††institutetext:  a Department of Physics, Bogazici University, 34342 Bebek,
Istanbul, Turkey
b Steklov Mathematical Institute of Russ. Acad. Sci., Gubkina str. 8, 119991
Moscow, Russia
c Department of Mathematics, Khazar University, Mehseti St. 41, AZ1096, Baku,
Azerbaijan
d Department of Electrical and Electronics Engineering, Bogazici University,
34342 Bebek, Istanbul, Turkey

# On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on
$S_{b}^{3}/\mathbb{Z}_{r}$

Ilmar Gahramanova,b,c, Batuhan Keskina,d, Dilara Kosvaa and Mustafa
Mullahasanoglua<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study Bailey pairs construction for hyperbolic hypergeometric integral
identities acquired via the duality of lens partitions functions for the
three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories on
$S_{b}^{3}/\mathbb{Z}_{r}$. The novel Bailey pairs are constructed for the
star-triangle relation, the star-star relation, and the pentagon identity. The
first two of them are integrability conditions for the Ising-type integrable
lattice models. The last one corresponds to the representation of the basic
$2-3$ Pachner move for triangulated 3-manifolds.

###### Keywords:

Bailey pairs, hyperbolic hypergeometric functions, star-triangle relation,
star-star relation, pentagon identity, supersymmetric duality.

##  1 Introduction

In recent years, the remarkable concept of hypergeometric identities sits at
the intersection of diverse studies such as exact results in supersymmetric
gauge theories Gang:2019juz ; Benini:2011nc ; Imamura:2012rq ; Imamura:2013qxa
; Nieri:2015yia ; Alday:2012au ; Yamazaki:2013fva ; Honda:2016vmv ;
Nedelin:2016gwu ; Spiridonov:2009za ; Spiridonov2014 ; Krattenthaler:2011da ;
Gahramanov:gka ; Dolan:2011rp  and their mathematical structures interacting
with various fields in mathematics, see, e.g., Gahramanov:2015tta ;
Tachikawa:2017byo ; Spiridonov:2019kto ; Gahramanov:2022qge , star-triangle
relation (Yang-Baxter equation) Spiridonov:2010em ; Yamazaki:2012cp ;
Kels:2015bda ; Yagi:2015lha ; Gahramanov:2015cva ; Gahramanov:2016ilb ;
Sarkissian:2018ppc ; Eren:2019ibl ; de-la-Cruz-Moreno:2020xop ;
Bozkurt:2020gyy  or star-star relation Yamazaki:2015voa ; Kels:2017toi ;
Catak:2021coz ; Mullahasanoglu:2021xyf  for spin lattice models, knot theory
Kashaev:2012cz , pentagon identities Kashaev:2012cz ; Kashaev:2014rea ;
kashaev2014euler ; Gahramanov:2013rda ; Gahramanov:2014ona ;
Gahramanov:2016wxi ; Bozkurt:2018xno ; Jafarzade:2018yei ; Dede:2022ofo ,
Bailey pairs spiridonov2004bailey ; Gahramanov:2015cva ; Brunner:2017lhb ;
Spiridonov_2019 ; Gahramanov:2021pgu , quantum algebras Hadasz:2013bwa ;
Bozkurt:2020gyy ; Fan:2021bwt ; Apresyan:2022erh , etc.

In this work, we consider certain three-dimensional $\mathcal{N}=2$
supersymmetric dualities on $S^{3}_{b}/\mathbb{Z}_{r}$. These dualities have
been studied via the gauge/YBE correspondence, which connects dualities with
integrable models in statistical models, see for a comprehensive review
Gahramanov:2017ysd ; Yamazaki:2018xbx .

Here we consider three-dimensional $SU(2)$ gauge theory with six flavors
$S^{3}_{b}/\mathbb{Z}_{r}$. The corresponding star-triangle relation for this
theory was constructed in Gahramanov:2016ilb . The reduction of the gauge
symmetry to $U(1)$ via the gauge symmetry breaking gives another solution to
the star-triangle relation which is realized as the generalized Faddeev-Volkov
model Bozkurt:2020gyy ; Sarkissian:2018ppc , since the $r=1$ case gives the
Faddeev-Volkov model Bazhanov:2007mh ; Bazhanov:2007vg  corresponding to
$\mathcal{N}=2$ supersymmetric dual theories on $S^{3}_{b}$ Hama:2011ea . The
star-star relations baxter:1997ssr  of these two models are constructed, and
for the factorized interaction-round-a-face (IRF) spin models, IRF-type YBE
are presented in Catak:2021coz ; Mullahasanoglu:2021xyf . The integral
identity obtained by supersymmetric duality with the gauge group $U(1)$ can
also be written as a pentagon identity Bozkurt:2020gyy .

We present new Bailey pairs for the hyperbolic hypergeometric integral
identities. The construction of Bailey pairs for the star-triangle relation
leads to acquiring the vertex-type111In the vertex-type models, the spins
(continuous and discrete) are located at the edges, and the interactions of
the spins are via vertices. YBE via the Coxeter relations Gahramanov:2015cva .
Bailey pairs for the star-star relations and the pentagon identity mentioned
above are also constructed.

The organization of the rest of this paper is as follows. In Section 2, we
briefly recall the mathematical tools and introduce integral identities
resulting from the equality of the partition functions of $\mathcal{N}=2$
supersymmetric gauge theories on $S^{3}_{b}/\mathbb{Z}_{r}$. In Section 3, we
present the star-triangle relations, the star-star relations, and the pentagon
identities obtained via the supersymmetric dualities. In Section 4, we
construct Bailey pairs for each integral identity considered in previous
sections. In section 5, we conclude our results and present some further
studies. The appendix consists of the construction of the Bailey pair for the
star-triangle relation.

##  2 Seiberg dualities on $S_{b}^{3}/\mathbb{Z}_{r}$

In this section, we consider Seiberg dualities Seiberg:1994pq  in three-
dimensional $\mathcal{N}=2$ theories Intriligator:1996ex ; Aharony:1997bx .
One of the evidences222The equality of the superconformal indices is also
evidence for duality, see e.g. Krattenthaler:2011da ; Kapustin:2011jm ;
Gahramanov:2013rda ; Gahramanov:2016wxi  for dualities is the equality of
partition functions333Supersymmetric partition functions are studied on sphere
(e.g. Kapustin:2010xq ), squashed sphere (e.g. Dolan:2011rp ; Gahramanov:gka ;
Amariti:2015vwa ), and squashed lens space (e.g. Benini:2011nc ;
Imamura:2012rq ; Imamura:2013qxa ).

