S-move matrix in the NS sector of
$N=1$ super Liouville field theory

Elena<EMAIL_ADDRESS>and Gor Sarkissian1,2
<EMAIL_ADDRESS><EMAIL_ADDRESS>
1Yerevan Physics Institute,

Alikhanian Br. 2, 0036 Yerevan, Armenia

and

2 Bogoliubov Laboratory of Theoretical Physics, JINR,

Dubna, Moscow region, 141980 Russia

###### Abstract

In this paper we calculate matrix of modular transformations of the one-point
toric conformal blocks in the Neveu-Schwarz sector of $N=1$ super Liouville
field theory. For this purpose we use explicit expression for this matrix as
integral of product of certain elements of fusion matrix. This integral is
computed using the chain of integral identities for supersymmetric hyperbolic
gamma functions derived by the degeneration of the integrals of parafermionic
elliptic gamma functions.

###### Contents

  1. 1 Introduction
  2. 2 Basics on $N=1$ super Lioville field theory
  3. 3 Elements of S-matrix for three NS primaries
  4. 4 Elements of S-matrix for two NS and one ÑS primaries
  5. 5 Conclusion
  6. A Properties of supersymmetric hyperbolic and double gamma functions
  7. B Parafermionic integral identities I
  8. C Parafermionic integral identities II
  9. D Parafermionic integral identities III
  10. E Parafermionic integral identities IV

##  1 Introduction

One of the basic objects in two-dimensional conformal field theory is the
$S$-matrix of modular transformation of one-point toric conformal blocks. The
$S$-matrix $S_{\beta_{1}\beta_{2}}(\beta_{3})$ satisfies the following Moore-
Seiberg duality identity [14, 13]:

$\displaystyle
S_{\beta_{1}\beta_{2}}(\beta_{3})\sum_{\beta_{4}}F_{\beta_{3},\beta_{4}}\left[\begin{array}[]{cc}\beta_{2}&\alpha_{1}\\\
\beta_{2}&\alpha_{2}\end{array}\right]e^{-2\pi
i(\Delta_{\beta_{4}}-\Delta_{\beta_{2}})}F_{\beta_{4},\beta_{5}}\left[\begin{array}[]{cc}\alpha_{2}&\alpha_{1}\\\
\beta_{2}&\beta_{2}\end{array}\right]$ (5)
$\displaystyle=\sum_{\beta_{6}}F_{\beta_{3},\beta_{6}}\left[\begin{array}[]{cc}\beta_{1}&\alpha_{1}\\\
\beta_{1}&\alpha_{2}\end{array}\right]e^{-\pi
i(\Delta_{\alpha_{1}}+\Delta_{\alpha_{2}}-\Delta_{\beta_{5}})}F_{\beta_{1},\beta_{5}}\left[\begin{array}[]{cc}\alpha_{1}&\alpha_{2}\\\
\beta_{6}&\beta_{6}\end{array}\right]S_{\beta_{6}\beta_{2}}(\beta_{5})\,.$
(10)

Here
$F_{\sigma_{5},\sigma_{6}}\left[\begin{array}[]{cc}\sigma_{2}&\sigma_{1}\\\
\sigma_{3}&\sigma_{4}\end{array}\right]$ is fusion matrix and
$\Delta_{\sigma}$ is conformal weight. Setting in eq. (5)
$\beta_{1}=\beta_{3}=0$ one obtains an explicit expression of the $S$-matrix
in terms of the fusion matrix and conformal weights [3, 13]:

$\displaystyle
S_{0\beta_{2}}\sum_{\beta_{4}}F_{0,\beta_{4}}\left[\begin{array}[]{cc}\beta_{2}&\alpha\\\
\beta_{2}&\alpha\end{array}\right]e^{-2\pi
i\Delta_{\beta_{4}}}F_{\beta_{4},\beta_{5}}\left[\begin{array}[]{cc}\alpha&\alpha\\\
\beta_{2}&\beta_{2}\end{array}\right]$ (15) $\displaystyle=e^{-\pi
i(2\Delta_{\alpha}-\Delta_{\beta_{5}}+2\Delta_{\beta_{2}})}F_{0,\beta_{5}}\left[\begin{array}[]{cc}\alpha&\alpha\\\
\alpha&\alpha\end{array}\right]S_{\alpha\beta_{2}}(\beta_{5})$ (18)

or this can be written also in the form [3]

$\displaystyle
S_{0\beta_{2}}\sum_{\beta_{4}}F_{0,\beta_{4}}\left[\begin{array}[]{cc}\beta_{2}&\alpha\\\
\beta_{2}&\alpha\end{array}\right]e^{2\pi
i\Delta_{\beta_{4}}}F_{\beta_{4},\beta_{5}}\left[\begin{array}[]{cc}\alpha&\alpha\\\
\beta_{2}&\beta_{2}\end{array}\right]$ (23) $\displaystyle=e^{\pi
i(2\Delta_{\alpha}+2\Delta_{\beta_{2}})}F_{0,\beta_{5}}\left[\begin{array}[]{cc}\alpha&\alpha\\\
\alpha&\alpha\end{array}\right]S_{\alpha\beta_{2}}(\beta_{5})\,.$ (26)

Here we used that $S_{\alpha\beta_{2}}(\beta_{5})$ in the limit $\beta_{5}\to
0$ becomes matrix of the modular transformation of characters
$S_{\alpha\beta_{2}}$ and $S_{0\beta_{2}}$ in (23) is the corresponding
element of this matrix. The formula (23) was used in [29] to calculate
$S_{\beta_{6}\beta_{2}}(\beta_{5})$ in the Liouville field theory. The
calculations are very involved and use sequence of integral identities for
various products of hyperbolic gamma functions. Important point here is that
all the necessary identities can be derived by some chains of the reductions
from the elliptic beta integral [23] and symmetry relations of so called
elliptic $V$-function [24, 25]. This becomes possible since as shown in [18,
17] the hyperbolic gamma function is an asymptotic form of the elliptic gamma
function.

In this paper we use the formula (23) to calculate $S$-matrix of modular
transformation of one-point toric conformal block in $N=1$ super Liouville
field theory (SLFT). For this purpose we should elaborate “supersymmetric
counterpart” of the mentioned integral identities. At this point we would like
to point out that if the essential part of the fusion matrix in the Liouville
field theory is integral of product of hyperbolic gamma functions $S_{b}(x)$,
defined in appendix A, then the essential part of the fusion matrix in $N=1$
super Liouville field theory is sum of integrals of product of so called
Neveu-Schwarz and Ramond hyperbolic gamma functions:

$S_{1}(y)\equiv S_{\rm NS}(y)\equiv S_{b}\left({y\over
2}\right)S_{b}\left({y\over 2}+{Q\over 2}\right),$ (27) $S_{0}(y)\equiv S_{\rm
R}(y)\equiv S_{b}\left({y\over 2}+{b\over 2}\right)S_{b}\left({y\over
2}+{b^{-1}\over 2}\right)\,,$ (28)

where $Q=b+1/b$. To derive the necessary integral formulas for them first we
note that the functions (27) and (28) are particular case of so called
parafermionic hyperbolic gamma functions [20], defined for any
$r\in\mathbb{N}$:

$\displaystyle\Lambda(y,m)=\prod_{k=0}^{m-1}S_{b}\left({y\over
r}+b\left(1-{m\over r}\right)+Q{k\over r}\right)$
$\displaystyle\makebox[20.00003pt]{}\times\prod_{k=0}^{r-m-1}S_{b}\left({y\over
r}+{m\over br}+Q{k\over r}\right)\,,$ (29)

where $0\leq m\leq r$. Obviously, for $r=2$ one obtains the functions (27) and
(28):

$\Lambda(y,0)=S_{1}(y),$ (30) $\Lambda(y,1)=S_{0}(y)\,.$ (31)

In paper [27], rarefied elliptic gamma functions are constructed and
corresponding generalizations of the elliptic beta integral and elliptic
V-function are studied. It was checked also in [20] that the functions (1) are
asymptotic forms of the rarefied elliptic gamma functions. Therefore we can
get the necessary identities using the same chains of the reductions as in the
case $r=1$. Some of the necessary integral identities were derived in this way
in papers [1, 20]. Others are obtained in appendices B-E .

The paper is organized as follows. In section 2, we review some basic facts on
$N=1$ Super Liouville field theory. In section 3, we calculate
$S_{\beta_{1}\beta_{2}}(\beta_{3})$ in the case when all $\beta_{i}$,
$i=1,2,3$ are NS primaries. The corresponding result is given by formulae (90)
and (91). In section 4, we calculate $S_{\beta_{1}\beta_{2}}(\beta_{3})$ in
the case when $\beta_{i}$, $i=1,2$ are NS primaries and $\beta_{3}$ is ÑS
primary. It is given by (107) and (108). In appendices A-E the necessary
special functions, their properties and integral identities are reviewed.

##  2 Basics on $N=1$ super Lioville field theory

$N=1$ super Liouville field theory is defined on a two-dimensional surface
with metric $g_{ab}$ by the Lagrangian density:

${\cal L}={1\over 2\pi}g_{ab}\partial_{a}\varphi\partial_{b}\varphi+{1\over
2\pi}(\psi\bar{\partial}\psi+\bar{\psi}\partial\bar{\psi})+2i\mu
b^{2}\bar{\psi}\psi e^{b\varphi}+2\pi\mu^{2}b^{2}e^{2b\varphi}\,.$ (32)

The energy-momentum tensor and the superconformal current are

$\displaystyle T=-{1\over
2}(\partial\varphi\partial\varphi-Q\partial^{2}\varphi+\psi\partial\psi)\,,$
(33) $\displaystyle G=i(\psi\partial\varphi-Q\partial\psi)\,,$ (34)

where $Q=b+{1\over b}$. The superconformal algebra is

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n}+{c_{SL}\over
12}m(m^{2}-1)\delta_{m+n}\,,$ (35) $\displaystyle[L_{m},G_{k}]={m-2k\over
2}G_{m+k}\,,$ (36) $\displaystyle\\{G_{k},G_{l}\\}=2L_{l+k}+{c\over
3}\left(k^{2}-{1\over 4}\right)\delta_{k+l}\,,$ (37)

with the central charge

$c_{SL}={3\over 2}+3Q^{2}\,.$ (38)

Here $k$ and $l$ take integer values for the Ramond algebra and half-integer
values for the Neveu-Schwarz algebra. Since in the Neveu-Schwarz sector of
$N=1$ SLFT one has besides the Virasoro generators $L_{m}$, also supercurrent
generators $G_{k}$ with half-integer $k$, descendant fields are broken into
two representations of the Virasoro algebra of integer and half-integer
levels. Therefore there are in the Neveu-Schwarz sector two types of Virasoro
primary fields, which we call NS and ÑS[4]. The first type of primary fields
is associated with the vertex operators $N_{\alpha}=e^{\alpha\varphi}$, and
has conformal dimensions

$\Delta_{\alpha}={1\over 2}\alpha(Q-\alpha)\,.$ (39)

The physical states have $\alpha={Q\over 2}+iP$.

