since
hook numbers of the Young diagrams of $sp(N)$ representations are always
positive. Bulk of the calculations are given in appendix C.2, here we outline
the main steps.

Using the expression of modular $\mathcal{S}$ and $\mathcal{T}$ matrices we
write down the modified two point correlation function (3.1) as,

$\begin{split}\widetilde{\mathcal{W}}_{\mathcal{R}_{1}\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},Sp(N),N,k)&=\frac{e^{\frac{ip\pi
N(2N+1)}{12}}}{2^{2N(N-1)}}\sum_{y_{i}(\mathcal{R})}\left(\frac{\det[\sin(N\theta_{j}(\mathcal{R}_{1})y_{i}(\mathcal{R}))]}{D[\theta(\mathcal{R}_{1})]}\right)\\\
&\quad\times\left(\frac{\det[\sin(N\theta_{j}(\mathcal{R}_{2})y_{i}(\mathcal{R}))]}{D[\theta(\mathcal{R}_{2})]}\right)e^{-\frac{i\pi
p\lambda N}{2}\sum_{i=1}^{N}y_{i}(\mathcal{R})^{2}}\end{split}$ (4.16)

where

$\displaystyle\displaystyle y_{i}(\mathcal{R})=\frac{h_{i}(\mathcal{R})}{N}$
(4.17)

and

$D[\theta(\mathcal{R}_{1})]=\prod_{i=1}^{N}\sin(\theta_{i}(\mathcal{R}_{1}))\prod_{i<j}^{N}\sin\Big{(}\frac{\theta_{i}(\mathcal{R}_{1})+\theta_{j}(\mathcal{R}_{1})}{2}\Big{)}\sin\Big{(}\frac{\theta_{i}(\mathcal{R}_{1})-\theta_{j}(\mathcal{R}_{1})}{2}\Big{)}.$
(4.18)

We follow the same procedure, what we did for $U(N)$ CS theory and define
(after a wick rotation $p\to-ip$)
$\tilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},Sp(N),N,k)$

$\widetilde{\mathcal{W}}_{\mathcal{R}_{1}\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},Sp(2N),N,k)=\frac{1}{2^{2N(N-1)}}e^{\frac{p\pi
N(2N+1)}{12}}\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},Sp(2N),N,k).$
(4.19)

Following (3.22), in the large $N$ limit we define a similar function function
$F(\sigma^{sp}_{1/2},A)$ where $\sigma^{sp}_{1/2}(\theta)$ are eigenvalue
distributions corresponding to $\mathcal{R}_{1/2}$ respectively. Segregating
the pure $\sigma^{sp}_{1/2}$ dependent part from $F(\sigma^{sp}_{1/2},A)$, we
define a function $S(\sigma^{sp}_{1/2},A)$ given in (C.13) and show that
$S[\sigma_{1/2}^{sp},A]$ satisfies,

$\begin{split}\frac{\partial S}{\partial
A}=\frac{1}{2}\int{\sigma}^{sp}_{a}\theta)\Big{[}\Big{(}\frac{1}{2}\frac{\partial}{\partial\theta}\frac{\delta
S}{\delta{\sigma}^{sp}_{a}(\theta)}\Big{)}^{2}-4\frac{\pi^{2}}{3}{\sigma}^{sp}_{a}(\theta)^{2}\Big{]}d\theta\end{split}$
(4.20)

where,

$\displaystyle\displaystyle A=\pi p\lambda.$ (4.21)

Thus we see that $S[\sigma^{sp}_{a},A]/4$ (Hopf link invariant in $Sp(N)$ CS
theory) satisfies the same Hamilton-Jacobi equation and hence the saddle point
is governed by an $Sp(N)$ free collective field theory equations.

The real dominant Young diagram in the large $N$ limit can also be obtained by
studying the Hamilton’s equations. The real dominant representation, if it
exists, is given by the inverse of $Sp(N)$ fluid density
$\sigma^{sp}(\theta,t)$ at some intermediate time when the fluid velocity is
zero. The detailed calculation for dominant representation is given in
appendix D.1.

##  5 Discussion

In this paper we find that the computation of two point correlation function
in $U(N)$ and $Sp(N)$ CS theory in $S^{3}/\mathbb{Z}_{p}$ (which renders
invariant for the Hopf link) in the large $N$ limit boils down to finding
solutions of continuity and Navier-Stokes equations of an incompressible one
dimensional fluid evolving from $t=0$ to $t=A$ with the initial and final
densities corresponding to the representations $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$. The Hopf link invariant
$\mathrm{W}^{(0)}_{\mathcal{R}_{1}\mathcal{R}_{2}}(\lambda)$ in the large $N$
limit satisfies the Hamilton-Jacobi equation where $p\lambda$ plays the role
of time and Hamiltonian is given by $U(N)$ (or $Sp(N)$) free collective field
theory Hamiltonian. The invariant for the Hopf link turns out to be equal to
the on shell action. Using the method developed in [3] we finally show that
invariants for other torus knots (2,$k$=odd) and links (2,$k$=even) can be
obtained from the invariants for the Hopf link and unknot.

We further discuss the large $N$ phase structure of two point correlators in
CS theory. Whether two point function undergoes a phase transition or not
depends on the evolution of the fluid. The absolute value of the fluid
velocity at a given point $\theta$ decreases with time as the fluid starts
evolving from $t=0$. The existence of a real dominant representation depends
on whether the absolute value of velocity can reach zero at some intermediate
time for all $-\pi\leq\theta\leq\pi$. If one start with a class of
$\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ such that corresponding
$\sigma_{1}(\theta)$ and $\sigma_{2}(\theta)$ are gapped then the density of
the fluid starts spreading from its initial distribution $\sigma_{1}(\theta)$.
The spreading is maximum when the velocity is zero (if such solution exists)
and then starts shrinking and goes to its final distribution
$\sigma_{2}(\theta)$ at $t=A$. The maximum spreading of fluid density (at
$t=t^{*}$) depends on the choice of $\mathcal{R}_{1/2}$ and $\lambda$. If the
maximum spread lies between $-\pi$ and $\pi$ then there is no phase transition
in $\mathrm{W}_{\mathcal{R}_{1}\mathcal{R}_{2}}^{(0)}$. However,
$\mathrm{W}_{\mathcal{R}_{1}\mathcal{R}_{2}}^{(0)}$ observes a third order
phase transition otherwise. The CS theory enjoys _level-rank_ duality. A
theory with rank $N$ and level $k$ is dual to a theory with level and rank
exchanged. The two point correlation function in the dual theory also admits
the similar fluid structure. It would be interesting to understand the
relation between the fluid and its dual fluid in the large $N$ limit.