The three-dimensional $\mathcal{N}=2$ partition functions on the squashed lens
space $S^{3}_{b}/\mathbb{Z}_{r}$ have been computed via dimensional reduction
of the four-dimensional lens superconformal index Benini:2011nc ;
Yamazaki:2013fva ; Eren:2019ibl  and via the supersymmetric localization
technique Imamura:2012rq ; Imamura:2013qxa . Such theories have been studied
in Nieri:2015yia ; Gahramanov:2016ilb ; Bozkurt:2020gyy ; Catak:2021coz ;
Mullahasanoglu:2021xyf .

###  2.1 3d $SU(2)$ duality

Our starting point is a three-dimensional $\mathcal{N}=2$ $SU(2)$ gauge theory
with six fundamentals and six anti-fundamentals. The confined dual theory
consists of only fifteen chiral multiplets in the totally antisymmetric tensor
representation of the flavor group. The equality of the partition functions
can be written as an integral identity in terms of hyperbolic hypergeometric
functions444For the integral identity written in terms of the improved double
sine function, see, e.g., Gahramanov:2016ilb ,

$\displaystyle\begin{aligned}
\sum_{y=0}^{[r/2]}\epsilon(y)\int_{-\infty}^{\infty}&\frac{\prod_{i=1}^{6}\gamma^{(2)}(-i(a_{i}\pm
z)-i\omega_{1}(u_{i}\pm y);-i\omega_{1}r,-i\omega)}{\gamma^{(2)}(\pm 2iz\pm
2i\omega_{1}y;-i\omega_{1}r,-i\omega)}\\\
\times&\frac{\gamma^{(2)}(-i(a_{i}\pm z)-i\omega_{2}(r-(u_{i}\pm
y));-i\omega_{2}r,-i\omega)}{\gamma^{(2)}(\pm 2iz-i\omega_{2}(r\pm
2y);-i\omega_{2}r,-i\omega)}\frac{dz}{2r\sqrt{-\omega_{1}\omega_{2}}}\\\
=\prod_{1\leq i<j\leq
6}&\gamma^{(2)}(-i(a_{i}+a_{j})-i\omega_{1}(u_{i}+u_{j});-i\omega_{1}r,-i\omega)\\\
&\times\gamma^{(2)}(-i(a_{i}+a_{j})-i\omega_{2}(r-(u_{i}+u_{j}));-i\omega_{2}r,-i\omega)\;,\end{aligned}$
(1)

with the balancing conditions555When the balancing condition is taken
$\sum_{i=1}^{6}u_{i}=mr$ where $m$ is an integer, there should be a sign
factor $e^{\frac{\pi i}{2}mr^{2}(m-1)(2m-1)}$ on the left-hand side of the
integral identity (1), seem Gahramanov:2016ilb  . are
$\sum^{6}_{i=1}a_{i}=\omega$ and $\sum_{i=1}^{6}u_{i}=0$, where we introduced
$\omega:=\omega_{1}+\omega_{2}$. Also, $\epsilon(0)=\epsilon({[r/2]})=1$ and
$\epsilon(y)=2$ otherwise. Here the function
$\gamma^{(2)}(z;\omega_{1},\omega_{2})$ is the so-called hyperbolic gamma
function666Different versions of the hyperbolic gamma function can be seen as
the double sine function Kharchev:2001rs ; Ponsot:2000mt , the non-compact
quantum dilogarithm Faddeev:1999fe ; Faddeev:2000if ; Ponsot:2000mt ;
Volkov:2005zrq ; Bazhanov:2010kz , the modified $q$-gamma function Spiridonov-
essays , etc. Ruijsenaars:1997:FOA ; van2007hyperbolic ; Andersen:2014aoa
which is the main tool in this study. One of the several representations777For
various integral representations, see, e.g. Faddeev:1995nb ;
woronowicz2000quantum  and one can also introduce the infinite product
representation
$\displaystyle\gamma^{(2)}(z;\omega_{1},\omega_{2})=e^{\frac{\pi
i}{2}B_{2,2}(z;\omega_{1},\omega_{2})}\frac{(e^{-2\pi
i\frac{z}{\omega_{2}}}\tilde{q};\tilde{q})}{(e^{-2\pi
i\frac{z}{\omega_{1}}};q)}\;,$ (2) where parameters are $\tilde{q}=e^{2\pi
i\omega_{1}/\omega_{2}}$ and $q=e^{-2\pi i\omega_{2}/\omega_{1}}$ and the
Bernoulli polynomial is  $\displaystyle
B_{2,2}(z;\omega_{1},\omega_{2})=\frac{z^{2}-z(\omega_{1}+\omega_{2})}{\omega_{1}\omega_{2}}+\frac{\omega_{1}^{2}+3\omega_{1}\omega_{2}+\omega_{2}^{2}}{6\omega_{1}\omega_{2}}.$
(3) of this special function is the following888One can list many areas of
study for this function, but we mention fewer examples from the areas of
mathematical and theoretical physics such as knot theory hikami2001hyperbolic
; Hikami2007 ; hikami2014braiding ; Chan:2017qnw  supersymmetric gauge theory
Teschner:2012em  integrable models of statistical mechanics Bazhanov:2007mh ;
Bazhanov:2007vg  special functions van2007hyperbolic .

$\displaystyle\gamma^{(2)}(z;\omega_{1},\omega_{2})=\exp{\left(-\int_{0}^{\infty}\frac{dx}{x}\left[\frac{\sinh{x(2z-\omega_{1}-\omega_{2})}}{2\sinh{(x\omega_{1})}\sinh{(x\omega_{2})}}-\frac{2z-\omega_{1}-\omega_{2}}{2x\omega_{1}\omega_{2}}\right]\right)}\;,$
(4)

where $Re(\omega_{1}),Re(\omega_{2})>0$ and
$Re(\omega_{1}+\omega_{2})>Re(z)>0$. We will mainly use the reflection
property of the hyperbolic gamma function

$\displaystyle\gamma^{(2)}(z;\omega_{1},\omega_{2})\gamma^{(2)}(\omega_{1}+\omega_{2}-z;\omega_{1},\omega_{2})=1\>.$
(5)

and the following shorthand notation

$\displaystyle\gamma^{(2)}(\pm
z;\omega_{1},\omega_{2})=\gamma^{(2)}(z;\omega_{1},\omega_{2})\gamma^{(2)}(-z;\omega_{1},\omega_{2})\>.$
(6)

Note that the case $r=1$ (see, e.g., van2007hyperbolic ) of the integral
identity (1) corresponds to the duality of supersymmetric gauge theories on
$S^{3}_{b}$.