The second type is given by the supercurrent descendant:

$\tilde{N}_{\alpha}=G_{-1/2}N_{\alpha}\,.$ (40)

It has the conformal dimension

$\Delta_{\tilde{\alpha}}={1\over 2}\alpha(Q-\alpha)+{1\over 2}\,.$ (41)

The degenerate states are given by the momenta:

$\alpha_{m,n}={1\over 2b}(1-m)+{b\over 2}(1-n)$ (42)

with even $m-n$ in the NS sector and odd $m-n$ in the R sector. For future use
we need matrix of modular transformation of the NS characters. The
corresponding characters for generic $P$ which have no null-states are

$\chi_{P}^{NS}(\tau)=\sqrt{\theta_{3}(q)\over\eta(q)}{q^{P^{2}/2}\over\eta(\tau)}\,,$
(43)

where $q=\exp(2\pi i\tau)$ and

$\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n})\,.$ (44)

Modular transformation of characters (43) is well-known:

$\chi_{P}^{NS}(\tau)=\int\chi_{P^{\prime}}^{NS}(-1/\tau)e^{-2i\pi
PP^{\prime}}dP^{\prime}\,.$ (45)

For the degenerate representations, the characters are given by those of the
corresponding Verma modules subtracted by those of null submodules:

$\chi_{m,n}^{NS}=\chi_{{1\over 2}(nb+mb^{-1})}^{NS}-\chi_{{1\over 2}(nb-
mb^{-1})}^{NS}\,.$ (46)

Modular transformations of (46) is

$\chi_{m,n}^{NS}(\tau)=\int\chi_{P}^{NS}(-1/\tau)2\sinh(\pi mP/b)\sinh(\pi
nbP)dP\,.$ (47)

Given that the identity field is specified by $(m,n)=(1,1)$ one finds the
vacuum component of the matrix of modular transformation:

$S_{0\alpha}=\sin\pi b^{-1}(\alpha-Q/2)\sin\pi b(\alpha-Q/2)\,.$ (48)

For brevity the $N_{\alpha}$ and $\tilde{N}_{\alpha}$ primary fields will be
denoted by $\alpha$ and $\tilde{\alpha}$ correspondingly. Therefore, if for
example we take in the relation (23) $\alpha$, $\beta_{2}$ and $\beta_{5}$ of
NS type, it takes the form

$\displaystyle S_{0\beta_{2}}\Bigg{[}\int{d\beta_{4}\over
i}F_{0,\beta_{4}}\left[\begin{array}[]{cc}\beta_{2}&\alpha\\\
\beta_{2}&\alpha\end{array}\right]e^{2\pi
i\Delta_{\beta_{4}}}F_{\beta_{4},\beta_{5}}\left[\begin{array}[]{cc}\alpha&\alpha\\\
\beta_{2}&\beta_{2}\end{array}\right]$ (53)
$\displaystyle+\int{d\beta_{4}\over
i}F_{0,\tilde{\beta_{4}}}\left[\begin{array}[]{cc}\beta_{2}&\alpha\\\
\beta_{2}&\alpha\end{array}\right]e^{2\pi
i\Delta_{\tilde{\beta_{4}}}}F_{\tilde{\beta_{4}},\beta_{5}}\left[\begin{array}[]{cc}\alpha&\alpha\\\
\beta_{2}&\beta_{2}\end{array}\right]\Bigg{]}=$ (58) $\displaystyle e^{\pi
i(2\Delta_{\alpha}+2\Delta_{\beta_{2}})}F_{0,\beta_{5}}\left[\begin{array}[]{cc}\alpha&\alpha\\\
\alpha&\alpha\end{array}\right]S_{\alpha\beta_{2}}(\beta_{5})\,.$ (61)

As we mentioned before $S_{\alpha\beta_{2}}(\beta_{5})$ in the limit
$\beta_{5}\to 0$ should coincide with the matrix of modular transformations of
the characters, and $S_{0\beta_{2}}$ is given by (48).

The elements of the fusion matrix in the Neveu-Schwarz sector of $N=1$ SLFT
are given by the formulae [9, 10]:

$F_{\alpha_{s},\alpha_{t}}\left[\begin{array}[]{cc}\alpha_{3}&\alpha_{2}\\\
\alpha_{4}&\alpha_{1}\end{array}\right]={{\cal
N}_{NS}(\alpha_{s},\alpha_{2},\alpha_{1}){\cal
N}_{NS}(\alpha_{4},\alpha_{3},\alpha_{s})\over{\cal
N}_{NS}(\alpha_{t},\alpha_{3},\alpha_{2}){\cal
N}_{NS}(\alpha_{4},\alpha_{t},\alpha_{1})}\big{\\{}\,{}^{\alpha_{1}}_{\alpha_{2}}\,{}^{\alpha_{3}}_{{\alpha}_{4}}\,|\,{}^{\alpha_{s}}_{\alpha_{t}}\big{\\}}_{11}^{11}\,,$
(62)
$F_{\alpha_{s},\tilde{\alpha_{t}}}\left[\begin{array}[]{cc}\alpha_{3}&\alpha_{2}\\\
\alpha_{4}&\alpha_{1}\end{array}\right]={{\cal
N}_{NS}(\alpha_{s},\alpha_{2},\alpha_{1}){\cal
N}_{NS}(\alpha_{4},\alpha_{3},\alpha_{s})\over{\cal
N}_{R}(\alpha_{t},\alpha_{3},\alpha_{2}){\cal
N}_{R}(\alpha_{4},\alpha_{t},\alpha_{1})}\big{\\{}\,{}^{\alpha_{1}}_{\alpha_{2}}\,{}^{\alpha_{3}}_{{\alpha}_{4}}\,|\,{}^{\alpha_{s}}_{\alpha_{t}}\big{\\}}_{00}^{11}\,,$
(63)

$F_{\tilde{\alpha_{s}},\alpha_{t}}\left[\begin{array}[]{cc}\alpha_{3}&\alpha_{2}\\\
\alpha_{4}&\alpha_{1}\end{array}\right]={{\cal
N}_{R}(\alpha_{s},\alpha_{2},\alpha_{1}){\cal
N}_{R}(\alpha_{4},\alpha_{3},\alpha_{s})\over{\cal
N}_{NS}(\alpha_{t},\alpha_{3},\alpha_{2}){\cal
N}_{NS}(\alpha_{4},\alpha_{t},\alpha_{1})}\big{\\{}\,{}^{\alpha_{1}}_{\alpha_{2}}\,{}^{\alpha_{3}}_{{\alpha}_{4}}\,|\,{}^{\alpha_{s}}_{\alpha_{t}}\big{\\}}_{11}^{00}\,,$
(64)
$F_{\tilde{\alpha_{s}},\tilde{\alpha_{t}}}\left[\begin{array}[]{cc}\alpha_{3}&\alpha_{2}\\\
\alpha_{4}&\alpha_{1}\end{array}\right]={{\cal
N}_{R}(\alpha_{s},\alpha_{2},\alpha_{1}){\cal
N}_{R}(\alpha_{4},\alpha_{3},\alpha_{s})\over{\cal
N}_{R}(\alpha_{t},\alpha_{3},\alpha_{2}){\cal
N}_{R}(\alpha_{4},\alpha_{t},\alpha_{1})}\big{\\{}\,{}^{\alpha_{1}}_{\alpha_{2}}\,{}^{\alpha_{3}}_{{\alpha}_{4}}\,|\,{}^{\alpha_{s}}_{\alpha_{t}}\big{\\}}_{00}^{00}\,,$
(65)

where normalization coefficients are described by the relations:

$\displaystyle{\cal N}_{NS}(\alpha_{3},\alpha_{2},\alpha_{1})=$ (66)
$\displaystyle{\Gamma_{NS}(2\alpha_{1})\Gamma_{NS}(2\alpha_{2})\Gamma_{NS}(2Q-2\alpha_{3})\over\Gamma_{NS}(2Q-\alpha_{1}-\alpha_{2}-\alpha_{3})\Gamma_{NS}(Q-\alpha_{1}-\alpha_{2}+\alpha_{3})\Gamma_{NS}(\alpha_{2}+\alpha_{3}-\alpha_{1})\Gamma_{NS}(\alpha_{3}+\alpha_{1}-\alpha_{2})}\,,$
$\displaystyle{\cal N}_{R}(\alpha_{3},\alpha_{2},\alpha_{1})=$ (67)
$\displaystyle{\Gamma_{NS}(2\alpha_{1})\Gamma_{NS}(2\alpha_{2})\Gamma_{NS}(2Q-2\alpha_{3})\over\Gamma_{R}(2Q-\alpha_{1}-\alpha_{2}-\alpha_{3})\Gamma_{R}(Q-\alpha_{1}-\alpha_{2}+\alpha_{3})\Gamma_{R}(\alpha_{2}+\alpha_{3}-\alpha_{1})\Gamma_{R}(\alpha_{3}+\alpha_{1}-\alpha_{2})}\,,$

where the functions $\Gamma_{NS}(x)$, $\Gamma_{R}(x)$ are described in
appendix A, and $6j$-symbols of the quantum algebra ${\rm U}_{q}({\rm
osp}(1|2))$ are [10]:

$\big{\\{}\,{}^{\alpha_{1}}_{\alpha_{3}}\,{}^{\alpha_{2}}_{{\overline{\alpha}_{4}}}\,|\,{}^{\alpha_{s}}_{\alpha_{t}}\big{\\}}_{\nu_{1}\nu_{2}}^{\nu_{3}\nu_{4}}=|S_{1}(2\alpha_{t})|^{2}{S_{\nu_{4}}(\alpha_{s}+\alpha_{2}-\alpha_{1})S_{\nu_{1}}(\alpha_{1}+\alpha_{t}-\alpha_{4})\over
S_{\nu_{2}}(\alpha_{t}+\alpha_{2}-\alpha_{3})S_{\nu_{3}}(\alpha_{3}+\alpha_{s}-\alpha_{4})}I_{\alpha_{s},\alpha_{t}}\left[\begin{array}[]{cc}\alpha_{3}&\alpha_{2}\\\
\alpha_{4}&\alpha_{1}\end{array}\right]_{\nu_{1}\nu_{2}}^{\nu_{3}\nu_{4}},$
(68)

where ${\overline{\alpha}_{4}}=Q-\alpha_{4}$, $\nu_{i}=0,1$, $i=1,2,3,4$ ,
$\sum_{i=1}^{4}\nu_{i}=0\,\,{\rm mod}\>2$, and