The study of large $N$ correlation functions is important in many aspects. It
sets up the platform to check the generalised volume conjecture [19, 20, 18,
21]. Further, based on the Gopakumar-Vafa conjecture [36] it was observed in
[37] that the CS invariants are mapped to topological string amplitudes on
Riemann surfaces with boundaries in the topological string theory side. The
knot invariants were reformulated in terms of new invariants (integer
invariants) capturing the spectrum of M2 branes ending on M5 branes. The
results were checked explicitly for unknot. Large $N$ analysis of this
observation and its generalisation to other links were considered in [9]. Our
analysis to compute invariants for a class of knots and links will be useful
to further investigate the connection between CS theory and topological string
theory beyond partition function.

As we mentioned in section 2 that in $S^{3}$ there exists a canonical framing
$\mathcal{K}=\mathcal{S}$. In this framing the two point correlation function
for the Hopf link is given by $\mathcal{S}_{\mathcal{R}_{1}\mathcal{R}_{2}}$
and does not show phase transition for any $\mathcal{R}_{1/2}$. In Seifert
framing the same quantity is given by (3.11). The sum is over the integrable
representations. Using the properties of $\mathcal{S}$ and $\mathcal{T}$
matrices one can show that these two expressions are related to each other up
to a phase factor. The question is why we see a phase transition in two point
function in the double scaling limit. If one takes the $N,k\rightarrow\infty$
limit without any restriction, then the sum in (3.11) runs over all possible
Young diagrams with any number of rows and any number of columns. However,
here we are considering a particular limit $N,k\rightarrow\infty$ keeping
$N/k$ fixed. Under this condition the sum becomes restricted - one does not
sum over all possible Young diagrams. Therefore we do not expect that in the
double scaling limit the above equality holds. This was also the reason behind
the phase transition in CS theory in $S^{3}$ studied in [30]. The framing
dependence of correlation functions in CS theory bears a mining in the
topological string theory side. Framing is related to inherent ambiguity in
the open topological string amplitude related to the IR geometry of the
D-brane [38, 39]. It would be interesting to study the topological string
amplitudes in the same double scaling limit and understand the connection
better. We keep the problem for future.

Acknowledgments : The work of SD is supported by the _MATRICS_ (grant no.
_MTR/ 2019/ 000390_ , the Department of Science and Technology, Government of
India). SD acknowledges the Simons Associateship of the Abdus Salam ICTP,
Trieste, Italy. We are indebted to people of India for their unconditional
support toward the researches in basic science.

##  Appendix A Framing dependence of partition function

Partition function of CS theory in three dimensions depends on choice of
framing [1]. In this appendix we discuss the framing dependence in details.

Canonical quantisation of CS theory on a three manifold $\mathcal{M}$ with a
boundary $\Sigma$ produces the "physical Hilbert space" $\mathcal{H}(\Sigma)$.
In [1] Witten constructed $\mathcal{H}(\Sigma)$ in terms of conformal blocks
of WZW model on $\Sigma$ with a gauge group $G$ and and level $k$. One can
explicitly construct the Hilbert space for $\Sigma=S^{2}$ and
$\Sigma=\mathbb{T}^{2}$. For $\Sigma=S^{2}$ the Hilbert space is trivial
(dimension one). However for $\Sigma=\mathbb{T}^{2}$ the space of conformal
blocks has one-to-one correspondence with integrable representations of
$\mathbf{g}_{k}$. There is a natural choice of basis for
$\mathcal{H}(\Sigma=\mathbb{T}^{2})$. The basis vectors are given by
integrable representations $|\mathcal{R}_{a}\big{>}$ of $\mathbf{g}_{k}$ :
these are the states associated with the partition function of CS theory in a
solid torus $\mathbf{T}^{2}$ with a WL in representation $\mathcal{R}_{a}$
along the non-contractible cycle. Therefore the Hilbert space is finite
dimensional and spanned by these integrable representations.

In order to write the partition function of CS theory on a generic three
manifold $\mathcal{M}$ we split the the manifold into two parts
$\mathcal{X}_{L}$ and $\mathcal{X}_{R}$ sharing a common boundary $\Sigma$.
The path integral of CS theory on $\mathcal{X}_{R}$ corresponds to a vector
$|\phi\big{>}$ in $\mathcal{H}(\Sigma)$. Since the boundary $\Sigma$ of
$\mathcal{X}_{L}$ has an opposite orientation of that of $\mathcal{X}_{R}$ the
path integral on $\mathcal{X}_{L}$ is mapped to a vector
$\big{<}\psi|\in\mathcal{H}^{*}(\Sigma)$ where $\mathcal{H}^{*}(\Sigma)$ is
dual of $\mathcal{H}(\Sigma)$. Since the manifold $\mathcal{M}$ can be
obtained by gluing $\mathcal{X}_{L}$ and $\mathcal{X}_{R}$ along $\Sigma$, the
partition function of CS theory on $\mathcal{M}$ is therefore given by [1]

$\displaystyle\displaystyle\mathcal{Z}(\mathcal{M})=\langle\psi|\phi\rangle.$
(A.1)

Using this result Witten showed that one can write the partition function and
correlation functions of CS theory in $S^{3}$ and other generic three
manifolds from the partition function and WLs in $S^{2}\times S^{1}$.

In order to understand the prescription in detail let us start with CS theory
on a three manifold $\mathcal{M}$. We consider a WL
$\mathcal{W}_{\mathcal{R}_{a}}=\text{Tr}_{\mathcal{R}_{a}}U_{\mathrm{K}}$ in
$\mathcal{M}$ along a knot $\mathrm{K}$, where
$U_{\mathrm{K}}=P\exp\left(\int_{\mathrm{K}}A\right)$ and $\mathcal{R}_{a}$ is
an integrable representation of $\mathbf{g}_{k}$. We take a tubular
neighbourhood of the knot $\mathrm{K}$ which is a solid torus $\mathbf{T}^{2}$
such that $\partial\mathbf{T^{2}}=\mathbb{T}^{2}$ is a torus. We take the
$\mathbf{T}^{2}$ out of the manifold and hence the three manifold
$\mathcal{M}$ is now a connected sum of $\mathcal{X}_{R}=\mathbf{T}^{2}$ with
a WL inserted and the reminder $\mathcal{X}_{L}$. Note both $\mathcal{X}_{L}$
and $\mathcal{X}_{R}$ have common boundary $\mathbb{T}^{2}$. Following the
work of Verlinde [40] Witten showed that the path integral over
$\mathbf{T}^{2}$ with a WL $\mathcal{W}_{\mathcal{R}_{a}}$ along the non-
contractible cycle of $\mathbf{T}_{2}$ is mapped to a state
$|\mathcal{R}_{a}\big{>}$ in $\mathcal{H}(\mathbb{T}^{2})$. Thus, following
(A.1) we see that the expectation value of WL $\mathcal{W}_{\mathcal{R}_{a}}$
in representation $R_{a}$ in $\mathcal{M}$ can be written as

$\displaystyle\displaystyle\langle\mathcal{W}_{\mathcal{R}_{a}}\rangle_{\mathcal{M}}=\big{<}\psi|\mathcal{R}_{a}\big{>}$
(A.2)

where $\big{<}\psi|\in\mathcal{H}^{*}(T^{2})$ is path integral over
$\mathcal{X}_{L}$. For $\mathcal{R}_{a}$ to be a trivial representation, the
WL is equal to $1$. Therefor $\big{<}\psi|0\big{>}$ is the partition function
on $\mathcal{M}$.

