M2-branes and ${\mathfrak {q}}$-Painlevé equations

Abstract

In this paper we investigate a novel connection between the effective theory of M2-branes on $({\mathbb {C}}^2/{\mathbb {Z}}_2\times {\mathbb {C}}^2/{\mathbb {Z}}_2)/{\mathbb {Z}}_k$ and the ${\mathfrak {q}}$-deformed Painlevé equations, by proposing that the grand canonical partition function of the corresponding four-nodes circular quiver ${\mathcal {N}}=4$ Chern–Simons matter theory solves the ${\mathfrak {q}}$-Painlevé VI equation. We analyse how this describes the moduli space of the topological string on local $\text {dP}_5$ and, via geometric engineering, five dimensional $N_f=4$ $\text {SU}(2)$ ${\mathcal {N}}=1$ gauge theory on a circle. The results we find extend the known relation between ABJM theory, ${\mathfrak {q}}$-Painlevé $\text {III}_3$, and topological strings on local ${{\mathbb {P}}}^1\times {{\mathbb {P}}}^1$. From the mathematical viewpoint the quiver Chern–Simons theory provides a conjectural Fredholm determinant realisation of the ${\mathfrak {q}}$-Painlevé VI $\tau$-function. We provide evidence for this proposal by analytic and numerical checks and discuss in detail the successive decoupling limits down to $N_f=0$, corresponding to ${\mathfrak {q}}$-Painlevé III$_3$.

Appendices

A Quantum dilogarithm and other special functions

In the following we assume $|{\mathfrak {q}}|\ne 1$. We recall the definition of ${\mathfrak {q}}-$numbers,

$$\begin{aligned}{}[u]=\frac{1-{\mathfrak {q}}^{u}}{1-{\mathfrak {q}}}, \end{aligned}$$

(A.1)

and the infinite multiple ${\mathfrak {q}}$-Pochhammer symbol

$$\begin{aligned}{} & {} (z;{\mathfrak {q}}_1,\dots ,{\mathfrak {q}}_k)_{\infty }\nonumber \\{} & {} \quad =\exp {\left( -\sum \limits _{p=1}^\infty \frac{z^p}{p}\frac{1}{1-{\mathfrak {q}}_1^p}\cdots \frac{1}{1-{\mathfrak {q}}_k^p} \right) } {\mathop {=}\limits ^{|{\mathfrak {q}}|<1}}\prod _{l_1,\dots ,l_k=0}^{\infty }\left( 1-z {\mathfrak {q}}_1^{l_1}\cdots {\mathfrak {q}}_k^{l_k}\right) .\qquad \quad \end{aligned}$$

(A.2)

This is defined for $|z|<1$, but can be analytically continued to $z\in {\mathbb {C}}$ using the latter equality. We extend the definition to an empty symbol $(z;)_\infty :=1-z$. For any $k\ge 1$ we then have the relations

$$\begin{aligned} \frac{( z;{\mathfrak {q}}_1,\dots ,{\mathfrak {q}}_k)_{\infty }}{({\mathfrak {q}}_1 z;{\mathfrak {q}}_1,\dots ,{\mathfrak {q}}_k)_{\infty }}= ( z;{\mathfrak {q}}_2,\dots ,{\mathfrak {q}}_k)_{\infty }. \end{aligned}$$

(A.3)

Let us introduce the ${\mathfrak {q}}$-Gamma and ${\mathfrak {q}}$-Barnes G functions

$$\begin{aligned} \Gamma _{{\mathfrak {q}}}(u)=\frac{({\mathfrak {q}};{\mathfrak {q}})_\infty }{({\mathfrak {q}}^u;{\mathfrak {q}})_\infty }(1-{\mathfrak {q}})^{u-1},\quad G_{{\mathfrak {q}}}(u)=\frac{({\mathfrak {q}}^u;{\mathfrak {q}},{\mathfrak {q}})_\infty }{({\mathfrak {q}};{\mathfrak {q}},{\mathfrak {q}})_\infty }({\mathfrak {q}};{\mathfrak {q}})_\infty ^{u-1}(1-{\mathfrak {q}})^{-\frac{(u-1)(u-2)}{2}},\nonumber \\ \end{aligned}$$

(A.4)

where $|{\mathfrak {q}}|<1$, which satisfy the ${\mathfrak {q}}$-analogues of the usual properties of Gamma and Barnes G functions,

$$\begin{aligned} \Gamma _{{\mathfrak {q}}}(u+1)=[u]\Gamma _{{\mathfrak {q}}}(u),\quad G_{{\mathfrak {q}}}(u+1)=\Gamma _{{\mathfrak {q}}}(u) G_{{\mathfrak {q}}}(u) \end{aligned}$$

(A.5)

and are both equal to one at $u=1$ and log-convex [80]. These can also be easily analytically continued to $|{\mathfrak {q}}|>1$ using

$$\begin{aligned} \Gamma _{{\mathfrak {q}}}(u)={\mathfrak {q}}^{\frac{(u-1)(u-2)}{2}}\Gamma _{{\mathfrak {q}}^{-1}}(u), \quad G_{{\mathfrak {q}}}(u) = {\mathfrak {q}}^{\frac{(u-1)(u-2)(u-3)}{6}}G_{{\mathfrak {q}}^{-1}}(u). \end{aligned}$$

(A.6)

Let us finally introduce the quantum dilogarithm function $\Phi _b\left( z\right)$ as [39]

$$\begin{aligned} \Phi _b\left( z\right) =\frac{ \left( e^{2\pi b\left( z+\frac{i}{2}\left( b+\frac{1}{b}\right) \right) };e^{2\pi ib^2}\right) _\infty }{ \left( e^{\frac{2\pi }{b}\left( z-\frac{i}{2}\left( b+\frac{1}{b}\right) \right) };e^{-\frac{2\pi i}{b^2}}\right) _\infty }. \end{aligned}$$

(A.7)

Note that $\Phi _b\left( z\right)$ satisfies the following recursive relations

$$\begin{aligned} \frac{\Phi _b\left( z+ib\right) }{\Phi _b\left( z\right) }=\frac{1}{1+e^{\pi ib^2}e^{2\pi bz}},\quad \frac{\Phi _b\left( z+\frac{i}{b}\right) }{\Phi _b\left( z\right) }=\frac{1}{1+e^{\frac{\pi i}{b^2}}e^{\frac{2\pi z}{b}}}, \end{aligned}$$