$\displaystyle
I_{\alpha_{s},\alpha_{t}}\left[\begin{array}[]{cc}\alpha_{3}&\alpha_{2}\\\
\alpha_{4}&\alpha_{1}\end{array}\right]_{\nu_{1}\nu_{2}}^{\nu_{3}\nu_{4}}=(-1)^{\nu_{3}\nu_{2}+\nu_{4}}\int_{-\textup{i}\infty}^{\textup{i}\infty}\sum_{\nu=0}^{1}(-1)^{\nu(\nu_{2}+\nu_{4})}S_{1+\nu+\nu_{3}}(y+\gamma_{1}^{\circ})\qquad\qquad$
(71) $\displaystyle\makebox[20.00003pt]{}\times
S_{1+\nu+\nu_{4}}(y+\gamma_{2}^{\circ})S_{1+\nu+\nu_{3}}(y+\gamma_{3}^{\circ})S_{1+\nu+\nu_{4}}(y+\gamma_{4}^{\circ})S_{\nu}(-y+\beta_{1}^{\circ})$
$\displaystyle\makebox[20.00003pt]{}\times
S_{\nu+\nu_{2}+\nu_{3}}(-y+\beta_{2}^{\circ})S_{\nu+\nu_{2}+\nu_{3}}(-y+\beta_{3}^{\circ})S_{\nu}(-y+\beta_{4}^{\circ}){dy\over
2\textup{i}}$
$\displaystyle=\int_{-\textup{i}\infty}^{\textup{i}\infty}\sum_{\nu=0}^{1}\Lambda(y+\gamma_{1}^{\circ},\nu_{3}+\nu)\Lambda(y+\gamma_{2}^{\circ},\nu+\nu_{4})\Lambda(y+\gamma_{3}^{\circ},\nu+\nu_{3})$
$\displaystyle\makebox[20.00003pt]{}\times\Lambda(y+\gamma_{4}^{\circ},\nu-\nu_{4})\Lambda(-y+\beta_{1}^{\circ},-\nu-1)\Lambda(-y+\beta_{2}^{\circ},-\nu+1+\nu_{2}-\nu_{3})$
$\displaystyle\makebox[20.00003pt]{}\times\Lambda(-y+\beta_{3}^{\circ},-\nu+1-\nu_{2}-\nu_{3})\Lambda(-y+\beta_{4}^{\circ},-\nu-1){dy\over
2\textup{i}}\,,$

where

$\begin{aligned} &\gamma_{1}^{\circ}=-Q/2+\alpha_{3}-\alpha_{4}\,,\\\
&\gamma_{2}^{\circ}=-Q/2+\alpha_{1}-\alpha_{2}\,,\\\
&\gamma_{3}^{\circ}=Q/2-\alpha_{3}-\alpha_{4}\,,\\\
&\gamma_{4}^{\circ}=Q/2-\alpha_{1}-\alpha_{2}\,,\end{aligned}\qquad\begin{aligned}
&\beta_{1}^{\circ}=Q/2+\alpha_{s}\,,\\\
&\beta_{2}^{\circ}=Q/2-\alpha_{t}+\alpha_{4}+\alpha_{2}\,,\\\
&\beta_{3}^{\circ}=-Q/2+\alpha_{t}+\alpha_{4}+\alpha_{2}\,,\\\
&\beta_{4}^{\circ}=3Q/2-\alpha_{s}\,.\end{aligned}$ (72)

The second equality in (71) was established in [1]. Let us explain briefly how
it arises. First recall the following periodicity and reflection properties of
$\Lambda(y,\nu)$[20, 21]:

$\Lambda(y,m+2k)=(-1)^{mk+r{k(k-1)\over 2}}\Lambda(y,m)\,,$ (73)
$\Lambda(Q-x,-\nu)\Lambda(x,\nu)=(-1)^{\nu}\,.$ (74)

For $r=2$ the relation (73) implies

$\Lambda(y,m+2k)=(-1)^{mk}\Lambda(y,m)\,.$ (75)

The functions $S_{\nu}(x)$ are supposed to be periodic in the discrete
variable: $S_{\nu}(x)=S_{\nu+2}(x)$. This together with (75) implies

$\Lambda(y,2m;b^{-1},b)=S_{1}(y),\quad\Lambda(y,2m+1;b^{-1},b)=(-1)^{m}S_{0}(y)\,.$
(76)

Repeatedly using (76) one can arrive to the second equality in (71). One can
check that the expressions (62)-(65) are invariant under the reflection
$\alpha\to Q-\alpha$ of any of six variables entering in the fusion matrix.
Inserting (62)-(65) in (53) we derive:

$S^{SL}_{\alpha\beta_{2}}(\beta_{5})={{\cal
N}_{NS}(\alpha,\beta_{5},\alpha)\over{\cal
N}_{NS}(\beta_{2},\beta_{5},\beta_{2})}S_{\alpha\beta_{2}}(\beta_{5})$ (77)

where $S_{\alpha\beta_{2}}(\beta_{5})$ is computed by the same formula (53)
where inserted only corresponding elements of $6j$-symbols of the quantum
algebra ${\rm U}_{q}({\rm osp}(1|2))$ appearing in the l.h.s of (62)-(65):

$S_{0\beta_{2}}I_{\alpha\beta_{2}}(\beta_{5})=e^{\pi
i(2\Delta_{\alpha}+2\Delta_{\beta_{2}})}\big{\\{}\,{}^{\alpha}_{\alpha}\,{}^{\alpha}_{\alpha}\,|\,{}^{0}_{\beta_{5}}\big{\\}}_{11}^{11}\,,S_{\alpha\beta_{2}}(\beta_{5})$
(78)

where

$\displaystyle I_{\alpha\beta_{2}}(\beta_{5})=\Bigg{[}\int{d\beta_{4}\over
i}\big{\\{}\,{}^{\alpha}_{\alpha}\,{}^{\beta_{2}}_{\beta_{2}}\,|\,{}^{0}_{\beta_{4}}\big{\\}}_{11}^{11}e^{2\pi
i\Delta_{\beta_{4}}}\big{\\{}\,{}^{\beta_{2}}_{\alpha}\,{}^{\alpha}_{\beta_{2}}\,|\,{}^{\beta_{4}}_{\beta_{5}}\big{\\}}_{11}^{11}$
(79) $\displaystyle+\int{d\beta_{4}\over
i}\big{\\{}\,{}^{\alpha}_{\alpha}\,{}^{\beta_{2}}_{\beta_{2}}\,|\,{}^{0}_{\beta_{4}}\big{\\}}_{00}^{11}e^{2\pi
i\Delta_{\tilde{\beta_{4}}}}\big{\\{}\,{}^{\beta_{2}}_{\alpha}\,{}^{\alpha}_{\beta_{2}}\,|\,{}^{\beta_{4}}_{\beta_{5}}\big{\\}}_{11}^{00}\Bigg{]}\,.$

##  3 Elements of S-matrix for three NS primaries

Now we turn to the calculation of the matrix $S_{\alpha\beta_{2}}(\beta_{5})$
in the case when all the entering primaries of the NS type. For this purpose
we use the formula (78). Start to calculate the left hand side integral (79).
The first term in (79) is the following element of the $6j$-symbol:
$\big{\\{}\,{}^{\alpha}_{\alpha}\,{}^{\beta_{2}}_{\beta_{2}}\,|\,{}^{0}_{\beta_{4}}\big{\\}}_{11}^{11}$.

Setting in (68)-(71) $\nu_{1}=\nu_{2}=\nu_{3}=\nu_{4}=1$, and
$\alpha_{1}=\alpha_{2}=\alpha$, $\alpha_{3}=\beta_{2}$,
$\alpha_{4}=Q-\beta_{2}$, $\alpha_{t}=\beta_{4}$, $\alpha_{s}=\delta\to 0$ we
obtain

$\displaystyle\big{\\{}\,{}^{\alpha}_{\alpha}\,{}^{\beta_{2}}_{\beta_{2}}\,|\,{}^{\delta}_{\beta_{4}}\big{\\}}_{11}^{11}=|S_{1}(2\beta_{4})|^{2}{S_{1}(\delta)S_{1}(\alpha+\beta_{4}+\beta_{2}-Q)\over
S_{1}(\beta_{4}+\alpha-\beta_{2})S_{1}(2\beta_{2}-Q)}$ (80)
$\displaystyle\times\int_{-\textup{i}\infty}^{\textup{i}\infty}\sum_{\nu=0}^{1}(-1)^{\nu+1}\Lambda(y-3Q/2+2\beta_{2},1+\nu)\Lambda(y-Q/2,\nu+1)\Lambda(y+Q/2-2\alpha,\nu-1)\qquad$
$\displaystyle\times\Lambda(-y+Q/2+\delta,-\nu-1)\Lambda(-y+3Q/2-\beta_{4}-\beta_{2}+\alpha,-\nu+1)$
$\displaystyle\times\Lambda(-y+Q/2+\beta_{4}-\beta_{2}+\alpha,-\nu-1){dy\over
2i}$
$\displaystyle=|S_{1}(2\beta_{4})|^{2}S_{1}(\delta)S_{1}(\alpha+\beta_{4}+\beta_{2}-Q)\Lambda(\delta,0)\Lambda(Q-\beta_{2}-\beta_{4}+\alpha,2)\Lambda(-2\alpha,-2)\,.$

To compute integral here we used formula (129) in appendix A for $r=2$.

Now we should compute the second term in (79):
$\big{\\{}\,{}^{\beta_{2}}_{\alpha}\,{}^{\alpha}_{\beta_{2}}\,|\,{}^{\beta_{4}}_{\beta_{5}}\big{\\}}_{11}^{11}$.
Setting in (68) $\nu_{1}=\nu_{2}=\nu_{3}=\nu_{4}=1$, and
$\alpha_{1}=\beta_{2}$, $\alpha_{2}=\alpha$, $\alpha_{3}=\alpha$,
$\alpha_{4}=Q-\beta_{2}$, $\alpha_{t}=\beta_{5}$, $\alpha_{s}=\beta_{4}$ we
have

$\displaystyle\big{\\{}\,{}^{\beta_{2}}_{\alpha}\,{}^{\alpha}_{\beta_{2}}\,|\,{}^{\beta_{4}}_{\beta_{5}}\big{\\}}_{11}^{11}=|S_{1}(2\beta_{5})|^{2}{S_{1}(\alpha+\beta_{4}-\beta_{2})S_{1}(2\beta_{2}+\beta_{5}-Q)\over
S_{1}(\beta_{4}+\alpha+\beta_{2}-Q)S_{1}(\beta_{5})}$ (81)
$\displaystyle\times\int_{-\textup{i}\infty}^{\textup{i}\infty}\sum_{\nu=0}^{1}\Lambda(-y+Q/2+\beta_{4},-\nu-1)\Lambda(-y+3Q/2-\beta_{4},-\nu-1)F(y,\nu+1)(-1)^{\nu-1}{dy\over
2i}\,,$

where

$\displaystyle
F(y,\nu)=\Lambda(y-3Q/2+\beta_{2}+\alpha,\nu)\Lambda(y-Q/2+\beta_{2}-\alpha,\nu)$
(82)
$\displaystyle\times\Lambda(y-Q/2-\alpha+\beta_{2},\nu)\Lambda(y+Q/2-\alpha-\beta_{2},\nu)$
$\displaystyle\times\Lambda(-y+3Q/2-\beta_{5}-\beta_{2}+\alpha,-\nu+2)\Lambda(-y+Q/2+\beta_{5}-\beta_{2}+\alpha,-\nu)\,.$

Writing in this way we separated in integral appearing in (81)
$\beta_{4}$-dependent terms and collected in $F(y,\nu)$ all
$\beta_{4}$-independent terms. The next element of the fusion matrix to be
computed is
$\big{\\{}\,{}^{\alpha}_{\alpha}\,{}^{\beta_{2}}_{\beta_{2}}\,|\,{}^{\delta}_{\beta_{4}}\big{\\}}_{00}^{11}$.
Putting in (68) $\nu_{1}=\nu_{2}=0$, $\nu_{3}=\nu_{4}=1$, and
$\alpha_{1}=\alpha_{2}=\alpha$, $\alpha_{3}=\beta_{2}$,
$\alpha_{4}=Q-\beta_{2}$, $\alpha_{t}=\beta_{4}$, $\alpha_{s}=\delta\to 0$ one
has

$\displaystyle\big{\\{}\,{}^{\alpha}_{\alpha}\,{}^{\beta_{2}}_{\beta_{2}}\,|\,{}^{\delta}_{\beta_{4}}\big{\\}}_{00}^{11}=|S_{1}(2\beta_{4})|^{2}{S_{1}(\delta)S_{0}(\alpha+\beta_{4}+\beta_{2}-Q)\over
S_{0}(\beta_{4}+\alpha-\beta_{2})S_{1}(2\beta_{2}-Q)}$ (83)
$\displaystyle\times\int_{-\textup{i}\infty}^{\textup{i}\infty}\sum_{\nu=0}^{1}(-1)^{\nu+1}\Lambda(y-3Q/2+2\beta_{2},1+\nu)\Lambda(y-Q/2,\nu+1)\Lambda(y+Q/2-2\alpha,\nu-1)\qquad$
$\displaystyle\times\Lambda(-y+Q/2+\delta,-\nu-1)\Lambda(-y+3Q/2-\beta_{4}-\beta_{2}+\alpha,-\nu)\Lambda(-y+Q/2+\beta_{4}-\beta_{2}+\alpha,-\nu){dy\over
2i}$
$\displaystyle=|S_{1}(2\beta_{4})|^{2}S_{1}(\delta)S_{0}(\alpha+\beta_{4}+\beta_{2}-Q)\Lambda(\delta,0)\Lambda(Q-\beta_{2}-\beta_{4}+\alpha,1)\Lambda(-2\alpha,-2)\,.$

