${\mathcal {N}}$ = $2^*$ Gauge Theory, Free Fermions on the Torus and Painlevé VI

Abstract

In this paper we study the extension of Painlevé/gauge theory correspondence to circular quivers by focusing on the special case of SU(2) ${\mathcal {N}}=2^*$ theory. We show that the Nekrasov–Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of $SL_2$ flat connections on the one-punctured torus. This is achieved by reformulating the Riemann–Hilbert problem associated to the latter in terms of chiral conformal blocks of a free-fermionic algebra. This viewpoint provides the exact solution of the renormalization group flow of the SU(2) ${\mathcal {N}}=2^*$ theory on self-dual $\Omega$-background and, in the Seiberg–Witten limit, an elegant relation between the IR and UV gauge couplings.

Appendices

Elliptic and Theta Functions

For elliptic and theta functions we use the notations of [76]. Our torus has periods $(1,\tau )$, and the theta function that we use are

$$\begin{aligned} \begin{aligned}&\theta _1(z|\tau )\equiv -i\sum _{n\in {\mathbb {Z}}}(-1)^n q^{(n+\frac{1}{2})^2/2} e^{2\pi iz(n+\frac{1}{2})},\\&\theta _2(z|\tau )\equiv \sum _{n\in {\mathbb {Z}}} q^{(n+\frac{1}{2})^2/2} e^{2\pi iz(n+\frac{1}{2})},\\&\theta _3(z|\tau )\equiv \sum _{n\in {\mathbb {Z}}} q^{n^2/2} e^{2\pi iz n},\\&\theta _4(z|\tau )\equiv \sum _{n\in {\mathbb {Z}}}(-1)^n q^{n^2/2} e^{2\pi izn}, \end{aligned} \end{aligned}$$

(A.1)

where

$$\begin{aligned} q=e^{2\pi i\tau }. \end{aligned}$$

(A.2)

A prime denotes a derivative with respect to z, and when the theta function or its derivatives are evaluated at $z=0$, we simply denote it by $\theta _\nu (\tau )$ or $\theta _1'(\tau )$, e.t.c. Transformations of $\theta _1$ under elliptic transformations are

$$\begin{aligned} \theta _1(z+1|\tau )=-\theta _1(z|\tau ), \quad \theta _1(z+\tau |\tau )=-q^{-1}e^{-2\pi i z}\theta _1(z|\tau ). \end{aligned}$$

(A.3)

In the main text we use also Weierstrass $\wp$ and $\zeta$. $\wp$ is a doubly periodic function with a single double pole at $z=0$, that can be written in terms of $\theta _1$ as

$$\begin{aligned} \wp (z|\tau )=-\partial _z^2\log \theta _1(z|\tau )-2\eta _1(\tau )=\zeta '(z|\tau ), \end{aligned}$$

(A.4)

where

$$\begin{aligned} \eta _1(\tau )=-\frac{1}{6}\frac{\theta _1'''(\tau )}{\theta _1'(\tau )}. \end{aligned}$$

(A.5)

Weierstrass’ $\zeta$ function is minus the primitive of $\wp$. It has only one simple pole at $z=0$, and is quasi-elliptic:

$$\begin{aligned} \zeta (z|\tau )= & {} 2\eta _1(\tau )z+\partial _z\log \theta _1(z|\tau ), \end{aligned}$$

(A.6)

$$\begin{aligned} \zeta (z+1|\tau )= & {} \zeta (z|\tau )+ 2\eta _1(\tau ), \qquad \zeta (z+\tau |\tau )=\zeta (z|\tau )+ 2\tau \eta _1(\tau )-2\pi i. \end{aligned}$$

(A.7)

Finally, we use Dedekind’s $\eta$ function, defined as

$$\begin{aligned} \eta (\tau )=q^{1/24}\prod _{n=1}^\infty (1-q^n). \end{aligned}$$

(A.8)

It is related to the function $\theta _1$ by

$$\begin{aligned} \eta (\tau )=\left( \frac{\theta _1'(\tau )}{2\pi } \right) ^{1/3}. \end{aligned}$$

(A.9)

Because of the periodicities (A.3), the elliptic transformations of the Lamé function

$$\begin{aligned} x(u,z)=\frac{\theta _1(z-u|\tau )\theta _1'(\tau )}{\theta _1(z|\tau )\theta _1(u|\tau )} \end{aligned}$$

(A.10)

are given by

$$\begin{aligned} x(u,z+1)=x(u,z), \quad x(u,z+\tau )=e^{2\pi i u}x(u,z). \end{aligned}$$

(A.11)

The product of Lamé functions satisfies the following identities:

$$\begin{aligned} x(u,z)x(-u,z)=\wp (z)-\wp (u), \quad x(u,z)y(-u,z)-y(u,z)x(-u,z)=\wp '(u), \end{aligned}$$

(A.12)

where $y(u,z)=\partial _ux(u,z)$, that are used in computing the isomonodromic Hamiltonian $H_\tau$ from the Lax matrix. Further, to show that the zero-curvature Eq. (3.11) is the compatibility condition for the system (3.12), one has to use the property

$$\begin{aligned} 2\pi i\partial _\tau x(u,z)+\partial _z\partial _u x(u,z)=0. \end{aligned}$$

(A.13)

The following theta-function identities are used in the study of the autonomous limit:

$$\begin{aligned} \partial _z\frac{\theta _2(z|\tau )}{\theta _3(z|\tau )}=-\,\pi \theta _4^2(\tau )\frac{\theta _1(z|\tau )\theta _4(z|\tau )}{\theta _3(z|\tau )^2}, \end{aligned}$$