(A.8)

and that its asymptotic behavior is

$$\begin{aligned} \Phi _{b}\left( z\right) \sim {\left\{ \begin{array}{ll} \exp \left( i\pi z^{2}+\frac{\pi i}{12}\left( b^{2}+b^{-2}\right) \right) &{} \left( \textrm{Re}\left[ z\right] \rightarrow \infty \right) \\ 1 &{} \left( \textrm{Re}\left[ z\right] \rightarrow -\infty \right) \end{array}\right. }. \end{aligned}$$

(A.9)

B Fermi gas formalism

In this appendix, we address the study of the matrix model (3.4) in the Fermi gas formalism. First of all, we introduce our notations for one-dimensional quantum mechanics. We denote the canonical position operators as ${{\widehat{x}}}$ and define the canonical momentum operator ${{\widehat{p}}}$ normalized as $[{{\widehat{x}}},{{\widehat{p}}}]=i\hbar$ with $\hbar =2\pi k$. We denote a position eigenstate as $|x\rangle$ and a momentum eigenstate as $|p\rangle \!\rangle$, which are normalized as

$$\begin{aligned}&{{\widehat{x}}}|x\rangle =x|x\rangle ,\quad \langle x|y\rangle =2\pi \delta \left( x-y\right) , \end{aligned}$$

(B.1)

$$\begin{aligned}&{{\widehat{p}}}|p\rangle \!\rangle =p|p\rangle \!\rangle ,\quad \langle \!\langle p|p'\rangle \!\rangle =2\pi \delta \left( p-p'\right) ,\quad \langle x|p\rangle \!\rangle =\frac{1}{\sqrt{k}}e^{\frac{ixp}{2\pi k}}. \end{aligned}$$

(B.2)

For later purposes, we also introduce a symbol $t_{\zeta ,n,r}$ defined as

$$\begin{aligned} t_{\zeta ,n,r}=2\pi \zeta +2\pi i\left( \frac{n+1}{2}-r\right) . \end{aligned}$$

(B.3)

Our goal is to derive the expressions for the matrix model (3.5) and ${{\widehat{\rho }}}_k^{-1}$ (3.11). Our derivation extends the one in [23] to the present case with rank deformations and generic Fayet-Iliopoulos parameters. In the following we assume that the FI parameters $\zeta _{i}$ are real, we use the parametrization

$$\begin{aligned} N_{1}=N+M_{1},\quad N_{2}=N+M,\quad N_{3}=N+M_{2},\quad N_{4}=N, \end{aligned}$$

(B.4)

and assume that N, $M_{1}$, $M_{2}$ and M are non-negative integers as in the main text. We also assume $M_{1}\ge M$ and $M_{2}\ge M$.

The integrand of the matrix model, after shifting all of the integration variables as $\mu \rightarrow \frac{\mu }{k}$, can be divided into two parts as

$$\begin{aligned}{} & {} Z_k\left( N;M_1,M_2,M,\zeta _1,\zeta _2\right) \nonumber \\{} & {} \quad =\frac{1}{N_{2}!N_{4}!}\int _{-\infty }^\infty \prod _{n=1}^{N_{4}}\frac{\textrm{d}\mu _{n}}{2\pi k}\prod _{n=1}^{N_{2}}\frac{\textrm{d}\nu _{n}}{2\pi k}Y_{N_{4},N_{1},N_{2}}\left( 0,\zeta _{1};\mu ,\nu \right) \nonumber \\{} & {} \qquad \left( Y_{N_{4},N_{3},N_{2}}\left( \zeta _{2},0;\mu ,\nu \right) \right) ^{*}, \end{aligned}$$

(B.5)

where

$$\begin{aligned}&Y_{N_{4},\tilde{N},N_{2}}\left( \zeta ,\zeta ';\mu ,\nu \right) \nonumber \\&\quad =\frac{i^{-\frac{\tilde{N}^{2}}{2}}}{\tilde{N}!}\int _{-\infty }^\infty \prod _{n=1}^{\tilde{N}}\frac{\textrm{d}\lambda _{n}}{2\pi k}e^{\frac{i}{4\pi k}\sum _{n=1}^{\tilde{N}}\lambda _{n}^{2}}e^{-\frac{i\zeta }{k}\left( \sum _{n=1}^{N_{4}}\mu _{n}-\sum _{n=1}^{\tilde{N}}\lambda _{n}\right) -\frac{i\zeta '}{k}\left( \sum _{n=1}^{\tilde{N}}\lambda _{n}-\sum _{n=1}^{N_{2}}\nu _{n}\right) }\nonumber \\&\qquad \times \frac{\prod _{m<m'}^{N_{4}}2\sinh \frac{\mu _{m}-\mu _{m'}}{2k}\prod _{n<n'}^{\tilde{N}}2\sinh \frac{\lambda _{n}-\lambda _{n'}}{2k}}{\prod _{m=1}^{N_{4}}\prod _{n=1}^{\tilde{N}}2\cosh \frac{\mu _{m}-\lambda _{n}}{2k}}\frac{\prod _{m<m'}^{\tilde{N}}2\sinh \frac{\lambda _{m}-\lambda _{m'}}{2k}\prod _{n<n'}^{N_{2}}2\sinh \frac{\nu _{n}-\nu _{n'}}{2k}}{\prod _{m=1}^{\tilde{N}}\prod _{n=1}^{N_{2}}2\cosh \frac{\lambda _{m}-\nu _{n}}{2k}}. \end{aligned}$$

(B.6)

We first focus on $Y_{N_{4},\tilde{N},N_{2}}$ and rewrite it in the operator formalism. By combining the Cauchy determinant formula and the Vandermonde determinant formula [81]

$$\begin{aligned}&\frac{\prod _{m<m'}^{K}2\sinh \frac{\mu _{m}-\mu _{m'}}{2k}\prod _{n<n'}^{K+L}2\sinh \frac{\lambda _{n}-\lambda _{n'}}{2k}}{\prod _{m=1}^{K}\prod _{n=1}^{K+L}2\cosh \frac{\mu _{m}-\lambda _{n}}{2k}} =\det \left( \begin{array}{c} \left[ \left( -1\right) ^{L}\frac{e^{\frac{L}{2k}\left( \mu _{m}-\lambda _{n}\right) }}{2\cosh \frac{\mu _{m}-\lambda _{n}}{2k}}\right] _{m,n}^{K\times \left( K+L\right) }\\ \left[ e^{\frac{1}{k}\left( \frac{L+1}{2}-r\right) \lambda _{n}}\right] _{r,n}^{L\times \left( K+L\right) } \end{array}\right) , \end{aligned}$$