Now, before gluing the solid torus with $\mathcal{X}_{L}$ one can also make a
diffeomorphism on the boundary of $\mathbf{T}^{2}$. Such operation (scooping
out $\mathbf{T}^{2}$ from $\mathcal{M}$ $\rightarrow$ apply diffeomorphism on
the boundary $\rightarrow$ gluing back with $\mathcal{X}_{L}$) generates a new
manifold $\tilde{\mathcal{M}}$. Let us first understand this with the help of
a simple example. Suppose our $\mathcal{M}=S^{2}\times S^{1}$. The manifold
$S^{2}\times S^{1}$ can be written as a connected sum of two solid tori. To
understand this one can think that a solid torus is a _disc_ times a _circle_
: $\mathbf{T}^{2}=D\times S^{1}$. When we glue two disc at the boundary we get
the two manifold $S^{2}$. Therefore, when we glue two solid tori along their
boundaries without any diffeomorphism we get $S^{2}\times S^{1}$. Thus we see
that when we scoop out a solid torus from $S^{2}\times S^{1}$ the reminder is
also a solid torus. Consider now a solid torus $\mathbf{T}^{2}$ embedded in
$S^{3}$ which is $R^{3}\cup\infty$. $\mathbf{T}^{2}$ is invariant under
inversion. The exterior of this $\mathbf{T}^{2}$ is another solid torus
$\mathbf{T^{\prime}}^{2}$ (as we have identified the points at infinity).
However, there is a difference. The contractible cycle in $\mathbf{T}^{2}$ is
mapped to the non-contractible cycle in $\mathbf{T^{\prime}}^{2}$ and vice
versa. Therefore these two tori are related to each other by $S$ modular
transformation on the boundary $\mathbb{T}^{2}$. Thus when we glue
$\mathbf{T}^{2}$ and $\mathbf{T^{\prime}}^{2}$ we get $S^{3}$. Now starting
from $\mathcal{M}=S^{2}\times S^{1}$ we split the manifold in two solid tori.
Then we perform an $S$ modular transformation (i.e. inverting the torus) on
the boundary of one of them (say the right one) and then glue them again. This
surgery produces $S^{3}$ from $S^{2}\times S^{1}$.

Consider now a CS theory in a solid torus with a WL in representation
$\mathcal{R}_{a}$ along the non-contractible circle. The path integral maps to
$|\mathcal{R}_{a}\big{>}\in\mathcal{H}(\mathbb{T}^{2})$. If $K$ is the
diffeomorphism that acts on the boundary of the solid torus, the path integral
on $\mathbf{T}^{2}$ changes and hence is mapped to a different state in
$\mathcal{H}(\mathbb{T}^{2})$. The new state can be written as

$\displaystyle\displaystyle|\chi\big{>}=\sum_{\mathcal{R}_{b}}\mathcal{K}_{a}^{\
b}|\mathcal{R}_{b}\big{>}.$ (A.3)

The matrix
$\mathcal{K}\in\text{Hom}(\mathcal{H}(\mathbb{T}^{2}),\mathcal{H}(\mathbb{T}^{2}))$
depends on the diffeomorphism $K$ that acts on the torus. If we now glue this
solid torus with $\mathcal{X}_{L}$ we get a new manifold $\tilde{\mathcal{M}}$
with a WL in $\mathcal{R}_{a}$. The expectation value of the WL in
representation $\mathcal{R}_{a}$ in $\tilde{\mathcal{M}}$ can be written in
terms of expectation value of WLs in $\mathcal{M}$

$\displaystyle\displaystyle\langle
W_{R_{a}}\rangle_{\tilde{\mathcal{M}}}=\sum_{\mathcal{R}_{b}}\mathcal{K}_{a}^{\
b}\langle W_{\mathcal{R}_{b}}\rangle_{\mathcal{M}}.$ (A.4)

Hence the partition function on $\tilde{\mathcal{M}}$ can be obtained by
considering $R_{a}$ to be a trivial representation

$\displaystyle\displaystyle
Z(\tilde{\mathcal{M}})=\sum_{\mathcal{R}_{b}}\mathcal{K}_{0}^{\ b}\langle
W_{R_{b}}\rangle_{\mathcal{M}}.$ (A.5)

This is a very powerful relation.

The partition function of Chern-Simons theory on $S^{2}\times S^{1}$ can be
calculated using (A.5). As explained above the manifold $S^{2}\times S^{1}$
can be written as a connected sum of two solid tori. As mentioned earlier,
partition function in $\mathbf{T}^{2}$ without any WL is mapped to $|0\big{>}$
in $\mathcal{H}(\mathbb{T}^{2})$. Thus we get partition function of CS theory
on $S^{2}\times S^{1}$ is given by

$\displaystyle\displaystyle Z(S^{2}\times S^{1})=\big{<}0|0\big{>}=1.$ (A.6)

Similarly when we have a non-trivial WL in $S^{2}\times S^{1}$ we can split
the manifold into two solid tori with one torus containing the WL along the
con-contractible cycle. Hence the expectation value of a WL in $S^{2}\times
S^{1}$ is given by

$\displaystyle\displaystyle\langle W_{\mathcal{R}_{a}}\rangle_{S^{2}\times
S^{1}}=\big{<}0|\mathcal{R}_{a}\big{>}=\delta_{0\mathcal{R}_{a}}$ (A.7)

Our goal is to generate $S^{3}$ from $S^{2}\times S^{1}$ by surgery. Starting
from $S^{2}\times S^{1}$ we split the manifold in two solid torus. Then we
perform an $S$ modular transformation on the boundary $\mathbb{T}^{2}$ and
then glue them again. This surgery produces $S^{3}$. The diffeomorphism $K$ is
the modular transformation $S$ and hence the corresponding $\mathcal{K}$
matrix is the modular transform matrix $\mathcal{S}$ in
$\mathcal{H}(\mathbb{T}^{2})$. Therefore following (A.5) and (A.6) we find the
CS partition function on $S^{3}$ is given by

$\displaystyle\displaystyle
Z(S^{3})=\sum_{\mathcal{R}_{b}}\mathcal{S}_{0\mathcal{R}_{b}}\delta_{0\mathcal{R}_{b}}=\mathcal{S}_{00}.$
(A.8)

However, instead of choosing $\mathcal{K}=\mathcal{S}$ if we choose
$\mathcal{K}=\mathcal{T}\mathcal{S}\mathcal{T}=\mathcal{S}\mathcal{T}^{-1}\mathcal{S}$,
this also produces $S^{3}$ but in a different framing called _Seifert
framing_. In this framing the partition function is given by

$\displaystyle\displaystyle
Z_{SF}(S^{3})=\sum_{\mathcal{R}}\mathcal{S}_{0\mathcal{R}}^{2}\mathcal{T}_{\mathcal{R}\mathcal{R}}^{-1}$
(A.9)

which is same as $Z(S^{3})$ up to a phase.