The last element of the fusion matrix which we need to compute is
$\big{\\{}\,{}^{\beta_{2}}_{\alpha}\,{}^{\alpha}_{\beta_{2}}\,|\,{}^{\beta_{4}}_{\beta_{5}}\big{\\}}_{11}^{00}$.
Now setting in (68) $\nu_{1}=\nu_{2}=1$, $\nu_{3}=\nu_{4}=0$, and
$\alpha_{1}=\beta_{2}$, $\alpha_{2}=\alpha$, $\alpha_{3}=\alpha$,
$\alpha_{4}=Q-\beta_{2}$, $\alpha_{t}=\beta_{5}$, $\alpha_{s}=\beta_{4}$ one
gets:

$\displaystyle\big{\\{}\,{}^{\beta_{2}}_{\alpha}\,{}^{\alpha}_{\beta_{2}}\,|\,{}^{\beta_{4}}_{\beta_{5}}\big{\\}}_{11}^{00}=|S_{1}(2\beta_{5})|^{2}{S_{0}(\alpha+\beta_{4}-\beta_{2})S_{1}(2\beta_{2}+\beta_{5}-Q)\over
S_{0}(\beta_{4}+\alpha+\beta_{2}-Q)S_{1}(\beta_{5})}$ (84)
$\displaystyle\times\int_{-\textup{i}\infty}^{\textup{i}\infty}\sum_{\nu=0}^{1}\Lambda(-y+Q/2+\beta_{4},-\nu-1)\Lambda(-y+3Q/2-\beta_{4},-\nu-1)F(y,\nu){dy\over
2i}\,,$

were the function $F(y,\nu)$ was defined in (82).

Since $\beta_{4}$ has the form $\beta_{4}=Q/2+iP$ for some real $P$, one can
write

$\displaystyle|S_{1}(2\beta_{4})|^{2}=S_{1}(2\beta_{4})S_{1}(2Q-2\beta_{4})={1\over
S_{1}(Q-2\beta_{4})S_{1}(-Q+2\beta_{4})}$ $\displaystyle={1\over
S_{1}(2(Q/2-\beta_{4}))S_{1}(2(-Q/2+\beta_{4}))}={1\over
S_{1}(2z)S_{1}(-2z)}\,,$

where $z=-Q/2+\beta_{4}$. Recalling also formulae for conformal dimensions
(39) and (41) we obtain

$\displaystyle I_{\alpha\beta_{2}}(\beta_{5})$ (85)
$\displaystyle=C\int_{-\textup{i}\infty}^{\textup{i}\infty}{dy\over
i}F(y,0)\int_{0}^{\textup{i}\infty}{dz\over
2i}\sum_{m=0}^{1}{\Lambda(Q/2+\alpha-\beta_{2}\pm z,\pm m)\Lambda(-y+Q\pm
z,\pm m)\over\Lambda(\pm 2z,\pm 2m)}e^{-i\pi z^{2}}(-1)^{m}$
$\displaystyle-C\int_{-\textup{i}\infty}^{\textup{i}\infty}{dy\over
i}F(y,1)\int_{0}^{\textup{i}\infty}{dz\over
2i}\sum_{m=0}^{1}{\Lambda(Q/2+\alpha-\beta_{2}\pm z,1\pm m)\Lambda(-y+Q\pm
z,\pm m)\over\Lambda(\pm 2z,\pm 2m)}e^{-i\pi z^{2}}(-1)^{m}\,,$
$z=\beta_{4}-{Q\over 2}\,,\qquad
C=2S_{1}(\delta)\Lambda(\delta,0)\Lambda(-2\alpha,-2){|S_{1}(2\beta_{5})|^{2}S_{1}(2\beta_{2}+\beta_{5}-Q)\over
S_{1}(\beta_{5})}e^{Q^{2}\pi i/4}\,.$ (86)

Here we used the shorthand notation $\Lambda(y\pm x;m\pm
n)\equiv\Lambda(y+x;m+n)\Lambda(y-x;m-n)$. The integrals appearing in the
third and the forth lines of (85) are particular cases of the integral (136)
in appendix A for $r=2$. Hence using (136) in appendix A for $r=2$ one has

$\displaystyle
I_{\alpha\beta_{2}}(\beta_{5})=C\int_{-\textup{i}\infty}^{\textup{i}\infty}{dy\over
i}F(y,0)\Lambda(3Q/2+\alpha-\beta_{2}-y,0)e^{-{i\pi\over
4}(-Q/2+\alpha-\beta_{2}+y)^{2}}e^{{i\pi\over
4}(3Q/2+\alpha-\beta_{2}-y)Q}\qquad$ (87)
$\displaystyle-C\int_{-\textup{i}\infty}^{\textup{i}\infty}{dy\over
i}F(y,1)\Lambda(3Q/2+\alpha-\beta_{2}-y,1)e^{-{i\pi\over
4}(-Q/2+\alpha-\beta_{2}+y)^{2}}e^{{i\pi\over
4}(3Q/2+\alpha-\beta_{2}-y)Q}e^{-{i\pi\over 4}}e^{{i\pi\over 2}}$
$\displaystyle=C^{\prime}\int_{-\textup{i}\infty}^{\textup{i}\infty}{dy\over
i}\sum_{\nu=0}^{1}\Lambda(y-3Q/2+\beta_{2}+\alpha,\nu)\Lambda(y-Q/2-\alpha+\beta_{2},\nu)\Lambda(y+Q/2-\alpha-\beta_{2},\nu)\qquad$
$\displaystyle\times\Lambda(-y+3Q/2-\beta_{5}-\beta_{2}+\alpha,-\nu+2)\Lambda(-y+Q/2+\beta_{5}-\beta_{2}+\alpha,-\nu)$
$\displaystyle\times e^{-{i\pi\over 4}y^{2}}e^{-{i\pi\over
2}y(\alpha-\beta_{2})}e^{-{i\pi\nu\over 4}}e^{{3i\pi\nu\over 2}}\,,$

where

$C^{\prime}=Ce^{-{i\pi\over 4}(-Q/2+\alpha-\beta_{2})^{2}}e^{{i\pi\over
4}(3Q/2+\alpha-\beta_{2})Q}\,.$ (88)

Changing sign of $y$ and using that for $\nu=0,1$ hold the relations:
$\Lambda(y,-\nu)=(-1)^{\nu}\Lambda(y,\nu)$ and
$\Lambda(y,2-\nu)=\Lambda(y,\nu)$ we can write the last integral in (87) in
the form

$\displaystyle
I_{\alpha\beta_{2}}(\beta_{5})=C^{\prime}\int_{-\textup{i}\infty}^{\textup{i}\infty}{dy\over
i}\sum_{\nu=0}^{1}\Lambda(-y-3Q/2+\beta_{2}+\alpha,-\nu)\Lambda(-y-Q/2-\alpha+\beta_{2},-\nu)\qquad$
$\displaystyle\times\Lambda(-y+Q/2-\alpha-\beta_{2},-\nu)\Lambda(y+3Q/2-\beta_{5}-\beta_{2}+\alpha,\nu)\Lambda(y+Q/2+\beta_{5}-\beta_{2}+\alpha,\nu)$
$\displaystyle\times e^{-{i\pi\over 4}y^{2}}e^{{i\pi\over
2}y(\alpha-\beta_{2})}e^{-{i\pi\nu\over 4}}e^{{3i\pi\nu\over 2}}\,.$

It is easy to see that this integral has the same structure as the integral on
the left hand side of relation (150) in appendix C for $r=2$, therefore we can
write for him taking as $\gamma_{1}=Q/2+\beta_{5}-\beta_{2}+\alpha$ and
$\gamma_{2}=3Q/2-\beta_{5}-\beta_{2}+\alpha$

$\displaystyle I_{\alpha\beta_{2}}(\beta_{5})=C^{\prime}e^{{i\pi\over
4}((\alpha-\beta_{2})^{2}+Q^{2}/4-Q(Q/2+\beta_{5}+2\alpha-2\beta_{2}))}$ (89)
$\displaystyle\times\Lambda(2\alpha-\beta_{5};0)\Lambda(Q-\beta_{5};0)\Lambda(2Q-2\beta_{2}-\beta_{5};0)$
$\displaystyle\times\int_{-i\infty}^{i\infty}\sum_{m=0}^{1}{\Lambda(\pm
y-Q+\beta_{2}+\alpha+\beta_{5}/2;\pm m)\Lambda(\pm
y-\alpha+\beta_{2}+\beta_{5}/2;\pm m)\over\Lambda(2y;2m)\Lambda(-2y;-(2m))}$
$\displaystyle\times\Lambda(\pm y+Q-\alpha-\beta_{2}+\beta_{5}/2;\pm
m)\Lambda(\pm y+\alpha-\beta_{2}+\beta_{5}/2;\pm m)e^{-i\pi y^{2}}e^{-i\pi
m^{2}}{dy\over 2i}\,.$

Note that integral in the r.h.s. of (89) coincides with the integral in the
r.h.s. of (146) in appendix B for $r=2$. Thus we can write for the latter

$\displaystyle\int_{-i\infty}^{i\infty}\sum_{m=0}^{1}\Lambda(\pm
y-Q/2+\beta_{2}+\beta_{5}/2;\pm m)\Lambda(\pm y+Q/2-\beta_{2}+\beta_{5}/2;\pm
m)e^{2i\pi y(-Q/2+\alpha)}{dy\over i}=$
$\displaystyle\Lambda(2\alpha-\beta_{5};0)\Lambda(-2\alpha-\beta_{5}+2Q;0)e^{-i\pi\left({1\over
4}\beta_{5}^{2}-(-Q/2+\alpha)^{2}-(Q/2-\beta_{2})^{2}\right)}$
$\displaystyle\times\int_{-i\infty}^{i\infty}\sum_{m=0}^{1}{\Lambda(\pm
y-Q+\beta_{2}+\alpha+\beta_{5}/2;\pm m)\Lambda(\pm
y-\alpha+\beta_{2}+\beta_{5}/2;\pm m)\over\Lambda(2y;2m)\Lambda(-2y;-(2m))}$
$\displaystyle\times\Lambda(\pm y+Q-\alpha-\beta_{2}+\beta_{5}/2;\pm
m)\Lambda(\pm y+\alpha-\beta_{2}+\beta_{5}/2;\pm m)e^{-i\pi y^{2}}e^{-i\pi
m^{2}}{dy\over 2i}\,.$

Collecting all we obtain:

$\displaystyle S_{\alpha\beta_{2}}(\beta_{5})=S_{0\beta_{2}}{e^{-{i\pi\over
2}\Delta_{\beta_{5}}}\over S_{1}(\beta_{5})}\quad$ (90)
$\displaystyle\times\int_{-i\infty}^{i\infty}\sum_{m=0}^{1}\Lambda(\pm
y-Q/2+\beta_{2}+\beta_{5}/2;\pm m)\Lambda(\pm y+Q/2-\beta_{2}+\beta_{5}/2;\pm
m)e^{2i\pi y(-Q/2+\alpha)}{dy\over i}$

or using (76):