(B.7)

$$\begin{aligned}&\frac{\prod _{m<m'}^{K+L}2\sinh \frac{\lambda _{m}-\lambda _{m'}}{2k}\prod _{n<n'}^{K}2\sinh \frac{\nu _{n}-\nu _{n'}}{2k}}{\prod _{m=1}^{K+L}\prod _{n=1}^{K}2\cosh \frac{\lambda _{m}-\nu _{n}}{2k}}\nonumber \\&=\det \left( \begin{array}{cc} \left[ \left( -1\right) ^{L}\frac{e^{-\frac{L}{2k}\left( \lambda _{m}-\nu _{n}\right) }}{2\cosh \frac{\lambda _{m}-\nu _{n}}{2k}}\right] _{m,n}^{\left( K+L\right) \times K}&\left[ e^{\frac{1}{k}\left( \frac{L+1}{2}-r\right) \lambda _{m}}\right] _{m,r}^{\left( K+L\right) \times L}\end{array}\right) , \end{aligned}$$

(B.8)

we can rewrite the third line of (B.6) as the determinant of the product of two matrices. The notation for the operator formalism is in (B.1),(B.2). The elements of the matrices can be rewritten in the operator formalism. We will make use of the following identities

(B.9)

(B.10)

where in the second line, as we shifted the integration contour from ${\mathbb {R}}$ to ${\mathbb {R}}+i\pi L$, we obtain a summation over the resulting residues. Fortunately, the contribution to the determinant of the latter vanishes being a linear combination of rows. Indeed it is evident from (B.7),(B.8) that the sum of the residues is a linear combination of the lower (or right) elements.

The third line of $Y_{N_{4},\tilde{N},N_{2}}$ is now written in the operator formalism. To write all of factors in the operator formalism, we multiply the first matrix by a Fresnel factor $e^{\frac{i}{4\pi k}\sum _{n=1}^{\bar{N}}\lambda _n^2}$. We also include the first FI factor, depending on $\zeta$, in the first matrix and the second FI factor, depending on $\zeta '$, in the second matrix. After performing the similarity transformations

(B.11)

we obtain

(B.12)

where $\tilde{M}=N_4-\tilde{N}$ and $t_{\zeta ,n,r}$ is defined in (B.3).

We now return to the matrix model (B.5). Upon the similarity transformation

(B.13)

(B.14)

(B.15)

and by using the formulae

(B.16)

we find that the matrix models can be written as

$$\begin{aligned}&Z_{k}\left( N;M_{1},M_{2},M,\zeta _{1},\zeta _{2}\right) \nonumber \\&\quad =\frac{1}{N_{2}!N_{4}!}e^{i\Theta _{k}\left( M_{1},M_{2},M,\zeta _{1},\zeta _{2}\right) }\int _{-\infty }^\infty \prod _{n=1}^{N_{4}}\frac{\textrm{d}\mu _{n}}{2\pi k}\prod _{n=1}^{N_{2}}\frac{\textrm{d}\nu _{n}}{2\pi k}\tilde{Y}_{N_{4},N_{1},N_{2}}\left( 0,\zeta _{1};\mu ,\nu \right) \nonumber \\&\qquad \left( \tilde{Y}_{N_{4},N_{3},N_{2}}\left( \zeta _{2},0;\mu ,\nu \right) \right) ^{*}, \end{aligned}$$

(B.17)

where

(B.18)

and

$$\begin{aligned}&\Theta _{k}\left( M_{1},M_{2},M,\zeta _{1},\zeta _{2}\right) \nonumber \\&\quad =\theta _{k}\left( M_{1},0\right) +\theta _{k}\left( M_{1}-M,\zeta _{1}\right) -\theta _{k}\left( M_{2},\zeta _{2}\right) -\theta _{k}\left( M_{2}-M,0\right) , \end{aligned}$$

(B.19)

$$\begin{aligned}&\theta _{k}\left( M,\zeta \right) =\frac{\pi }{k}\left[ \frac{1}{12}\left( M^{3}-M\right) -M\zeta ^{2}\right] . \end{aligned}$$

(B.20)

$\tilde{Y}_{N_{4},\tilde{N},N_{2}}$ can be computed as follows. Since the second determinant is an anti-symmetric function of $\lambda _{m}$, we can simplify the first determinant by using

$$\begin{aligned}&\frac{1}{N!}\int _{-\infty }^\infty d^{N}\lambda \det \left( \left[ g_{m}\left( \lambda _{n}\right) \right] _{m,n}^{N\times N}\right) f\left( \lambda _{1},\lambda _{2},\ldots ,\lambda _{N}\right) \nonumber \\&\quad =\int _{-\infty }^\infty d^{N}\lambda \prod _{n}^{N}g_{n}\left( \lambda _{n}\right) f\left( \lambda _{1},\lambda _{2},\ldots ,\lambda _{N}\right) , \end{aligned}$$

(B.21)

which holds for any anti-symmetric function $f\left( \varvec{\lambda }\right)$. We decompose the other determinant by using (B.10),(B.9) and (B.7),(B.8) backwards. Now we can perform the integration by using the delta functions coming form the inner products of the position operators. After a short computation, we obtain

$$\begin{aligned} \tilde{Y}_{N_{4},\tilde{N},N_{2}}\left( \zeta ,\zeta ';\mu ,\nu \right)&= i^{-\frac{\tilde{N}^{2}}{2}+\frac{\tilde{M}^{2}}{2}}e^{\frac{2\pi i\zeta \zeta '\tilde{M}}{k}}e^{-\frac{i\zeta '}{k}\left( \sum _{n=1}^{N_{4}}\mu _{n}-\sum _{n=1}^{N_{2}}\nu _{n}\right) }Z_{k}^{\left( \text {CS}\right) }\left( \tilde{M}\right) \nonumber \\&\times \prod _{n=1}^{N_{4}}\frac{\prod _{r=1}^{\tilde{M}}2\sinh \frac{\mu _{n}+t_{\zeta ,\tilde{M},r}}{2k}}{2\cosh \frac{\mu _{n}+2\pi \zeta -i\pi \tilde{M}}{2}}\prod _{n=1}^{N_{2}}\frac{1}{\prod _{r=1}^{\tilde{M}}2\cosh \frac{\nu _{n}+t_{\zeta ,\tilde{M},r}}{2k}}\nonumber \\&\times \frac{\prod _{m<m'}^{N_{4}}2\sinh \frac{\mu _{m}-\mu _{m'}}{2k}\prod _{n<n'}^{N_{2}}2\sinh \frac{\nu _{n}-\nu _{n'}}{2k}}{\prod _{m=1}^{N_{4}}\prod _{n=1}^{N_{2}}2\cosh \frac{\mu _{m}-\nu _{n}}{2k}}, \end{aligned}$$