###  2.2 3d $U(1)$ duality

One obtains the following integral identity via breaking the gauge symmetry
Bozkurt:2020gyy  (see also Spiridonov:2010em ; Sarkissian:2018ppc ) from
$SU(2)$ to $U(1)$ in the duality (1)

$\displaystyle\begin{aligned} \sum_{y=0}^{[r/2]}\epsilon(y)e^{\frac{\pi
i}{2}C}\int_{-\infty}^{\infty}\prod_{i=1}^{3}&\gamma^{(2)}(-i(a_{i}-z)-i\omega_{1}(u_{i}-y);-i\omega_{1}r,-i\omega)\\\
\times&\gamma^{(2)}(-i(a_{i}-z)-i\omega_{2}(r-(u_{i}-y));-i\omega_{2}r,-i\omega)\\\
\times&\gamma^{(2)}(-i(b_{i}+z)-i\omega_{1}(v_{i}+y);-i\omega_{1}r,-i\omega)\\\
\times&\gamma^{(2)}(-i(b_{i}+z)-i\omega_{2}(r-(v_{i}+y));-i\omega_{2}r,-i\omega)\frac{dz}{r\sqrt{-\omega_{1}\omega_{2}}}\\\
=\prod_{i,j=1}^{3}&\gamma^{(2)}(-i(a_{i}+b_{j})-i\omega_{1}(u_{i}+v_{j});-i\omega_{1}r,-i\omega)\\\
\times&\gamma^{(2)}(-i(a_{i}+b_{j})-i\omega_{2}(r-(u_{i}+v_{j}));-i\omega_{2}r,-i\omega))\;,\end{aligned}$
(7)

where the balancing conditions are $\sum_{i=1}^{3}a_{i}+b_{i}=\omega$ and
$\sum_{i=1}^{3}u_{i}+v_{i}=0$ and the sign factor is
$C=-2y+\sum_{i=1}^{3}(u_{i}-v_{i})$. It is possible to shift the discrete
parameters $u_{i}$ and $v_{i}$ and obtain a new balancing condition
$\sum_{i=1}^{3}u_{i}+v_{i}=r$. In this case $C=0$ in (7).

##  3 Integrability conditions and the basic 2-3 Pachner move

###  3.1 Star-triangle relation

In the transfer matrix method Baxter:1982zz  for Ising-type models, it is
sufficient to write a star-triangle relation to obtain the integrability
property of the lattice spin model. Here we are interested in Ising-like
models with discrete $m_{i}$ and continuous $x_{i}$ spin variables. We denote
the discrete and continuous spins together in the form of
$\sigma_{i}:=(x_{i},m_{i})$.

The Boltzmann weights of the models discussed here have the reflection
property $W(\sigma_{i},\sigma_{j})=W(\sigma_{j},\sigma_{i})$ and the crossing
symmetry
$\overline{W}_{\alpha_{i},\tilde{\alpha}_{j}}(\sigma_{i},\sigma_{j})=W_{\eta-\alpha_{i},\beta-\tilde{\alpha}_{j}}(\sigma_{i},\sigma_{j})$,
which means that one can write vertical interactions in terms of horizontal
interactions. Hence, we can write the star-triangle relation as the following

$\displaystyle\begin{aligned} \sum_{m_{0}}\int
dx_{0}\>S(\sigma_{0})\>W_{\alpha_{1},\tilde{\alpha}_{1}}(\sigma_{1},\sigma_{0})W_{\alpha_{2},\tilde{\alpha}_{2}}(\sigma_{2},\sigma_{0})W_{\alpha_{3},\tilde{\alpha}_{3}}(\sigma_{3},\sigma_{0})\makebox[100.00015pt]{}\\\
\makebox[60.00009pt]{}=\mathcal{R}\>W_{\eta-\alpha_{1},\beta-\tilde{\alpha}_{1}}(\sigma_{1},\sigma_{2})W_{\eta-\alpha_{2},\beta-\tilde{\alpha}_{2}}(\sigma_{1},\sigma_{3})W_{\eta-\alpha_{3},\beta-\tilde{\alpha}_{3}}(\sigma_{2},\sigma_{3})\>,\end{aligned}$
(8)

where the constraints on the spectral parameters are
$\alpha_{1}+\alpha_{2}+\alpha_{3}=\eta$ (note that these are continuous and in
our cases $\eta=-\frac{\omega}{2}$) and
$\tilde{\alpha}_{1}+\tilde{\alpha}_{2}+\tilde{\alpha}_{3}=\beta$ (note that
these are discrete and in our cases $\beta=0$) with crossing parameters $\eta$
and $\beta$. The functions $S(\sigma_{0})$ and $\mathcal{R}$ are the self-
interaction contribution and the spin-independent functions, respectively.

The identity (1) turns to the star-triangle relation999In Gahramanov:2016ilb ,
it is firstly appeared as normalized and with only continuous spectral
parameters. when new variables

$\displaystyle\begin{aligned}
a_{i}&=-\alpha_{i}+x_{i}\;,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\;a_{i+3}=-\alpha_{i}-x_{i}\;,\\\
u_{i}&=-\tilde{\alpha}_{i}+y_{i}\;,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\;u_{i+3}=-\tilde{\alpha}_{i}-y_{i}\>,\end{aligned}$
(9)

are introduced. The Boltzmann weights can be written as

$\displaystyle W_{\alpha_{i},\tilde{\alpha}_{i}}(x_{i},x_{j}$
$\displaystyle,y_{i},y_{j})=$ (10)
$\displaystyle\gamma^{(2)}(-i(-\alpha_{i}+x_{i}\pm
x_{j})-i\omega_{1}(-\tilde{\alpha}_{i}+y_{i}\pm
y_{j});-i\omega_{1}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(-\alpha_{i}+x_{i}\pm
x_{j})-i\omega_{2}(r-(-\tilde{\alpha}_{i}+y_{i}\pm
y_{j}));-i\omega_{2}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(-\alpha_{i}-x_{i}\pm
x_{j})-i\omega_{1}(-\tilde{\alpha}_{i}-y_{i}\pm
y_{j});-i\omega_{1}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(-\alpha_{i}-x_{i}\pm
x_{j})-i\omega_{2}(r-(-\tilde{\alpha}_{i}-y_{i}\pm
y_{j}));-i\omega_{2}r,-i\omega)\;.$