(A.14)

$$\begin{aligned} \frac{\theta _1(2Q|2\tau _0)^2}{\theta _3(2Q|2\tau _0)^2}= \frac{\theta _2(2\tau _0)^2}{\theta _4(2\tau _0)^2}-\frac{\theta _3(2\tau _0)^2}{\theta _4(2\tau _0)^2}\frac{\theta _2(2Q|2\tau _0)^2}{\theta _3(2Q|2\tau _0)^2}, \nonumber \\ \frac{\theta _4(2Q|2\tau _0)^2}{\theta _3(2Q|2\tau _0)^2}= \frac{\theta _3(2\tau _0)^2}{\theta _4(2\tau _0)^2}-\frac{\theta _2(2\tau _0)^2}{\theta _4(2\tau _0)^2}\frac{\theta _2(2Q|2\tau _0)^2}{\theta _3(2Q|2\tau _0)^2}. \end{aligned}$$

(A.15)

Fusion and Braiding of Degenerate Fields

In this paper monodromies of the fundamental solution have been computed by braiding and fusion of degenerate fields with Virasoro primaries. Fusion matrices linearly relate different sets of blocks, and analytic continuation of a conformal block along any arbitrary contour can be decomposed in elementary braiding and fusion rules [52] (Fig. 4).

Fig. 4

Braiding of a degenerate field and a primary

In this paper we only need the following braiding move, that corresponds to the following analytic continuation operation on the matrix elements¹⁷:

$$\begin{aligned} \langle \varvec{a}' |V_{\varvec{m}}(y) \phi _s(\gamma _y\cdot z)|\varvec{a}\rangle =i\sum _{s'=\pm } \langle \varvec{a}' |\phi _{s'}(z) V_m(y) |\varvec{a}\rangle e^{i\pi s'a'}F_{s's}(a',m,a)e^{-i\pi sa},\nonumber \\ \end{aligned}$$

(B.1)

where $\gamma _y\cdot z$ is the analytic continuation of the degenerate field along a contour going around y in the positive direction, on the sphere from zero to infinity. F is the fusion matrix, that with our normalization conventions takes the form

$$\begin{aligned} F(a',m,a)=\left( \begin{array}{cc} \frac{\cos \pi (m+a+a')}{\sin 2\pi a} &{} \frac{\cos \pi (m+a'-a)}{\sin 2\pi a} \\ -\frac{\cos \pi (m+a-a')}{\sin 2\pi a} &{} -\frac{\cos \pi (m-a-a')}{\sin 2\pi a} \end{array} \right) . \end{aligned}$$

(B.2)

Periodicity of the Tau Functions

Equation (3.14) is invariant under the transformation $Q\mapsto Q+\frac{n}{2} +\frac{\tau k}{2}$, where $k,n\in {\mathbb {Z}}$. We denote this transformation by $\delta _{\frac{n}{2},\frac{k}{2}}$. One might ask the following question: what are the transformation properties of the tau function and dual partition functions under $\delta _{\frac{n}{2},\frac{k}{2}}$?

The tau function after the transformation is defined by

$$\begin{aligned} 2\pi i\partial _\tau \log (\delta _{\frac{n}{2},\frac{k}{2}}{\mathcal T})= & {} (2\pi i)^2(\partial _\tau Q+k/2)^2-m^2\wp (2Q)-2m^2\eta _1(\tau )\nonumber \\= & {} 2\pi i\partial _\tau {\mathcal T}+(2\pi i)^2(k^2/4+k\partial _\tau Q). \end{aligned}$$

(C.1)

Therefore

$$\begin{aligned} \delta _{\frac{n}{2},\frac{k}{2}}{\mathcal T}=C_{\frac{n}{2},\frac{k}{2}}\cdot q^{k^2/4}e^{2\pi i kQ}\mathcal T \end{aligned}$$

(C.2)

Using now (3.55) we compute transformations of $Z_0^D$, $Z_{1/2}^D$:

$$\begin{aligned} \delta _{\frac{n}{2},\frac{k}{2}}Z_{0}^D= & {} C_{\frac{n}{2},\frac{k}{2}}\eta (\tau )^{-1}\theta _3(2Q+n+k\tau |2\tau )q^{k^2/4}e^{2\pi ikQ}\mathcal T\nonumber \\= & {} \left[ \begin{array}{l} k\in 2{\mathbb {Z}}: C_{\frac{n}{2},\frac{k}{2}}\eta (\tau )^{-1}\theta _3(2Q\tau |2\tau )\mathcal T=C_{\frac{n}{2},\frac{k}{2}}Z_{0}^D\\ k\in 1+2{\mathbb {Z}}: C_{\frac{n}{2},\frac{k}{2}}\eta (\tau )^{-1}\theta _2(2Q\tau |2\tau )\mathcal T=C_{\frac{n}{2},\frac{k}{2}}Z_{1/2}^D. \end{array} \right. \end{aligned}$$

(C.3)

In this way we see that the dual partition functions have much better behaviour than the tau function $\mathcal T$. As we will see later in Appendix D, such shifts of parameters correspond to simple shifts of the initial data:

$$\begin{aligned} \delta _{\frac{n}{2},\frac{k}{2}}(\eta ,a)=(\eta +2\pi n,a+\frac{k}{2}). \end{aligned}$$

(C.4)

Therefore transformations of the dual partition functions look as follows:

$$\begin{aligned} \delta _{\frac{n}{2},\frac{k}{2}}Z^D_{1/2}=e^{-i\eta \frac{k}{2}}\cdot \left[ \begin{array}{l} k\in 2{\mathbb {Z}}: Z^D_{1/2}\\ k\in 1+2{\mathbb {Z}}: Z^D_{0}. \end{array} \right. \end{aligned}$$

(C.5)

Therefore we can conclude that $C_{\frac{n}{2},\frac{k}{2}}=e^{-i\eta \frac{k}{2}}$, so $\delta _{\frac{n}{2},\frac{k}{2}}\mathcal T=q^{k^2/4}e^{ik(2\pi Q-\frac{\eta }{2})}\mathcal T$.