(B.22)

where

$$\begin{aligned} Z_{k}^{\left( \text {CS}\right) }\left( L\right)&=\frac{1}{k^{\frac{L}{2}}}\prod _{j<j'}^{L}2\sin \frac{\pi }{k}\left( j'-j\right) , \end{aligned}$$

(B.23)

is the partition function of $\textrm{U}\left( L\right) _{k}$ pure Chern–Simons theory. We again use the determinant formula (B.7),(B.8) and the operator formula (B.10),(B.9) for the factor at the third line, and we include the FI factors and the factors in the second line into the matrix. As a result, we obtain

(B.24)

where

$$\begin{aligned} S_{L}\left( x\right) =i^{L}\frac{\prod _{r=1}^{L}2\sinh \frac{x-2\pi i\left( \frac{L+1}{2}-r\right) }{2k}}{2\cosh \frac{x+i\pi L}{2}},\quad C_{L}\left( x\right) =\frac{1}{\prod _{r=1}^{L}2\cosh \frac{x-2\pi i\left( \frac{L+1}{2}-r\right) }{2k}}. \end{aligned}$$

(B.25)

By using the recursive formula for the quantum dilogarithm functions (A.8), (B.25) can be written in terms of the quantum dilogarithm as ($b=\sqrt{k}$)

$$\begin{aligned} S_{L}\left( x\right) =e^{\frac{k-L}{2k}x}\frac{\Phi _{ b}\left( \frac{x}{2\pi b}-\frac{iL}{2b}+\frac{i}{2} b\right) }{\Phi _{ b}\left( \frac{x}{2\pi b}+\frac{iL}{2b}-\frac{i}{2} b\right) },\quad C_{L}\left( x\right) =e^{\frac{L}{2k}x}\frac{\Phi _{ b}\left( \frac{x}{2\pi b}+\frac{iL}{2 b}\right) }{\Phi _{ b}\left( \frac{x}{2\pi b}-\frac{iL}{2 b}\right) }. \end{aligned}$$

(B.26)

By substituting (B.24) into (B.17), we finally arrive at

(B.27)

where

$$\begin{aligned} \widehat{D}_{1}^{\text {VI}}&=e^{-\frac{i\zeta _{1}}{k}\widehat{x}}S_{M_1}\left( {{\widehat{x}}}\right) \frac{1}{2\cosh \frac{\widehat{p}-i\pi M}{2}}e^{\frac{i\zeta _{1}}{k}\widehat{x}}C_{M_1}\left( {{\widehat{x}}}\right) , \end{aligned}$$

(B.28)

$$\begin{aligned} \widehat{d}_{1}^{\text {VI}}&=e^{\frac{i\zeta _{1}}{k}\widehat{x}}C_{M_1}\left( {{\widehat{x}}}\right) ,\end{aligned}$$

(B.29)

$$\begin{aligned} \widehat{D}_{2}^{\text {VI}}&=C_{M_2}\left( {{\widehat{x}}}+2\pi \zeta _2\right) \frac{1}{2\cosh \frac{\widehat{p}+\pi iM}{2}}S_{M_2}\left( {{\widehat{x}}}+2\pi \zeta _2\right) ,\end{aligned}$$

(B.30)

$$\begin{aligned} \widehat{d}_{2}^{\text {VI}}&=C_{M_2}\left( {\widehat{x}}+2\pi \zeta _2\right) . \end{aligned}$$

(B.31)

Let us perform a short digression on the formulas which can be used to rephrase our final result (B.27) in a more concise way, though we do not use them in the main text. Note that by using the formula

$$\begin{aligned}{} & {} \frac{1}{N!}\int _{-\infty }^\infty d^{N}\nu \det \left( \left[ f_{m}\left( \nu _{n}\right) \right] _{m,n}^{N\times N}\right) \det \left( \left[ g_{n}\left( \nu _{m}\right) \right] _{m,n}^{N\times N}\right) \nonumber \\ {}{} & {} \quad =\det \left( \left[ \int _{-\infty }^\infty \textrm{d}\nu f_{m}\left( \nu \right) g_{n}\left( \nu \right) \right] _{m,n}^{N\times N}\right) , \end{aligned}$$

(B.32)

the partition function for $M>0$ (B.27), which is written as a $\left( 2N+M\right)$ dimensional integral, can be further reduced to a N dimensional integral:

(B.33)

which implies that the grand partition function (3.2) can be written as [81]

(B.34)

1.1 B.1 $M=0$ case

When $M=0$, the matrix model (B.33) simplifies to

(B.35)

where

$$\begin{aligned}&\widehat{\rho }_{k}\left( M_{1},M_{2},0,\zeta _{1},\zeta _{2}\right) \nonumber \\&\quad =\left. {{\widehat{D}}}_1^{\text {VI}} {{\widehat{D}}}_2^{\text {VI}} \right| _{M=0}\nonumber \\&\quad = S_{M_{1}}\left( \widehat{x}\right) \frac{1}{2\cosh \frac{\widehat{p}+2\pi \zeta _{1}}{2}}C_{M_{1}}\left( \widehat{x}\right) C_{M_{2}}\left( \widehat{x}+2\pi \zeta _{2}\right) \frac{1}{2\cosh \frac{\widehat{p}}{2}}S_{M_{2}}\left( \widehat{x}+2\pi \zeta _{2}\right) . \end{aligned}$$

(B.36)

This is the same as (3.7). For this expression, we can relate the density matrix to the quantum curve [39, 82]. The important relations are

$$\begin{aligned}&C_{L}^{-1}\left( \widehat{x}\right) e^{\pm \frac{1}{2}\widehat{p}}S_{L}^{-1}\left( \widehat{x}\right) \nonumber \\&\quad =e^{\mp \frac{1}{2}i\pi L}e^{\pm \frac{1}{2}\widehat{p}}\frac{\Phi _{ b}\left( \frac{\widehat{x}}{2\pi b}-\frac{iL}{2 b}\pm \frac{i}{2} b\right) }{\Phi _{ b}\left( \frac{\widehat{x}}{2\pi b}+\frac{iL}{2 b}\pm \frac{i}{2} b\right) }\frac{\Phi _{ b}\left( \frac{\widehat{x}}{2\pi b}+\frac{iL}{2b}-\frac{i}{2} b\right) }{\Phi _{ b}\left( \frac{\widehat{x}}{2\pi b}-\frac{iL}{2b}+\frac{i}{2} b\right) }e^{-\frac{1}{2}\widehat{x}}\nonumber \\&\quad =e^{\pm \frac{1}{2}\widehat{p}}\left( e^{\pm \frac{1}{2}i\pi L}e^{\frac{1}{2}\widehat{x}}+e^{\mp \frac{1}{2}i\pi L}e^{-\frac{1}{2}\widehat{x}}\right) , \end{aligned}$$

