# Painlevé/CFT correspondence on a torus

Harini Desiraju
<EMAIL_ADDRESS>
School of Mathematics and Statistics F07, The University of Sydney, NSW 2006,
Australia,

###### Abstract

This note details the relationship between the isomonodromic tau-function and
conformal blocks, on a torus with one simple pole. It is based on the author’s
talk at ICMP 2021.

###### Contents

  1. 1 A quick overview of the story on a Riemann sphere
  2. 2 Painlevé/CFT correspondence on the one-point torus
    1. 2.1 Monodromy problem
    2. 2.2 Fredholm determinant
    3. 2.3 Minor expansion and combinatorial expression
    4. 2.4 Relation to conformal blocks

The seminal paper [GIL12] by Gamayun, Iorgov, Lisovyy introduced the relation
between the Painlevé VI equation and conformal field theory (CFT) on a sphere
with four punctures. Since then, such a correspondence has been extended to
include other Painlevé equations [GIL13, Bon+17]. A rigorous approach to the
Painlevé/CFT correspondence comes through the formulation of isomonodromic
tau-functions as Fredholm determinants [GL18, CGL19, GL18a]. In a recent paper
[DDG20] we rigorously proved the Painlevé/CFT correspondence on the torus with
any number of simple poles, using Fredholm determinants. This note provides a
detailed treatment of the Painlevé/CFT correspondence for the case of a one-
point torus.

##  1 A quick overview of the story on a Riemann sphere

Painlevé equations were first formulated in the 1900s by Fuchs [Fuc05],
Gambier [Gam10], Painlevé [Pai06], and Picard [Pic89] in their quest to find
second order differential equations whose solutions are such that their only
movable singularities are poles. There are 6 such equations and they take the
form

$\displaystyle\frac{d^{2}u}{dt^{2}}=F\left(u,\frac{du}{dt},t\right)$ (1.1)

with $F$ rational in $u,\frac{du}{dt}$, analytic in $t$. The solutions of
these equations are transcendental in nature and can be viewed as nonlinear
analogues to special functions. A century later, these equations found a
renewed purpose in mathematical physics due to the discovery [FN80, Gar12,
JMU81] that the Painlevé equations are integrable, i.e they have an associated
Lax pair

$\displaystyle\frac{\partial Y(z,t)}{\partial
z}=Y(z,t)L_{1}(z,t),\qquad\frac{\partial Y(z,t)}{\partial
t}=Y(z,t)L_{2}(z,t),$ (1.2)

where the matrices $L_{1},L_{2}$ are meromorphic matrix valued functions on an
$n$-puntured Riemann sphere $C_{0,n}$ depending on $z,t,u(t),du(t)/dt$, and
the consistency condition

$\displaystyle\frac{\partial^{2}Y(z,t)}{\partial t\partial
z}=\frac{\partial^{2}Y(z,t)}{\partial t\partial z}\Rightarrow\frac{\partial
L_{1}}{\partial t}-\frac{\partial L_{2}}{\partial z}-[L_{1},L_{2}]=0$ (1.3)

gives back the Painlevé equation (1.1). Moreover, the solution $Y(z,t)$ is
multivalued with the monodromy data (Stokes data included) dictated by the
order of the singularities on the Riemann sphere $C_{0,n}$. The Painlevé
equation therefore dictates the monodromy preserving deformations of the
system (1.2), and in this sense, the variable $t$ is called the isomonodromic
time variable.

Painlevé equations also have an associated Hamiltonian structure [JMU81,
Oka86, FW01] 111See [DLM] eq. 32.6 for a comprehensive list

$\displaystyle H:=\frac{1}{2}\operatorname{Res}\operatorname{tr}L_{1}^{2}$
(1.4)

such that (1.1) are the equations of motion of the Hamiltonian above. Here
$\operatorname{Res}$ means the residue at all the singularities in $z$ of the
Lax matrix $A(z,t)$.

A central object in the study of integrable systems is the so-called tau-
function, which not only generates the Hamiltonians governing its dynamics,
but also contains important information, in the present case, about the
Painlevé transcendent such as the singular locus, and the asymptotic data.
Refer to [HB21] for a thorough treatment of the subject.

######  Definition 1.

Isomonodromic tau-function can be formulated as a generator of the Hamiltonian
(1.4) as [JMU81, IP18]

$\displaystyle\partial_{t}\log{\mathcal{T}}:=H.$ (1.5)

The solutions of Painlevé equations $u(x)$, known as the Painlevé
transcendents, are then typically described by the logarithmic derivatives of
the isomonodromic tau-function. For example, for Painlevé II,

$\displaystyle u_{PII}(t)^{2}=-\frac{d^{2}}{dt^{2}}\log{\mathcal{T}},$ (1.6)

and for the other Painlevé equations, the analogue of the above relations may
involve some factors of $t$. Due to the Riemann-Hilbert correspondence, which
provides a map between the initial data of the Painlevé equation and the
associated monodromy data obtained through the solution $Y(z,t)$ of the system
(2.2), the tau-function can be formulated in terms of the monodromy data
[Mal83, Ber17]. The zeros of such tau-function determine where the Riemann-
Hilbert map ceases to be bijective. Furthermore, as a consequence of the
relations such as (1.6), the Painlevé transcendents can be expressed as a
function of the monodromy data and isomonodromic time. Such an expression is
key to understanding the relation to conformal blocks as will be shown in this
note for the genus 1 case.

In late 1970s Jimbo, Miwa, Sato (dubbed the Kyoto school) formulated the tau-
function as a 2-point correlation function of what they called holonomic
fields in a series of papers starting from [SMJ77], in a first step towards
giving a field theoretic interpretation of Painlevé equations. This program
came to full fruition for the Painlevé equations III, V, VI, only in mid 2010s
with the papers of Gamayun, Iorgov, Lisovyy [GIL12, GIL13] that established a
connection between some Painlevé equations and 2 dimensional CFTs, and was
more rigorously proven in [CGL19, GL18]. The crucial technique in [GL18,
CGL19] was that the tau-function of three (III, V, VI) of the six (I-VI)
Painlevé equations could be written as a Fredholm determinant of some operator
[GL18, CGL19], and its minor expansion is a combinatorial expression in terms
of some Virasoro conformal blocks $\mathcal{B}$,

$\displaystyle{\mathcal{T}}=\sum_{n\in\mathbb{Z}}e^{\pi
in\eta}\mathcal{B}(\sigma+n,t).$ (1.7)

For example, the above relation states that the tau-function of the Painlevé
VI equation can be expressed in terms of Virasoro conformal blocks
$\mathcal{B}$ of a CFT with central charge $c=1$ that are explicitly
determined by hypergeometric functions. The expression (1.7) can equivalently,
due to the Alday-Gaiotto-Tachikawa (AGT) correspondence [AGT10], be written in
terms of Nekrasov-Okounkov partition functions of a supersymmetric gauge
theory with so called self-dual ($\epsilon_{1}+\epsilon_{2}=0$) omega
background, which are combinatorial sums over random partitions. The conformal
block expansions of the Painlevé equations I, II, IV are conjectured in
[Bon+17] The Fredholm determinant representation for Painlevé II equation was
formulated by the author in [Des19, Des21]. A few comments are in order:

  * 1.

the expression of tau-functions as Fredholm determinants and as conformal
blocks is remarkable due to the transcendental nature of the solutions,

  * 2.

the ratio of Painlevé transcendents at the critical points, known as the
connection constant is related to the fusion kernel of the associated
conformal blocks,

  * 3.

the zero locus of the Fredholm determinant uniquely determines the poles of
the transcendent, also known as the Malgrange divisor.

With all of the above discussion in mind, we detail the extension of the
relationship between Painlevé type equations and CFTs on the torus with one
Fuchsian singularity $C_{1,1}$. It will be shown that the isomonodromic tau-
function of a special case of the elliptic Painlevé VI equation can be
written, first as a Fredholm determinant, and then as a Fourier series of
conformal blocks. The discussion in the following section holds in general for
a torus with any number of Fuchsian singularities and the corresponding Lax
matrices that are $N\times N$, as shown in the paper of the author along with
P. Gavrylenko, F. Del Monte [DDG20].

### Acknowledgements

The author thanks F. Del Monte, P. Gavrylenko, for useful discussions and
suggestions.

##  2 Painlevé/CFT correspondence on the one-point torus

Our goal in the paper [DDG20] was two fold:

  * 1.

to explicitly formulate isomonodromic tau-functions on a torus and thereby
find a way to understand the solutions of isomonodromic equations on a torus
explicitly,

  * 2.

to make express the isomonodromic tau-functions in terms of Nekrasov-Okounkov
functions or equivalently conformal blocks.