$\displaystyle S_{\alpha\beta_{2}}(\beta_{5})=S_{0\beta_{2}}{e^{-{i\pi\over
2}\Delta_{\beta_{5}}}\over S_{\rm NS}(\beta_{5})}\quad$ (91)
$\displaystyle\times\int_{-i\infty}^{i\infty}\Big{[}S_{\rm NS}(\pm
y-Q/2+\beta_{2}+\beta_{5}/2)S_{\rm NS}(\pm y+Q/2-\beta_{2}+\beta_{5}/2)$
$\displaystyle+S_{\rm R}(\pm y-Q/2+\beta_{2}+\beta_{5}/2)S_{\rm R}(\pm
y+Q/2-\beta_{2}+\beta_{5}/2)\Big{]}e^{2i\pi y(-Q/2+\alpha)}{dy\over i}\,.$

This expression has number of properties analogous to that of in usual
Liouville field theory [29]. First we obtain easily

$S_{\alpha\beta_{2}}(\beta_{5})=S_{Q-\alpha,\beta_{2}}(\beta_{5})=S_{\alpha,Q-\beta_{2}}(\beta_{5})\,.$
(92)

Using formula (E) in apendix E for $r=2$ one can establish

$\displaystyle S_{\alpha\beta_{2}}(Q-\beta_{5})=S_{\rm
NS}(-Q+2\alpha+\beta_{5})S_{\rm NS}(Q-2\alpha+\beta_{5})$ (93)
$\displaystyle\times S_{\rm NS}(2Q-\beta_{5}-2\beta_{2})S_{\rm
NS}(-\beta_{5}+2\beta_{2})S_{\alpha\beta_{2}}(\beta_{5})\,.$

Combining (92) and (93) with definition of the normalization factors (66) and
using formulae (126) in apendix A we obtain that

$S^{SL}_{\alpha\beta_{2}}(\beta_{5})={{\cal
N}_{NS}(\alpha,\beta_{5},\alpha)\over{\cal
N}_{NS}(\beta_{2},\beta_{5},\beta_{2})}S_{\alpha\beta_{2}}(\beta_{5})$ (94)

is invariant under reflection of all three variables.

Now let us check that (91) is self-consistent in the limit $\beta_{5}\to 0$.
In this limit using formulae in appendix A one has that ${1\over S_{\rm
NS}(\beta_{5})}\sim\pi\beta_{5}\to 0$, and thus we should pick up only
diverging part of integral. This diverging part originates due to pinching of
integration contour in this limit between poles of $S_{\rm
NS}(y-Q/2+\beta_{2}+\beta_{5}/2)$ and $S_{\rm
NS}(-y+Q/2-\beta_{2}+\beta_{5}/2)$ and also between poles of $S_{\rm
NS}(-y-Q/2+\beta_{2}+\beta_{5}/2)$ and $S_{\rm
NS}(y+Q/2-\beta_{2}+\beta_{5}/2)$. Analyzing poles of $S_{R}(x)$ reviewed in
appendix A we see that the second term in (91) has no diverging part. The
diverging part caused by pinching is pole with coefficient given by the sum of
residues:

$\displaystyle{2\pi i\over i\pi^{2}\beta_{5}}S_{\rm NS}(-Q+2\beta_{2})S_{\rm
NS}(Q-2\beta_{2})(e^{2i\pi(Q/2-\beta_{2})(-Q/2+\alpha)}+e^{-2i\pi(Q/2-\beta_{2})(-Q/2+\alpha)})$
$\displaystyle=-{1\over\pi\beta_{5}}{\cos\pi(Q/2-\beta_{2})(-Q/2+\alpha)\over\sin{\pi\over
b}(Q/2-\beta_{2})\sin\pi b(Q/2-\beta_{2})}\,,$

where the product $S_{\rm NS}(-Q+2\beta_{2})S_{\rm NS}(Q-2\beta_{2})$ was
computed using formulae (120) and (121) in appendix A. Recalling (48) and (45)
we obtain that indeed

$\ \mathrel{\mathop{\lim}\limits_{\beta_{5}\to
0}}S^{SL}_{\alpha\beta_{2}}(\beta_{5})=S_{\alpha\beta_{2}}\,.$ (95)

##  4 Elements of S-matrix for two NS and one ÑS primaries

Here we consider the case when $\alpha,\beta_{2}$ are NS primaries and
$\beta_{5}$ is ÑS primary. In this case the relation (53) takes the form

$S^{SL}_{\alpha\beta_{2}}(\tilde{\beta_{5}})={{\cal
N}_{R}(\alpha,\beta_{5},\alpha)\over{\cal
N}_{R}(\beta_{2},\beta_{5},\beta_{2})}S_{\alpha\beta_{2}}(\tilde{\beta_{5}})\,,$
(96)

where $S_{\alpha\beta_{2}}(\tilde{\beta_{5}})$ is given by the relation:

$S_{0\beta_{2}}I_{\alpha\beta_{2}}(\tilde{\beta_{5}})=e^{\pi
i(2\Delta_{\alpha}+2\Delta_{\beta_{2}})}\big{\\{}\,{}^{\alpha}_{\alpha}\,{}^{\alpha}_{\alpha}\,|\,{}^{\delta}_{\beta_{5}}\big{\\}}_{00}^{11}\;S_{\alpha\beta_{2}}(\tilde{\beta_{5}})$
(97)

and

$\displaystyle
I_{\alpha\beta_{2}}(\tilde{\beta_{5}})=\Bigg{[}\int{d\beta_{4}\over
i}\big{\\{}\,{}^{\alpha}_{\alpha}\,{}^{\beta_{2}}_{\beta_{2}}\,|\,{}^{\delta}_{\beta_{4}}\big{\\}}_{11}^{11}\;e^{2\pi
i\Delta_{\beta_{4}}}\;\big{\\{}\,{}^{\beta_{2}}_{\alpha}\,{}^{\alpha}_{\beta_{2}}\,|\,{}^{\beta_{4}}_{\beta_{5}}\big{\\}}_{00}^{11}$
(98) $\displaystyle+\int{d\beta_{4}\over
i}\big{\\{}\,{}^{\alpha}_{\alpha}\,{}^{\beta_{2}}_{\beta_{2}}\,|\,{}^{\delta}_{\beta_{4}}\big{\\}}_{00}^{11}\;e^{2\pi
i\Delta_{\tilde{\beta_{4}}}}\;\big{\\{}\,{}^{\beta_{2}}_{\alpha}\,{}^{\alpha}_{\beta_{2}}\,|\,{}^{\beta_{4}}_{\beta_{5}}\big{\\}}_{00}^{00}\Bigg{]}\,.$

Here $\delta$ should be taken to $0$. The first terms in the first and second
lines
$\big{\\{}\,{}^{\alpha}_{\alpha}\,{}^{\beta_{2}}_{\beta_{2}}\,|\,{}^{\delta}_{\beta_{4}}\big{\\}}_{11}^{11}$
and
$\big{\\{}\,{}^{\alpha}_{\alpha}\,{}^{\beta_{2}}_{\beta_{2}}\,|\,{}^{\delta}_{\beta_{4}}\big{\\}}_{00}^{11}$
are the same as before and computed in (80) and (83) respectively.

To compute the second fusion matrix element in the first line we set in (68)
$\nu_{1}=\nu_{2}=0$, $\nu_{3}=\nu_{4}=1$, and $\alpha_{1}=\beta_{2}$,
$\alpha_{2}=\alpha$, $\alpha_{3}=\alpha$, $\alpha_{4}=Q-\beta_{2}$,
$\alpha_{t}=\beta_{5}$, $\alpha_{s}=\beta_{4}$ and receive

$\displaystyle\big{\\{}\,{}^{\beta_{2}}_{\alpha}\,{}^{\alpha}_{\beta_{2}}\,|\,{}^{\beta_{4}}_{\beta_{5}}\big{\\}}_{00}^{11}=|S_{1}(2\beta_{5})|^{2}{S_{1}(\alpha+\beta_{4}-\beta_{2})S_{0}(2\beta_{2}+\beta_{5}-Q)\over
S_{1}(\beta_{4}+\alpha+\beta_{2}-Q)S_{0}(\beta_{5})}$
$\displaystyle\times\int_{-\textup{i}\infty}^{\textup{i}\infty}\sum_{\nu=0}^{1}\Lambda(-y+Q/2+\beta_{4},-\nu-1)\Lambda(-y+3Q/2-\beta_{4},-\nu-1)F_{2}(y,\nu+1)(-1)^{\nu-1}{dy\over
2i}\,,$

where we defined

$\displaystyle
F_{2}(y,\nu)=\Lambda(y-3Q/2+\beta_{2}+\alpha,\nu)\Lambda(y-Q/2+\beta_{2}-\alpha,\nu)$
(100)
$\displaystyle\times\Lambda(y-Q/2-\alpha+\beta_{2},\nu)\Lambda(y+Q/2-\alpha-\beta_{2},\nu)$
$\displaystyle\times\Lambda(-y+3Q/2-\beta_{5}-\beta_{2}+\alpha,-\nu+1)\Lambda(-y+Q/2+\beta_{5}-\beta_{2}+\alpha,-\nu+1)\,.$

To derive the second fusion matrix element in the second line we set in (68):
$\nu_{1}=\nu_{2}=0$, $\nu_{3}=\nu_{4}=0$, and $\alpha_{1}=\beta_{2}$,
$\alpha_{2}=\alpha$, $\alpha_{3}=\alpha$, $\alpha_{4}=Q-\beta_{2}$,
$\alpha_{t}=\beta_{5}$, $\alpha_{s}=\beta_{4}$ and obtain

$\displaystyle\big{\\{}\,{}^{\beta_{2}}_{\alpha}\,{}^{\alpha}_{\beta_{2}}\,|\,{}^{\beta_{4}}_{\beta_{5}}\big{\\}}_{00}^{00}=|S_{1}(2\beta_{5})|^{2}{S_{0}(\alpha+\beta_{4}-\beta_{2})S_{0}(2\beta_{2}+\beta_{5}-Q)\over
S_{0}(\beta_{4}+\alpha+\beta_{2}-Q)S_{0}(\beta_{5})}$ (101)
$\displaystyle\times\int_{-\textup{i}\infty}^{\textup{i}\infty}\sum_{\nu=0}^{1}\Lambda(-y+Q/2+\beta_{4},-\nu-1)\Lambda(-y+3Q/2-\beta_{4},-\nu-1)F_{2}(y,\nu){dy\over
2i}\,.$

Collecting all, and repeating the same steps as in the first case and using
the integral (136) in appendix A, we obtain

$\displaystyle
I_{\alpha\beta_{2}}(\tilde{\beta}_{5})=C^{\prime\prime}\int_{-\textup{i}\infty}^{\textup{i}\infty}{dy\over
i}\sum_{\nu=0}^{1}\Lambda(-y-3Q/2+\beta_{2}+\alpha,-\nu)\Lambda(-y-Q/2-\alpha+\beta_{2},-\nu)\qquad$
(102)
$\displaystyle\times\Lambda(-y+Q/2-\alpha-\beta_{2},-\nu)\Lambda(y+3Q/2-\beta_{5}-\beta_{2}+\alpha,\nu+1)$
$\displaystyle\times\Lambda(y+Q/2+\beta_{5}-\beta_{2}+\alpha,\nu+1)e^{-{i\pi\over
4}y^{2}}e^{{i\pi\over 2}y(\alpha-\beta_{2})}e^{-{i\pi\nu\over
4}}e^{{i\pi\nu\over 2}}\,,$

where

$\displaystyle
C^{\prime\prime}=S_{1}(\delta)\Lambda(\delta,0)\Lambda(-2\alpha,-2){|S_{1}(2\beta_{5})|^{2}S_{0}(2\beta_{2}+\beta_{5}-Q)\over
S_{0}(\beta_{5})}$ (103) $\displaystyle\times e^{-{i\pi\over
4}(-Q/2+\alpha-\beta_{2})^{2}}e^{{i\pi\over
4}(3Q/2+\alpha-\beta_{2})Q}e^{Q^{2}\pi i/4}\,.$