(B.37)

$$\begin{aligned}&S_{L}^{-1}\left( \widehat{x}\right) e^{\pm \frac{1}{2}\widehat{p}}C_{L}^{-1}\left( \widehat{x}\right) \nonumber \\&\quad =e^{\pm \frac{1}{2}i\pi L}e^{-\frac{1}{2}\widehat{x}}\frac{\Phi _{ b}\left( \frac{\widehat{x}}{2\pi b}+\frac{iL}{2 b}-\frac{i}{2} b\right) }{\Phi _{ b}\left( \frac{\widehat{x}}{2\pi b}-\frac{iL}{2 b}+\frac{i}{2} b\right) }\frac{\Phi _{ b}\left( \frac{\widehat{x}}{2\pi b}-\frac{iL}{2 b}\mp \frac{i}{2} b\right) }{\Phi _{ b}\left( \frac{\widehat{x}}{2\pi b}+\frac{iL}{2 b}\mp \frac{i}{2} b\right) }e^{\pm \frac{1}{2}\widehat{p}}\nonumber \\&\quad =\left( e^{\mp \frac{1}{2}i\pi L}e^{\frac{1}{2}\widehat{x}}+e^{\pm \frac{1}{2}i\pi L}e^{-\frac{1}{2}\widehat{x}}\right) e^{\pm \frac{1}{2}\widehat{p}}, \end{aligned}$$

(B.38)

where we used the Baker–Campbell–Hausdorff formula $e^{\alpha \widehat{x}}e^{\beta \widehat{p}}=e^{2\pi i\alpha \beta k}e^{\beta \widehat{p}}e^{\alpha \widehat{x}}$ and (A.8). By using these relations, we obtain

$$\begin{aligned}&S_{L}\left( \widehat{x}\right) \frac{1}{2\cosh \frac{\widehat{p}}{2}}C_{L}\left( \widehat{x}\right) \nonumber \\&\quad =\left[ e^{\frac{1}{2}\widehat{p}}\left( e^{\frac{1}{2}i\pi L}e^{\frac{1}{2}\widehat{x}}+e^{-\frac{1}{2}i\pi L}e^{-\frac{1}{2}\widehat{x}}\right) +e^{-\frac{1}{2}\widehat{p}}\left( e^{-\frac{1}{2}i\pi L}e^{\frac{1}{2}\widehat{x}}+e^{\frac{1}{2}i\pi L}e^{-\frac{1}{2}\widehat{x}}\right) \right] ^{-1}, \end{aligned}$$

(B.39)

$$\begin{aligned}&C_{L}\left( \widehat{x}\right) \frac{1}{2\cosh \frac{\widehat{p}}{2}}S_{L}\left( \widehat{x}\right) \nonumber \\&\quad =\left[ \left( e^{-\frac{1}{2}i\pi L}e^{\frac{1}{2}\widehat{x}}+e^{\frac{1}{2}i\pi L}e^{-\frac{1}{2}\widehat{x}}\right) e^{\frac{1}{2}\widehat{p}}+\left( e^{\frac{1}{2}i\pi L}e^{\frac{1}{2}\widehat{x}}+e^{-\frac{1}{2}i\pi L}e^{-\frac{1}{2}\widehat{x}}\right) e^{-\frac{1}{2}\widehat{p}}\right] ^{-1}. \end{aligned}$$

(B.40)

The inverse of the density matrix is the product of the above two quantum curves. Therefore, we finally obtain

$$\begin{aligned}&\widehat{\rho }_{k}^{-1}\left( M_{1},M_{2},0,\zeta _{1},\zeta _{2}\right) \nonumber \\&\quad =\left[ \left( e^{-\frac{1}{2}i\pi M_2+\pi \zeta _2}e^{\frac{1}{2}\widehat{x}}+e^{\frac{1}{2}i\pi M_2-\pi \zeta _2}e^{-\frac{1}{2}\widehat{x}}\right) e^{\frac{1}{2}\widehat{p}}\right. \nonumber \\&\left. \qquad +\left( e^{\frac{1}{2}i\pi M_2+\pi \zeta _2}e^{\frac{1}{2}\widehat{x}}+e^{-\frac{1}{2}i\pi M_2-\pi \zeta _2}e^{-\frac{1}{2}\widehat{x}}\right) e^{-\frac{1}{2}\widehat{p}}\right] \nonumber \\&\qquad \times \left[ e^{\pi \zeta _1}e^{\frac{1}{2}\widehat{p}}\left( e^{\frac{1}{2}i\pi M_1}e^{\frac{1}{2}\widehat{x}}+e^{-\frac{1}{2}i\pi M_1}e^{-\frac{1}{2}\widehat{x}}\right) +e^{-\pi \zeta _1}e^{-\frac{1}{2}\widehat{p}}\left( e^{-\frac{1}{2}i\pi M_1}e^{\frac{1}{2}\widehat{x}}+e^{\frac{1}{2}i\pi M_1}e^{-\frac{1}{2}\widehat{x}}\right) \right] \nonumber \\&\quad =e^{\frac{\pi i(M_{1}-M_{2})}{2}+\pi (\zeta _{1}+\zeta _{2})}e^{{\widehat{x}}+{\widehat{p}}}+[e^{\frac{\pi i(-M_{1}-M_{2})}{2}+\pi (\zeta _{1}+\zeta _{2})+\pi ik}+e^{\frac{\pi i(M_{1}+M_{2})}{2}+\pi (\zeta _{1}-\zeta _{2})-\pi ik}]e^{{\widehat{p}}}\nonumber \\&\qquad +e^{\frac{\pi i(-M_{1}+M_{2})}{2}+\pi (\zeta _{1}-\zeta _{2})}e^{-{\widehat{x}}+{\widehat{p}}}\nonumber \\&\qquad +[e^{\frac{\pi i(-M_{1}-M_{2})}{2}+\pi (-\zeta _{1}+\zeta _{2})}+e^{\frac{\pi i(M_{1}+M_{2})}{2}+\pi (\zeta _{1}+\zeta _{2})}]e^{{\widehat{x}}}\nonumber \\&\qquad +e^{\frac{\pi i(-M_{1}+M_{2})}{2}+\pi (-\zeta _{1}-\zeta _{2})}+e^{\frac{\pi i(-M_{1}+M_{2})}{2}+\pi (\zeta _{1}+\zeta _{2})}\nonumber \\&\qquad +e^{\frac{\pi i(M_{1}-M_{2})}{2}+\pi (-\zeta _{1}+\zeta _{2})}+e^{\frac{\pi i(M_{1}-M_{2})}{2}+\pi (\zeta _{1}-\zeta _{2})}\nonumber \\&\qquad +[e^{\frac{\pi i(-M_{1}-M_{2})}{2}+\pi (\zeta _{1}-\zeta _{2})}+e^{\frac{\pi i(M_{1}+M_{2})}{2}+\pi (-\zeta _{1}-\zeta _{2})}]e^{-{\widehat{x}}}\nonumber \\&\qquad +e^{\frac{\pi i(-M_{1}+M_{2})}{2}+\pi (-\zeta _{1}+\zeta _{2})}e^{{\widehat{x}}-{\widehat{p}}}+[e^{\frac{\pi i(-M_{1}-M_{2})}{2}+\pi (-\zeta _{1}-\zeta _{2})+\pi ik}\nonumber \\&\qquad +e^{\frac{\pi i(M_{1}+M_{2})}{2}+\pi (-\zeta _{1}+\zeta _{2})-\pi ik}]e^{-{\widehat{p}}}\nonumber \\&\qquad +e^{\frac{\pi i(M_{1}-M_{2})}{2}+\pi (-\zeta _{1}-\zeta _{2})}e^{-{\widehat{x}}-{\widehat{p}}}. \end{aligned}$$