The spin-independent weight function $\mathcal{R}$ depending only on spectral
parameters is the same for both models

$\displaystyle\mathcal{R}=\prod_{j=1}^{3}\gamma^{(2)}(2i\alpha_{j}+2i\omega_{1}\tilde{\alpha}_{j};-i\omega_{1}r,-i\omega)\gamma^{(2)}(2i\alpha_{j}-i\omega_{2}(r+2\tilde{\alpha}_{j});-i\omega_{2}r,-i\omega)\;.$
(11)

However, a self-interaction contribution is not trivial in the $SU(2)$ model
(the identity (1)) and has the following form

$\displaystyle S(\sigma_{0})=\frac{\epsilon(n)}{\gamma^{(2)}(\pm 2iu\pm
2i\omega_{1}n;-i\omega_{1}r,-i\omega)\gamma^{(2)}(\pm 2iu-i\omega_{2}(r\pm
2n);-i\omega_{2}r,-i\omega)}\>,$ (12)

where $\epsilon(n)$ disappears if one changes boundaries of the summation as
in Bozkurt:2020gyy  (see Appendix in Eren:2019ibl ).

The same procedure is applied to (7)101010This star-triangle relation for
generalized Faddeev-Volkov model appeared in Sarkissian:2018ppc ;
Bozkurt:2020gyy  has only continuous spectral parameters.  by re-defining
variables as

$\displaystyle\begin{aligned}
a_{i}&=-\alpha_{i}+x_{i}\;,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\;b_{i}=-\alpha_{i}-x_{i}\;,\\\
u_{i}&=-\tilde{\alpha}_{i}+y_{i}\;,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\;v_{i}=-\tilde{\alpha}_{i}-y_{i}\>,\end{aligned}$
(13)

then, the Boltzmann weights become

$\displaystyle\begin{aligned}
W_{\alpha_{i},\tilde{\alpha}_{i}}(x_{i},x_{j}&,y_{i},y_{j})=\\\ &e^{-\pi
i(y_{i}+y_{j})}\gamma^{(2)}(-i(-\alpha_{i}+x_{i}-x_{j})-i\omega_{1}(-\tilde{\alpha}_{i}+y_{i}-y_{j});-i\omega_{1}r,-i\omega)\\\
&\times\gamma^{(2)}(-i(-\alpha_{i}+x_{i}-x_{j})-i\omega_{2}(r-(-\tilde{\alpha}_{i}+y_{i}-y_{j}));-i\omega_{2}r,-i\omega)\\\
&\times\gamma^{(2)}(-i(-\alpha_{i}-x_{i}+x_{j})-i\omega_{1}(-\tilde{\alpha}_{i}-y_{i}+y_{j});-i\omega_{1}r,-i\omega)\\\
&\times\gamma^{(2)}(-i(-\alpha_{i}-x_{i}+x_{j})-i\omega_{2}(r-(-\tilde{\alpha}_{i}-y_{i}+y_{j}));-i\omega_{2}r,-i\omega)\;,\end{aligned}$
(14)

where the exponent term vanishes if we change the balancing condition
mentioned in (7). This model has no self-interaction term and the spin-
independent function is the same as (11).

###  3.2 Star-star relation

One can obtain another fundamental integrability condition in statistical
mechanics which is the star-star relation111111For the star-star relation in
the context of supersymmetric partition functions for dualities, and
symmetries of beta hypergeometric integrals, see Spiridonov:2008zr ;
Dimofte:2012pd ; Kels:2017toi . baxter:1997ssr  in the existence of star-
triangle relation. In some lattice spin models, Boltzmann weights satisfy the
star-star relation but not the star-triangle relation, see e.g. baxter:1997ssr
; Bazhanov:2011mz ; Bazhanov:2013bh .

The star-star relation has the following form

$\displaystyle\begin{aligned} &\sum_{m_{0}}\int
dx_{0}W_{\alpha_{1},\tilde{\alpha}_{1}}\left(\sigma_{1},\sigma_{0}\right)W_{\alpha_{2},\tilde{\alpha}_{2}}\left(\sigma_{0},\sigma_{2}\right)W_{\alpha_{3},\tilde{\alpha}_{3}}\left(\sigma_{3},\sigma_{0}\right)W_{\alpha_{4},\tilde{\alpha}_{4}}\left(\sigma_{0},\sigma_{4}\right)\\\
&\quad\quad\quad\quad=\frac{W_{2\eta-\alpha_{1}-\alpha_{2},2\beta-\tilde{\alpha}_{1}-\tilde{\alpha}_{2}}(\sigma_{1},\sigma_{2})W_{2\eta-\alpha_{1}-\alpha_{4},2\beta-\tilde{\alpha}_{1}-\tilde{\alpha}_{4}}(\sigma_{1},\sigma_{4})}{W_{2\eta-\alpha_{3}-\alpha_{4},\beta-\tilde{\alpha}_{3}-\tilde{\alpha}_{4}}(\sigma_{4},\sigma_{3})W_{2\eta-\alpha_{2}-\alpha_{3},2\beta-\tilde{\alpha}_{2}-\tilde{\alpha}_{3}}(\sigma_{2},\sigma_{3})}\\\
&\quad\quad\quad\quad\times\sum_{m_{0}}\int
dx_{0}W_{\alpha_{3},\tilde{\alpha}_{3}}\left(\sigma_{0},\sigma_{1}\right)W_{\alpha_{4},\tilde{\alpha}_{4}}\left(\sigma_{2},\sigma_{0}\right)W_{\alpha_{1},\tilde{\alpha}_{1}}\left(\sigma_{0},\sigma_{3}\right)W_{\alpha_{2},\tilde{\alpha}_{2}}\left(\sigma_{4},\sigma_{0}\right)\>,\end{aligned}$
(15)

where the spectral parameters satisfy the conditions
$\sum_{i=1}^{4}\alpha_{i}=2\eta$ and
$\sum_{i=1}^{4}\tilde{\alpha}_{i}=2\beta$.