We now demonstrate the above points using the simplest example of a torus with
one simple pole: consider a special case of the elliptic Painlevé VI equation:
the equation of motion of the 2-particle non-autonomous Calogero-Moser system
(in the center of mass reference frame),

$\displaystyle(2\pi
i)^{2}\frac{d^{2}Q(\tau)}{d\tau^{2}}=m^{2}\wp^{\prime}(2Q(\tau)|\tau).$ (2.1)

The modular parameter $\tau$ assumes the role of the isomonodromic time and
brings in an additional factor of $2\pi i$ w.r.t to the sphere case in the
introduction. The above equation (2.1) is integrable in the sense that it has
an associated Lax pair

$\begin{split}\partial_{z}\mathcal{Y}_{CM}(z,\tau)=\mathcal{Y}_{CM}(z,\tau)L_{1}(z,\tau);\quad
2\pi
i\partial_{\tau}\mathcal{Y}_{CM}(z,\tau)=\mathcal{Y}_{CM}(z,\tau)L_{2}(z,\tau),\end{split}$
(2.2)

where

$\displaystyle L_{1}(z,\tau)=2\pi
i\frac{dQ(\tau)}{d\tau}\sigma_{3}+{m}\frac{\theta_{1}^{\prime}(0|\tau)}{\theta_{1}(z|\tau)}\left(\begin{array}[]{cc}0&\frac{\theta_{1}(z+2Q(\tau)|\tau)}{\theta_{1}(-2Q(\tau)|\tau)}\\\
\frac{\theta_{1}(z-2Q(\tau)|\tau)}{\theta_{1}(2Q(\tau)|\tau)}&0\end{array}\right),$
(2.5) $\displaystyle
L_{2}(z,\tau)=\left(\begin{array}[]{cc}0&my(-2Q(\tau),z)\\\
my(2Q(\tau),z)&0\end{array}\right),$ (2.8)

and the zero curvature equation (1.3) gives back (2.1). The notation for the
solution $\mathcal{Y}_{CM}$ as opposed to $Y_{CM}$ will be addressed in
definition 2. The toric analogue to the Hamiltonian (1.4) of the system above
is the following a-cycle integral

$H(\tau)=\oint_{a}dz\frac{1}{2}\operatorname{tr}L_{1}^{2}(z,\tau)=\left(2\pi
i\frac{dQ(\tau)}{d\tau}\right)^{2}-{m}^{2}\wp(2Q(\tau)|\tau)+4\pi
i{m}^{2}\partial_{\tau}\log\eta(\tau),$ (2.9)

where $\eta(\tau)$ is Dedekind’s eta function

$\eta(\tau):=\left(\frac{\theta_{1}^{\prime}(0|\tau)}{2\pi}\right)^{1/3}.$

The tau-function is therefore

$\displaystyle 2\pi i\partial_{\tau}\log\mathcal{T}_{CM}(\tau):=H(\tau).$
(2.10)

In the form above, the tau-function depends on the initial data of the
equation (2.1). Expressing the tau-function in terms of the monodromy data of
the problem reveals the connection to conformal blocks as we show in
subsequent sections.

###  2.1 Monodromy problem

The Lax matrix (2.5) has a simple pole at $z=0$ and is multivalued:

$\displaystyle L_{1}(z+1,\tau)=L_{1}(z,\tau),$ $\displaystyle
L_{1}(z+\tau,\tau)=e^{-2\pi iQ(\tau)\sigma_{3}}L_{1}(z,\tau)e^{2\pi
iQ(\tau)\sigma_{3}}.$ (2.11)

In turn, the monodromies around the A, B-cycles, and the puncture at $z=0$
read

$\displaystyle\begin{array}[]{c}\mathcal{Y}(z+1,\tau)=M_{A}\mathcal{Y}(z,\tau),\qquad\mathcal{Y}(z+\tau,\tau)=M_{B}\mathcal{Y}(z,\tau)e^{2\pi
iQ(\tau)\sigma_{3}},\\\ \\\ \mathcal{Y}(e^{2\pi
i}z,\tau)=M_{0}\mathcal{Y}(z,\tau),\end{array}$ (2.15)

and the monodromy matrices $M_{A},M_{B},M_{0}$ satisfy the constraint

$M_{0}=M_{A}^{-1}M_{B}^{-1}M_{A}M_{B}.$ (2.16)

Explicitly, the monodromies around the A-cycle, and the puncture $z=0$ can be
imposed to be

$\displaystyle M_{A}=e^{2\pi i{a}\sigma_{3}},\qquad M_{0}\sim e^{2\pi
i{m}\sigma_{3}},$ (2.17)

where $\sim$ means in the conjugacy class, $a\notin\mathbb{Z}+\frac{1}{2}$,
$m\in\mathbb{C}$ and

$\displaystyle M_{B}=\left(\begin{array}[]{cc}\frac{\sin\pi(2a-m)}{\sin 2\pi
a}e^{-i\nu/2}&\frac{\sin\pi m}{\sin 2\pi a}\\\ \\\ -\frac{\sin\pi m}{\sin 2\pi
a}&\frac{\sin\pi(2a+m)}{\sin 2\pi a}e^{i\nu/2}\end{array}\right).$ (2.21)

The parameters $a,\nu$ are called the monodromy data.

###  2.2 Fredholm determinant

There are three key steps in writing the tau-function (2.10) as a Fredholm
determinant:

  * 1.

pants decomposition,

  * 2.

Cauchy operators,

  * 3.

determinant of the Cauchy operators.

#### 1\. Pants decomposition

The pants decomposition of a one-point torus, i.e cutting the torus along its
a-cycle, gives us a 3-point sphere (pair of pants).

$M_{0}$$\mathscr{T}$$\mathscr{A}$$\mathcal{C}_{in}$$\mathcal{C}_{out}$ (a)
Pants decomposition of $C_{1,1}$

$\mathscr{T}$$M_{0}$$M_{A}$$\mathcal{C}_{in}$$M_{B}^{-1}M_{A}M_{B}$$\mathcal{C}_{out}$
(b) Trinion

Figure 1:

The pants decomposition induces a homomorphism $\phi$ [Gol09] such that the
monodromy around $\mathcal{C}_{in},\mathcal{C}_{out}$ denoted by $M_{in}$,
$M_{out}$ respectively are

$\displaystyle\phi(M_{A})=:M_{in},$ $\displaystyle\phi(M_{0})=M_{0},$
$\displaystyle\phi(M_{B}^{-1}M_{A}^{-1}M_{B})=:M_{out},$ (2.22)

and the monodromy constraint

$\displaystyle
M_{A}M_{0}M_{B}^{-1}M_{A}^{-1}M_{B}=1=(M_{A})M_{0}(M_{B}^{-1}M_{A}M_{B})^{-1}:=M_{in}M_{0}M_{out}.$
(2.23)

######  Definition 2.

The solution $\mathcal{Y}_{CM}(z)$, of the linear problems (2.2), can be
redefined on the boundary spaces

$\displaystyle\mathcal{H}_{in}:=L^{2}\left(\mathcal{C}_{in}\right),$
$\displaystyle\mathcal{H}_{out}:=L^{2}\left(\mathcal{C}_{out}\right),$ (2.24)

such that $Y_{CM}(z)$, has diagonal monodromies $M_{A},M_{A}^{-1}$
respectively around the boundary circles $\mathcal{C}_{in}$ and
$\mathcal{C}_{out}$ in Figure 1 as

$\displaystyle
Y_{CM}(z)|_{\mathcal{C}_{in}}:=\mathcal{Y}_{CM}(z)|_{\mathcal{C}_{in}}\in\mathcal{H}_{in},$
$\displaystyle
Y_{CM}(z)|_{\mathcal{C}_{out}}:=M_{B}^{-1}\mathcal{Y}_{CM}(z)|_{\mathcal{C}_{out}}\in\mathcal{H}_{out}.$
(2.25)

We now associate the following linear system to the pair of pants fig. 1b: the
$2\times 2$ matrix valued function $\widetilde{Y}(z)$ lives on a cylinder with
3 punctures at $-i\infty,0,i\infty$ respectively

$\displaystyle\partial_{z}\widetilde{Y}(z)=-2\pi
i\left(\widetilde{A}_{in}+\frac{\widetilde{A}_{0}}{1-e^{2\pi
iz}}\right)\widetilde{Y}(z),$ $\displaystyle\widetilde{A}_{in}\sim
a\sigma_{3},$ $\displaystyle\widetilde{A}_{0}\sim m\sigma_{3},$ (2.26)

with $A_{in},A_{0}$ ensuring that the monodromies around
$\mathcal{C}_{in},\mathcal{C}_{out}$ hold true. The local solutions of the
3-point sphere

$\displaystyle\widetilde{Y}_{in}(z):=\widetilde{Y}(z)|_{\mathcal{C}_{in}},$
$\displaystyle\widetilde{Y}_{out}(z):=e^{2\pi
i\bm{{\nu}}}\sigma_{1}\widetilde{Y}_{in}(-z)\sigma_{1},$ (2.27)

are described by the Wronskians of the hypergeometric functions, where

$\begin{gathered}\widetilde{Y}_{in}(z)=(1-e^{-2\pi
iz})^{m}\operatorname{diag}(e^{2\pi iaz},e^{-2\pi iaz})\times\\\
\times\begin{pmatrix}{}_{2}F_{1}(m,m-2a,-2a,e^{-2\pi
iz})&-\frac{m}{2a}{}_{2}F_{1}(1+m,m-2a,1-2a,e^{-2\pi iz})\\\ \frac{me^{-2\pi
iz}}{2a+1}{}_{2}F_{1}(1+m,1+m+2a,2+2a,e^{-2\pi
iz})&{}_{2}F_{1}(m,1+m+2a,1+2a,e^{-2\pi iz})\end{pmatrix}.\end{gathered}$
(2.28)

is the local behavior of the solution to the associated linear system (2.26)
for $z\rightarrow-i\infty$, normalized so that the monodromy around $-i\infty$
is $e^{2\pi ia\sigma_{3}}$. The above function has a well-defined series
expansion in $e^{-2\pi iz}$, is convergent for $|e^{-2\pi iz}|<1$, and
${{}_{2}F_{1}}$ are hypergeometric functions, We will soon see that the tau-
function (2.10) will be completely determined by the functions
$\widetilde{Y}_{in},\widetilde{Y}_{out}$.

#### 2\. Cauchy operators

Let us introduce two Cauchy operators $\mathcal{P}_{\oplus}$,
$\mathcal{P}_{\Sigma}$ respectively in terms of the solutions of the 3-point
sphere $\widetilde{Y}(z)$ (2.26) and the linear systems on the one-point torus
$Y_{CM}(z,\tau)$ (2.25). We define

$\mathcal{H}:=\mathcal{H}_{in}\oplus\mathcal{H}_{out}=\mathcal{H}_{+}\oplus\mathcal{H}_{-}$

with $\mathcal{H}_{in}$, $\mathcal{H}_{out}$ defined in (2.24) and
$\mathcal{H}_{\pm}$ defining the positive and negative Fourier modes.