Asymptotic Calculation of the Tau Function

1.1 Algorithm of computations

We start from the following ansatz for the solution of the non-autonomous Calogero equation:

$$\begin{aligned} Q=\alpha \tau +\beta +\frac{1}{2\pi i}\sum _{n=0}^{\infty }\sum _{k=-n}^\infty c_{n,k} q^n e^{4\pi i k(\alpha \tau +\beta )}=\alpha \tau +\beta + \frac{1}{2\pi i}X. \end{aligned}$$

(D.1)

Series expansion of $\wp (z|\tau )+2\eta _1(\tau )=-\partial _z^2\log \theta _1(z|\tau )$ looks as follows:

$$\begin{aligned} \frac{1}{(2\pi i)^2}(-\partial _z^2\log \theta _1(z|\tau ))=\sum _{n=0}^\infty \sum _{k=1}^\infty k q^{kn}e^{2\pi ikz}+ \sum _{n=1}^\infty \sum _{k=1}^\infty k q^{kn}e^{-2\pi ikz}. \end{aligned}$$

(D.2)

Rewrite now Eq. (3.14) using last two formulas and introducing notation $s=e^{2\pi i(\alpha \tau +\beta )}$. We also introduce single formal parameter of expansion $\epsilon$ in the following way: $q\mapsto q\cdot \epsilon ^2$, $s\mapsto s\cdot \sqrt{\epsilon }$.

$$\begin{aligned}&\sum _{n=0}^\infty \sum _{k=-n}^\infty (n+2\alpha k)^2 c_{n,k} q^n s^{2k}\epsilon ^{2n+k}\nonumber \\&\quad = m^2\sum _{n=0}^\infty \sum _{k=1}^\infty k^2 q^{kn} s^{2k} e^{k X(q,s)}\epsilon ^{k(2n+1)}- m^2\sum _{n=1}^\infty \sum _{k=1}^\infty k^2 q^{kn} s^{-2k}e^{-kX(q,s)}\epsilon ^{k(2n-1)}.\nonumber \\ \end{aligned}$$

(D.3)

One can see that powers of $\epsilon$ in the r.h.s. are at least one, therefore higher-order coefficients $c_{n,k}$ become functions of the lower-order ones, and thus equation can be solved order-by-order starting from $c_{0,0}=0$¹⁸.

After this is done, we need to compute the logarithm of the isomonodromic tau function:

$$\begin{aligned} \log \mathcal T= & {} \alpha ^2\log q + \sum _{n,k} q^n s^k b_{n,k}=\alpha ^2\log q+Y(q,s)\nonumber \\&\sum _{n,k}(n+2\alpha k)q^ns^kb_{n,k} =\left( \sum _{n,k}(n+2\alpha k)q^ns^{2k}c_{n,k}\right) ^2\nonumber \\&-m^2\left( \sum _{n=0}^\infty \sum _{k=1}^\infty k q^{kn}s^{2k} e^{k X(q,s)}+ \sum _{n=1}^\infty \sum _{k=1}^\infty k q^{kn}s^{-2k} e^{-k X(q,s)} \right) . \end{aligned}$$

(D.4)

Then we compute the two dual partition functions:

$$\begin{aligned} Z^D_{0}= & {} q^{\alpha ^2-1/24}\sum _{n=1}^\infty p(n)q^n\cdot \sum _{n=-\infty }^\infty q^{n^2}s^{2n}e^{2nX(q,s)}\cdot e^{Y(q,s)+2\alpha X(q,s)}=\sum _{n,k}z_{n,k}q^{n+\alpha ^2-1/{24}} s^{2k},\nonumber \\ Z^D_{1/2}= & {} q^{\alpha ^2-1/24}\sum _{n=1}^\infty p(n)q^n\cdot \sum _{n=-\infty }^\infty q^{(n+\frac{1}{2})^2}s^{2n+1}e^{(2n+1)X(q,s)}\cdot e^{Y(q,s)+2\alpha X(q,s)}\nonumber \\= & {} \sum _{n,k}z'_{n,k}q^{n+\alpha ^2+5/{24}} s^{2k-1}. \end{aligned}$$

(D.5)

1.2 Results

We solved the equation asymptotically up to $\epsilon ^6$.

Important information about the function $f(q,s)=\sum f_{n,k}q^n s^{2k}$ is encoded in the list of non-zero coefficients $f_{n,k}$: we will denote such coefficients by points (k, n) in the integer plane (sometimes it will be shifted by some fractional numbers which we neglect). We call the set of such points support of the function f.

First we show the support of the solution X(q, s) in Fig. 5. In this picture the gray region contains all coefficients that were computed in the asymptotic expansion. Dotted lines show monomials with the same order of $\epsilon$. The full support is bounded from below by the lines $n=0$ and $n=-k$.

Fig. 5

Support of X(q, s)

The first few terms of the expansion look as follows:

$$\begin{aligned} c_{0,1}= & {} \frac{m^2}{4 \alpha ^2},\quad c_{1,-1}=-\frac{m^2}{(2 \alpha -1)^2},\quad c_{0,2}=\frac{m^2 \left( 8 \alpha ^2+m^2\right) }{32 \alpha ^4},\nonumber \\ c_{1,0}= & {} -\frac{(4 \alpha -1) m^4}{2 \alpha ^2 (2 \alpha -1)^2},\quad c_{2,-2}=-\frac{m^2\left( 8 \alpha ^2-8 \alpha +m^2+2\right) }{2 (2 \alpha -1)^4},\quad \ldots \end{aligned}$$

(D.6)

The support of the dual partition functions is shown in Fig. 6.

Fig. 6

Support of $Z_{0}^D$ (left) and of $Z_{1/2}^D$ (right).

We see that some non-trivial cancellation happened and a lot of coefficients that naively might be non-zero (denoted by small dots) actually vanish.