(B.41)

This is the quantum curve associated to the (2,2) model for $M=0$.

C Proof of (4.7): ${{\widehat{\rho }}}_k\left( M_1,M_2,0,\zeta _1,\zeta _2\right) \sim {{\widehat{\rho }}}_k\left( M_1,M_2,0,\zeta _2,\zeta _1\right)$

First we notice the following identity

$$\begin{aligned}&e^{\frac{M}{2k}{{\widehat{x}}}} \frac{\Phi _b\left( \frac{{\widehat{x}}}{2\pi b}+\frac{iM}{2b}\right) }{\Phi _b\left( \frac{{\widehat{x}}}{2\pi b}-\frac{iM}{2b}\right) } \frac{1}{2\cosh \frac{{\widehat{p}}}{2}} e^{\left( \frac{1}{2}-\frac{M}{2k}\right) {{\widehat{x}}}} \frac{\Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}-\frac{iM}{2b}+\frac{ib}{2}\right) }{\Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}+\frac{iM}{2b}-\frac{ib}{2}\right) }\nonumber \\&\quad = e^{\left( \frac{1}{2}-\frac{M}{2k}\right) {{\widehat{p}}}} \frac{\Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}-\frac{iM}{2b}+\frac{ib}{2}\right) }{\Phi _b\left( \frac{{\widehat{p}}}{2\pi b}+\frac{iM}{2b}-\frac{ib}{2}\right) } \frac{1}{2\cosh \frac{{{\widehat{x}}}}{2}} e^{\frac{M}{2k}{{\widehat{p}}}} \frac{\Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}+\frac{iM}{2b}\right) }{\Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}-\frac{iM}{2b}\right) }. \end{aligned}$$

(C.1)

This identity can be proved by calculating the inverse of both sides. The inverse of the left-hand side can be calculated as

$$\begin{aligned}&\left( e^{\frac{M}{2k}{{\widehat{x}}}} \frac{\Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}+\frac{iM}{2b}\right) }{\Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}-\frac{iM}{2b}\right) } \frac{1}{2\cosh \frac{{{\widehat{p}}}}{2}} e^{\left( \frac{1}{2}-\frac{M}{2k}\right) {{\widehat{x}}}} \frac{\Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}-\frac{iM}{2b}+\frac{ib}{2}\right) }{\Phi _b\left( \frac{{\widehat{x}}}{2\pi b}+\frac{iM}{2b}-\frac{ib}{2}\right) } \right) ^{-1}\nonumber \\&\quad = \frac{\Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}+\frac{iM}{2b}-\frac{ib}{2}\right) }{\Phi _b\left( \frac{{\widehat{x}}}{2\pi b}-\frac{iM}{2b}+\frac{ib}{2}\right) } e^{\left( -\frac{1}{2}+\frac{M}{2k}\right) {{\widehat{x}}}} \left( e^{\frac{{{\widehat{p}}}}{2}}+e^{-\frac{{{\widehat{p}}}}{2}}\right) \frac{\Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}-\frac{iM}{2b}\right) }{\Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}+\frac{iM}{2b}\right) } e^{-\frac{M}{2k}{{\widehat{x}}}}. \end{aligned}$$

(C.2)

By moving $e^{\pm \frac{{{\widehat{p}}}}{2}}$ to the right by using the formula $e^{\pm \frac{{{\widehat{p}}}}{2}}f\left( {\widehat{x}}\right) e^{\mp \frac{{{\widehat{p}}}}{2}}=f\left( {{\widehat{x}}}\mp \pi ik\right)$, we are left with ratios of $\Phi _b$ which can be simplified by the recursive formula (A.8). We finally obtain

$$\begin{aligned}&\left( e^{\frac{M}{2k}{{\widehat{x}}}} \frac{\Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}+\frac{iM}{2b}\right) }{\Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}-\frac{iM}{2b}\right) } \frac{1}{2\cosh \frac{{{\widehat{p}}}}{2}} e^{\left( \frac{1}{2}-\frac{M}{2k}\right) {{\widehat{x}}}} \frac{\Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}-\frac{iM}{2b}+\frac{ib}{2}\right) }{\Phi _b\left( \frac{{\widehat{x}}}{2\pi b}+\frac{iM}{2b}-\frac{ib}{2}\right) } \right) ^{-1}\nonumber \\&\quad = \left( e^{-\frac{\pi iM}{2}}e^{\frac{{\widehat{x}}}{2}}+e^{\frac{\pi iM}{2}}e^{-\frac{{\widehat{x}}}{2}}\right) e^{\frac{{{\widehat{p}}}}{2}} +\left( e^{\frac{\pi iM}{2}}e^{\frac{{{\widehat{x}}}}{2}}+e^{-\frac{\pi iM}{2}}e^{-\frac{{{\widehat{x}}}}{2}}\right) e^{-\frac{{\widehat{p}}}{2}}. \end{aligned}$$