  * •

The operator $\mathcal{P}_{\oplus}:\mathcal{H}\rightarrow\mathcal{H}$ is
defined as

$\displaystyle\left(\mathcal{P}_{\oplus}f\right)(z)=\int_{\mathcal{C}_{in}\cup\mathcal{C}_{out}}dw\,\frac{\widetilde{Y}(z)\widetilde{Y}(w){{}^{-1}}}{1-e^{-2\pi
i(z-w)}}f(w)$
$\displaystyle=\int_{\mathcal{C}}dw\,\frac{\widetilde{Y}(z)\widetilde{Y}(w){{}^{-1}}}{1-e^{-2\pi
i(z-w)}}f(w).$ (2.29)

Note that the above is the Cauchy operator in cylindrical coordinates. For
$z\sim w$

$\frac{1}{1-e^{-2\pi i(z-w)}}=\frac{1}{2\pi i(z-w)}+\frac{1}{2}+\frac{2\pi
i}{12}(z-w)+\mathcal{O}\left((z-w)^{2}\right).$ (2.30)

It is straightforward to verify that $\mathcal{P}_{\oplus}$ is indeed
projective, i.e $\mathcal{P}_{\oplus}^{2}=\mathcal{P}_{\oplus}$, and
$\textrm{ker}\leavevmode\nobreak\ \mathcal{P}_{\oplus}=\mathcal{H}_{-}$.

A function $f(z)\in\mathcal{H}$ has the following splitting in terms of
positive and negative Fourier modes respectively

$\displaystyle f(z)=\begin{pmatrix}f_{in,-}\\\
f_{out,+}\end{pmatrix}\oplus\begin{pmatrix}f_{in,+}\\\
f_{out,-}\end{pmatrix}.$ (2.31)

The operator $\mathcal{P}_{\oplus}$ then has the following explicit form

$\displaystyle(\mathcal{P}_{\oplus}f)(z)=\begin{pmatrix}(\mathcal{P}_{\oplus}f)_{in,-}\\\
(\mathcal{P}_{\oplus}f)_{out,+}\end{pmatrix}\oplus\begin{pmatrix}(\mathcal{P}_{\oplus}f)_{in,+}\\\
(\mathcal{P}_{\oplus}f)_{out,-}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}f_{in,-}\\\
f_{out,+}\end{pmatrix}\oplus\begin{pmatrix}{\sf a}&{\sf b}\\\ {\sf c}&{\sf
d}\end{pmatrix}\begin{pmatrix}f_{in,-}\\\ f_{out,+}\end{pmatrix},$ (2.32)

where ${{\sf a}},{{\sf b}},{{\sf c}},{{\sf d}}$ are the components of
$\mathcal{P}_{\oplus}$ with respect to the decomposition
$\mathcal{H}=\mathcal{H}_{in}\oplus\mathcal{H}_{out}$:

$\displaystyle({{\sf
a}}g)(z)=\oint_{\mathcal{C}_{in}}dw\frac{\widetilde{Y}_{in}(z)\widetilde{Y}_{in}(w)^{-1}-\mathbb{1}}{1-e^{-2\pi
i(z-w)}}g(w),\,\,z\in\mathcal{C}_{in},$ (2.33) $\displaystyle({{\sf
b}}g)(z)=\oint_{\mathcal{C}_{out}}dw\frac{\widetilde{Y}_{in}(z)\widetilde{Y}_{out}(w)^{-1}}{1-e^{-2\pi
i(z-w)}}g(w),\,\,z\in\mathcal{C}_{in},$ (2.34) $\displaystyle({{\sf
c}}g)(z)=\oint_{\mathcal{C}_{in}}dw\frac{\widetilde{Y}_{out}(z)\widetilde{Y}_{in}(w)^{-1}}{1-e^{-2\pi
i(z-w)}}g(w),\,\,z\in\mathcal{C}_{out},$ (2.35) $\displaystyle({{\sf
d}}g)(z)=\oint_{\mathcal{C}_{out}}dw\frac{\widetilde{Y}_{out}(z)\widetilde{Y}_{out}(w)^{-1}-\mathbb{1}}{1-e^{-2\pi
i(z-w)}}g(w),\,\,z\in\mathcal{C}_{out}.$ (2.36)

The functions $\widetilde{Y}_{in},\widetilde{Y}_{out}$ are the local solutions
of the three-point problem (2.27) around $\mp i\infty$.

  * •

The operator $\mathcal{P}_{\Sigma}:\mathcal{H}\rightarrow\mathcal{H}$ is
defined in terms of the redefined solutions of the linear system (2.25) as

$\displaystyle\left(\mathcal{P}_{\Sigma}f\right)(z)=\int_{\mathcal{C}_{in}\cup\mathcal{C}_{out}}\frac{dw}{2\pi
i}\,Y_{CM}(z,\tau)\Xi(z,w;\tau)Y_{CM}(w,\tau)^{-1}f(w)$
$\displaystyle\equiv\int_{\mathcal{C}}\frac{dw}{2\pi
i}\,Y_{CM}(z,\tau)\Xi(z,w;\tau)Y_{CM}(w,\tau)^{-1}f(w),$ (2.37)

where the twisted Cauchy kernel

$\displaystyle\Xi(z,w;\tau)=\left(\begin{array}[]{cc}\frac{\theta_{1}(z-w+Q-\rho)\theta_{1}^{\prime}(0)}{\theta_{1}(z-w)\theta_{1}(Q-\rho)}&0\\\
0&-\frac{\theta_{1}(z-w-Q-\rho)\theta_{1}^{\prime}(0)}{\theta_{1}(z-w)\theta_{1}(Q+\rho)}\end{array}\right),$
(2.40)

and

$\displaystyle\Xi(z+\tau,w;\tau)=e^{-2\pi iQ\sigma_{3}+2\pi
i\rho}\Xi(z,w;\tau),$ $\displaystyle\Xi(z,w+\tau;\tau)=\Xi(z,w;\tau)e^{2\pi
iQ\sigma_{3}-2\pi i\rho},$ (2.41)

where $\rho$ is a parameter encoding a $U(1)$ B-cycle monodromy of the twisted
Cauchy kernel.

######  Remark 1.

Cauchy kernels on a torus are not unique but are determined by their divisor.
In the present case, the Cauchy kernel has one simple pole at $z=w$ and is
explicitly dependent on the transcendent $Q(\tau)$. On the contrary, there are
other Cauchy kernels that do not depend on the solutions to the integrable
equations being studied but have an extra pole. Such kernels have been studied
in [Ber21, Rod13, BI88].

  * •

For $z\sim w$,

$\begin{split}\Xi(z,w;\tau)=\frac{\mathbb{1}}{z-w}&+\text{diag}\left[\frac{\theta_{1}^{\prime}(Q-\rho)}{\theta_{1}(Q-\rho)},-\frac{\theta_{1}^{\prime}(Q+\rho)}{\theta_{1}(Q+\rho)}\right]\\\
&+\frac{1}{2}(z-w)\,\text{diag}\left[\frac{\theta_{1}^{\prime\prime}(Q-\rho)}{\theta_{1}(Q-\rho)},\frac{\theta_{1}^{\prime\prime}(Q+\rho)}{\theta_{1}(Q+\rho)}\right]\\\
&-\frac{\mathbb{1}}{6}(z-w)\frac{\theta_{1}^{\prime\prime\prime}}{\theta_{1}^{\prime}}+\mathcal{O}\left((z-w)^{2}\right).\end{split}$
(2.42)

As before, one can easily verify that
$\mathcal{P}_{\Sigma}^{2}=\mathcal{P}_{\Sigma}$, and the space of functions on
the annulus $\mathcal{H}_{\mathscr{A}}=\textrm{ker}\mathcal{P}_{\Sigma}$
(assuming $\mathcal{P}_{\Sigma}$ is invertible).

  * •

Moreover,
$\mathcal{P}_{\oplus}\mathcal{P}_{\Sigma}=\mathcal{P}_{\Sigma},\quad\mathcal{P}_{\Sigma}\mathcal{P}_{\oplus}=\mathcal{P}_{\oplus}$
Therefore, the space of functions on the trinion $\mathscr{T}$ in figure 1 is
defined as

$\displaystyle\mathcal{H}_{\mathscr{T}}:=\textrm{im}\leavevmode\nobreak\
\mathcal{P}_{\oplus}=\textrm{im}\leavevmode\nobreak\ \mathcal{P}_{\Sigma},$
(2.43)

such that the Hilbert space

$\displaystyle\mathcal{H}=\mathcal{H}_{\mathscr{T}}\oplus\mathcal{H}_{\mathscr{A}}$
(2.44)

with $\mathcal{H}_{\mathscr{A}}$ being the space of functions on the annulus
$\mathscr{A}$ in figure 1.

#### 3\. Determinant of Cauchy operators

As a last step to expressing the tau-function (2.10) as a Fredholm
determinant, let us define the operator $K$ such that

$\displaystyle\det\left[\mathbb{1}-K(\tau)\right]:=\det_{\mathcal{H}_{+}}\left[\mathcal{P}_{\Sigma,+}^{-1}\mathcal{P}_{\oplus,+}\right]=\det\left[\mathbb{1}-\begin{pmatrix}\nabla^{-1}{{\sf
c}}&\nabla^{-1}{{\sf d}}\nabla\\\ {{\sf a}}&{{\sf
b}}\nabla\end{pmatrix}\right],\quad\mathcal{P}_{\cdot,+}:=\mathcal{P}_{\cdot}|_{\mathcal{H}_{+}},$
(2.45)

and $\nabla$ is a shift operator defined as

$\displaystyle\nabla g(z)=e^{2\pi i\rho}g(z-\tau).$ (2.46)

With some straightforward analysis (refer to the proof of proposition 1 in
[DDG20]) one can see that the operator $K(\tau)$ depends solely on the
operators (2.33)-(2.36) shifted by the action of the operator $\nabla$.