Using the hyperbolic hypergeometric integral identity (1) (the star-triangle
relation), one can obtain the following integral identity
Mullahasanoglu:2021xyf  presented as a star-star relation

$\displaystyle\begin{aligned}
\sum_{y=0}^{[r/2]}\epsilon(y)\int_{-\infty}^{\infty}\frac{\prod_{i=1}^{8}\gamma^{(2)}(-i(a_{i}\pm
z)-i\omega_{1}(u_{i}\pm y);-i\omega_{1}r,-i\omega)}{\gamma^{(2)}(\pm 2iz\pm
i\omega_{1}2y;-i\omega_{1}r,-i\omega)}\\\ \times\frac{\gamma^{(2)}(-i(a_{i}\pm
z)-i\omega_{2}(r-(u_{i}\pm y));-i\omega_{2}r,-i\omega)}{\gamma^{(2)}(\pm
2iz-i\omega_{2}(r\pm
2y);-i\omega_{2}r,-i\omega)}\frac{dz}{r\sqrt{-\omega_{1}\omega_{2}}}\\\
=\frac{\prod_{1\leq i<j\leq
4}\gamma^{(2)}(-i(a_{i}+a_{j})-i\omega_{1}(u_{i}+u_{j});-i\omega_{1}r,-i\omega)}{\prod_{5\leq
i<j\leq
8}\gamma^{(2)}(-i(\tilde{a}_{i}+\tilde{a}_{j})-i\omega_{1}(\tilde{u}_{i}+\tilde{u}_{j});-i\omega_{1}r,-i\omega)}\\\
\times\frac{\prod_{1\leq i<j\leq
4}\gamma^{(2)}(-i(a_{i}+a_{j})-i\omega_{2}(r-(u_{i}+u_{j}));-i\omega_{2}r,-i\omega)}{\prod_{5\leq
i<j\leq
8}\gamma^{(2)}(-i(\tilde{a}_{i}+\tilde{a}_{j})-i\omega_{2}(r-(\tilde{u}_{i}+\tilde{u}_{j}));-i\omega_{2}r,-i\omega)}\\\
\times\sum_{m=0}^{[r/2]}\epsilon(m)\int_{-\infty}^{\infty}\frac{\prod_{i=1}^{8}\gamma^{(2)}(-i(\tilde{a}_{i}\pm
x)-i\omega_{1}(\tilde{u}_{i}\pm m);-i\omega_{1}r,-i\omega)}{\gamma^{(2)}(\pm
2ix\pm i\omega_{1}2m;-i\omega_{1}r,-i\omega)}\\\
\times\frac{\gamma^{(2)}(-i(\tilde{a}_{i}\pm
x)-i\omega_{2}(r-(\tilde{u}_{i}\pm
m));-i\omega_{2}r,-i\omega)}{\gamma^{(2)}(\pm 2ix-i\omega_{2}(r\pm
2m);-i\omega_{2}r,-i\omega)}\frac{dx}{r\sqrt{-\omega_{1}\omega_{2}}}\>,\end{aligned}$
(16)

with the balancing conditions $\sum_{i=1}^{8}a_{i}=2\omega$ and
$\sum_{i=1}^{8}u_{i}=0$, and parameters are identified as

$\displaystyle\begin{aligned}
\tilde{a}_{i}&=a_{i}+s,&\tilde{u}_{i}&=u_{i}+p,&\text{if}\;\;\;i=1,2,3,4\>,\\\
\tilde{a}_{i}&=a_{i}-s,&\tilde{u}_{i}&=u_{i}-p,&\text{if}\;\;\;i=5,6,7,8\>,\end{aligned}$
(17)

where

$\displaystyle\begin{aligned}
s&=\frac{1}{2}\left(\omega-\sum_{i=1}^{4}a_{i}\right)=\frac{1}{2}\left(-\omega_{1}-\omega_{2}+\sum_{i=5}^{8}a_{i}\right)\>,\\\
p&=-\frac{1}{2}\left(\sum_{i=1}^{4}u_{i}\right)=\frac{1}{2}\left(\sum_{i=5}^{8}u_{i}\right)\>.\end{aligned}$
(18)

The following identity is the star-star relation Catak:2021coz  of the
generalized Faddeev-Volkov model

$\displaystyle\begin{aligned}
\sum_{y=0}^{[r/2]}\epsilon(y)\int_{-\infty}^{\infty}\prod_{i=1}^{4}\gamma^{(2)}(-i(a_{i}-z)-i\omega_{1}(u_{i}-y);-i\omega_{1}r,-i\omega)\\\
\times\gamma^{(2)}(-i(a_{i}-z)-i\omega_{2}(r-(u_{i}-y));-i\omega_{2}r,-i\omega)\\\
\times\gamma^{(2)}(-i(b_{i}+z)-i\omega_{1}(v_{i}+y);-i\omega_{1}r,-i\omega)\\\
\times\gamma^{(2)}(-i(b_{i}+z)-i\omega_{2}(r-(v_{i}+y));-i\omega_{2}r,-i\omega)\frac{dz}{r\sqrt{-\omega_{1}\omega_{2}}}\\\
=\frac{e^{\frac{\pi i}{2}\sum_{i=1}^{2}(u_{i}-v_{i})}}{e^{\frac{\pi
i}{2}\sum_{i=3}^{4}(\tilde{u_{i}}-\tilde{v_{i}})}}\frac{\prod_{i,j=1}^{2}\gamma^{(2)}(-i(a_{i}+b_{j})-i\omega_{1}(u_{i}+v_{j});-i\omega_{1}r,-i\omega)}{\prod_{i,j=3}^{4}\gamma^{(2)}(-i(\tilde{a_{i}}+\tilde{b_{i}})-i\omega_{1}(\tilde{u_{i}}+\tilde{v_{i}});-i\omega_{1}r,-i\omega)}\\\
\times\frac{\prod_{i,j=1}^{2}\gamma^{(2)}(-i(a_{i}+b_{j})-i\omega_{2}(r-(u_{i}+v_{j}));-i\omega_{2}r,-i\omega)}{\prod_{i,j=3}^{4}\gamma^{(2)}(-i(\tilde{a_{i}}+\tilde{b_{i}})-i\omega_{2}(r-(\tilde{u_{i}}+\tilde{v_{i}}));-i\omega_{2}r,-i\omega)}\\\
\times\sum_{m=0}^{[r/2]}\epsilon(m)\int_{-\infty}^{\infty}\prod_{i=1}^{4}\gamma^{(2)}(-i(\tilde{a_{i}}-x)-i\omega_{1}(\tilde{u_{i}}-m);-i\omega_{1}r,-i\omega)\\\
\times\gamma^{(2)}(-i(\tilde{a_{i}}-x)-i\omega_{2}(r-(\tilde{u_{i}}-m));-i\omega_{2}r,-i\omega)\\\
\times\gamma^{(2)}(-i(\tilde{b_{i}}+x)-i\omega_{1}(\tilde{v_{i}}+m);-i\omega_{1}r,-i\omega)\\\
\times\gamma^{(2)}(-i(\tilde{b_{i}}+x)-i\omega_{2}(r-(\tilde{v_{i}}+m));-i\omega_{2}r,-i\omega)\frac{dx}{r\sqrt{-\omega_{1}\omega_{2}}}\>,\end{aligned}$
(19)