Partition function for CS theory on a generic Lens space
$S^{3}/\mathbb{Z}_{p}$ (a Seifert manifold with $g=0$) can be obtained from
$S^{2}\times S^{1}$ by choosing
$\mathcal{K}=\left(\mathcal{T}\mathcal{S}\mathcal{T}\right)^{p}=\left(\mathcal{S}\mathcal{T}^{-1}\mathcal{S}\right)^{p}$.
The partition function is given by (B.1) with $g=0$.

In order to get the partition function of Hopf link in $S^{3}/\mathbb{Z}_{p}$
we can start with CS theory on $S^{2}\times S^{1}$ with two WLs. We consider a
solid torus along one WL and split the manifold with two parts scooping out
the solid torus. The reminder is also a solid torus with the other loop.
Partition functions on these two tori are given by $|\mathcal{R}_{a}\big{>}$
and $\big{<}\mathcal{R}_{b}|$. Hence CS partition function with two WLs is
given by

$\displaystyle\displaystyle Z(S^{2}\times
S^{1},\mathcal{R}_{a}\mathcal{R}_{b})=\big{<}\mathcal{R}_{b}|\mathcal{R}_{a}\big{>}=\delta_{\mathcal{R}_{a}\mathcal{R}_{b}}.$
(A.10)

Before gluing if we give an inversion on the right torus by choosing
$\mathcal{K}=\left(\mathcal{S}\mathcal{T}^{-1}\mathcal{S}\right)^{p}$ we get
Hopf link in $S^{3}/\mathbb{Z}_{p}$ given by (3.11).

We now want to find the correlation of three WLs in representations
$\mathcal{R}1$, $\mathcal{R}_{2}$ and $\mathcal{R}_{3}$ in $S^{2}\times
S^{1}$. We can split $S^{2}\times S^{1}$ with three WLs into two solid tori :
one contains a WL in $\mathcal{R}_{1}$ and the other torus contains two WLs in
$\mathcal{R}_{2}$ and $\mathcal{R}_{3}$. Suppose the CS partition function in
$\mathbf{T}^{2}$ with two WLs along non-contractible circle is mapped to a
state $|\mathcal{R}_{2}\mathcal{R}_{3}\big{>}$ in
$\mathcal{H}(\mathbb{T}^{2})$. Since $\mathcal{R}_{i}$s correspond to primary
fields in WZW we have

$\displaystyle\displaystyle|\mathcal{R}_{2}\mathcal{R}_{3}\big{>}=\sum_{\mathcal{R}}\mathcal{N}^{\mathcal{R}}_{\mathcal{R}_{1}\mathcal{R}_{2}}|\mathcal{R}\big{>}$
(A.11)

where $\mathcal{N}^{\mathcal{R}}_{\mathcal{R}_{1}\mathcal{R}_{2}}$ is the
Verlinde numbers. Hence

$\displaystyle\displaystyle Z(S^{2}\times
S^{1},\mathcal{R}_{1},\mathcal{R}_{2},\mathcal{R}_{3})=\mathcal{N}^{\mathcal{R}_{1}}_{\mathcal{R}_{2}\mathcal{R}_{3}}.$
(A.12)

One can now use the result (A.11) to write the $n$-point correlation function
in $S^{2}\times S^{1}$ given by (2.3) for $g=0$ with the help of Verlinde
formula

$\displaystyle\displaystyle\mathcal{N}^{\mathcal{R}_{1}}_{\mathcal{R}_{2}\mathcal{R}_{3}}=\sum_{\mathcal{R}}\frac{\mathcal{S}_{\mathcal{R}\mathcal{R}_{2}}\mathcal{S}_{\mathcal{R}\mathcal{R}_{3}}\mathcal{S}^{*}_{\mathcal{R}\mathcal{R}_{1}}}{\mathcal{S}_{0\mathcal{R}}}.$
(A.13)

Starting from (2.3) we take a solid torus out from $S^{2}\times S^{1}$ with
any of the $n$ WLs inside, apply an inversion on the torus and then put that
back inside. In this process we generate an $S^{3}$ with the link, shown in
fig.2 inside and the invariant is given by (2.2).

##  Appendix B Review of $U(N)_{k}$ Chern-Simons theory on Seifert Manifold

The partition function for $U(N)$ CS theory on Seifert manifold is given by

$\displaystyle\displaystyle\mathcal{Z}_{N,k}=\sum_{\mathcal{R}}\mathcal{S}_{0\mathcal{R}}^{2-2g}\mathcal{T}_{\mathcal{R}\mathcal{R}}^{-p}.$
(B.1)

Since the affine Lie algebra $u(N)_{k}$ is the quotient of $su(N)_{k}\times
u(1)_{N(k+N)}$ by $\mathbb{Z}_{N}$, the $u(N)$ representations can be
expressed in terms of $su(N)$ representations (denoted by $R$) and eigenvalues
of $u(1)$ generator $Q$ : $\mathcal{R}=(R,Q)$, where $Q=r(R)\ \text{mod}\ N$
and $r(R)$ is the number of boxes in $R$. The trivial representation
$\mathcal{R}=0$ means both $R=0$ and $Q=0$. The modular transform matrix
$\mathcal{S}_{\mathcal{R}\mathcal{R}^{\prime}}$ for $u(N)_{k}$ in terms of
$su(N)$ representations and the $u(1)$ charges is given by [26, 41, 29]

$\displaystyle\mathcal{S}_{\mathcal{R}\mathcal{R}^{\prime}}=\frac{(-i)^{\frac{N(N-1)}{2}}}{(k+N)^{\frac{N}{2}}}e^{-\frac{2\pi
iQQ^{\prime}}{N(N+k)}}\det M(R,R^{\prime})$ (B.2)

where, $M(R,R^{\prime})$ is a $N\times N$ matrix with elements,

$\displaystyle\displaystyle M_{ij}(R,R^{\prime})=\exp\left[\frac{2\pi
i}{k+N}\phi_{i}(R)\phi_{j}(R^{\prime})\right],$ (B.3)
$\displaystyle\displaystyle\phi_{i}(R)=l_{i}-\frac{r(R)}{N}-i+\frac{1}{2}(N+1)$
(B.4)

and $l_{i}$’s are the number of boxes in $i^{th}$ row in $R$. The modular
transformation matrix $\mathcal{T}_{\mathcal{R}\mathcal{R}^{\prime}}$ is given
by