######  Theorem 1.

The logarithmic derivative of the Fredholm determinant (2.45) gives back the
isomonodromic tau-function (2.10)

$\begin{split}2\pi i\partial_{\tau}\log\det\left[\mathbb{1}-K\right]=2\pi
i&\partial_{\tau}\log\mathcal{T}_{CM}-(2\pi i)^{2}{a}^{2}-\frac{(2\pi
i)^{2}}{6}\\\ &+2\pi
i\frac{d}{d\tau}\log\left(\frac{\theta_{1}(Q-{\rho})\theta_{1}(Q+{\rho})}{\eta(\tau)^{2}}\right).\end{split}$
(2.47)

###### Sketch of the proof.

Take the logarithmic derivative of the determinant and express the Cauchy
operators in their integral form (refer to Theorem 1 in [DDG20] for a detailed
proof). ∎

One of the merits of the determinant representation is that the transcendent
$Q(\tau)$ of the elliptic form of Painlevé VI can be explicitly expressed in
terms of the determinant

$\displaystyle\frac{\theta_{3}(2Q(\tau)|2\tau)}{\theta_{2}(2Q(\tau)|2\tau)}=ie^{3i\pi\tau/2}\frac{\det\left(\mathbb{1}-K|_{\rho=\frac{1}{4}+\frac{\tau}{2}}\right)}{\det\left(\mathbb{1}-K|_{\rho=\frac{1}{4}}\right)},$
(2.48)

which is derived using the identity

$\displaystyle\theta_{1}(x-y|\tau)\theta_{1}(x+y|\tau)=\theta_{3}(2x|2\tau)\theta_{2}(2y|2\tau)-\theta_{2}(2x|2\tau)\theta_{3}(2y|2\tau),$
(2.49)

where

$\displaystyle\theta_{2}(z|\tau)$
$\displaystyle=\sum_{n\in\mathbb{Z}}q^{(n+1/2)^{2}}/2e^{2\pi iz(n+1/2)},$
$\displaystyle\theta_{3}(z|\tau)$
$\displaystyle=\sum_{n\in\mathbb{Z}}q^{n^{2}}/2e^{2\pi izn}.$ (2.50)

The above equation can be obtained from equations 3.56 and F.3 in [Bon+20]).
In this sense, the Fredholm determinant is the ’true’ tau-function associated
to the equation (2.1).

###  2.3 Minor expansion and combinatorial expression

The minor expansion of the Fredholm determinant (2.45) contains the conformal
block structure, and it is useful to make the residue matrix around the
puncture $z=0$ rank-1222The reason behind this is computational and is
detailed in [GL18, DDG20] through a simple gauge transformation to make the
precise connection with the Nekrasov-Okounkov functions (see remark (2)).

The Lax matrix $L_{1}$ in (2.2) has a rank-2 residue at the puncture $z=0$:

$L_{1}(z)=\left(\begin{array}[]{cc}P&mx(2Q,z)\\\
mx(-2Q,z)&-P\end{array}\right)=\frac{m\sigma_{1}}{z}+\mathcal{O}(1).$ (2.51)

The scalar gauge transformation

$\displaystyle
L_{1}(z)\rightarrow\widetilde{L}_{1}:=L_{1}(z)-\lambda(z)^{-1}\partial_{z}\lambda(z),$
$\displaystyle\lambda(z)=\theta_{1}(z)^{m},$ (2.52)

makes the residue at the puncture rank-1

$\widetilde{L}_{1}(z)=\left(\begin{array}[]{cc}P-m\frac{\theta_{1}^{\prime}(z)}{\theta_{1}(z)}&mx(2Q,z)\\\
mx(-2Q,z)&-P-m\frac{\theta_{1}^{\prime}(z)}{\theta_{1}(z)}\end{array}\right)=\frac{m}{z}\left(\begin{array}[]{cc}-1&1\\\
1&-1\end{array}\right)+\mathcal{O}(1).$ (2.53)

The above gauge transformation, through the periodicity of theta functions,
induces an additional scalar factor $g_{B}(z)$ to the B-cycle monodromy and
the A-cycle monodromy remains unchanged:

$\displaystyle\lambda(z+\tau)=\theta_{1}(z+\tau)^{m}=e^{-2\pi
i\left(z+\frac{\tau+1}{2}\right)m}\lambda(z):=g_{B}(z)\lambda(z),$ (2.54)
$\displaystyle\lambda(z+1)=\theta_{1}(z+1)^{m}=e^{i\pi
m}\lambda(z):=g_{A}\lambda(z).$ (2.55)

The monodromy around $z=0$ also gains a scalar factor

$g_{1}=e^{-2\pi im}$ (2.56)

due to the $z$-dependence of $g_{B}(z)$. All in all, monodromy matrices around
$\mathcal{C}_{in},\mathcal{C}_{out}$ are

$\displaystyle M_{in}=e^{2\pi
i\,\text{diag}\left(a-\frac{m}{2},-a-\frac{m}{2}\right)}$ $\displaystyle
M_{out}=e^{2\pi i\,\text{diag}\left(a+\frac{m}{2},-a+\frac{m}{2}\right)},$
(2.57)

and the parameter $\rho$ transforms to

$\displaystyle\widetilde{\rho}=\rho-\frac{m(\tau+1)}{2}.$ (2.58)

We now define the monodromy exponents $\vec{\sigma}_{in}$,
$\vec{\sigma}_{out}$ as

$\displaystyle\vec{\sigma}_{in}:=\left(a-\frac{m}{2},-a-\frac{m}{2}\right),$
$\displaystyle\vec{\sigma}_{out}:=\left(a+\frac{m}{2},-a+\frac{m}{2}\right).$
(2.59)

The tau-function $\widetilde{\mathcal{T}}_{CM}$ associated to the Lax matrix
(2.53) is then

$2\pi
i\partial_{\tau}\log\widetilde{\mathcal{T}}_{CM}:=\frac{1}{2}\oint_{A}dz\operatorname{tr}\widetilde{L}_{1}^{2}.$
(2.60)

######  Proposition 1.

The tau-function $\mathcal{T}_{CM}$ in (2.47) is related to the tau-function
$\widetilde{\mathcal{T}}_{CM}$ in (2.60) associated to the rank-1 Lax matrix
(2.52) as

$\mathcal{T}_{CM}(\tau)=\widetilde{\mathcal{T}}_{CM}(\tau)\left(\eta(\tau)e^{\frac{i\pi\tau}{6}}\right)^{-2m^{2}},$
(2.61)

where $m$ is the monodromy exponent at the puncture and $\tau$ is the
isomonodromic time.

###### Sketch of the proof.

The equality (2.61) can be proved by taking the logarithmic derivative w.r.t
on both sides. Refer to section 4.2 in [DDG20] for the complete proof. ∎

The construction in the section 2.2 holds for the rank-1 case with the
appropriate shifts in monodromy parameters and the parameter $\rho$. The
relation equivalent to (2.47) is

$\begin{split}2\pi i&\partial_{\tau}\log\widetilde{\mathcal{T}}_{CM}=2\pi
i\partial_{\tau}\log\det\left[\mathbb{1}-\widetilde{K}\right]+(2\pi
i)^{2}\left({a}^{2}+\frac{m^{2}}{4}\right)+\frac{(2\pi i)^{2}}{6}\\\ &+2\pi
i\frac{d}{d\tau}\log\left(\frac{\eta(\tau)^{2}}{\theta_{1}\left(Q-{\rho}+\frac{m(\tau+1)}{2}\right))\theta_{1}\left(Q+{\rho}-\frac{m(\tau+1)}{2}\right)}\right).\end{split}$
(2.62)

Now we perform the minor expansion for the rank-1 analogue of the Fredholm
determinant $\det\left[\mathbb{1}-\widetilde{K}\right]$.

There are two main steps to write the minor expansion as Nekrasov-Okounkov
functions

  * 1.

Basis expansion

  * 2.

Combinatorial expression

#### Basis expansion

The first step in the minor expansion is noting that kernels of the operators
${\sf a},{\sf b},{\sf c},{\sf d}$ in (2.36) are

$\begin{gathered}\begin{split}{\sf
a}(z,w)=\frac{\mathbb{1}-\widetilde{Y}_{in}(z)\widetilde{Y}_{in}(w)^{-1}}{1-e^{-2\pi
i(z-w)}},&&z,w\in\mathcal{C}_{in},\\\ {\sf
b}(z,w)\,=\frac{\widetilde{Y}_{in}(z)\widetilde{Y}_{out}(w)^{-1}}{1-e^{-2\pi
i(z-w)}},&&z\in\mathcal{C}_{in},\,w\in\mathcal{C}_{out}\\\ {\sf
c}(z,w)=-\frac{\widetilde{Y}_{out}(z)\widetilde{Y}_{in}(w)^{-1}}{1-e^{-2\pi
i(z-w)}},&&z\in\mathcal{C}_{out},\,w\in\mathcal{C}_{in}\\\ {\sf
d}(z,w)=\frac{\widetilde{Y}_{out}(z)\widetilde{Y}_{out}(w)^{-1}-\mathbb{1}}{1-e^{-2\pi
i(z-w)}},&&z,w\in\mathcal{C}_{out}.\end{split}\end{gathered}$ (2.63)

The solution to the three-point problem $\widetilde{Y}$ is multivalued333For
the ease of notation we the keep the notation of the kernels ${\sf a},{\sf
b},{\sf c},{\sf d}$. on $\mathcal{C}_{in},\mathcal{C}_{out}$, with monodromy
determined by $M_{in},M_{out}$ defined in (2.59), consequently the basis is
monodromy dependent, and the matrix elements of the kernels above are