Values of the first non-trivial coefficients are given by

$$\begin{aligned} z_{0,0}= & {} 1,\quad z'_{0,0}=1,\quad z'_{0,1}=1-\frac{m^2}{4\alpha ^2},\quad z_{1,-1}=\frac{ (m^2-(1-2 \alpha ))^2}{(1-2 \alpha )^2},\nonumber \\ z_{1,1}= & {} \frac{(m^2-4 \alpha ^2)^2 (m^2-(2 \alpha +1)^2)}{16 \alpha ^4 (2 \alpha +1)^2},\quad \ldots \end{aligned}$$

(D.7)

We also found experimentally that normalized values of all other non-trivial coefficients can be given in terms of a single function, which will be identified with the toric conformal block:

$$\begin{aligned} \mathcal B(a,m,q)= & {} 1+q \left( \frac{(m^2-1) m^2}{2 a^2}+1\right) +q^2 \left( \frac{3 (m^2-4) (m^2-1)^2 m^2}{16 (a^2-\frac{1}{4})^2}\right. \nonumber \\&\left. -\frac{(m^2-3) (m^2-1) (m^2+2) m^2}{4 (a^2-\frac{1}{4})}+\frac{(m^2-1) (m^4-m^2+2) m^2}{4 a^2}+2\right) \nonumber \\&+\,q^3 \left( \frac{(m^2-9) (m^2-4)^2 (m^2-1)^2 m^2}{36 (a^2-1)^2}+\frac{(m^2-4) (m^2-1)^2 (2 m^4-2 m^2+9) m^2}{48 (a^2-\frac{1}{4})^2}\right. \nonumber \\&-\frac{(m^2-4) (m^2-1) (11 m^6-106 m^4+131 m^2+108) m^2}{216 (a^2-1)}\nonumber \\&+\,\frac{(m^2-1) (3 m^8-22 m^6+65 m^4-46 m^2+24) m^2}{24 a^2}\nonumber \\&\left. -\frac{(m^2-1) (8 m^8-24 m^6+15 m^4+m^2-162) m^2}{108 (a^2-\frac{1}{4})}+3\right) +O(q^3)=\sum _{n=0}^\infty \mathcal B_n(a,m)q^n . \nonumber \\ \end{aligned}$$

(D.8)

Namely, the ratios of coefficients are

$$\begin{aligned} z_{n,0}/z_{0,0}= & {} \mathcal B_n(\alpha ,m),\quad n=0,1,2,3,\quad z_{2,1}/z_{1,1}=\mathcal B_1(\alpha +1,m),\quad n=0,1,2,\nonumber \\ \frac{z_{n+1,-1}}{z_{1,-1}}= & {} \mathcal B_n(\alpha -1,m),\quad \frac{z_{3,1}}{z_{1,1}}=\mathcal B_2(\alpha -1,m),\quad \frac{z'_{-1,3}}{z'_{-1,2}}=\mathcal B_1(a-\frac{3}{2},m)\nonumber \\ \frac{z'_{n,0}}{z'_{0,0}}= & {} \mathcal B_n(\alpha -\frac{1}{2},m),\quad n=0,1,2,3,\quad \frac{z'_{n,1}}{z'_{0,1}}=\mathcal B_n(\alpha +\frac{1}{2},m),\quad n=0,1,2.\nonumber \\ \end{aligned}$$

(D.9)

We see that the latter formula is in complete agreement with (1.2) if $\mathcal B$ is a conformal block, so we check that it actually coincides with the AGT formula

$$\begin{aligned} \mathcal B(a,m,q)=\prod _{n=1}^\infty (1-q^n)^{1-2m^2} \sum _{Y_+,Y_-}q^{|Y_+|+|Y_-|}\prod _{\epsilon ,\epsilon '=\pm } \frac{N_{Y_\epsilon ,Y_{\epsilon '}}(m+(\epsilon -\epsilon ') a)}{N_{Y_\epsilon ,Y_{\epsilon '}}((\epsilon -\epsilon ') a)}\nonumber \\ \end{aligned}$$

(D.10)

where Nekrasov factors are given by

$$\begin{aligned} N_{\lambda ,\mu }(x)=\prod _{s\in \lambda }(x+a_\lambda (s)+l_\mu (s)+1) \prod _{t\in \mu }(x-l_\lambda (t)-a_\mu (t)-1). \end{aligned}$$

(D.11)

1.3 Asymptotic computation of monodromies

At the moment we have two different parameterization: one in terms of $(\alpha ,\beta )$, initial data of the equation, and another in terms of the monodromy data $(a,\eta )$. We need to know the explicit identification between them. To compute this identification we use the fact that the evolution is isomonodromic, so monodromies can be computed in the limit $\tau \rightarrow +i\infty$.

$$\begin{aligned} \theta _1(z|\tau )=2q^{1/8}\left( \sin \pi z-q \sin 3\pi z+\cdots \right) . \end{aligned}$$

(D.12)

One can take just the first term of expansion until it becomes smaller than the first correction. This occurs when $\sin \pi z\approx e^{2 \pi i\tau }\sin {3\pi z}$, so for $z=\pm \tau$. We will choose two copies of the A-cycle with $\mathrm{Im} z=\pm \mathrm{Im} \tau /2$ and work in the region between them. So for all computations to be consistent we need to have $-1/4<\mathrm{Re}\alpha <1/4$ (one can easily overcome this constraint taking more terms of expansion). Also convergence of the series (D.1) requires $\alpha >0$. So for simplicity we just take $\alpha$ to have a sufficiently small positive real part.