$\displaystyle\displaystyle\mathcal{T}_{\mathcal{R}\mathcal{R}^{\prime}}$
$\displaystyle=e^{2\pi
i(h_{R}-\frac{c}{24})}\delta_{\mathcal{R}\mathcal{R}^{\prime}},\
h_{R}=\frac{1}{2}\frac{C_{2}(\mathcal{R})}{k+N},\ c=\frac{N(Nk+1)}{k+N}\ \ \ $
(B.5)

where $\mathcal{C}_{2}(\mathcal{R})$ is the quadratic Casimir of $u(N)_{k}$.
The $u(N)$ representations $\mathcal{R}$ can be characterised by extended YDs
by introducing the number of boxes in $i^{th}$ row $n_{i}=l_{i}+s$, for $1\leq
i\leq N-1$ and $n_{N}=s$ where $s\in\mathbb{Z}$. $n_{i}$s can be negative and
the corresponding YDs will have _anti-boxes_ [42]. In terms of these $n_{i}$s
the quadratic Casimir $C_{2}(\mathcal{R})$ is given by

$\displaystyle\displaystyle
C_{2}(\mathcal{R})=\sum_{i=1}^{N}n_{i}(n_{i}-2i+N+1).$ (B.6)

_For an integrable representation $\mathcal{R}$ of $u(N)_{k}$_

$\displaystyle\displaystyle-\frac{k}{2}\leq n_{N}\leq\cdots\leq
n_{1}\leq\frac{k}{2}.$ (B.7)

and hence

$\displaystyle\displaystyle-\frac{k}{2}<h_{N}<\cdots<h_{1}\leq\frac{k}{2}+N-1.$
(B.8)

In terms of the variables $\theta_{i}$ introduced in (3.3), the CS partition
function (B.1) in $S^{3}/\mathbb{Z}_{p}$ in the continuum limit is given by

$\displaystyle\displaystyle\begin{split}\mathcal{Z}_{N,k}^{p}&=\int[d\theta]e^{-(N+K)^{2}S_{eff}[\sigma]},\quad\text{where}\\\
S_{eff}[\sigma]&=\frac{p\lambda}{\pi}\int\sigma(\theta)\left(\frac{\theta^{2}}{4}-\frac{\pi^{2}}{12}\right)d\theta+\frac{\pi
p\lambda(1-\lambda)}{12}\\\
&-\frac{\lambda^{2}}{2}\int\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\sigma(\theta)\sigma(\theta^{\prime})\log\left[4\sin^{2}\left(\frac{\theta-\theta^{\prime}}{2}\right)\right]d\theta
d\theta^{\prime}.\end{split}$ (B.9)

The saddle point equation for $\sigma(\theta)$, obtained from the effective
action is given by

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\sigma(\theta^{\prime})\cot\left(\frac{\theta-\theta^{\prime}}{2}\right)d\theta^{\prime}=\frac{p}{2\pi\lambda}\theta.$
(B.10)

In the large $N$ limit we have to solve this equation for $\sigma(\theta)$ in
presence of this constraint (3.6).

The unitary matrix model (B.9) has a gapped phase in the large $k,N$ limit and
the eigenvalue distribution is given by [43, 44],

$\displaystyle\sigma(\theta)=\frac{p}{2\pi^{2}\lambda}\tanh^{-1}\left[\sqrt{1-\frac{e^{-\frac{2\pi\lambda}{p}}}{\cos^{2}\frac{\theta}{2}}}\right],\quad\text{for}\
\
-2\cos^{-1}e^{-\frac{\pi\lambda}{p}}<\theta<2\cos^{-1}e^{-\frac{\pi\lambda}{p}}.$
(B.11)

It turns out that the solution saturates the upper bound (3.6) at
$\lambda=p/\pi\log\cosh(\pi/p)\equiv\lambda^{*}$ [44] and is not valid beyond
that.

_Cap-gap phase :_ It was observed in [30] that for $\lambda>\lambda^{*}$ the
saddle point equation admits a _cap-gap_ solution. The eigenvalue density
$\sigma(\theta)$ for $\lambda>p/\pi\log\cosh(\pi/p)$ is given by

$\displaystyle\displaystyle\sigma(\theta)=\Bigg{\\{}{\frac{1}{2\pi\lambda}\hskip
8.5359pt\text{for}\
-\theta_{2}<\theta<\theta_{2}\hfill\atop\hat{\sigma}(\theta)\ \ \text{for}\
-\theta_{1}<\theta<-\theta_{2}\ \text{and}\ \theta_{2}<\theta<\theta_{1}.}$
(B.12)

where,

$\displaystyle\displaystyle\hat{\sigma}(\theta)$ $\displaystyle=$
$\displaystyle\frac{|\sin\phi|}{\pi^{2}\lambda}\frac{\sqrt{(\sin^{2}\frac{\phi}{2}-\sin^{2}\frac{\theta_{2}}{2})(\sin^{2}\frac{\theta_{1}}{2}-\sin^{2}\frac{\phi}{2})}}{\sqrt{(1+\cos\theta_{2})(1-\cos\theta_{1})}}\Bigg{[}\frac{4\left(\Pi(n_{2},m_{2})-\sin^{2}\frac{\phi}{2}K(m_{2})\right)}{\sin^{2}\phi}$
(B.13)
$\displaystyle\qquad\frac{2p\left(\cos^{2}\frac{\theta_{1}}{2}\Pi(\psi,n_{1},m_{1})-\cos^{2}\frac{\phi}{2}F(\psi,m_{1})\right)}{(1+\cos\phi)(\cos\phi-\cos\theta_{1})}\Bigg{]}.$

The constants $m_{1},m_{2},n_{1},n_{2},\psi$ are given in [30].

##  Appendix C Explicit derivation of Hamilton-Jacobi equation

###  C.1 $U(N)$ Chern-Simons theory

Replacing
$\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}=e^{N^{2}F_{N}}$ in
(3.20) and using the relation

$\begin{split}\frac{1}{D[\theta^{(a)}]}\frac{\partial^{2}D[\theta^{(a)}]}{\partial\theta_{k}^{(a)^{2}}}=N\frac{\partial
U_{k}}{\partial\theta_{k}^{(a)}}+N^{2}U_{k}^{2}\end{split}$ (C.1)

for

$\begin{split}U_{k}=\frac{1}{N}\frac{\partial\log[D[\theta^{(a)}]}{\partial\theta_{k}^{(a)}}=\frac{1}{2N}\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{N}\cot\Big{(}\frac{\theta_{k}^{(a)}-\theta_{j}^{(a)}}{2}\Big{)}\end{split}$
(C.2)

we get

$\begin{split}2\frac{\partial F_{N}}{\partial
A}=\frac{1}{N}\sum_{k=1}^{N}\frac{\partial^{2}F_{N}}{\partial\theta_{k}^{(a)^{2}}}+\frac{1}{N}\sum_{K=1}^{N}\Big{(}N\frac{\partial
F_{N}}{\partial\theta_{k}^{(a)}}\Big{)}^{2}+\frac{2}{N}\sum_{k=1}^{N}U_{k}N\frac{\partial
F_{N}}{\partial\theta_{k}^{(a)}}+\frac{1}{N^{2}}\sum_{k=1}^{N}\frac{\partial
U_{k}}{\partial\theta_{k}^{(a)}}+\frac{1}{N}\sum_{k=1}^{N}U_{k}^{2}\end{split}$
(C.3)