$\begin{split}{\sf
a}_{\alpha,\beta}(z,w)=\sum_{-r,s\in\mathbb{Z}^{\prime}_{+}}{\sf
a}^{-r;\alpha}_{s;\beta}e^{2\pi
iz\left(\frac{1}{2}-r+{\sigma}_{in}^{(\alpha)}\right)}e^{2\pi
iw\left(-\frac{1}{2}-s-{\sigma}_{in}^{(\beta)}\right)},\\\ {\sf
b}_{\alpha,\beta}(z,w)=\sum_{r,s\in\mathbb{Z}^{\prime}_{+}}{\sf
b}^{-r;\alpha}_{-s;\beta}e^{2\pi
iz\left(\frac{1}{2}-r+{\sigma}_{in}^{(\alpha)}\right)}e^{2\pi
iw\left(-\frac{1}{2}+s-{\sigma}_{out}^{(\beta)}\right)},\\\ {\sf
c}_{\alpha,\beta}(z,w)=\sum_{r,s\in\mathbb{Z}^{\prime}_{+}}{\sf
c}^{r;\alpha}_{s;\beta}e^{2\pi
iz\left(\frac{1}{2}+r+{\sigma}_{out}^{(\alpha)}\right)}e^{2\pi
iw\left(-\frac{1}{2}-s-{\sigma}_{in}^{(\beta)}\right)},\\\ {\sf
d}_{\alpha,\beta}(z,w)=\sum_{r,s\in\mathbb{Z}^{\prime}_{+}}{\sf
d}^{r;\alpha}_{-s;\beta}e^{2\pi
iz\left(\frac{1}{2}+r+{\sigma}_{out}^{(\alpha)}\right)}e^{2\pi
iw\left(-\frac{1}{2}+s-{\sigma}_{out}^{(\beta)}\right)},\end{split}$ (2.64)

with $\alpha,\beta=1,2$, the Fourier indices are denoted by $r,s$, and
$\mathbb{Z}^{\prime}_{+}$ is the set of positive half-integers 444The explicit
form of the Fourier coefficients ${\sf a}^{-r;\alpha}_{s;\beta},{\sf
b}^{-r;\alpha}_{-s;\beta},{\sf c}^{r;\alpha}_{s;\beta},{\sf
d}^{r;\alpha}_{-s;\beta}$ are not relevant for the purposes of this note but
are computed in [GL18]. . A submatrix of ${\sf a},{\sf b},{\sf c},{\sf d}$, of
size $i\times j$, is denoted by two unordered sets
$\\{(r,\alpha)_{1},\dots,(r,\alpha)_{i}\\}\in
2^{\mathbb{Z}^{\prime}_{+}\times\\{1,...,N\\}}$ and
$\\{(s,\beta)_{1},\dots,(s,\beta)_{j}\\}\in
2^{\mathbb{Z}^{\prime}_{+}\times\\{1,2\\}}$. The minors can then be denoted by
the multi indices of positive (negative) Fourier modes $I$ ($J$). The minor
expansion of the determinant is then indexed by $I,J$ as

$\displaystyle\det\left[\mathbb{1}-\widetilde{K}\right]=\sum_{\left(I,J\right)}\det\left(\begin{array}[]{cc}\left({\sf
a}\right)^{I}_{J}&\left({\sf b}\nabla\right)^{I}_{I}\\\ \\\
\left(\nabla^{-1}{\sf c}\right)^{J}_{J}&\left(\nabla^{-1}{\sf
d}\nabla\right)^{J}_{I}\end{array}\right).$ (2.68)

#### Combinatorial expression

The multi-indices $(I,J)$ can be viewed as the positions $\sf h(m^{(\alpha)})$
and $\sf p(m^{(\alpha)})$ of ’holes’ and ’particles’ respectively, of a
coloured Maya diagram $\sf m^{(\alpha)}$, where $\alpha=1,2$. Each particle
(hole) carries a positive (negative) unit charge, so that the total charge
associated to every Maya diagram is

$\sf Q({\sf m}^{(\alpha)}):=|p(m^{(\alpha)})|-|h(m^{(\alpha)})|.$ (2.69)

Using the notation

$\displaystyle\vec{\sf m}:=\left(\sf{m}^{(1)},\sf{m}^{(2)}\right),$
$\displaystyle\vec{\sf Q}:=\left(\sf{Q}^{(1)},\sf{Q}^{(2)}\right),$ (2.70)

the total charge is

$\begin{gathered}{\sf Q}:=\sf{Q}^{(1)}+\sf{Q}^{(2).}\end{gathered}$ (2.71)

Furthermore, there is a one-one correspondence between Maya diagrams and
charged Young diagrams $\left(\sf Y,\sf Q\right)$. Consequently, the minors
can be labeled by $2$-tuples of charged partitions $\left(\vec{{\sf
Y}},\vec{{\sf Q}}\right)$. Explicitly, the minor

$\displaystyle\det\left(\begin{array}[]{cc}\left({\sf
a}\right)^{I}_{J}&\left({\sf b}\nabla\right)^{I}_{I}\\\ \\\
\left(\nabla^{-1}{\sf c}\right)^{J}_{J}&\left(\nabla^{-1}{\sf
d}\nabla\right)^{J}_{I}\end{array}\right)$ (2.75)
$\displaystyle=\det\left(\begin{array}[]{cc}\left({\sf
a}\right)^{I}_{J}&\left({\sf b}\right)^{I}_{I}\\\ \\\ \left({\sf
c}\right)^{J}_{J}&\left({\sf
d}\right)^{J}_{I}\end{array}\right)\leavevmode\nobreak\
\textrm{exp}\left\\{\sum_{(r,\alpha)\in J}\left[-2\pi
i\left(\rho-\frac{\tau}{2}-m\tau\right)+2\pi
i\tau\left(r+\sigma_{1}^{(\alpha)}\right)\right]\right\\}$ (2.79)
$\displaystyle\times\exp\left\\{\sum_{(s,\beta)\in I}\left[2\pi
i\left(\rho-\frac{\tau}{2}-m\tau\right)+2\pi
i\tau\left(s-\sigma_{1}^{(\beta)}\right)\right]\right\\}$
$\displaystyle=\det\left(\begin{array}[]{cc}\left({\sf
a}\right)^{I}_{J}&\left({\sf b}\right)^{I}_{I}\\\ \\\ \left({\sf
c}\right)^{J}_{J}&\left({\sf d}\right)^{J}_{I}\end{array}\right)e^{2\pi
i\tau\left[\frac{1}{2}\left({\vec{\sf
Q}}+{\vec{\sigma}}_{1}\right)^{2}-\frac{1}{2}{\vec{\sigma}}_{1}^{2}+|\vec{\sf
Y}|\right]-2\pi i\left(\rho-\frac{\tau}{2}-m\tau\right){\sf Q}}.$ (2.83)

Here are few more details about the above computation.

  * 1.

To obtain the second line of (2.83), note that $\sigma_{1}$ is the monodromy
exponent on $\mathcal{C}_{in}$, and the monodromy exponent on
$\mathcal{C}_{out}$ is $\sigma_{1}+m$.

  * 2.

To obtain the last line in (2.83), we use the following equalities:

$\displaystyle\sum_{l}r_{l}+\sum_{k}s_{k}=\frac{\sf Q(m)^{2}}{2}+|\vec{\sf
Y}|,\qquad\\#r-\\#s=\sf Q(m),$ (2.84)

which can be read off from Figure 2 noting that the $r$’s and $s$’s are to the
left and right sides of the axis respectively.

Figure 2: Pictorial proof of (2.84)

As an example555Since $s\in I$, the hole positions in the corresponding Maya
diagram $\sf{m}$ are ${\sf h(m)}=\left\\{-s_{1},...,-s_{k}\right\\}$, and
since $r\in J$, the particle positions are ${\sf
p(m)}=\left\\{r_{1},...,r_{l}\right\\}$.  , in the Figure 2, ${\sf
p(m)}=\left\\{\frac{13}{2},\frac{9}{2},\frac{3}{2}\right\\}$, ${\sf
h(m)}=\left\\{\frac{-3}{2}\right\\}$. $|Y|$ is the $\\#$boxes in the Young
diagram which in the present example is 12. The charge $\sf Q(m)=2$, $\sum r$
is the blue area and $\sum s$ is the red area in the Figure 2.

######  Remark 2.

  * 1.

The determinant in (2.83) is often called the trinion partition function
$Z^{{{\sf Y}},{{\sf Q}}}_{{{\sf Y}},{{\sf Q}}}\left(\mathscr{T}\right)$:

$\displaystyle Z^{{{\sf Y}},{{\sf Q}}}_{{{\sf Y}},{{\sf
Q}}}\left(\mathscr{T}\right)=Z^{I,J}_{I,J}\left(\mathscr{T}\right):=\det\left(\begin{array}[]{cc}\left({\sf
a}\right)^{I}_{J}&\left({\sf b}\right)^{I}_{I}\\\ \\\ \left({\sf
c}\right)^{J}_{J}&\left({\sf d}\right)^{J}_{I}\end{array}\right).$ (2.88)

  * 2.

The expressions such as (2.83) are valid for linear systems of any rank.
However, the trinion partition functions (2.88) are explicitly known in terms
of Nekrasov-Okounkov functions only in the case where the Lax matrix residues
are of rank-1 [GL18, GIL18].