where the balancing conditions are $\sum_{i=1}^{4}a_{i}+b_{i}=2\omega$ and
$\sum_{i=1}^{4}u_{i}+v_{i}=0$, and we used the following choice of parameters,

$\displaystyle\begin{aligned}
\tilde{a}_{i}&=a_{i}+s,&\tilde{b}_{i}&=b_{i}+s,&\tilde{u}_{i}&=u_{i}+p,&\tilde{v}_{i}&=v_{i}+p,&\text{if}\;\;\;i=1,2\>,\\\
\tilde{a}_{i}&=a_{i}-s,&\tilde{b}_{i}&=b_{i}-s,&\tilde{u}_{i}&=u_{i}-p,&\tilde{v}_{i}&=v_{i}-p,&\text{if}\;\;\;i=3,4\>,\end{aligned}$
(20)

where

$\displaystyle\begin{aligned}
s&=\frac{1}{2}(\omega+a_{1}+a_{2}+b_{1}+b_{2})=\frac{1}{2}(\omega-
a_{3}-a_{4}-b_{3}-b_{4})\>,\\\
p&=-\frac{1}{2}(u_{1}+u_{2}+v_{1}+v_{2})=\frac{1}{2}(u_{3}+u_{4}+v_{3}+v_{4})\>.\end{aligned}$
(21)

###  3.3 The pentagon identity

Pentagon relation Kashaev:2014rea ; kashaev2014euler  has a meaning of the 2-3
Pachner move pachner1991pl  for triangulated three-dimensional manifolds and
can be formally written as

$\displaystyle\mathcal{B}\mathcal{B}\mathcal{B}=\mathcal{B}\mathcal{B}\>.$
(22)

Here we write the equation (7) as an integral pentagon identity
Bozkurt:2020gyy . It can be interpreted as a topological invariant of
corresponding 3–manifold via 3d–3d correspondence Dimofte:2011ju ;
Dimofte:2011py  building bridges between three dimensional $\mathcal{N}=2$
supersymmetric gauge theories and triangulated 3-manifolds.

Then the hyperbolic hypergeometric solution to the pentagon identity can be
obtained by the following definition

$\displaystyle\begin{aligned}
\mathcal{B}(z_{1},u_{1};z_{2},u_{2})=\frac{\gamma^{(2)}(-iz_{1}-i\omega_{1}u_{1};-i\omega_{1}r,-i\omega)\gamma^{(2)}(-iz_{1}-i\omega_{2}(r-u_{1});-i\omega_{2}r,-i\omega))}{\gamma^{(2)}(-i(z_{1}+z_{2})-i\omega_{1}(u_{1}+u_{2});-i\omega_{1}r,-i\omega))}\\\
\frac{\gamma^{(2)}(-iz_{2}-i\omega_{1}u_{2};-i\omega_{1}r,-i\omega)\gamma^{(2)}(-iz_{2}-i\omega_{2}(r-u_{2});-i\omega_{2}r,-i\omega)}{\gamma^{(2)}(-i(z_{1}+z_{2})-i\omega_{2}(r-u_{1}-u_{2});-i\omega_{2}r,-i\omega)}\;,\end{aligned}$
(23)

and the equation (7) turns to the integral pentagon identity

$\displaystyle\begin{aligned}
\frac{1}{r\sqrt{-\omega_{1}\omega_{2}}}\sum_{y=0}^{\lfloor
r/2\rfloor}e^{\frac{\pi
iC}{2}}\int_{-\infty}^{\infty}dz\prod_{i=1}^{3}\mathcal{B}(a_{i}-z,u_{i}-y;b_{i}+z,v_{i}+y)\\\
=\mathcal{B}(a_{1}+b_{2},u_{1}+v_{2};a_{2}+b_{3},u_{2}+v_{3})\mathcal{B}(a_{1}+b_{3},u_{1}+v_{3};a_{2}+b_{1},u_{2}+v_{1})\>,\end{aligned}$
(24)

where the sign factor and the balancing conditions are the same as in (7).

##  4 Bailey pairs

Influenced by Rogers’ work in proving combinatorial identities which are now
known as the Rogers-Ramanujan identities, W.N. Bailey introduced the following
lemma to abstract the notions underlying the proofs bailey1948identities ,

######  Lemma 4.1.

If the series $\\{\alpha\\}_{n\geq 0}$, $\\{\beta\\}_{n\geq 0}$,
$\\{\delta\\}_{n\geq 0}$, $\\{\gamma\\}_{n\geq 0}$, $\\{u\\}_{n\geq 0}$ and
$\\{v\\}_{n\geq 0}$ satisfy

$\beta_{n}=\sum_{r=0}^{n}\alpha_{r}u_{n-r}v_{n+r}\;,$

and

$\gamma_{n}=\sum_{r=n}^{\infty}\delta_{r}u_{r-n}v_{r+n}\;,$

then

$\sum_{n=0}^{\infty}\alpha_{n}\gamma_{n}=\sum_{n=0}^{\infty}\beta_{n}\delta_{n}\;.$

The proof relies on a simple rearrangement of the series and is trivial as
noted by Bailey, hence will be omitted. Theorem $\eqref{bailey transform}$ is
commonly referred to as the Bailey lemma and for specific choices of the
mentioned series, various identities in mathematics can be derived
slater1966generalized ; bressoud1981 ; dbsearsbaileytransform ; Schilling1997
.