(C.3)

In the same way the inverse of the right-hand side of (C.1) can be simplified as

$$\begin{aligned}&\left( e^{\left( \frac{1}{2}-\frac{M}{2k}\right) {{\widehat{p}}}} \frac{\Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}-\frac{iM}{2b}+\frac{ib}{2}\right) }{\Phi _b\left( \frac{{\widehat{p}}}{2\pi b}+\frac{iM}{2b}-\frac{ib}{2}\right) } \frac{1}{2\cosh \frac{{{\widehat{x}}}}{2}} e^{\frac{M}{2k}{{\widehat{p}}}} \frac{\Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}+\frac{iM}{2b}\right) }{\Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}-\frac{iM}{2b}\right) } \right) ^{-1}\nonumber \\&\quad = e^{\frac{{{\widehat{x}}}}{2}}\left( e^{-\frac{\pi iM}{2}}e^{\frac{{{\widehat{p}}}}{2}}+e^{\frac{\pi iM}{2}}e^{-\frac{{{\widehat{p}}}}{2}}\right) +e^{-\frac{{\widehat{x}}}{2}}\left( e^{\frac{\pi iM}{2}}e^{\frac{{\widehat{p}}}{2}}+e^{-\frac{\pi iM}{2}}e^{-\frac{{{\widehat{p}}}}{2}}\right) . \end{aligned}$$

(C.4)

This is identical to the inverse (C.3) of the left-hand side of (C.1).

By using the identity (C.1) we can show (4.7), namely ${{\widehat{\rho }}}_k\left( M_1,M_2,0,\zeta _1,\zeta _2\right) \sim {{\widehat{\rho }}}_k\left( M_1,M_2,0,\zeta _2,\zeta _1\right)$, as follows. First we reorder the terms in ${{\widehat{\rho }}}$ (4.1) cyclically

$$\begin{aligned}&{{\widehat{\rho }}}_k\left( M_1,M_2,0,\zeta _1,\zeta _2\right) \nonumber \\&\quad \sim e^{\pi \zeta _2} e^{\left( -\frac{i\zeta _1}{k}+1-\frac{M_1+M_2}{2k}\right) {\widehat{x}}} e^{\frac{i\zeta _2}{k}{{\widehat{p}}}} \frac{ \Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}-\frac{iM_2}{2b}+\frac{ib}{2}\right) }{ \Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}+\frac{iM_2}{2b}-\frac{ib}{2}\right) } \frac{1}{2\cosh \frac{{{\widehat{p}}}}{2}} \frac{ \Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}+\frac{iM_2}{2b}\right) }{ \Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}-\frac{iM_2}{2b}\right) }\nonumber \\&\qquad \times e^{-\frac{i\zeta _2}{k}{{\widehat{p}}}} e^{\left( \frac{i\zeta _1}{k}+\frac{M_1+M_2}{2k}\right) {{\widehat{x}}}} \frac{ \Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}+\frac{iM_1}{2b}\right) }{ \Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}+-\frac{iM_1}{2b}\right) } \frac{1}{2\cosh \frac{{{\widehat{p}}}}{2}} \frac{ \Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}-\frac{iM_1}{2b}+\frac{ib}{2}\right) }{ \Phi _b\left( \frac{{{\widehat{x}}}}{2\pi b}+\frac{iM_1}{2b}-\frac{ib}{2}\right) }. \end{aligned}$$

(C.5)

Now to each line we can apply the identity (C.1). After combining the exponential factors together and cyclically reordering the terms, we obtain

$$\begin{aligned}&{{\widehat{\rho }}}_k\left( M_1,M_2,0,\zeta _1,\zeta _2\right) \nonumber \\&\quad \sim e^{\pi \zeta _2} e^{\left( -\frac{i\zeta _1}{k}+1-\frac{M_1+M_2}{2k}\right) {\widehat{x}}} e^{\frac{i\zeta _2}{k}{{\widehat{p}}}} e^{\left( -\frac{1}{2}+\frac{M_2}{2k}\right) {{\widehat{x}}}} e^{\frac{M_2}{2k}{{\widehat{p}}}} \frac{ \Phi _b\left( \frac{{\widehat{p}}}{2\pi b}+\frac{iM_2}{2b}\right) }{ \Phi _b\left( \frac{{\widehat{p}}}{2\pi b}-\frac{iM_2}{2b}\right) }\nonumber \\&\qquad \frac{1}{2\cosh \frac{{\widehat{x}}}{2}} e^{\left( \frac{1}{2}-\frac{M_2}{2k}\right) {{\widehat{p}}}} \frac{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}-\frac{iM_2}{2b}+\frac{ib}{2}\right) }{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}+\frac{iM_2}{2b}-\frac{ib}{2}\right) }\nonumber \\&\qquad \times e^{-\frac{M_2}{2k}{{\widehat{x}}}} e^{-\frac{i\zeta _2}{k}{{\widehat{p}}}} e^{\left( \frac{i\zeta _1}{k}+\frac{M_1+M_2}{2k}\right) {{\widehat{x}}}} e^{-\frac{M_1}{2k}{{\widehat{x}}}} e^{\left( \frac{1}{2}-\frac{M_1}{2k}\right) {{\widehat{p}}}} \frac{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}-\frac{iM_1}{2b}+\frac{ib}{2}\right) }{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}+\frac{iM_1}{2b}-\frac{ib}{2}\right) }\nonumber \\&\qquad \frac{1}{2\cosh \frac{{{\widehat{x}}}}{2}} e^{\frac{M_1}{2k}{\widehat{p}}} \frac{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}+\frac{iM_1}{2b}\right) }{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}-\frac{iM_1}{2b}\right) }\nonumber \\&\qquad \times e^{\left( -\frac{1}{2}+\frac{M_1}{2k}\right) {{\widehat{x}}}}\nonumber \\&\sim e^{\pi \zeta _1} e^{\left( -\frac{i\zeta _2}{k}+1-\frac{M_1+M_2}{2k}\right) {\widehat{p}}} \frac{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}-\frac{iM_1}{2b}+\frac{ib}{2}\right) }{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}+\frac{iM_1}{2b}-\frac{ib}{2}\right) } \frac{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}-\frac{iM_2}{2b}+\frac{ib}{2}+\frac{\zeta _1}{b}\right) }{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}+\frac{iM_2}{2b}-\frac{ib}{2}+\frac{\zeta _1}{b}\right) } \frac{1}{2\cosh \frac{{{\widehat{x}}}}{2}}\nonumber \\&\qquad \times e^{\left( \frac{i\zeta _2}{k}+\frac{M_1+M_2}{2k}\right) {{\widehat{p}}}} \frac{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}+\frac{iM_1}{2b}\right) }{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}-\frac{iM_1}{2b}\right) } \frac{ \Phi _b\left( \frac{{\widehat{p}}}{2\pi b}+\frac{iM_2}{2b}+\frac{\zeta _1}{b}\right) }{ \Phi _b\left( \frac{{{\widehat{p}}}}{2\pi b}-\frac{iM_2}{2b}+\frac{\zeta _1}{b}\right) } \frac{1}{2\cosh \frac{{{\widehat{x}}}}{2}}. \end{aligned}$$