Using the expressions for $Z^{{\sf Y},{\sf Q}}_{{\sf Y},{\sf Q}}(\mathscr{T})$
computed in [GL18, GIL18] for the rank-1 case, the determinant (2.88) is

$\displaystyle\det\left(\begin{array}[]{cc}\left({\sf
a}\right)^{I}_{J}&\left({\sf b}\right)^{I}_{I}\\\ \\\ \left({\sf
c}\right)^{J}_{J}&\left({\sf d}\right)^{J}_{I}\end{array}\right)=e^{2\pi
i\vec{{\sf Q}}\cdot\vec{\nu}}\frac{Z_{pert}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf
Q}+m\right)}{Z_{pert}\left(\vec{a},\vec{a}+m\right)}Z_{inst}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf Q}+m|\vec{\sf Y},\vec{\sf Y}\right),$ (2.92)

where $\vec{a}=(a,-a)$ and the functions

$Z_{pert}\left(\sigma,{\mu}\right):=\prod_{\alpha,\beta=1}^{N}\frac{G\left(1+\sigma^{(\alpha)}-\mu^{(\beta)}\right)}{G\left(1+\sigma^{(\alpha)}-\sigma^{(\beta)}\right)},$
(2.93)

$G(x)$ being the Barnes’ G-function, and

$\displaystyle Z_{inst}\left({\sigma},{\mu}|\vec{{\sf
Y}},\vec{\sf{W}}\right):=\prod_{\alpha,\beta=1}^{N}\frac{Z_{\textrm{\sf
bif}}\left(\sigma^{(\alpha)}-\mu^{(\beta)}|{\sf
Y}^{(\alpha)},\sf{W}^{(\beta)}\right)}{Z_{\textrm{\sf
bif}}\left(\sigma^{(\alpha)}-\sigma^{(\beta)}|{\sf Y}^{(\alpha)},{\sf
Y}^{(\beta)}\right)},$ (2.94)

with

$Z_{\sf bif}\left(x|{\sf Y}^{\prime},{\sf Y}\right):=\prod_{\Box\in{\sf
Y}}\left(x+1+a_{{\sf Y}^{\prime}}(\Box)+l_{{\sf
Y}}(\Box)\right)\prod_{\Box^{\prime}\in{\sf Y}^{\prime}}\left(x-1-a_{{\sf
Y}}(\Box^{\prime})-l_{{\sf Y}^{\prime}}(\Box^{\prime})\right).$ (2.95)

Note that, $\sigma,\mu\in\mathbb{C}^{2}$, $\vec{{\sf
Y}},\vec{\sf{W}}\in\mathbb{Y}^{2}$, and $a_{{\sf Y}}(\Box)$ and $l_{{\sf
Y}}(\Box)$ denote respectively the arm and leg length of the box $\Box$ in the
Young diagram ${\sf Y}$, as in figure 3.

$s$$a(s)$$l(s)$ Figure 3: Arm and leg length

######  Theorem 2.

The combinatorial expression of the tau-function $\mathcal{T}_{CM}$ reads

$\begin{split}\mathcal{T}_{CM}&=\frac{\left(\eta(\tau)e^{-\frac{i\pi\tau}{12}}\right)^{2-2m^{2}}e^{-2\pi
i\left[\rho-\frac{\tau}{2}\left(m+\frac{1}{2}\right)-\frac{m}{2}\right]}}{\theta_{1}\left(Q+\rho-\frac{m(\tau+1)}{2}\right)\theta_{1}\left(Q-\rho+\frac{m(\tau+1)}{2}\right)}\sum_{\vec{{\sf
Q}}}\sum_{\vec{{\sf Y}}\in\mathbb{Y}^{2}}e^{2\pi
i\tau\left[\frac{1}{2}(\vec{{\sf Q}}+\vec{a})^{2}+|\vec{{\sf Y}}|\right]}\\\
&\times e^{2\pi i\left[\vec{{\sf Q}}\cdot\vec{\nu}-{\sf
Q}\left(\rho-\frac{m(\tau+1)}{2}-\frac{\tau}{2}\right)\right]}\frac{Z_{pert}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf
Q}+m\right)}{Z_{pert}\left(\vec{a},\vec{a}+m\right)}Z_{inst}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf Q}+m|\vec{\sf Y},\vec{\sf Y}\right).\\\ \end{split}$
(2.96)

where $a,\nu$ are monodromy data, $t$ is the isomonodromic time, $m$ is the
parameter from the equation, $Z_{pert},Z_{inst}$ are combinatorial objects
defined in (2.93), (2.94).

###### Proof.

Combine (2.92), (2.83), (2.68), (2.62), (2.61) and integrate on both
sides666Refer to eq 2.45 in [DDG20] for the full computation.. ∎

###  2.4 Relation to conformal blocks

Due to the AGT correspondence [FL10], the expression (2.96), in terms of
Nekrasov-Okounkov functions can be put in the form (1.7) as a discrete Fourier
transform of conformal blocks with a few manipulations.

######  Proposition 2.

The tau-function can be expressed as a discrete Fourier transform of the
conformal blocks as

$\displaystyle\mathcal{T}_{CM}$
$\displaystyle=\frac{\eta(\tau)^{2}}{\theta_{1}\left(Q-\rho\right)\theta_{1}\left(Q+\rho\right)}\sum_{n,k\in\mathbb{Z}}e^{2\pi
i\nu n}q^{\left(k+n/2\right)^{2}}e^{4\pi
i(k+n/2)\left(\rho+1/2\right)}\mathcal{B}(a+n/2,m,q),$ (2.97)

where, $a,\nu$ are the monodromy data, $Q$ is the transcendent, $\tau$ is the
isomonodromic time, $q=e^{2\pi i\tau}$, $m$ is a parameter of the equation
(2.1).

###### Proof.

Starting with the expression of the tau-function in terms of the Nekrasov-
Okounkov functions (2.96)

$\begin{split}\mathcal{T}_{CM}&=\frac{\left(\eta(\tau)e^{-\frac{i\pi\tau}{12}}\right)^{2-2m^{2}}e^{-2\pi
i\left[\rho-\frac{\tau}{2}\left(m+\frac{1}{2}\right)-\frac{m}{2}\right]}}{\theta_{1}\left(Q+\rho-\frac{m(\tau+1)}{2}\right)\theta_{1}\left(Q-\rho+\frac{m(\tau+1)}{2}\right)}\sum_{\vec{{\sf
Q}}}\sum_{\vec{{\sf Y}}\in\mathbb{Y}^{2}}e^{2\pi
i\tau\left[\frac{1}{2}(\vec{{\sf Q}}+\vec{a})^{2}+|\vec{{\sf Y}}|\right]}\\\
&\times e^{2\pi i\left[\vec{{\sf Q}}\cdot\vec{\nu}-{\sf
Q}\left(\rho-\frac{m(\tau+1)}{2}-\frac{\tau}{2}\right)\right]}\frac{Z_{pert}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf
Q}+m\right)}{Z_{pert}\left(\vec{a},\vec{a}+m\right)}Z_{inst}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf Q}+m|\vec{\sf Y},\vec{\sf Y}\right).\\\ \end{split}$
(2.98)

To match with the notation in [Bon+20], we redefine ${{\sf
Q}}^{(1)}=n_{1},{{\sf Q}}^{(2)}=n_{2}$ in (2.70), i.e

$\vec{\sf Q}=\left(\begin{array}[]{c}n_{1}\\\ n_{2}\end{array}\right),\quad
n_{1}+n_{2}=k,\quad n_{1}-n_{2}=n.$ (2.99)

The terms in the exponents in the above expression (2.98)

$\displaystyle\left(\vec{\sf Q}+\vec{a}\right)^{2}$
$\displaystyle=\left(\begin{array}[]{cc}n_{1}+a&n_{2}-a\end{array}\right)\left(\begin{array}[]{c}n_{1}+a\\\
n_{2}-a\end{array}\right)$ (2.103) $\displaystyle=(n_{1}+a)^{2}+(n_{2}-a)^{2}$
$\displaystyle=n_{1}^{2}+n_{2}^{2}+2a^{2}+2a(n_{1}-n_{2})$
$\displaystyle=\frac{1}{2}\left[(n_{1}+n_{2})^{2}+(n_{1}-n_{2})^{2}\right]+2\left(a+\frac{(n_{1}-n_{2})}{2}\right)^{2}-\frac{(n_{1}-n_{2})^{2}}{2}$
$\displaystyle=\frac{k^{2}}{2}+2\left(a+\frac{n}{2}\right)^{2},$ (2.104)

and

$\displaystyle\vec{{\sf
Q}}\cdot\vec{\nu}=\left(\begin{array}[]{cc}n_{1}&n_{2}\end{array}\right)\left(\begin{array}[]{c}\nu\\\
-\nu\end{array}\right)=\nu(n_{1}-n_{2})=\nu n.$ (2.108)

Plugging in (2.104), (2.108) in (2.98)

$\begin{split}\mathcal{T}_{CM}&=\frac{\left(\eta(\tau)e^{-\frac{i\pi\tau}{12}}\right)^{2-2m^{2}}e^{-2\pi
i\left[\rho-\frac{\tau}{2}\left(m+\frac{1}{2}\right)-\frac{m}{2}\right]}}{\theta_{1}\left(Q+\rho-\frac{m(\tau+1)}{2}\right)\theta_{1}\left(Q-\rho+\frac{m(\tau+1)}{2}\right)}\sum_{\vec{{\sf
Q}}}\sum_{\vec{{\sf Y}}\in\mathbb{Y}^{2}}e^{2\pi
i\tau\left[\frac{1}{2}(\vec{{\sf Q}}+\vec{a})^{2}+|\vec{{\sf Y}}|\right]}\\\
&\times e^{2\pi i\left[\vec{{\sf Q}}\cdot\vec{\nu}-{\sf
Q}\left(\rho-\frac{m(\tau+1)}{2}-\frac{\tau}{2}\right)\right]}\frac{Z_{pert}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf
Q}+m\right)}{Z_{pert}\left(\vec{a},\vec{a}+m\right)}Z_{inst}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf Q}+m|\vec{\sf Y},\vec{\sf Y}\right)\\\
&=\frac{\left(\eta(\tau)e^{-\frac{i\pi\tau}{12}}\right)^{2-2m^{2}}e^{-2\pi
i\left[\rho-m\frac{\left(\tau+1\right)}{2}-\frac{\tau}{4}\right]}}{\theta_{1}\left(Q+\rho-\frac{m(\tau+1)}{2}\right)\theta_{1}\left(Q-\rho+\frac{m(\tau+1)}{2}\right)}\sum_{n,k\in\mathbb{Z}}e^{2\pi
i\tau\left[\frac{k^{2}}{4}+\left(a+\frac{n}{2}\right)^{2}\right]}e^{2\pi
i\left[\nu n-k\left(\rho-\frac{m(\tau+1)}{2}-\frac{\tau}{2}\right)\right]}\\\
&\times\sum_{\vec{{\sf Y}}\in\mathbb{Y}^{2}}e^{2\pi i\tau|\vec{{\sf
Y}}|}\frac{Z_{pert}\left(\vec{a}+\vec{\sf Q},\vec{a}+\vec{\sf
Q}+m\right)}{Z_{pert}\left(\vec{a},\vec{a}+m\right)}Z_{inst}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf Q}+m|\vec{\sf Y},\vec{\sf Y}\right).\end{split}$