Our approximation for x is then

$$\begin{aligned} x(u,z)\approx \frac{\pi \sin \pi (z-u)}{\sin \pi z\sin \pi u} =\frac{2\pi i}{e^{2\pi iu}-1}-\frac{2\pi i}{e^{2\pi iz}-1}. \end{aligned}$$

(D.13)

The first terms of the expansion of Q look as

$$\begin{aligned} Q(\tau )=\alpha \tau +\beta +\frac{m^2}{8\pi i\alpha ^2}e^{4\pi i(\alpha \tau +\beta )}+\cdots \end{aligned}$$

(D.14)

The leading behavior of the connection matrix is

$$\begin{aligned} L(z|\tau )=2\pi i\begin{pmatrix} \alpha +\frac{m^2}{2\alpha }e^{4\pi i(\alpha \tau +\beta )}&{} \frac{m}{e^{4\pi i(\alpha \tau +\beta )}-1}-\frac{m}{e^{2\pi iz}-1}\\ \frac{m}{e^{-4\pi i(\alpha \tau +\beta )}-1}-\frac{m}{e^{2\pi iz}-1}&{} -\alpha -\frac{m^2}{2\alpha }e^{4\pi i(\alpha \tau +\beta )}\end{pmatrix}. \end{aligned}$$

(D.15)

Further expanding up to first order in $e^{4\pi i\alpha \tau }$ we get

$$\begin{aligned} L(z|\tau )=2\pi i\begin{pmatrix} \alpha &{}-\frac{m e^{2\pi i z}}{e^{2\pi iz}-1}\\ -\frac{m}{e^{2\pi iz}-1}&{}-\alpha \end{pmatrix}+ 2\pi i e^{4\pi i(\alpha \tau +\beta )}\begin{pmatrix} \frac{m^2}{2\alpha }&{}-m\\ m&{} -\frac{m^2}{2\alpha }\end{pmatrix}. \end{aligned}$$

(D.16)

One may notice that there is an equality

$$\begin{aligned} L(z|\tau )=U_0L_0(z|\tau )U_0^{-1}+ o\left( e^{4\pi i(\alpha \tau +\beta )}\right) , \end{aligned}$$

(D.17)

where

$$\begin{aligned} U_0=1+\frac{m}{2\alpha }e^{4\pi i(\alpha \tau +\beta )}\sigma ^x,\quad L_0(z|\tau )=2\pi i\begin{pmatrix} \alpha &{}-\frac{m e^{2\pi i z}}{e^{2\pi iz}-1}\\ -\frac{m}{e^{2\pi iz}-1}&{}-\alpha \end{pmatrix}. \end{aligned}$$

(D.18)

This equality is a reminiscence of the isomonodromic deformation Eq. (3.12). The solution of the linear system in the region $z\sim \frac{\tau }{2}$ is given by

$$\begin{aligned} Y(z)= & {} (1-e^{2\pi iz})^mU_0\nonumber \\&\times \begin{pmatrix} {}_2F_1(m,m+2\alpha ,2\alpha ,e^{2\pi iz})&{} \frac{me^{2\pi i z}}{1-2\alpha }{}_2F_1(1+m,1+m-2\alpha ,2-2\alpha ,e^{2\pi iz})\\ \frac{m}{2\alpha }{}_2F_1(1+m,m+2\alpha ,1+2\alpha ,e^{2\pi iz})&{} {}_2F_1(m,1+m-2\alpha ,1-2\alpha ,e^{2\pi i z}) \end{pmatrix}\nonumber \\&\times \mathrm{diag}((-e^{2\pi i z})^\alpha ,(-e^{2\pi i z})^{-\alpha }). \end{aligned}$$

(D.19)

Now we compute the analytic continuation of this solution to the region $z\sim -\frac{\tau }{2}$ along imaginary line $\mathrm{Re} z=1/2$¹⁹:

$$\begin{aligned} Y(z)=\sigma ^x Y(-z) \sigma ^x \widetilde{M_B},\qquad \widetilde{M_B}= \begin{pmatrix} \frac{\Gamma (2\alpha )^2}{\Gamma (2\alpha +m)\Gamma (2\alpha -m)}&{} \frac{\Gamma (2-2\alpha )\Gamma (-1+2\alpha )}{\Gamma (m)\Gamma (1-m)}\\ \frac{\Gamma (1-2\alpha )\Gamma (2\alpha )}{\Gamma (1-m)\Gamma (m)}&{} \frac{\Gamma (1-2\alpha )^2}{\Gamma (1-m-2\alpha )\Gamma (1+m-2\alpha )} \end{pmatrix}. \nonumber \\ \end{aligned}$$

(D.21)

We wish to compute B-cycle monodromy. Its defining relation is

$$\begin{aligned} Y(\frac{\tau }{2}+x)=e^{2\pi i\varvec{Q}}Y(-\frac{\tau }{2}+x)M_B. \end{aligned}$$

(D.22)

Using (D.21) we get

$$\begin{aligned} Y(\frac{\tau }{2}+x)=e^{2\pi i\varvec{Q}}\sigma ^x Y(\frac{\tau }{2}-x)\sigma ^x \widetilde{M_B}M_B. \end{aligned}$$

(D.23)

We write this expression down keeping only the first orders:

$$\begin{aligned}&(1+\frac{m}{2\alpha }e^{4\pi i(\alpha \tau +\beta )}\sigma ^x) \begin{pmatrix} 1&{}0\\ \frac{m}{2\alpha }&{}1 \end{pmatrix} \times \text {diag}(e^{\pi i\alpha \tau },e^{-\pi i\alpha \tau }) \times \text {diag}((-e^{2\pi i x})^{\alpha },(-e^{2\pi ix})^{-\alpha })\nonumber \\&\quad = \text {diag}\left( e^{2\pi i(\alpha \tau +\beta )},e^{-2\pi i(\alpha \tau +\beta )}+\frac{m^2}{4\alpha ^2}e^{2\pi i(\alpha \tau +\beta )}\right) (1+\frac{m}{2\alpha }e^{4\pi i(\alpha \tau +\beta )}\sigma ^x) \begin{pmatrix} 1&{}\frac{m}{2\alpha }\\ 0&{}1 \end{pmatrix}\nonumber \\&\qquad \times \text {diag}(e^{-\pi i\alpha \tau },e^{\pi i\alpha \tau }) \times \text {diag}((-e^{2\pi i x})^{\alpha },(-e^{2\pi ix})^{-\alpha })\times \widetilde{M_B} M_B. \end{aligned}$$