Following Bailey’s work, Andrews geandrews  formulated a method of deriving
infinitely many identities from a known one iteratively. Given two sequences
of functions $\\{\alpha\\}_{k}$ and $\\{\beta\\}_{k}$ for
$k\in\\{0,\dots,n\\}$, with the relation

$\beta_{k}=F_{k}(\alpha_{0},\dots,\alpha_{n})\;,$ (25)

one can construct the functions

$\displaystyle\beta_{k}^{(i)}$
$\displaystyle=G_{k}\left(\beta_{0}^{(i-1)},\dots,\beta_{n}^{(i-1)}\right)\>,\hskip
28.45274pti\in\mathbb{Z}_{>0}\>,$ (26) $\displaystyle\alpha_{k}^{(i)}$
$\displaystyle=H_{k}\left(\alpha_{0}^{(i-1)},\dots,\alpha_{n}^{(i-1)}\right)\>,\hskip
28.45274pti\in\mathbb{Z}_{>0}\>,$ (27)

that also satisfy $\eqref{bailey def}$ for every $i\in\mathbb{Z}_{>0}$. Then
the functions $\\{\alpha^{(i)}\\}_{k}$, $\\{\beta^{(i)}\\}_{k}$ are called a
Bailey pair, and they form a chain of infinite length. This notion can be
generalized from chains to lattices and higher-dimensional chains, see, e.g.
milne1992al ; lilly1993thec ; milne1995consequences . Apart from their purely
mathematical implications, Bailey pairs are also used in superconformal field
theories Berkovich1996 ; Andrews1998scft  and exactly solvable models of
statistical mechanics. The latter will be the focus of our interest in the
next sections. For a detailed study of the history of the Bailey lemma, see
warnaarhistory .

###  4.1 Star-triangle relation

######  Definition 4.1.

The functions $\alpha(x,m;t,p)$ and $\beta(x,m;t,p)$, $x\in\mathbb{C}$,
$m\in\mathbb{Z}$ form an integral Bailey pair with respect to parameters
$t\in\mathbb{C}$ and $p\in\mathbb{Z}$ if the following relation is satisfied,

$\beta(z,m;t,p)=M(t,p)_{z,m;x,j}\alpha(x,j;t,p)\;,$ (28)

where $M(t,p)_{z,m;x,j}$ is an integral-sum operator that integrates and sums
over the continuous variable $x\in\mathbb{C}$ and the discrete variable
$j\in\mathbb{Z}$ of $\alpha(x,j;t,p)$, respectively.

Suppose we have another operator $D(s,q;y,l;x,k)$ of continuous
$s,y,x\in\mathbb{C}$ and discrete variables $q,l,k\in\mathbb{Z}$, such that it
satisfies the relation

$D(s,q;y,l;x,k)D(-s,-q;y,l;x,k)=1$ (29)

that we will refer to as the reflection relation, and $D(0,0;y,l;x,k)=1$.
Moreover, we assume that the operators $M$ and $D$ satisfy the ”star-triangle
relation”, given as

$\begin{gathered}M(s,q)_{w,k;z,m}D(s+t,q+p;y,l;z,m)M(t,p)_{z,m;x,j}=\\\
D(t,p;y,l;w,k)M(s+t,q+p)_{w,k;x,j}D(s,q;y,l;x,j)\>.\end{gathered}$ (30)

Utilizing $M$ and $D$ operators, the next lemma addresses the question of
forming infinitely many Bailey pairs after finding a particular one.

######  Lemma 4.2 (Bailey Lemma).

Suppose $\alpha(x,m;t,p)$ and $\beta(x,m;t,p)$ form an integral Bailey pair
with respect to $t\in\mathbb{C}$ and $p\in\mathbb{Z}$. Then, the sequences of
functions $\alpha^{\prime}(x,k;t+s,p+q)$ and $\beta^{\prime}(x,k;t+s,p+q)$,
$k\in\mathbb{Z}$, defined by

$\displaystyle\alpha^{\prime}(x,k;t+s,p+q)$
$\displaystyle=D(s,q;y,l;x,k)\alpha(x,k;t,p)\>,$ (31)
$\displaystyle\beta^{\prime}(x,k;t+s,p+q)$
$\displaystyle=D(-t,-p;y,l;x,k)M(s,q)_{x,k;z,m}D(s+t,p+q;y,l;z,m)\beta(z,m;t,p)\>,$
(32)

form a Bailey pair with respect to the new parameters $t+s$ and $p+q$ where
$s,y\in\mathbb{C}$, $q,l\in\mathbb{Z}$ are arbitrary and the operator
$D(s,q;y,l;x,k)$ is described as above.

###### Proof.

The proof follows easily from the definitions. We substitute
$\alpha^{\prime}(x,j;t+s,p+q)$ and $\beta^{\prime}(x,k;t+s,p+q)$ into the
relation defining a Bailey pair. We want to show,

$\beta^{\prime}(w,k;t+s,p+q)=M(t+s,p+q)_{w,k;x,j}\alpha^{\prime}(x,j;t+s,p+q)\>,$
(33) $\displaystyle
D(-t,-p;y,l;w,k)M(s,q)_{w,k;z,m}D(s+t,p+q;y,l;z,m)\beta(z,m;t,p)=$ (34)
$\displaystyle M(s+t,p+q)_{w,k;x,j}D(s,q;y,l;x,j)\alpha(x,j;t,p)\;.$ (35)

Using the reflection relation displayed above for $D$ operators, the problem
reduces to the star-triangle relation,

$\begin{gathered}M(s,q)_{w,k;z,m}D(s+t,q+p;y,l;z,m)M(t,p)_{z,m;x,j}=\\\
D(t,p;y,l;w,k)M(s+t,q+p)_{w,k;x,j}D(s,q;y,l;x,j)\>,\end{gathered}$ (36)

which we have assumed to be true for $M$ and $D$ operators. ∎

We will be constructing such $M$ and $D$ operators satisfying the star-
triangle relation in virtue of the integral identities $\eqref{SU2identity}$
on $SU(2)$ and $\eqref{U1identity}$ on $U(1)$ in the following sections to
construct Bailey pairs. We should mention that there is no systematic way of
constructing Bailey pairs.