The integral (102) is of the form (150) in appendix C, which yields

$\displaystyle
I_{\alpha\beta_{2}}(\tilde{\beta}_{5})=C^{\prime\prime}e^{{i\pi\over
4}((\alpha-\beta_{2})^{2}+Q^{2}/4-Q(Q/2+\beta_{5}+2\alpha-2\beta_{2}))}e^{i\pi\over
4}$ (104)
$\displaystyle\times\Lambda(2\alpha-\beta_{5};1)\Lambda(Q-\beta_{5};1)\Lambda(2Q-2\beta_{2}-\beta_{5};1)\int_{-i\infty}^{i\infty}\sum_{m=0}^{1}$
$\displaystyle{\Lambda(\pm y-Q+\beta_{2}+\alpha+\beta_{5}/2;\pm m-\theta(\mp
1))\Lambda(\pm y-\alpha+\beta_{2}+\beta_{5}/2;\pm m-\theta(\mp
1))\over\Lambda(2y;2m+1)\Lambda(-2y;-2m-1)}$ $\displaystyle\times\Lambda(\pm
y+Q-\alpha-\beta_{2}+\beta_{5}/2;\pm m-\theta(\mp 1))\Lambda(\pm
y+\alpha-\beta_{2}+\beta_{5}/2;\pm m+1+\theta(\pm 1))$ $\displaystyle\times
e^{-i\pi y^{2}}{dy\over 2i}\,.$

Here $\theta(x)$ is Heaviside step function: $\theta(-1)=0$ and $\theta(1)=1$.
As before the functions with double sign in arguments are used as shorthand
for their product where the first term is taken with upper signs everywhere
and the second term is taken with lower signs everywhere. For example the
first term in integrand reads

$\displaystyle\Lambda(\pm y-Q+\beta_{2}+\alpha+\beta_{5}/2;\pm m-\theta(\mp
1))\equiv$ (105)
$\displaystyle\Lambda(y-Q+\beta_{2}+\alpha+\beta_{5}/2;m)\Lambda(-y-Q+\beta_{2}+\alpha+\beta_{5}/2;-m-1)\,.$

And finally we express the r.h.s. of (104) via (146) in appendix B

$\displaystyle\int_{-i\infty}^{i\infty}\sum_{m=0}^{1}\Lambda(y-Q/2+\beta_{2}+\beta_{5}/2;m-1)\Lambda(y+Q/2-\beta_{2}+\beta_{5}/2;m+1)$
(106)
$\displaystyle\Lambda(-y-Q/2+\beta_{2}+\beta_{5}/2;-m)\Lambda(-y+Q/2-\beta_{2}+\beta_{5}/2;-m)e^{2i\pi
y(-Q/2+\alpha)}e^{i\pi m}{dy\over i}=$
$\displaystyle\Lambda(2\alpha-\beta_{5};1)\Lambda(-2\alpha-\beta_{5}+2Q;-1)e^{-i\pi\left({1\over
4}\beta_{5}^{2}-(\alpha-Q/2)^{2}-(Q/2-\beta_{2})^{2}\right)}e^{i\pi\over
2}\int_{-i\infty}^{i\infty}\sum_{m=0}^{1}$ $\displaystyle{\Lambda(\pm
y-Q+\beta_{2}+\alpha+\beta_{5}/2;\pm m-\theta(\mp 1))\Lambda(\pm
y-\alpha+\beta_{2}+\beta_{5}/2;\pm m-\theta(\mp
1))\over\Lambda(2y;2m+1)\Lambda(-2y;-2m-1)}$ $\displaystyle\times\Lambda(\pm
y+Q-\alpha-\beta_{2}+\beta_{5}/2;\pm m-\theta(\mp 1))\Lambda(\pm
y+\alpha-\beta_{2}+\beta_{5}/2;\pm m+1+\theta(\pm 1))$ $\displaystyle e^{-i\pi
y^{2}}{dy\over 2i}\,.$

Collecting all we derive

$\displaystyle
S_{\alpha\beta_{2}}(\tilde{\beta}_{5})=-S_{0\beta_{2}}{e^{-{i\pi\over
2}(\Delta_{\beta_{5}}+1/2)}\over S_{0}(\beta_{5})}\quad$ (107)
$\displaystyle\times\int_{-i\infty}^{i\infty}\sum_{m=0}^{1}\Lambda(y-Q/2+\beta_{2}+\beta_{5}/2;m-1)\Lambda(y+Q/2-\beta_{2}+\beta_{5}/2;m+1)\quad$
$\displaystyle\times\Lambda(-y-Q/2+\beta_{2}+\beta_{5}/2;-m)\Lambda(-y+Q/2-\beta_{2}+\beta_{5}/2;-m)e^{2i\pi
y(-Q/2+\alpha)}e^{i\pi m}{dy\over i}\,,$

or using (76):

$\displaystyle
S_{\alpha\beta_{2}}(\tilde{\beta}_{5})=S_{0\beta_{2}}{e^{-{i\pi\over
2}\Delta_{\beta_{5}}}\over S_{\rm R}(\beta_{5})}\quad$ (108)
$\displaystyle\times\int_{-i\infty}^{i\infty}\Big{[}S_{\rm
NS}(y-Q/2+\beta_{2}+\beta_{5}/2)S_{\rm NS}(y+Q/2-\beta_{2}+\beta_{5}/2)\quad$
$\displaystyle\times S_{\rm R}(-y-Q/2+\beta_{2}+\beta_{5}/2)S_{\rm
R}(-y+Q/2-\beta_{2}+\beta_{5}/2)$ $\displaystyle+S_{\rm
R}(y-Q/2+\beta_{2}+\beta_{5}/2)S_{\rm R}(y+Q/2-\beta_{2}+\beta_{5}/2)\quad$
$\displaystyle\times S_{\rm NS}(-y-Q/2+\beta_{2}+\beta_{5}/2)S_{\rm
NS}(-y+Q/2-\beta_{2}+\beta_{5}/2)\big{]}e^{2i\pi y(-Q/2+\alpha)}{dy\over
i}\,.$

Note the following properties of this expression. First we obtain easily

$S_{\alpha\beta_{2}}(\tilde{\beta_{5}})=S_{Q-\alpha,\beta_{2}}(\tilde{\beta_{5}})=S_{\alpha,Q-\beta_{2}}(\tilde{\beta_{5}})\,.$
(109)

Using formula (E) in apendix E one can establish

$\displaystyle S_{\alpha\beta_{2}}(\widetilde{Q-\beta_{5}})=S_{\rm
R}(-Q+2\alpha+\beta_{5})S_{\rm R}(Q-2\alpha+\beta_{5})$ (110)
$\displaystyle\times S_{\rm R}(2Q-\beta_{5}-2\beta_{2})S_{\rm
R}(-\beta_{5}+2\beta_{2})S_{\alpha\beta_{2}}(\tilde{\beta}_{5})\,.$

Combining (109) and (110) with definition of the normalization factors (67)
and using formulae (126) in apendix A we obtain that

$S^{SL}_{\alpha\beta_{2}}(\tilde{\beta}_{5})={{\cal
N}_{R}(\alpha,\beta_{5},\alpha)\over{\cal
N}_{R}(\beta_{2},\beta_{5},\beta_{2})}S_{\alpha\beta_{2}}(\tilde{\beta}_{5})$
(111)

is invariant under the reflection of all three variables.

##  5 Conclusion

In this paper we calculated the matrix $S_{\alpha\beta_{2}}(\beta_{5})$ in two
cases. These results allow us to conjecture that in all cases the integral
part of the matrix $S_{\alpha\beta_{2}}(\beta_{5})$ will be given by the
following function with the various choices of discrete variables
$k_{a},t_{a}$, $a=1,2$ and $N$:

$\displaystyle
F_{\alpha\beta_{2}}(\beta_{5})(\underline{k},\underline{t},N)=\int_{-i\infty}^{i\infty}\sum_{m=0}^{1}\Lambda(y-Q/2+\beta_{2}+\beta_{5}/2;k_{1}+m)$
$\displaystyle\times\Lambda(y+Q/2-\beta_{2}+\beta_{5}/2;k_{2}+m)\times\Lambda(-y-Q/2+\beta_{2}+\beta_{5}/2;t_{1}-m)$
$\displaystyle\times\Lambda(-y+Q/2-\beta_{2}+\beta_{5}/2;t_{2}-m)e^{2i\pi
y(-Q/2+\alpha)}e^{i\pi mN}dy\,.$

This conjecture will be studied in future works. Another interesting problem
is the role of S-move matrix in the AGT correspondence. Remember that
logarithm of one-point toric block in semi-classical limit yields Seiberg-
Witten prepotential of ${\cal N}=2^{*}$ theory [7, 8, 12]. Therefore
asymptotic form of the S-move matrix of Liouville field theory implements
electromagnetic duality of SW prepotential. For Liouville field theory this
was studied in [8, 12, 11]. We are planning to pursue these issues for S-move
matrix in $N=1$ SLFT using relation between the Nekrasov partition function of
${\cal N}=2^{*}$ theory on $\mathbb{C}^{2}/\mathbb{Z}_{2}$ and $N=1$ SLFT one-
point toric blocks found in [5, 6].

Acknowledgements. The work of Elena Apresyan was supported by Armenian SCS
grants 21AG‐1C024 and 20TTAT-QTa009. We would like also to thank V. P.
Spiridonov for many useful discussions.

##  Appendix A Properties of supersymmetric hyperbolic and double gamma
functions

The function $S_{b}(y)$ has the integral representation

$S_{b}(y)=\exp\left(-\int_{0}^{\infty}\left({\sinh(2y-b-b^{-1})x\over
2\sinh(bx)\sinh(b^{-1}x)}-{2y-b-b^{-1}\over 2x}\right)\right){dx\over x}\,,$
(112)

and obeys the equations:

${S_{b}(y+b)\over S_{b}(y)}=2\sin\pi by\,,\quad{S_{b}(y+b^{-1})\over
S_{b}(y)}=2\sin{\pi y\over b}.$ (113)

Let us mention for further use the following of properties of the the function
$\Lambda(y;m;\omega_{1},\omega_{2})$. The function
$\Lambda(y;m;\omega_{1},\omega_{2})$ has the asymptotics [15, 20]:

$\
\mathrel{\mathop{\lim}\limits_{y\to\infty}}\Lambda(y;m)=e^{\mp{\textup{i}\pi\over
2}\left({1\over r}B_{2,2}(y,b,b^{-1})+{m^{2}\over r}-m+{r\over 6}-{1\over
6r}\right)},\quad{\rm for}\quad\mathfrak{Im}(y)\to\pm\infty\,,$ (114)

where $B_{2,2}(y,b,b^{-1})$ is the second order Bernoulli polynomial

$B_{2,2}(y;b,b^{-1})=\left(y-\frac{Q}{2}\right)^{2}-\frac{b^{2}+b^{-2}}{12}\,.$
(115)

The poles and zeros of the function $\Lambda(y;m;\omega_{1},\omega_{2})$ are
given by the relations [15, 21]:

$u^{(m)}_{\rm poles}\in\\{-n_{1}b-n_{2}b^{-1}\\},\;u^{(m)}_{\rm
zeros}\in\\{(n_{1}+1)b+(n_{2}+1)b^{-1}\\},$ (116)

where $n_{1}-n_{2}=m\ {\rm mod}\ r,\;n_{1},n_{2}\in{\mathbb{Z}}_{\geq 0}$.