(D.24)

By using also the relation

$$\begin{aligned} \sigma ^x \widetilde{M_B} \sigma ^x = \widetilde{M_B}^{-1} \end{aligned}$$

(D.25)

we can then express $M_B$ as

$$\begin{aligned} M_B=\sigma ^x\widetilde{M_B}\sigma ^x\times \text {diag}(e^{-2\pi i\beta },e^{2\pi i\beta })+O(e^{4\pi i\alpha \tau }). \end{aligned}$$

(D.26)

We see that to our precision $M_B$ is actually constant when $\tau \rightarrow i\infty$, its value is given by

$$\begin{aligned} M_B= \begin{pmatrix} \frac{\Gamma (1-2\alpha )^2}{\Gamma (1-m-2\alpha )\Gamma (1+m-2\alpha )}e^{-2\pi i\beta }&{} \frac{\sin \pi m}{\sin \pi \alpha }e^{2\pi i\beta }\\ -\frac{\sin \pi m}{\sin \pi \alpha }e^{-2\pi i\beta }&{} \frac{\Gamma (2\alpha )^2}{\Gamma (2\alpha +m)\Gamma (2\alpha -m)}e^{2\pi i\beta } \end{pmatrix}. \end{aligned}$$

(D.27)

To compare this with the monodromy matrix (3.28) computed by braiding we now change normalization

$$\begin{aligned} e^{2\pi i\beta }=e^{2\pi i\beta '}/r, \quad M_B'=\text {diag}(r^{1/2},r^{-1/2}) M_B \text {diag}(r^{-1/2},r^{1/2}), \end{aligned}$$

(D.28)

where

$$\begin{aligned} r=\frac{\Gamma (2\alpha )\Gamma (1-2\alpha -m)}{\Gamma (1-2\alpha )\Gamma (2\alpha -m)}. \end{aligned}$$

(D.29)

The new monodromy is

$$\begin{aligned} M_B'=\begin{pmatrix} \frac{\sin \pi (2\alpha -m)}{\sin 2\pi \alpha }e^{-2\pi i\beta '}&{} \frac{\sin \pi m}{\sin 2\pi \alpha }e^{2\pi i\beta '}\\ -\frac{\sin \pi m}{\sin 2\pi \alpha }e^{-2\pi i\beta '}&{} \frac{\sin \pi (2\alpha +m)}{\sin 2\pi \alpha }e^{2\pi i\beta '} \end{pmatrix}. \end{aligned}$$

(D.30)

Corresponding A-cycle monodromy is clearly given by the formula

$$\begin{aligned} M_A'=M_A=\begin{pmatrix}e^{2\pi i a}&{}0\\ 0&{}e^{-2\pi ia}\end{pmatrix}. \end{aligned}$$

(D.31)

These monodromies are related to those computed from CFT (3.28) by a conjugation with the matrix

$$\begin{aligned} C=\begin{pmatrix}0&{}-i e^{\pi i a}\\ i e^{-\pi i a}&{}\end{pmatrix}. \end{aligned}$$

(D.32)

We can in fact check explicitly that

$$\begin{aligned} M_A'= & {} CM_A^{(CFT)}C^{-1}=\begin{pmatrix}e^{2\pi ia}&{}0\\ 0&{}e^{-2\pi i a} \end{pmatrix},\nonumber \\ M_B'= & {} CM_B^{(CFT)}C^{-1}= \begin{pmatrix} \frac{\sin \pi (2 a-m)}{\sin 2\pi a}e^{-i\eta /2}&{} \frac{\sin \pi m}{\sin 2\pi a}e^{i\eta /2}\\ -\frac{\sin \pi m}{\sin 2\pi a}e^{-i\eta /2}&{} \frac{\sin \pi (2 a+m)}{\sin 2\pi a}e^{i\eta /2} \end{pmatrix}, \end{aligned}$$

(D.33)

after we made the identification

$$\begin{aligned} \alpha =a, \quad \eta =4\pi \beta '. \end{aligned}$$

(D.34)

1.4 Identification of parameters

One may check that the two asymptotic expansions have the following forms:

$$\begin{aligned} \eta (\tau )^{-1}\theta _3(2Q|\tau )\mathcal T(\tau )= & {} \sum _{n\in {\mathbb {Z}}} C_n e^{4\pi i n\beta } q^{(\alpha +n)^2} \mathcal B(\alpha +n,m,q)\nonumber \\ \eta (\tau )^{-1}\theta _2(2Q|\tau )\mathcal T(\tau )= & {} \sum _{n\in {\mathbb {Z}}+\frac{1}{2}} C_n e^{4\pi i n\beta } q^{(\alpha +n)^2} \mathcal B(\alpha +n,m,q).\nonumber \\ \end{aligned}$$

(D.35)

Where the structure constants are given by explicit formula

$$\begin{aligned} C_n= & {} \frac{G(1-m+2(\alpha +n))G(1-m-2(\alpha +n))}{G(1+2(\alpha +n))G(1-2(\alpha +n))} \frac{G(1+2\alpha )G(1-2\alpha )}{G(1-m+2\alpha )G(1-m-2\alpha )}\nonumber \\&\times \left( \frac{\Gamma (2\alpha )\Gamma (1-m-2\alpha )}{\Gamma (1-2\alpha )\Gamma (2\alpha -m)}\right) ^{2n}. \end{aligned}$$

(D.36)

We may check, in particular, that $C_0=C_{-\frac{1}{2}}=1$, which is consistent with experimental results. We also see that after the redefinition²⁰

$$\begin{aligned} e^{2 i\beta '}=e^{2i\beta }\frac{\Gamma (2\alpha ) \Gamma (1-m-2\alpha )}{\Gamma (1-2\alpha )\Gamma (2\alpha -m)}. \end{aligned}$$