Calculating

$\begin{split}\frac{1}{N^{2}}\sum_{k=1}^{N}\frac{\partial
U_{k}}{\partial\theta_{k}^{(a)}}=-\frac{1}{4N^{3}}\sum_{k=1}^{N}\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{N}\frac{1}{\sin^{2}\Big{(}\frac{\theta_{k}^{(a)}-\theta_{j}^{(a)}}{2}\Big{)}}\end{split}$
(C.4)

we note that the right hand side is _zero_ in the large $N$ limit when
$\theta_{k}^{(a)}\neq\theta_{j}^{(a)}$. It only gives a nonzero contribution
for $\theta_{k}^{(a)}\approx\theta_{j}^{(a)}$. In the large $N$ limit we
define continuous distribution functions $\sigma_{a}(\theta)$ for
$\\{\theta_{i}^{(a)}\\}$ given by (3.5)

$\displaystyle\displaystyle\theta_{i}^{(a)}=\theta^{(a)}(x).$ (C.5)

Thus
$\theta_{k}^{(a)}-\theta_{j}^{(a)}\approx\frac{|k-j|}{N\sigma_{(a)}(\theta_{k})}$,
also in this limit the _sum_ will be replaced by integration and all the
partial derivatives by corresponding functional derivatives

$\begin{split}\frac{1}{N}\sum_{k=1}^{N}\to\int\sigma_{a}(\theta)d\theta;\\\
N\frac{\partial}{\partial\theta_{k}^{(a)}}\to\frac{\partial}{\partial\theta}\frac{\delta}{\delta\sigma_{a}(\theta)}\end{split}$
(C.6)

Finally using the identity

$\sum\limits_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{N\to\infty}\frac{1}{(j-k)^{2}}=\frac{\pi^{2}}{3}$

the equation (C.3) reduced to (3.23). Here we have neglected the term
$\frac{\partial^{2}F_{N}}{\partial\theta_{k}^{(a)^{2}}}$ as it is
$\mathcal{O}(\frac{1}{N})$ in the large $N$ limit.

###  C.2 $Sp(N)$ Chern-Simons theory

The derivation of the Hamilton-Jacobi equation for $Sp(N)$ CS theory falls in
the same line as that of a $U(N)$ theory once we define the eigenvalue density
for the eigenvalues and their mirror images (4.11).

The function
$\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},Sp(N),N,k)$
is given by,

$\begin{split}\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}(S^{3}/\mathbb{Z}_{p},Sp(N),N,k)&=\sum_{y_{i}(\mathcal{R})}\frac{\det[\sin(N\theta_{j}(\mathcal{R}_{1})y_{i}(\mathcal{R}))]}{D[\theta(\mathcal{R}_{1})]}\\\
&\frac{\det[\sin(N\theta_{j}(\mathcal{R}_{2})y_{i}(\mathcal{R}))]}{D[\theta(\mathcal{R}_{2})]}e^{-\frac{AN}{2}\sum_{i=1}^{N}y_{i}(\mathcal{R})^{2}}.\end{split}$
(C.7)

the quantity $\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}$ in
(C.7) will satisfy (3.20) with $A=\pi p\lambda$. The quantity
$\log[D[\theta]]$ for $Sp(N)$ theory has the form

$\log[D[\theta]]=\frac{1}{4}\sum_{i=-N}^{N}\log[\sin\theta_{i}]+\frac{1}{4}\sum_{k=-N}^{N}\sum\limits_{\begin{subarray}{c}j=-N\\\
j\neq
k\end{subarray}}^{N}\log\Big{[}\sin\Big{(}\frac{\theta_{k}-\theta_{j}}{2}\Big{)}\Big{]}.$
(C.8)

For $Sp(N)$ theory we define the quantity $U_{k}$ as follows,

$U_{k}=\frac{1}{N}\frac{\partial\log[D[\theta^{(a)}]]}{\partial\theta_{k}^{(a)}}=\frac{1}{2N}\cot\theta_{k}^{(a)}+\frac{1}{2N}\sum\limits_{\begin{subarray}{c}j=-N\\\
j\neq
k\end{subarray}}^{N}\cot\Big{(}\frac{\theta_{k}^{(a)}-\theta_{j}^{(a)}}{2}\Big{)}$
(C.9)

and find that it satisfies

$\frac{1}{N^{2}}\frac{\partial
U_{k}}{\partial\theta_{k}^{(a)}}=-\frac{1}{4N^{3}}\sum\limits_{\begin{subarray}{c}j=-N\\\
j\neq
k\end{subarray}}^{N}\frac{1}{\sin^{2}(\frac{\theta_{k}^{(a)}-\theta_{j}^{(a)}}{2})}-\frac{3}{4N^{3}}\frac{1}{\sin^{2}\theta_{k}}$
(C.10)

In the large $N$ limit the only contribution will comes from those
$\theta_{j}$ which are close to $\theta_{k}$. Thus
$\theta_{k}^{(a)}-\theta_{j}^{(a)}\approx\frac{|k-j|}{2N\hat{\sigma}_{(a)}(\theta_{k})}$
and using the identity
$\lim_{N\to\infty}\sum\limits_{\begin{subarray}{c}j=-N\\\ j\neq
k\end{subarray}}^{N}\frac{1}{(j-k)^{2}}=\frac{\pi^{2}}{3}$ we find that

$\begin{split}\frac{1}{N}\sum_{k=1}^{N}\frac{\partial
U_{k}}{\partial\theta^{(a)}_{k}}\approx-\frac{1}{N^{2}}\sum_{k=1}^{N}4N^{2}\frac{\pi^{2}}{3}\hat{\sigma}_{(a)}(\theta_{k}).\end{split}$
(C.11)

Assuming $\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}$ is
dominated by a single representation in the large $N$ limit we use the ansatz
$\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}=e^{N^{2}F_{N}}$,where
$F_{N}$ can be written

$F_{N}=S_{N}-\frac{1}{N^{2}}\sum_{a=1}^{2}\Big{[}\frac{1}{4}\sum_{i=-N}^{N}\log[\sin\theta_{i}^{(a)}]+\frac{1}{4}\sum_{k=-N}^{N}\sum\limits_{\begin{subarray}{c}j=-N\\\
j\neq
k\end{subarray}}^{N}\log\Big{[}\sin\Big{(}\frac{\theta_{k}^{(a)}-\theta_{j}^{(a)}}{2}\Big{)}\Big{]}\Big{]}$
(C.12)

from above equation we can find two relation

$\displaystyle\begin{split}\frac{\partial
F_{N}}{\partial\theta_{k}^{(a)}}=\frac{\partial
S_{N}}{\partial\theta_{k}^{(a)}}-\frac{1}{N}U_{k}\\\ \frac{\partial
F_{N}}{\partial A}=\frac{\partial S_{N}}{\partial A}\end{split}$ (C.13)