Making the shift

$\displaystyle\rho-\frac{m(\tau+1)}{2}\rightarrow-\rho-1/2-\tau/2,$ (2.109)

the above expression reads

$\displaystyle\mathcal{T}_{CM}$
$\displaystyle=\frac{\left(\eta(\tau)e^{-\frac{i\pi\tau}{12}}\right)^{2-2m^{2}}e^{-2\pi
i\left[-\rho-\frac{1}{2}-\tau/2-\frac{\tau}{4}\right]}}{\theta_{1}\left(Q-\rho-\frac{1}{2}-\tau/2\right)\theta_{1}\left(Q+\rho+\frac{1}{2}+\tau/2\right)}\sum_{n,k\in\mathbb{Z}}e^{2\pi
i\tau\left[\frac{k^{2}}{4}+\left(a+\frac{n}{2}\right)^{2}\right]}e^{2\pi
i\left[\nu n-k\left(-\rho-\frac{1}{2}-\tau/2-\frac{\tau}{2}\right)\right]}$
$\displaystyle\times\sum_{\vec{{\sf Y}}\in\mathbb{Y}^{2}}e^{2\pi
i\tau|\vec{{\sf Y}}|}\frac{Z_{pert}\left(\vec{a}+\vec{\sf Q},\vec{a}+\vec{\sf
Q}+m\right)}{Z_{pert}\left(\vec{a},\vec{a}+m\right)}Z_{inst}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf Q}+m|\vec{\sf Y},\vec{\sf Y}\right).$ (2.110)

The following relation holds true for theta functions [DLM]

$\displaystyle\theta_{1}\left(z+(m+n\tau)\right)=(-1)^{m+n}e^{-i\pi\tau
n^{2}}e^{-2\pi inz}\theta_{1}(z),$ (2.111)

implying that for $m=n=-1/2$:

$\displaystyle\theta_{1}\left(Q-\rho-1/2-\tau/2\right)=(-1)^{-1}e^{-i\pi\tau/4}e^{i\pi(Q-\rho)}\theta_{1}(Q-\rho),$
(2.112)

and for $m=n=1/2$:

$\displaystyle\theta_{1}\left(Q+\rho+1/2+\tau/2\right)=(-1)e^{-i\pi\tau/4}e^{-i\pi(Q+\rho)}\theta_{1}(Q+\rho).$
(2.113)

Using the above expressions (2.112), (2.113), the expression (2.110)
simplifies as

$\displaystyle\mathcal{T}_{CM}$
$\displaystyle=\frac{\left(\eta(\tau)e^{-\frac{i\pi\tau}{12}}\right)^{2-2m^{2}}e^{2\pi
i\left[\rho+\frac{1}{2}+\frac{3\tau}{4}\right]}e^{i\pi\tau/2}e^{2i\pi\rho}}{\theta_{1}\left(Q-\rho\right)\theta_{1}\left(Q+\rho\right)}\sum_{n,k\in\mathbb{Z}}e^{2\pi
i\tau\left[\frac{k^{2}}{4}+\left(a+\frac{n}{2}\right)^{2}\right]}e^{2\pi
i\left[\nu n+k\left(\rho+\frac{1}{2}+\tau\right)\right]}$
$\displaystyle\times\sum_{\vec{{\sf Y}}\in\mathbb{Y}^{2}}e^{2\pi
i\tau|\vec{{\sf Y}}|}\frac{Z_{pert}\left(\vec{a}+\vec{\sf Q},\vec{a}+\vec{\sf
Q}+m\right)}{Z_{pert}\left(\vec{a},\vec{a}+m\right)}Z_{inst}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf Q}+m|\vec{\sf Y},\vec{\sf Y}\right)$
$\displaystyle=\frac{\left(\eta(\tau)e^{-\frac{i\pi\tau}{12}}\right)^{2-2m^{2}}}{\theta_{1}\left(Q-\rho\right)\theta_{1}\left(Q+\rho\right)}\sum_{n,k\in\mathbb{Z}}e^{2\pi
in\nu}e^{2\pi i\tau\left[\frac{k^{2}}{4}+k+1\right]}e^{2\pi
i(k+2)\left(\rho+\frac{1}{2}\right)}e^{2\pi
i\tau\left(a+\frac{n}{2}\right)^{2}}$ $\displaystyle\times\sum_{\vec{{\sf
Y}}\in\mathbb{Y}^{2}}e^{2\pi i\tau|\vec{{\sf
Y}}|}\frac{Z_{pert}\left(\vec{a}+\vec{\sf Q},\vec{a}+\vec{\sf
Q}+m\right)}{Z_{pert}\left(\vec{a},\vec{a}+m\right)}Z_{inst}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf Q}+m|\vec{\sf Y},\vec{\sf Y}\right).$ (2.114)

Now with the following change of index

$\displaystyle k\rightarrow 2k+n-2,$ (2.115)

the expression

$\displaystyle\frac{k^{2}}{4}+k+1$
$\displaystyle\rightarrow\frac{1}{4}(2k+n-2)^{2}+2k+n-2+1$
$\displaystyle=k^{2}+\frac{n^{2}}{4}+1+kn-2k-n+2k+n-1=\left(k+n/2\right)^{2}.$
(2.116)

With the above change in the index, the expression (2.114) reads

$\displaystyle\mathcal{T}_{CM}$
$\displaystyle=\frac{\left(\eta(\tau)e^{-\frac{i\pi\tau}{12}}\right)^{2-2m^{2}}}{\theta_{1}\left(Q-\rho\right)\theta_{1}\left(Q+\rho\right)}\sum_{n,k\in\mathbb{Z}}e^{2\pi
i\nu n}e^{2\pi i\tau\left(k+n/2\right)^{2}}e^{4\pi
i(k+n/2)\left(\rho+1/2\right)}e^{2\pi i\tau\left(a+\frac{n}{2}\right)^{2}}$
$\displaystyle\times\sum_{\vec{{\sf Y}}\in\mathbb{Y}^{2}}e^{2\pi
i\tau|\vec{{\sf Y}}|}\frac{Z_{pert}\left(\vec{a}+\vec{\sf Q},\vec{a}+\vec{\sf
Q}+m\right)}{Z_{pert}\left(\vec{a},\vec{a}+m\right)}Z_{inst}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf Q}+m|\vec{\sf Y},\vec{\sf Y}\right).$ (2.117)

The toric conformal block is defined as

$\displaystyle\operatorname{tr}_{\mathcal{V}_{a+n/2}}\left(q^{L_{0}}V_{m}(0)\right)$
$\displaystyle=\mathcal{B}(a+n/2,m,q):=\left(\eta(\tau)e^{-\frac{i\pi\tau}{12}}\right)^{-2m^{2}}e^{-\frac{i\pi\tau}{6}}e^{2\pi
i\tau\left(a+\frac{n}{2}\right)^{2}}$ $\displaystyle\times\sum_{\vec{{\sf
Y}}\in\mathbb{Y}^{2}}e^{2\pi i\tau|\vec{{\sf
Y}}|}\frac{Z_{pert}\left(\vec{a}+\vec{\sf Q},\vec{a}+\vec{\sf
Q}+m\right)}{Z_{pert}\left(\vec{a},\vec{a}+m\right)}Z_{inst}\left(\vec{a}+\vec{\sf
Q},\vec{a}+\vec{\sf Q}+m|\vec{\sf Y},\vec{\sf Y}\right).$ (2.118)

Finally in terms of the conformal blocks defined above,

$\displaystyle\mathcal{T}_{CM}$
$\displaystyle=\frac{\eta(\tau)^{2}}{\theta_{1}\left(Q-\rho\right)\theta_{1}\left(Q+\rho\right)}\sum_{n,k\in\mathbb{Z}}e^{2\pi
i\nu n}q^{\left(k+n/2\right)^{2}}e^{4\pi
i(k+n/2)\left(\rho+1/2\right)}\mathcal{B}(a+n/2,m,q).$ (2.119)

∎

The relation (2.119) can be simplified further. The series

$\displaystyle\sum_{n,k\in\mathbb{Z}}e^{2\pi i\nu
n}q^{\left(k+n/2\right)^{2}}e^{4\pi
i(k+n/2)\left(\rho+1/2\right)}\mathcal{B}(a+n/2,m,q)$ (2.120)

can be split into the sum over the odd and even values of $n$:

$\displaystyle\sum_{k\in\mathbb{Z},n\in 2\mathbb{Z}+1}e^{4\pi
i\nu(n+1/2)}q^{\left(k+n+1/2\right)^{2}}e^{4\pi
i(k+n+1/2)\left(\rho+1/2\right)}\mathcal{B}(a+n+1/2,m,q)$
$\displaystyle+\sum_{k\in\mathbb{Z},n^{\prime}\in 2\mathbb{Z}}e^{4\pi i\nu
n^{\prime}}q^{\left(k+n^{\prime}\right)^{2}}e^{4\pi
i(k+n^{\prime})\left(\rho+1/2\right)}\mathcal{B}(a+n^{\prime},m,q)$
$\displaystyle=$ $\displaystyle\sum_{n\in 2\mathbb{Z}+1}e^{4\pi
i\nu(n+1/2)}\mathcal{B}(a+n+1/2,m,q)\sum_{l=k+n\in\mathbb{Z}}q^{\left(l+1/2\right)^{2}}e^{4\pi
i(l+1/2)\left(\rho+1/2\right)}$ $\displaystyle+\sum_{n^{\prime}\in
2\mathbb{Z}}e^{4\pi i\nu
n^{\prime}}\mathcal{B}(a+n^{\prime},m,q)\sum_{l^{\prime}=k+n^{\prime}\in\mathbb{Z}}q^{l^{\prime
2}}e^{4\pi il^{\prime}\left(\rho+1/2\right)}$
$\displaystyle\mathop{=:}^{\eqref{series:theta23}}\Bigg{(}Z^{D}_{1/2}(\tau)\theta_{3}\left(2\rho|2\tau\right)-Z^{D}_{0}(\tau)\theta_{2}\left(2\rho|2\tau\right)\Bigg{)}.$
(2.121)

where $Z^{D}_{0}(\tau),Z^{D}_{1/2}(\tau)$ are called the dual partition
functions with integer and half-integer shifts (refer to [Bon+20] for further
details on the partition functions). The equation analogous to (2.48) is
obtained by using the relation (2.49) in (2.119) and a straightforward
simplification

$\displaystyle\frac{\theta_{3}\left(2Q|2\tau\right)}{\theta_{2}\left(2Q|2\tau\right)}=\frac{Z_{0}^{D}(\tau)}{Z_{1/2}^{D}(\tau)}.$
(2.122)

### Further remarks

  * •

Relation to Lamé equation

in a recent paper [BGG] the authors show the following

    * 1.

the scalar form of the linear system (2.2) at the zeros of the transcendent
gives the Lamé equation,

    * 2.

the accessory parameter of the Lamé equation is the Liouville action, i.e the
difference of the action between the $\tau_{\star}$ where $Q(\tau_{\star})=0$
and the asymptote $\tau\rightarrow i\infty$

further solidifying the Painlevé/CFT correspondence at the
$c\rightarrow\infty$ limit. A rigorous proof of this semiclassical limit will
be presented in an upcoming paper [DGP].

  * •

Connection constant

An immediate consequence of the Fredholm determinant representation for the
tau-function is the computation of the connection constant, which is
equivalent to studying the modular transformation of the toric conformal
block. On the torus, the connection constant corresponds to the choice of
pants decomposition along the A-cycle vs the B-cycle. In turn, this
corresponds to the ratio of the tau function in the limits $\tau\rightarrow
i\infty$ and $\tau\rightarrow 0$. In an upcoming paper [DDG] we evaluate this
constant.

## References

  * [AGT10] L F. Alday, D Gaiotto and Y Tachikawa  “Liouville Correlation Functions from Four-dimensional Gauge Theories”  In _Lett. Math. Phys._ 91, 2010, pp. 167–197  DOI: 10.1007/s11005-010-0369-5
  * [Ber17] M Bertola  “The Malgrange Form and Fredholm Determinants”  In _Symmetry, Integrability and Geometry: Methods and Applications_ 13.0, 2017, pp. 46–12  DOI: 10.3842/SIGMA.2017.046
  * [Ber21] M Bertola  “Padé approximants on Riemann surfaces and KP tau functions”  In _Analysis and Mathematical Physics_ 11.4 Springer, 2021, pp. 1–38
  * [BGG] M Bershtein, P Gavrylenko and A Grassi  “Quantum spectral problems and isomonodromic deformations”  arXiv:2105.00985
  * [BI88] R Bikbaev and A Its  “Asymptotics at $t\rightarrow\infty$ of the solution of the Cauchy problem for the Landau-Lifshitz equation”  In _Theoretical and Mathematical Physics_ 76.1 Springer, 1988, pp. 665–675
  * [Bon+17] G Bonelli et al.  “On Painlevé/gauge theory correspondence”  In _Letters in Mathematical Physics_ 107.12 Springer, 2017, pp. 2359–2413  arXiv:1612.06235v1
  * [Bon+20] G Bonelli, F Del Monte, P Gavrylenko and A Tanzini  “$\mathcal{N}=2^{*}$ gauge theory, free fermions on the torus and Painlevé VI”  In _Commun. Math. Phys._ 377.2, 2020, pp. 1381–1419  DOI: 10.1007/s00220-020-03743-y
  * [CGL19] M Cafasso, P Gavrylenko and O Lisovyy  “Tau functions as Widom constants”  In _Communications in Mathematical Physics_ 365.2 Springer, 2019, pp. 741–772  arXiv:1712.08546
  * [DDG] F Del Monte, H Desiraju and P Gavrylenko  “On the modularity of tau-functions”  arXiv:220X.XXXX
  * [DDG20] F Del Monte, H Desiraju and P Gavrylenko  “Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions”  In _arXiv preprint arXiv:2011.06292_ , 2020
  * [Des19] H Desiraju  “The $\tau$-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant”  In _Journal of Mathematical Physics_ 60.11 AIP Publishing LLC, 2019, pp. 113505  arXiv:1906.11517
  * [Des21] H Desiraju  “Fredholm determinant representation of the homogeneous Painlevé II $\tau$-function”  In _Nonlinearity_ 34.9 IOP Publishing, 2021, pp. 6507  arXiv:2008.01142
  * [DGP] H Desiraju, P Ghosal and A Prokhorov  “Proof of the Zamolodchikov conjecture on the one point torus”  arXiv:220X.XXXX
  * [DLM] DLMF  “NIST Digital Library of Mathematical Functions” F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds., http://dlmf.nist.gov/, Release 1.1.4 of 2022-01-15  URL: http://dlmf.nist.gov/
  * [FL10] VA Fateev and AV Litvinov  “On AGT conjecture”  In _Journal of High Energy Physics_ 2010.2 Springer, 2010, pp. 1–17
  * [FN80] H Flaschka and A Newell  “Monodromy-and spectrum-preserving deformations I”  In _Communications in Mathematical Physics_ 76.1 Springer, 1980, pp. 65–116  DOI: 10.1007/BF01197110
  * [Fuc05] R Fuchs  “Sur quelques équations différentielles linéaires du second ordre”  Gauthier-Villars, 1905
  * [FW01] PJ Forrester and NS Witte  “Application of the $\tau$-function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE”  In _Communications in Mathematical Physics_ 219.2 Springer, 2001, pp. 357–398  DOI: 10.1007/s002200100422
  * [Gam10] B Gambier  “Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes”  In _Acta Mathematica_ 33 Institut Mittag-Leffler, 1910, pp. 1–55
  * [Gar12] R Garnier  “Sur des équations différentielles du troisième ordre dont l’intégrale générale est uniforme et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale a ses points critiques fixes”  In _Annales scientifiques de lÉcole normale supérieure_ 29, 1912, pp. 1–126
  * [GIL12] O. Gamayun, N. Iorgov and O. Lisovyy  “Conformal field theory of Painlevé VI” [Erratum: JHEP10,183(2012)]  In _JHEP_ 10, 2012, pp. 038  DOI: 10.1007/JHEP10(2012)183, 10.1007/JHEP10(2012)038
  * [GIL13] O. Gamayun, N. Iorgov and O. Lisovyy  “How instanton combinatorics solves Painlevé VI, V and IIIs”  In _J. Phys._ A46, 2013, pp. 335203  DOI: 10.1088/1751-8113/46/33/335203
  * [GIL18] P Gavrylenko, N Iorgov and O Lisovyy  “On solutions of the Fuji-Suzuki-Tsuda system”  In _SIGMA_ 14, 2018, pp. 123  DOI: 10.3842/SIGMA.2018.123
  * [GL18] P. Gavrylenko and O. Lisovyy  “Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions”  In _Commun. Math. Phys._ 363, 2018, pp. 1–58  DOI: 10.1007/s00220-018-3224-7
  * [GL18a] P Gavrylenko and O Lisovyy  “Pure $SU(2)$ gauge theory partition function and generalized Bessel kernel”, 2018  arXiv:1705.01869
  * [Gol09] W. Goldman  “Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces”  In _arXiv e-prints_ , 2009, pp. arXiv:0901.1404  arXiv:0901.1404 [math.GT]
  * [HB21] J Harnad and F Balogh  “Tau functions and their applications”  Cambridge University Press, 2021
  * [IP18] A R Its and A Prokhorov  “On some Hamiltonian properties of the isomonodromic tau functions”  In _Reviews in Mathematical Physics_ 30.07 World Scientific, 2018, pp. 1840008
  * [JMU81] M. Jimbo, T. Miwa and K. Ueno  “Monodromy Preserving Deformations Of Linear Differential Equations With Rational Coefficients. 1.”  In _Physica_ D2, 1981, pp. 306–352  DOI: 10.1016/0167-2789(81)90013-0
  * [Mal83] B Malgrange  “Déformations isomonodromiques des singularités réguliéres”  In _Les rencontres physiciens-mathématiciens de Strasbourg-RCP25_ 31, 1983, pp. 1–26  URL: http://www.numdam.org/item/CIF_1982__17__A2_0/
  * [Oka86] K Okamoto  “Studies on the Painlevé equations. III: Second and fourth Painlevé equations, PII and PIV”  In _Mathematische Annalen_ 275.2, 1986, pp. 221–255
  * [Pai06] P Painlevé  “Sur les équations différentielles du second ordre à points critiques fixes”  In _CR Acad. Sci. Paris_ 143, 1906, pp. 1111–1117
  * [Pic89] E Picard  “Mémoire sur la théorie des fonctions algébriques de deux variables”  In _Journal de Mathématiques pures et appliquées_ 5, 1889, pp. 135–320
  * [Rod13] Y Rodin  “The Riemann boundary problem on Riemann surfaces”  Springer Science & Business Media, 2013
  * [SMJ77] M Sato, T Miwa and M Jimbo  “Studies on holonomic quantum fields, I”  In _Proceedings of the Japan Academy, Series A, Mathematical Sciences_ 53.1 The Japan Academy, 1977, pp. 6–10