(D.37)

Experimentally found tau functions may be rewritten as

$$\begin{aligned} \theta _3(2Q|\tau )\mathcal T(\tau )= & {} C(\alpha )^{-1}\sum _{n\in {\mathbb {Z}}} C(\alpha +n) e^{4\pi i n\beta '} q^{(\alpha +n)^2} \mathcal B(\alpha +n,m,q)\nonumber \\ \theta _2(2Q|\tau )\mathcal T(\tau )= & {} C(\alpha )^{-1}\sum _{n\in {\mathbb {Z}}+\frac{1}{2}} C(\alpha +n) e^{4\pi i n\beta '} q^{(\alpha +n)^2} \mathcal B(\alpha +n,m,q) \end{aligned}$$

(D.38)

where

$$\begin{aligned} C(\alpha )=\frac{G(1-m+2\alpha )G(1-m-2\alpha )}{G(1+2\alpha )G(1-2\alpha )}. \end{aligned}$$

(D.39)

Again, comparing with expressions (3.55) we find $a=\alpha$, $\eta =4\pi \beta '$, consistently with what we found in the asymptotic computation of the monodromy matrices.

A Self-consistency Check

In this appendix we compute the derivative with respect to $\tau$ of the conformal block, using the generalized Wick’s theorem, and provide a consistency check for the relation 3.32 between free fermion CFT and the linear system 3.4. Namely, we compute

$$\begin{aligned} F(w,y)_{\alpha \beta }= & {} 2\pi i\partial _\tau \left( Y(w)^{-1}\Xi (y-w)Y(y)\right) _{\alpha \beta }=2\pi i\partial _\tau \frac{\langle {\bar{\psi }}_\alpha (w) \psi _\beta (y) V_m(0)\rangle }{\langle V_m(0)\rangle }\nonumber \\= & {} 2\pi i\frac{\partial _\tau \langle {\bar{\psi }}_\alpha (w) \psi _\beta (y) V_m(0)\rangle }{\langle V_m(0)\rangle }- 2\pi i\frac{\langle {\bar{\psi }}_\alpha (w) \psi _\beta (y) V_m(0)\rangle }{\langle V_m(0)\rangle } \frac{\partial _\tau \langle V_m(0)\rangle }{\langle V_m(0)\rangle }. \nonumber \\ \end{aligned}$$

(E.1)

We then note that the derivative of any correlator with respect to the modular parameter $\tau$ can be realized by an integral over the A-cycle of an insertion of the energy-momentum tensor T(z):

$$\begin{aligned} 2\pi i\partial _\tau \langle {\mathcal {O}} \rangle =\oint _A\langle {\mathcal {O}}T(z)\rangle dz. \end{aligned}$$

(E.2)

Further, we can use the explicit expression of the free fermion energy-momentum tensor

$$\begin{aligned} T(z)=\frac{1}{2}\sum _\gamma \left( :\partial {\bar{\psi }}_\gamma (z) \psi _\gamma (z):+:\partial \psi _\gamma (z){\bar{\psi }}_\gamma (z):\right) \end{aligned}$$

(E.3)

where $:\ :$ denotes regular part of the OPE. Then,

$$\begin{aligned} F(w,y)_{\alpha \beta }= & {} \sum _\gamma \oint _A dz \frac{\langle :\partial {\bar{\psi }}_\gamma (z)\psi _\gamma (z):{\bar{\psi }}_\alpha (w) \psi _\beta (y) V_m(0)\rangle }{\langle V_m(0)\rangle }\nonumber \\&- \frac{\langle :\partial {\bar{\psi }}_\gamma (z)\psi _\gamma (z): V_m(0)\rangle \langle {\bar{\psi }}_\alpha (w) \psi _\beta (y) V_m(0)\rangle }{\langle V_m(0)\rangle ^2}. \end{aligned}$$

(E.4)

Here we added a total derivative to T(z) since it does not change correlator. Now we can compute this expression using the generalized Wick theorem [77]. The second term cancels the one containing pairing between ${\bar{\psi }}_\alpha (w)$ and $\psi _\beta (y)$. So finally we get only

$$\begin{aligned} F(w,y)_{\alpha \beta }= & {} \sum _\gamma \oint _A dz \frac{\langle \partial {\bar{\psi }}_\gamma (z)\psi _\beta (y) V_m(0)\rangle \langle \psi _\gamma (z){\bar{\psi }}_\alpha (w) V_m(0)\rangle }{\langle V_m(0)\rangle ^2}\nonumber \\= & {} -\sum _\gamma \oint _A dz \left( Y(w)^{-1}\Xi (z-w)Y(z)\right) _{\alpha \gamma } \partial _z\left( Y(z)^{-1}\Xi (y-z)Y(y)\right) _{\gamma \beta }\nonumber \\= & {} \sum _\gamma \oint _A dz \partial _z\left( Y(w)^{-1}\Xi (z-w)Y(z)\right) _{\alpha \gamma } \left( Y(z)^{-1}\Xi (y-z)Y(y)\right) _{\gamma \beta }\nonumber \\= & {} \oint _Adz\left( Y(w)^{-1}\left( \Xi (z-w)L(z)+ \partial _z\Xi (z-w)\right) \Xi (y-z)Y(y)\right) _{\alpha \beta }.\nonumber \\ \end{aligned}$$

(E.5)

Comparing (E.1) with (E.5) and using (3.12) we get the identity

$$\begin{aligned}&M(w)\Xi (y-w)-\Xi (y-w)M(y)+2\pi i\partial _\tau \Xi (y-w)\nonumber \\&\quad =\oint _A dz\left( \Xi (z-w)L(z)+\partial _z\Xi (z-w)\right) \Xi (y-z). \end{aligned}$$

(E.6)