Using the above two relation we can reduce the differential equation of
$\widetilde{\mathcal{Z}}_{\mathcal{R}_{1}\mathcal{R}_{2}}$ to differential
equation of $S_{N}$

$\frac{\partial S_{N}}{\partial
A}=\frac{1}{2N}\sum_{K=1}^{N}\Big{[}\Big{(}N\frac{\partial
S_{N}}{\partial\theta_{k}^{a}}\Big{)}^{2}+\frac{1}{N}\frac{\partial
U_{k}}{\partial\theta_{k}}\Big{]}$ (C.14)

In the continuum we assume that the density converges to a smooth function and
the sum and partial derivative replace by

$\lim_{N\to\infty}\frac{1}{2N}\sum_{j=-N}^{N}\to\int\sigma^{sp}_{(a)}(\theta)d\theta\qquad
2N\frac{\partial}{\partial\theta_{k}^{(a)}}\to\frac{\partial}{\partial\theta}\frac{\delta}{\delta{\sigma}^{sp}_{(a)}(\theta)}|_{\theta=\theta_{k}^{(a)}}$
(C.15)

Replacing the sum by integral and using (C.11) we get the continuum version of
(C.14) as given by (4.20). With redefinition of $S=4S^{\prime}$ the above
equation is same as (3.25) which can be mapped to a Hamilton-Jacobi equation
with Hamiltonian

$H[\sigma,\Pi]=\frac{1}{2}\int\sigma(\theta)^{2}\Big{[}\Big{(}\frac{\partial\Pi}{\partial\theta}\Big{)}^{2}-\frac{\pi^{2}}{3}\sigma(\theta)^{2}\Big{]}d\theta$
(C.16)

the Hamilton’s equation of motion is same as (3.27) with the boundary
condition $\sigma(t=0,\theta)=\hat{\sigma}_{1}(\theta)$ and
$\sigma(t=A,\theta)=\hat{\sigma}_{2}(\theta)$

##  Appendix D From eigenvalue density to Young tableau density

To find the the dominant Young tableau density we can use the large $N$
properties of the Itzykson-Zuber integral

$I_{N}(A,B)\equiv\int[dU]e^{N\text{Tr}(AUBU^{\dagger})}=\frac{\det||e^{Na_{k}b_{j}}||}{\Delta(a)\Delta(b)}.$
(D.1)

Here $A$ and $B$ are arbitrary hermitian matrices, $a_{k}$ and $b_{j}$ are
their eigenvalues and $\Delta(a)=\prod_{i<j}(a_{i}-a_{j})$. One nice property
of the integral is

$\frac{\det||e^{Na_{k}b_{j}}||}{\Delta(a)\Delta(b)}=e^{\frac{N}{2}\Big{[}\sum_{k=1}^{N}a_{k}^{2}+\sum_{j=1}^{N}b_{j}^{2}\Big{]}}\frac{\det||e^{-\frac{N}{2}(a_{k}-b_{j})^{2}}||}{\Delta(a)\Delta(b)}.$
(D.2)

Then one can define the quantity

$J_{N}(t,A,B)=\frac{1}{t^{\frac{N}{2}}}\frac{\det||e^{-\frac{N}{2t}(a_{k}-b_{j})^{2}}||}{\Delta(a)\Delta(b)}$
(D.3)

which satisfy the partial differential equation

$2N\frac{\partial J_{N}}{\partial
t}=\frac{1}{\Delta(a)}\sum_{k=1}^{N}\frac{\partial^{2}}{\partial
a_{k}^{2}}[\Delta(a)J_{N}].$ (D.4)

(D.4) is similar to (3.20) with

$U_{k}=\frac{1}{N}\sum\limits_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{N}\frac{1}{a_{k}-a_{j}}$ (D.5)

and like (3.24) we can also write

$\lim_{N\to\infty}\frac{1}{N^{2}}\log[J_{N}]=S_{J}[\alpha,\beta,t]-\frac{1}{2}\int\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\alpha(a)\alpha(a^{\prime})\log[a-a^{\prime}]-\frac{1}{2}\int\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\beta(b)\beta(b^{\prime})\log[b-b^{\prime}]$ (D.6)

with $\alpha(a)$ and $\beta(b)$ being the densities of $a$ and $b$.
$S_{J}[\alpha,\beta,t]$ satisfies the differential equation

$\frac{\partial S_{J}}{\partial
t}=\frac{1}{2}\int\alpha(a)\Big{[}\Big{(}\frac{\partial}{\partial
a}\frac{\delta
S_{J}}{\delta\alpha(a)}\Big{)}^{2}-\frac{\pi^{2}}{3}\alpha(a)^{2}\Big{]}da$
(D.7)

one can think of this as the Hamilton-Jacobi equation for the dynamical system
with the Hamiltonian

$H[\rho,\Pi]=\frac{1}{2}\int\rho(a)\Big{[}\Big{(}\frac{\partial}{\partial
a}\frac{\delta
S_{J}}{\delta\rho(a)}\Big{)}^{2}-\frac{\pi^{2}}{3}\rho(a)^{2}\Big{]}da$ (D.8)

We are interested to the solution which connect $\rho(t=0,a)=\alpha(a)$ and
$\rho(t=1,b)=\beta(b)$ within time $t=1$.The variational derivative at the end
of the trajectory is

$\frac{\delta S_{J}}{\delta\alpha(a)}=\Pi(t=0,a),\qquad\qquad\frac{\delta
S_{J}}{\delta\beta(b)}=-\Pi(t=1,b).$ (D.9)

The equation of motion which follow from above Hamiltonian can be transform in
a single equation (Hopf equation) of the function
$f_{J}(t,a)=\frac{\partial\Pi(a)}{\partial a}+i\pi\rho(a)$

$\frac{\partial f_{J}}{\partial t}+f_{J}\frac{\partial f_{J}}{\partial a}=0.$
(D.10)

The general solution of (D.10) can be written in parametric form

$\begin{split}x=R(\xi)+F(\xi)t,\quad f_{J}(t,x)=F(\xi).\end{split}$

If we introduce two analytic function

$\begin{split}G_{+}(x)=x+f_{J}(t=0,x),\quad
G_{-}(x)=x-f_{J}(t=1,x)\end{split}$

then one can show that

$G_{+}(G_{-}(x))=G_{-}(G_{+}(x))=x.$ (D.11)

After replacing $a_{k}=y_{k}^{\prime}=\tau y_{k}$ and $b_{j}=\theta_{j}$
followed by $\tau\to i$ we can transform Itzykson-Zuber integral to character
of $U_{N}$ group

$\det||e^{Na_{k}b_{j}}||\to\det||e^{iNy_{k}\theta_{j}}||=\chi_{R}(U)\Delta(e^{i\theta})$
(D.12)

also the densities of $y^{\prime}$ and $y$ are related by

$\rho_{\tau}(y)=-\frac{\partial x}{\partial
y^{\prime}}=\frac{1}{\tau}\rho(\frac{y}{\tau})|_{\tau=i}.$ (D.13)