(C.6)

The last expression coincides with ${{\widehat{\rho }}}$ (4.1) with the replacement $\left( \zeta _1,\zeta _2,{{\widehat{x}}},{{\widehat{p}}}\right) \rightarrow \left( \zeta _2,\zeta _1,{{\widehat{p}}},-{{\widehat{x}}}\right)$. Since the change of variables in ${{\widehat{x}}},{{\widehat{p}}}$ is a canonical transformation, which can be realized by a similarity transformation, we conclude that ${{\widehat{\rho }}}_k\left( M_1,M_2,0,\zeta _2,\zeta _1\right) \sim {{\widehat{\rho }}}_k\left( M_1,M_2,0,\zeta _1,\zeta _2\right)$.

D Weyl transformations

We give the list of $2\cdot 4!$ Weyl transformations discussed in section in terms of the generators, realized also as matrices, arranged by length. A positive integer $i_1...i_k$ represents the product $s_{i_1}\ldots s_{i_k}$:

$$\begin{aligned}&21343=\left( {\begin{matrix} 1 &{} 0 &{} -1 &{} 0 &{} 0 \\ 0 &{} 1 &{} -1 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ \end{matrix}} \right) ,\, 213413=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} -1 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} 1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) , 213423=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} 1 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} -1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\, \\&432134=\left( {\begin{matrix} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ \end{matrix}} \right) ,\, 452134=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} 1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} -1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ \end{matrix}} \right) ,\, 2134213=\left( {\begin{matrix} 0 &{} 1 &{} -1 &{} 0 &{} 0 \\ 1 &{} 0 &{} -1 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{matrix}} \right) ,\, \\&4532134=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} 1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} -1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ \end{matrix}} \right) ,\, 2345134=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} -1 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} 1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ \end{matrix}} \right) ,\\&\quad 1345234=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} 1 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} -1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ \end{matrix}} \right) ,\, \\&432134131=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} -1 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\, 452134131=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} 1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} -1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\\&\quad 432134232=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 1 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} -1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\, \\&452134232=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} 1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} -1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\, 4532134131=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} 1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} -1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\\&\quad 2345134131=\left( {\begin{matrix} 1 &{} 0 &{} -1 &{} 0 &{} 0 \\ 0 &{} 1 &{} -1 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ \end{matrix}} \right) ,\, \\&1345234131=\left( {\begin{matrix} 0 &{} 1 &{} -1 &{} 0 &{} 0 \\ 1 &{} 0 &{} -1 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ \end{matrix}} \right) ,\, 2345321341=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} -1 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ \end{matrix}} \right) ,\\&\quad 4532134232=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} 1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} -1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\, \\&2345134232=\left( {\begin{matrix} 0 &{} 1 &{} -1 &{} 0 &{} 0 \\ 1 &{} 0 &{} -1 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ \end{matrix}} \right) ,\, 1345234232=\left( {\begin{matrix} 1 &{} 0 &{} -1 &{} 0 &{} 0 \\ 0 &{} 1 &{} -1 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ \end{matrix}} \right) ,\\&\quad 1345321342=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 1 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} -1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ \end{matrix}} \right) ,\, \\&4321343213=\left( {\begin{matrix} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{matrix}} \right) ,\, 4521343213=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} 1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} -1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{matrix}} \right) ,\\&\quad 3213452134=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} -1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} 1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{matrix}} \right) ,\, \\&23453213431=\left( {\begin{matrix} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ \end{matrix}} \right) ,\, 13453213432=\left( {\begin{matrix} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ \end{matrix}} \right) ,\\&\quad 45321343213=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} 1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} -1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{matrix}} \right) ,\, \\&23451343213=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} 1 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} -1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ \end{matrix}} \right) ,\, 13452343213=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} -1 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} 1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ \end{matrix}} \right) ,\\&\quad 32134532134=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} -1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{matrix}} \right) ,\, \\&1345321342131=\left( {\begin{matrix} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ \end{matrix}} \right) ,\, 3213452134131=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} -1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} 1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\\&\quad 2345321342132=\left( {\begin{matrix} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ \end{matrix}} \right) ,\, \\&3213452134232=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} -1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} 1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\, 5432134521343=\left( {\begin{matrix} 1 &{} 0 &{} -1 &{} 0 &{} 0 \\ 0 &{} 1 &{} -1 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{matrix}} \right) ,\\&\quad 13453213432131=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} -1 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ \end{matrix}} \right) ,\, \\&32134532134131=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} -1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\, 23453213432132=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 1 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} -1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ \end{matrix}} \right) ,\\&\quad 32134532134232=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} -1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\, \\&32134521343213=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} -1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} 1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ \end{matrix}} \right) ,\, 54321345213413=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} -1 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} 1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\\&\quad 54321345213423=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} 1 \\ {1}/{2} &{} {1}/{2} &{} -1 &{} 0 &{} -1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\, \\&54321345432134=\left( {\begin{matrix} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{matrix}} \right) ,\, 321345321343213=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} -1 &{} 0 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ \end{matrix}} \right) ,\\&\quad , 543213452134213=\left( {\begin{matrix} 0 &{} 1 &{} -1 &{} 0 &{} 0 \\ 1 &{} 0 &{} -1 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ \end{matrix}} \right) ,\, \\&54321345432134131=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} -1 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ -{1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\, 54321345432134232=\left( {\begin{matrix} {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} 1 \\ {1}/{2} &{} {1}/{2} &{} 0 &{} 0 &{} -1 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ {1}/{2} &{} -{1}/{2} &{} 0 &{} 0 &{} 0 \\ \end{matrix}} \right) ,\\&\quad 543213454321343213=\left( {\begin{matrix} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ \end{matrix}} \right) . \end{aligned}$$