The function $\Lambda(y;m)$ satisfies the relation:

${\Lambda(y+b,m-1)\over\Lambda(y,m)}=-2\sin{\pi b(y-mb^{-1})\over
r},\;{\Lambda(y+b^{-1},m+1)\over\Lambda(y,m)}=2\sin{\pi(y+mb)\over rb}.$ (117)

Specializing these properties to the case $r=2$ we derive, that poles and
zeros of $S_{NS}(x)$ are given by the set

$u^{(0)}_{\rm poles}\in\\{-n_{1}b-n_{2}b^{-1}\\},\;u^{(0)}_{\rm
zeros}\in\\{(n_{1}+1)b+(n_{2}+1)b^{-1}\\},$ (118)

where $n_{1}-n_{2}=0\ {\rm mod}\ 2,\;n_{1},n_{2}\in{\mathbb{Z}}_{\geq 0}$ ,
and poles and zeros of $S_{R}(x)$ are given by the set

$u^{(1)}_{\rm poles}\in\\{-n_{1}b-n_{2}b^{-1}\\},\;u^{(1)}_{\rm
zeros}\in\\{(n_{1}+1)b+(n_{2}+1)b^{-1}\\},$ (119)

where $n_{1}-n_{2}=1\ {\rm mod}\ 2,\;n_{1},n_{2}\in{\mathbb{Z}}_{\geq 0}$.

The relations (117) for the case $r=2$ take the form:

${S_{NS}(y+b)\over S_{R}(y)}=-2\sin{\pi b(y-b^{-1})\over
2}\,,\quad{S_{NS}(y+b^{-1})\over S_{R}(y)}=2\sin{\pi(y+b)\over 2b}$ (120)

and

${S_{R}(y+b)\over S_{NS}(y)}=2\sin{\pi by\over 2}\,,\quad{S_{R}(y+b^{-1})\over
S_{NS}(y)}=2\sin{\pi y\over 2b}.$ (121)

We need also relations of sypersymmetric hyperbolic gamma functions with the
supersymmetric double gamma functions [10, 16].

Start with defintion of $\Gamma_{b}(x)$ function.

The function $\Gamma_{b}(x)$ is a close relative of the double Gamma function
studied in [2, 22]. It can be defined by means of the integral representation

$\log\Gamma_{b}(x)=\int_{0}^{\infty}{dt\over
t}\left({e^{-xt}-e^{-Qt/2}\over(1-e^{-bt})(1-e^{-t/b})}-{(Q-2x)^{2}\over
8e^{t}}-{Q-2x\over t}\right)\,.$ (122)

Important properties of $\Gamma_{b}(x)$ are

  1. 1.

Functional equation: $\Gamma_{b}(x+b)=\sqrt{2\pi}b^{bx-{1\over
2}}\Gamma^{-1}(bx)\Gamma_{b}(x)$.

  2. 2.

Analyticity: $\Gamma_{b}(x)$ is meromorphic, poles: $x=-nb-
mb^{-1},n,m\in\mathbb{Z}^{\geq 0}$.

  3. 3.

Self-duality: $\Gamma_{b}(x)=\Gamma_{1/b}(x)$.

The function $S_{b}(x)$ may be defined in terms of $\Gamma_{b}(x)$ as follows

$S_{b}(x)={\Gamma_{b}(x)\over\Gamma_{b}(Q-x)}\,.$ (123)

Similar role in the super Liouville theory play the functions:

$\Gamma_{1}(x)\equiv\Gamma_{\rm NS}(x)=\Gamma_{b}\left({x\over
2}\right)\Gamma_{b}\left({x+Q\over 2}\right)\,,$ (124)
$\Gamma_{0}(x)\equiv\Gamma_{\rm R}(x)=\Gamma_{b}\left({x+b\over
2}\right)\Gamma_{b}\left({x+b^{-1}\over 2}\right)\,.$ (125)

The sypersymmetric hyperbolic gamma functions are related to them by the
formulae similar to (123)

$S_{1}(x)\equiv S_{\rm NS}(x)={\Gamma_{\rm NS}(x)\over\Gamma_{\rm
NS}(Q-x)}\,,\quad S_{0}(x)\equiv S_{\rm R}(x)={\Gamma_{\rm
R}(x)\over\Gamma_{\rm R}(Q-x)}\,.$ (126)

##  Appendix B Parafermionic integral identities I

In paper [20] the rarefied hyperbolic beta integral has been evaluated‡‡‡We
put periods $\omega_{1}$ and $\omega_{2}$ there equal $b^{-1}$ and $b$
correspondingly and also $\epsilon=0$.:

$\displaystyle\int_{-i\infty}^{i\infty}\sum_{m=0}^{r-1}{\prod_{a=1}^{6}\Lambda(s_{a}\pm
y,n_{a}\pm m)\over\Lambda(\pm 2y,\pm 2m)}{dy\over i}=2r\prod_{1\leq a<b\leq
6}\Lambda(s_{a}+s_{b},n_{a}+n_{b})\,,\qquad$ (127)

where

$\sum_{a=1}^{6}n_{a}=0\,,\quad\sum_{a=1}^{6}s_{a}=Q\,.$ (128)

Here we used the shorthand notation $\Lambda(y\pm x;m\pm
n)\equiv\Lambda(y+x;m+n)\Lambda(y-x;m-n)$.

It is shown in [20], that in certain limit (127) yields

$\displaystyle\int_{-i\infty}^{i\infty}\sum_{m=0}^{r-1}(-1)^{m}\prod_{a=1}^{3}\Lambda(y+f_{a},n_{a}+m)\Lambda(-y+g_{a},l_{a}-m){dy\over
i}\quad\quad\quad$ (129)
$\displaystyle=(-1)^{n_{1}+n_{2}+n_{3}}r\prod_{a,b=1}^{3}\Lambda(f_{a}+g_{b},n_{a}+l_{b})\,,$

where

$\sum_{a=1}^{3}(n_{a}+l_{a})=0\,,\quad\sum_{a=1}^{3}(f_{a}+g_{a})=Q\,.$ (130)

We can show that taking the following limit in (127):

$s_{5}\to\textup{i}\infty,\qquad
s_{6}=Q-s_{5}-\sum_{j=1}^{4}s_{j}\to-\textup{i}\infty$ (131)

and using the asymptotics (114) one derives

$\displaystyle\int_{-i\infty}^{i\infty}\sum_{m=0}^{r-1}{\prod_{a=1}^{4}\Lambda(s_{a}\pm;n_{a}\pm
m)\over\Lambda(\pm 2y;\pm 2m)}{dy\over i}=2r{\prod_{1\leq a<b\leq
4}\Lambda(s_{a}+s_{b};n_{a}+n_{b})\over\Lambda(\sum_{i=1}^{4}s_{i};\sum_{i=1}^{4}n_{i})}\,.\qquad$
(132)

Taking in (132) further limit

$s_{4}\to\textup{i}\infty$ (133)

and again using the asymptotics (114) one gets

$\displaystyle\int_{-i\infty}^{i\infty}\sum_{m=0}^{r-1}{\prod_{a=1}^{3}\Lambda(s_{a}\pm;n_{a}\pm
m)\over\Lambda(\pm 2y;\pm 2m)}e^{-{i\pi\over r}y^{2}}e^{-{i\pi\over
r}m^{2}}{dy\over i}=2re^{{i\pi\over
r}(s_{1}s_{2}+s_{2}s_{3}+s_{1}s_{3})}\qquad$ (134) $\displaystyle\times
e^{{i\pi\over
r}(n_{1}n_{2}+n_{2}n_{3}+n_{1}n_{3})}\Lambda(s_{1}+s_{2};n_{1}+n_{2})\Lambda(s_{2}+s_{3};n_{2}+n_{3})\Lambda(s_{1}+s_{3};n_{1}+n_{3}).$

And finally taking the limit

$s_{3}\to\textup{i}\infty$ (135)

in (134) and again using the asymptotics (114) we obtain

$\displaystyle\int_{-i\infty}^{i\infty}\sum_{m=0}^{r-1}{\prod_{a=1}^{2}\Lambda(s_{a}\pm
y;n_{a}\pm m)\over\Lambda(\pm 2y;\pm 2m)}e^{-{2i\pi\over
r}y^{2}}e^{-{2i\pi\over r}m^{2}}{dy\over i}\quad\quad$ (136)
$\displaystyle=2re^{-{i\pi\over 2r}(s_{1}-s_{2})^{2}}e^{{i\pi\over
2r}(s_{1}+s_{2})Q}e^{-{i\pi\over 2r}(n_{1}-n_{2})^{2}}e^{{i\pi\over
2}(n_{1}+n_{2})}\Lambda(s_{1}+s_{2};n_{1}+n_{2}).$

##  Appendix C Parafermionic integral identities II

In paper [1] some parafermionic hypergeometric functions were defined. The
first is the parafermionic generalization of the Ruijsenaars function [19]:

$\displaystyle
J(\underline{\beta},\underline{l};\underline{\gamma},\underline{t})=\int_{-i\infty}^{i\infty}\sum_{m=0}^{r-1}\prod_{a=1}^{4}\Lambda(y+\gamma_{a};t_{a}+m)\Lambda(-y+\beta_{a};l_{a}-m){dy\over
ir}\ .\qquad$ (137)

Here $t_{a}\,,l_{a}\in\mathbb{Z}$. Parameters $\gamma_{j}$, $\beta_{j}$ and
$l_{j}$, $t_{j}$ satisfy the constraints:

$\sum_{j=1}^{4}(\gamma_{j}+\beta_{j})=2Q,\quad\sum_{j=1}^{4}(l_{j}+t_{j})=0.$
(138)

The second function is :

$\displaystyle
E_{\delta}(s_{a},n_{a})=\int_{-i\infty}^{i\infty}\sum_{m=0}^{r-1}{\prod_{a=1}^{6}\Lambda(y+s_{a};n_{a}+m+\delta)\Lambda(-y+s_{a};n_{a}-m)\over\Lambda(2y;2m+\delta)\Lambda(-2y;-(2m+\delta))}{dy\over
2ir}\,.\quad$ (139)

We proved in [1] the following relation between (137) and (139):

$\displaystyle
J(\underline{\beta},\underline{l};\underline{\gamma},\underline{t})=e^{i\pi(t_{4}+l_{4})}\prod_{i=1}^{3}\Lambda(\beta_{i}+\gamma_{4};l_{i}+t_{4})\Lambda(\gamma_{i}+\beta_{4};t_{i}+l_{4})$
(140) $\displaystyle\times
E_{\delta}(\beta_{1}+\xi,l_{1}-L_{1};\beta_{2}+\xi,l_{2}-L_{1};\beta_{3}+\xi,l_{3}-L_{1};$
$\displaystyle\gamma_{1}-\xi,t_{1}-L_{2};\gamma_{2}-\xi,t_{2}-L_{2};\gamma_{3}-\xi,t_{3}-L_{2})\,,\quad$
(141)

where

$L_{1}={1\over 2}(l_{1}+l_{2}+l_{3}+t_{4}+\delta)\,,$ (142) $L_{2}={1\over
2}(t_{1}+t_{2}+t_{3}+l_{4}+\delta)\,,$ (143) $\xi={1\over
2}(Q-\gamma_{4}-\sum_{j=1}^{3}\beta_{j}).$ (144)

The parameter $\delta=0,1$ should be determined from the requirement that
$L_{1}$ and $L_{2}$ are integer numbers. Let us now paparemeterize parameters
$\underline{\beta}$, $\underline{\gamma}$ in (140) in the following way:

$\begin{aligned} &\gamma_{1}=A-\alpha_{1}+Q/2\,,\\\
&\gamma_{2}=\beta_{2}=\alpha_{2}/2-\alpha_{3}+Q/4\,,\\\
&\gamma_{3}=\beta_{3}=\alpha_{2}/2+\alpha_{3}+Q/4\,,\end{aligned}\qquad\begin{aligned}
&\gamma_{4}=-A-\alpha_{2}+\alpha_{1}\,,\\\ &\beta_{1}=A+\alpha_{1}+Q/2\,,\\\
&\beta_{4}=-A-\alpha_{2}-\alpha_{1}\,.\end{aligned}$ (145)

Obviously parametrized in this way $\underline{\beta}$ and
$\underline{\gamma}$ satisfy (138).