We can plug in the explicit expression for L, M and see that the relation will hold iff the following relations are satisfied:

$$\begin{aligned}&2\pi i \partial _\tau x(Q,w-y)=p\partial _Qx(Q,w-y)-\partial _w\partial _Qx(Q,w-y)\nonumber \\&\quad =\oint _Adz\left[ px(Q,w-z)x(Q,z-y)-\partial _wx(Q,w-z) x(Q,z-y) \right] , \end{aligned}$$

(E.7)

$$\begin{aligned}&y(2Q,w)x(Q,y-w)-y(2Q,y)x(-Q,y-w)=\oint _Adz x(2Q,z)\nonumber \\&\qquad x(-Q,z-w)x(Q,y-z). \end{aligned}$$

(E.8)

To find the r.h.s. we need to compute two integrals:

$$\begin{aligned} I_1(w,y)= & {} \oint _A dz\ x(Q,w-z)x(Q,z-y),\nonumber \\ I_2(w,y)= & {} \oint _A dz\ x(q_1,z-w)x(q_2-q_1,z)x(q_2,y-z). \end{aligned}$$

(E.9)

$I_1$ and $\partial _w I_1$ contribute to diagonal elements, whereas $I_2$ defines off-diagonal ones.

First we consider the integral over the boundary of the cut torus:

$$\begin{aligned} I_1^\epsilon =\oint _{\partial {\mathbb {T}}}dz\ x(Q,w-z) x(Q+\epsilon ,z-y). \end{aligned}$$

(E.10)

The function inside the integral is not periodic under $z\rightarrow z+\tau$, but rather acquires a phase $e^{-2\pi i\epsilon }$. If we take the combination of two contour integrals over A-cycles shifted by $\tau$ in the opposite directions, they enter with opposite signs times the quasi-periodicity:

$$\begin{aligned} I_1^\epsilon =(1-e^{-2\pi i\epsilon })\oint _A dz\ x(Q,w-z) x(Q+\epsilon ,z-y). \end{aligned}$$

(E.11)

On the other hand, this integral can be computed by residues:

$$\begin{aligned} I_1^\epsilon =2\pi i\left[ x(Q+\epsilon ,w-y)-x(Q,w-y)\right] . \end{aligned}$$

(E.12)

Now we take a limit $\epsilon =0$:

$$\begin{aligned} I_1=2\pi i\lim _{\epsilon \rightarrow 0}\frac{x(Q+\epsilon ,w-y)-x(Q,w-y)}{1-e^{-2\pi i\epsilon }}=\partial _Q x(Q,w-y). \end{aligned}$$

(E.13)

By plugging this result in (E.7) we see that it is satisfied. The second integral can be computed in the same manner, noting that the integrand now has quasi-periodicity $e^{2\pi i\epsilon }$:

$$\begin{aligned} I_2^{\epsilon }= & {} \oint _{\partial {\mathbb {T}}}dz\ x(q_1,z-w)x(q_2-q_1+\epsilon ,z)x(q_2,y-z)\nonumber \\= & {} 2\pi i \left( x(q_1,y-w) x(q_2-q_1+\epsilon ,y) \right. \nonumber \\&\left. - x(q_1,-w)x(q_2,y) - x(q_2-q_1+\epsilon ,w)x(q_2,y-w)\right) . \end{aligned}$$

(E.14)

The expansion of this integral up to the first order yields $I_2$:

$$\begin{aligned} I_2(w,z)=y(2Q,w)x(Q,y-w)-y(2Q,y)x(-Q,y-w) \end{aligned}$$

(E.15)

because of which (E.8) is satisfied.

Other Determinantal Formulas

We report in this appendix some further identities that can be derived from the Fredholm determinant expression (4.10), that we recall here for convenience:

$$\begin{aligned} Z^D(\tau )=-N(a,m,a) q^{a^2+(\sigma +1/2)^2-1/12}e^{2\pi i\rho }\prod _{n=1}^\infty (1-q^n)^{-2\gamma ^2} \det \left( {\mathbb {I}}+K\right) . \end{aligned}$$

(F.1)

Notice that $\left. K\right| _{\rho \ \mapsto \rho +1/2}=-K$. Combining this observation with (3.51) and with periodicity properties of theta functions we find

$$\begin{aligned} Z_{1/2}^D(\tau )\eta (\tau )^{-1}\theta _3(2\rho |2\tau )= \frac{1}{2}N(a,m,a) q^{a^2-1/3}e^{2\pi i\rho } \left( \det ({\mathbb {I}}- K)-\det ({\mathbb {I}}+ K)\right) ,\nonumber \\ Z_{0}^D(\tau )\eta (\tau )^{-1}\theta _2(2\rho |2\tau )= \frac{1}{2}N(a,m,a) q^{a^2-1/3}e^{2\pi i\rho } \left( \det ({\mathbb {I}}- K)+\det ({\mathbb {I}}+ K)\right) , \nonumber \\ \end{aligned}$$

(F.2)

where we put $\gamma =0$ to simplify the formulas. For arbitrary $\gamma$ everything is the same.

Another option is to substitute $\rho =\frac{1}{4}$ and $\rho =\frac{1}{4}+\frac{\tau }{2}$ into (3.51) in order to cancel each of the two theta functions:

$$\begin{aligned} Z_{1/2}^D(\tau )\eta (\tau )^{-1}\theta _4(2\tau )= -i N(a,m,a) q^{a^2-1/3} \det \left( {\mathbb {I}}+\left. K\right| _{\rho =\frac{1}{4}}\right) ,\nonumber \\ Z_{0}^D(\tau )\eta (\tau )^{-1}\theta _4(2\tau )= N(a,m,a) q^{a^2+5/12} \det \left( {\mathbb {I}}+\left. K\right| _{\rho =\frac{1}{4}+\frac{\tau }{2}}\right) . \end{aligned}$$

(F.3)