With this we can define two type of velocity

$\begin{split}U(y)&=\frac{\partial}{\partial y}\frac{\delta
S_{J}[\rho_{\tau},\sigma]}{\delta\rho(y)},\quad
U_{\tau}(y)=\frac{\partial}{\partial y}\frac{\delta
S_{J}[\rho_{\tau},\sigma]}{\delta\rho_{\tau}(y)},\\\
V(\theta)&=-\frac{\partial}{\partial\theta}\frac{\delta
S_{J}[\rho_{\tau},\sigma]}{\delta\sigma(\theta)},\quad
v_{2}(\theta)=-\frac{\partial}{\partial y}\frac{\delta
S[\sigma_{1},\sigma_{2}]}{\delta\sigma_{2}(\theta)}\end{split}$ (D.14)

where the quantity $S[\sigma_{1},\sigma_{2}]$ is from (3.24). The above four
velocities are related to each other by

$\begin{split}U(y)=\tau U_{\tau}(\tau
y)|_{\tau=i},\quad-V(\theta)+\theta=-v_{2}(\theta)\end{split}$ (D.15)

from the large $N$ limit of $\chi_{R}(U)=e^{N^{2}\Xi[\rho,\sigma]}$ and using
on shell condition one can show

$\frac{\partial}{\partial
y}\frac{\delta\Xi[\rho,\sigma_{1}]}{\delta\rho(y)}=\tau U_{\tau}(\tau
y)|_{\tau=i}-y=\frac{A}{2}y.$ (D.16)

We can also define two analytic function with these variable

$\begin{split}G_{+}(x)=x+U_{\tau}(x)+i\pi\rho_{\tau}(x)=-\frac{A}{2}x+-\pi\rho_{\tau}(x)\\\
G_{-}(x)=x-V(x)-i\pi\sigma_{1}(x)=-v_{2}(x)-i\pi\sigma_{1}(x).\end{split}$
(D.17)

Hence from (3.45) with $\sigma_{1}=\sigma_{2}$, in this case
$t^{\ast}=\frac{A}{2}$ and at $t=A$ we have

$i\pi\sigma^{\ast}[\theta-\frac{A}{2}(v_{2}(\theta)+i\pi\sigma_{1}(\theta))]=v_{2}+i\pi\sigma_{1}(\theta)$
(D.18)

By using (D.13) and (D.17) we can show that

$\pi\rho(-\pi\sigma^{\ast}(\theta))=\theta$ (D.19)

###  D.1 $Sp(N)$

To find the Young-Tableau density for $Sp(N)$ group we can extend the above
procedure of $U(N)$ by a simple observation

$\det||\sinh{Na_{k}}b_{j}||=\frac{1}{2^{N}}e^{\frac{N}{2}[\sum_{k=1}^{N}a_{k}^{2}+\sum_{j=1}^{N}b_{j}^{2}]}\det||e^{-\frac{N}{2}(a_{k}-b_{j})^{2}}-e^{-\frac{N}{2}(a_{k}+b_{j})^{2}}||$
(D.20)

helps us to define

$J_{N}(t,a,b)=\frac{1}{t^{\frac{N}{2}}}\frac{\det||e^{-\frac{N}{2t}(a_{k}-b_{j})^{2}}-e^{-\frac{N}{2t}(a_{k}+b_{j})^{2}}||}{\Delta(a)\Delta(b)}$
(D.21)

which satisfy the same differential equation as (D.4), with
$\Delta(a)=\prod_{i=1}^{N}a_{i}\prod_{j<j}^{N}(a_{i}+a_{j})(a_{i}-a_{j})$.
Following the same procedure as depicted in (4.2) i.e. identifying
$a_{-k}=-a_{k}$ and same for $b_{k}$ also,we can write everything in terms of
this new $a_{k}$ as

$\log[\Delta(a)]=\frac{1}{4}\sum_{i=-N}^{N}\sum\limits_{\begin{subarray}{c}j=-N\\\
j\neq
k\end{subarray}}^{N}\log[a_{i}-a_{j}]+\frac{1}{4}\sum_{i=-N}^{N}\log[2a_{i}]$
(D.22)

with

$U_{k}=\frac{1}{N}\log[\Delta(a)]=\frac{1}{N}\sum\limits_{\begin{subarray}{c}j=-N\\\
j\neq k\end{subarray}}^{N}\frac{1}{a_{k}-a_{j}}+\frac{1}{2a_{k}}$ (D.23)

and

$\frac{1}{N}\frac{\partial U_{k}}{\partial
a_{k}}\approx-4\frac{\pi^{2}}{3}\alpha(a_{k})^{2}.$ (D.24)

Assuming that $J_{N}$ at the large $N$ is given by $J_{N}=e^{N^{2}F_{N}^{J}}$
and

$F_{N}^{J}=S_{N}^{J}-\frac{1}{N^{2}}(\log[\Delta(a)]+\log[\Delta(b)])$ (D.25)

then $S^{J}=\lim_{N\to\infty}S_{N}^{J}$ will satisfy the differential equation

$\frac{\partial S^{J}}{\partial
t}=\frac{1}{2}\int\alpha(a)\Big{[}\Big{(}\frac{1}{2}\frac{\partial}{\partial
a}\frac{\delta
S^{J}}{\delta\alpha(a)}\Big{)}^{2}-4\frac{\pi^{2}}{3}\alpha(a)^{2}\Big{]}da.$
(D.26)

After redefinition of of $S^{J}=\rightarrow 4\tilde{S}^{J}$ the above equation
reduces to the Hamilton-Jacobi like equation (D.7). Thus the function
$f_{J}(t,a)=v(t,a)+i\pi\rho(t,a)$ follows the Burger’s equation with the
boundary condition

$\begin{split}Im[f_{J}(t=0,a)]=\pi\alpha(a),\qquad
Re[f_{J}(t=0,a)]=\frac{\partial}{\partial
a}\frac{\delta{S}^{J}}{\delta\alpha(a)}\\\ Im[f_{J}(t=1,b)]=\pi\beta(b),\qquad
Re[f_{J}(t=1,b)]=-\frac{\partial}{\partial
b}\frac{\delta{S}^{J}}{\delta\beta(b)}\end{split}$ (D.27)

Similar to the $U(N)$ case, after replacing $a_{k}=y_{k}^{\prime}=\tau y_{k}$
and $b_{j}=\theta_{j}$ followed by $\tau\to i\tau$ we get

$\det||\sinh{Ny_{k}^{\prime}\theta_{j}}||\to\frac{\det||\sin{Ny_{k}\theta_{j}}||}{D[\theta]}(i)^{N}D[\theta].$
(D.28)

We can follow the same procedure as above and prove that when
${\sigma}_{1}(\theta)={\sigma}_{2}(\theta)$ the dominant young tableau density
satisfy

$\pi{\rho}[-\pi{\sigma}^{\ast}(\theta)]=\theta.$ (D.29)

## References

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