Now putting (145) in (140) and taking the limit $A\to\infty$, denoting
$l_{1}+t_{4}=T$, and at the end renaming indices $2,3\to 1,2$ for all
variables, what we can do since variables with indices $1,4$ disappear, we
obtain the identity

$\displaystyle\int_{-i\infty}^{i\infty}\sum_{m=0}^{r-1}\prod_{a=1}^{2}\Lambda(y+\gamma_{a};t_{a}+m)\Lambda(-y+\gamma_{a};l_{a}-m)e^{{4i\over
r}\pi y\alpha_{1}}e^{{i\pi m\over r}(2T+\sum_{a=1}^{2}(t_{a}+l_{a}))}dy=$
(146)
$\displaystyle\Lambda(2\alpha_{1}-\gamma_{1}-\gamma_{2}+Q;T)\Lambda(-2\alpha_{1}-\gamma_{1}-\gamma_{2}+Q;-T-\sum_{a=1}^{2}(t_{a}+l_{a}))e^{-i\pi\left[\sum_{a=1}^{2}(t_{a}+l_{a})+{2\over
r}\left(\gamma_{1}\gamma_{2}-\alpha_{1}^{2}\right)\right]}$
$\displaystyle\times\int_{-i\infty}^{i\infty}\sum_{m=0}^{r-1}{\prod_{a=1}^{2}\Lambda(y+\gamma_{a}-\alpha_{1};l_{a}-M_{1}+m+\delta)\Lambda(-y+\gamma_{a}-\alpha_{1};l_{a}-M_{1}-m)\over\Lambda(2y;2m+\delta)\Lambda(-2y;-(2m+\delta))}$
$\displaystyle\times\prod_{a=1}^{2}\Lambda(y+\gamma_{a}+\alpha_{1};t_{a}-M_{2}+m+\delta)\Lambda(-y+\gamma_{a}+\alpha_{1};t_{a}-M_{2}-m)e^{-{2i\pi\over
r}y^{2}}e^{-{2i\pi\over r}(m^{2}+m\delta)}{dy\over 2}$ $\displaystyle\times
e^{{i\pi\over
2r}\left[l_{1}^{2}+l_{2}^{2}+t_{1}^{2}+t_{2}^{2}-2M_{1}^{2}-2M_{2}^{2}+2(T+l_{1}+l_{2})^{2}-(t_{1}+t_{2})^{2}-(l_{1}+l_{2})^{2}\right]}\,,$

where

$M_{1}={1\over 2}(T+l_{1}+l_{2}+\delta)\,,$ (147) $M_{2}=\delta-M_{1}\,,$
(148)

and $\delta=0,1$ is determined from the requirement that $M_{1}$ and therefore
also $M_{2}$ are integer. This formula for $r=1$ case can be found in [26,
29].

##  Appendix D Parafermionic integral identities III

Now let us set in (140)

$\beta_{4}=-{\lambda\over 3}-2A+Q/2\,,\quad\gamma_{1}=-{\lambda\over
3}+A+Q/2\,,\quad\gamma_{2}=-{\lambda\over 3}+A+Q/2\,.$ (149)

Taking the limit $A\to\infty$, we obtain, after renaming at the end the
variables
$\gamma_{3},\gamma_{4},t_{3},t_{4}\to\gamma_{1},\gamma_{2},t_{1},t_{2}$, the
identity

$\displaystyle\int_{-i\infty}^{i\infty}\sum_{m=0}^{r-1}\prod_{a=1}^{2}\Lambda(y+\gamma_{a};t_{a}+m)\prod_{a=1}^{3}\Lambda(-y+\beta_{a};l_{a}-m)e^{{i\pi\over
r}(\lambda y-{y^{2}\over 2})}e^{{i\pi\over
r}\left[m(t_{1}+t_{2}+\sum_{i=1}^{3}l_{i}+{3r\over 2})-{m^{2}\over
2}\right]}dy\quad\quad\quad$ (150)
$\displaystyle=\int_{-i\infty}^{i\infty}\sum_{m=0}^{r-1}{\prod_{a=1}^{3}\Lambda(y+\beta_{a}+\eta;l_{a}-L_{1}+m+\delta)\Lambda(-y+\beta_{a}+\eta;l_{a}-L_{1}-m)\over\Lambda(2y;2m+\delta)\Lambda(-2y;-(2m+\delta))}$
$\displaystyle\times\Lambda(y+\gamma_{1}-\eta;t_{1}-L_{2}+m+\delta)\Lambda(-y+\gamma_{1}-\eta;t_{1}-L_{2}-m)e^{-{2i\pi\over
r}y^{2}}e^{-{2i\pi\over r}(m^{2}+m\delta)}{dy\over 2}$ $\displaystyle\times
e^{{i\pi\over 2r}(\lambda^{2}+Q^{2}/4-Q(\lambda+\gamma_{1}))}e^{{i\pi\over
2r}\left[4L_{2}^{2}-4L_{2}(t_{1}+\delta)+2\delta
t_{1}+t_{1}^{2}\right]}e^{{i\pi\over
2}\left[-t_{1}+2(l_{1}+l_{2}+l_{3})\right]}\prod_{a=1}^{3}\Lambda(\beta_{a}+\gamma_{2};l_{a}+t_{2})\,,$

where

$\sum_{i=1}^{2}\gamma_{i}+\sum_{i=1}^{3}\beta_{i}=Q/2+\lambda\,,$ (151)

and

$2\eta=Q-\gamma_{2}-\sum_{i=1}^{3}\beta_{i}=Q/2-\lambda+\gamma_{1}\,,$ (152)

$L_{1}={1\over 2}(l_{1}+l_{2}+l_{3}+t_{2}+\delta)\,,$ (153) $L_{2}=\delta-
L_{1}\,,$ (154)

and $\delta=0,1$ is determined from the requirement that $L_{1}$ and therefore
also $L_{2}$ are integer. For $r=1$ case this formula can be found in [29].

##  Appendix E Parafermionic integral identities IV

We proved also in [1] that (137) satisfies the transformation property

$\displaystyle
J(\underline{\beta},\underline{l};\underline{\gamma},\underline{t})=e^{\textup{i}\pi\left[t_{3}+t_{4}+l_{1}+l_{2}\right]}J(\underline{\tilde{\beta}},\underline{\tilde{l}};\underline{\tilde{\gamma}},\underline{\tilde{t}})\times$
(155)
$\displaystyle\prod_{j,k=1}^{2}\Lambda(\gamma_{j}+\beta_{k};t_{j}+l_{k};\omega_{1},\omega_{2})\prod_{j,k=3}^{4}\Lambda(\gamma_{j}+\beta_{k};t_{j}+l_{k};\omega_{1},\omega_{2})\,,$
$\displaystyle\tilde{\beta}_{a}=\beta_{a}+\Theta(a)\eta\,,\quad\tilde{l}_{a}=l_{a}-(\Theta(a)+1)K\,,\quad
a=1,2,3,4\,,$ (156)
$\displaystyle\tilde{\gamma}_{a}=\gamma_{a}+\Theta(a)\eta\,,\;\quad\tilde{t}_{a}=t_{a}-(\Theta(a)-1)K\,,\quad
a=1,2,3,4\,,$

where $\Theta(a)$ is the sign function taking the values

$\Theta(a)=1\,,\quad a=1,2\quad{\rm and}\quad\Theta(a)=-1\,,\quad a=3,4\,,$
(157)

and

$K={1\over 2}(t_{1}+t_{2}+l_{1}+l_{2})=-{1\over
2}(t_{3}+t_{4}+l_{3}+l_{4})\,,$ (158) $\eta={1\over
2}(Q-\gamma_{1}-\gamma_{2}-\beta_{1}-\beta_{2})\,.$ (159)

Setting in (155):

$\displaystyle\gamma_{1}=A-\alpha_{1}+Q/2\,,\quad\beta_{1}=A+\alpha_{1}+Q/2\,,$
(160)
$\displaystyle\gamma_{2}=-A-\alpha_{2}+\alpha_{1}+Q/2\,,\quad\beta_{2}=-A-\alpha_{1}-\alpha_{2}+Q/2\,,$
$\displaystyle\gamma_{3}=\beta_{3}=\alpha_{2}/2-\alpha_{3}\,,\quad\gamma_{4}=\beta_{4}=\alpha_{2}/2+\alpha_{3}\,,$

and denoting

$N={1\over 2}(t_{2}+l_{1}-t_{1}-l_{2})\,,$ (161)

we obtain in the limit $A\to\infty$

$\displaystyle\sum_{m\in\mathbb{Z}_{r}}\int_{-i\infty}^{i\infty}\Lambda(y+\alpha_{2}/2+\alpha_{3},t_{4}+m)\Lambda(-y+\alpha_{2}/2+\alpha_{3},l_{4}-m)$
$\displaystyle\times\Lambda(y+\alpha_{2}/2-\alpha_{3},t_{3}+m)\Lambda(-y+\alpha_{2}/2-\alpha_{3},l_{3}-m)e^{{4i\pi\over
r}y\alpha_{1}}e^{{2miN\pi\over r}}{dy\over i}=$ $\displaystyle
e^{{2iNK\pi\over
r}}e^{i\pi(t_{3}+t_{4}-N)}\Lambda(Q-2\alpha_{1}-\alpha_{2},K-N)\Lambda(Q+2\alpha_{1}-\alpha_{2},K+N)$
$\displaystyle\times\Lambda(\alpha_{2}-2\alpha_{3},t_{3}+l_{3})\Lambda(\alpha_{2}+2\alpha_{3},t_{4}+l_{4})\Lambda(\alpha_{2},t_{3}+l_{4})\Lambda(\alpha_{2},t_{4}+l_{3})$
$\displaystyle\times\sum_{m\in\mathbb{Z}_{r}}\int_{-i\infty}^{i\infty}\Lambda(y+(Q-\alpha_{2})/2+\alpha_{3},t_{4}+2K+m)\Lambda(-y+(Q-\alpha_{2})/2+\alpha_{3},l_{4}-m)$
$\displaystyle\times\Lambda(y+(Q-\alpha_{2})/2-\alpha_{3},t_{3}+2K+m)\Lambda(-y+(Q-\alpha_{2})/2-\alpha_{3},l_{3}-m)e^{{4i\pi\over
r}y\alpha_{1}}e^{{2miN\pi\over r}}{dy\over i}\,,$

where $K+N\in\mathbb{Z}$ and $K-N\in\mathbb{Z}$ .

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