####  4.1.1 $SU(2)$ gauge symmetry

Let us first construct the operators that will be used to satisfy $\eqref{star
triangle}$. We can basically build the operators via the characteristic
properties of hypergeometric functions.

$\displaystyle D(t,p;y,l;w,k)=\gamma^{(2)}(-i(t+y\pm
w+\omega\rho)-i\omega_{1}(p\pm k+r\sigma+l);-i\omega_{1}r,-i\omega)$ (37)
$\displaystyle\times\gamma^{(2)}(-i(t-y\pm w+\omega(1-\rho))-i\omega_{1}(p\pm
k+r(1-\sigma)-l);-i\omega_{1}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(t-y\pm w+\omega\rho)-i\omega_{2}(r-(p\pm
k+r\sigma+l));-i\omega_{2}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(t-y\pm
w+\omega(1-\rho))-i\omega_{2}(r-(p\pm
k+r(1-\sigma)-l));-i\omega_{2}r,-i\omega).$

Obviously, one can see that

$\displaystyle D(t,p;y,l;w,k)D(-t,-p;y,l;w,k)=1\;,$ (38)

and

$\displaystyle D(0,0;y,l;w,k)=1\;.$ (39)

Now we can construct the integral sum operator as follows

$\displaystyle M(t,p)_{z,m;x,j}=$
$\displaystyle\frac{1}{C(t,p)}\sum_{j=0}^{[r/2]}\int_{-\infty}^{\infty}\gamma^{(2)}(-i(-t+z\pm
x)-i\omega_{1}(m-p\pm j);-i\omega_{1}r,-i\omega)$ (40)
$\displaystyle\times\gamma^{(2)}(-i(-t+z\pm x)-i\omega_{2}(r-(m-p\pm
j));-i\omega_{2}r,-i\omega)$ $\displaystyle\times\gamma^{(2)}(-i(-t-z\pm
x)-i\omega_{1}(-m-p\pm j);-i\omega_{1}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(-t-z\pm x)-i\omega_{r}(r-(-m-p\pm
j));-i\omega_{2}r,-i\omega)\frac{[\mathrm{d}_{j}{x}]}{2r\sqrt{-\omega_{1}\omega_{2}}}\>,$

where the measure of this integral operator is written as the following

$\displaystyle[\mathrm{d}_{j}{x}]=\frac{\epsilon(j)\mathrm{d}x}{\gamma^{(2)}(\pm
2ix\pm 2i\omega_{1}j;-i\omega_{1}r,-i\omega)\gamma^{(2)}(\pm
2ix-i\omega_{2}(r\pm 2j);-i\omega_{2}r,-i\omega)}\>,$ (41)

and the contribution of the spin-independent function is

$\displaystyle C(t,p)=$
$\displaystyle\gamma^{(2)}(-i(-2t)-i\omega_{1}(-2p);-i\omega_{1}r,-i\omega)$
(42)
$\displaystyle\times\gamma^{(2)}(-i(-2t)-i\omega_{2}(r-(-2p));-i\omega_{2}r,-i\omega)\;.$

Using these operators and the identity $\eqref{star triangle}$, one can find
the parameters as follows

$\displaystyle{a}_{1,2}$ $\displaystyle=-s\pm w,$ $\displaystyle{a}_{3}$
$\displaystyle=s+t+y+\omega\rho\>,$ (43) $\displaystyle{a}_{4}$
$\displaystyle=s+t-y+\omega(1-\rho),$ $\displaystyle{a}_{5,6}$
$\displaystyle=-t\pm x\>,$ $\displaystyle{u}_{1,2}$ $\displaystyle=-q\pm k,$
$\displaystyle u_{3}$ $\displaystyle=q+p+l+r\sigma\>,$ $\displaystyle{u}_{4}$
$\displaystyle=q+p-l+r(1-\sigma),$ $\displaystyle{u}_{5,6}$
$\displaystyle=-p\pm m\>,$

####  4.1.2 $U(1)$ gauge symmetry

For $U(1)$ gauge symmetry we need to redefine the operators such that

$\displaystyle
D(t,p;y,l;w,k)=\gamma^{(2)}(-i(t+y+w+\omega\rho)-i\omega_{1}(p+k+r\sigma+l);-i\omega_{1}r,-i\omega)$
(44)
$\displaystyle\times\gamma^{(2)}(-i(t-y-w+\omega(1-\rho))-i\omega_{1}(p-k+r(1-\sigma)-l);-i\omega_{1}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(t-y+w+\omega\rho)-i\omega_{2}(r-(p+k+r\sigma+l));-i\omega_{2}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(t-y-w+\omega(1-\rho))-i\omega_{2}(r-(p-k+r(1-\sigma)-l));-i\omega_{2}r,-i\omega)\>.$

Obviously, one can again catch the reflection property of the operator such
that

$\displaystyle D(t,p;y,l;w,k)D(-t,-p;y,l;w,k)=1\;,$ (45)

and similarly

$\displaystyle D(0,0;y,l;w,k)=1\;.$ (46)

Also for the integral-sum operator, we need a slight change such that

$\displaystyle M(t,p)_{z,m;x,j}=$
$\displaystyle\frac{1}{C(t,p)}\sum_{j=0}^{[r/2]}\epsilon(j)\int_{-\infty}^{\infty}\gamma^{(2)}(-i(-t+z+x)-i\omega_{1}(m-p+j);-i\omega_{1}r,-i\omega)$
(47)
$\displaystyle\times\gamma^{(2)}(-i(-t+z+x)-i\omega_{2}(r-(m-p+j));-i\omega_{2}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(-t-z-x)-i\omega_{1}(-m-p-j);-i\omega_{1}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(-t-z-x)-i\omega_{r}(r-(-m-p-j));-i\omega_{2}r,-i\omega)$
$\displaystyle\times\frac{dx}{2r\sqrt{-\omega_{1}\omega_{2}}}\>,$

where $C(t,p)$ has the same definition in (42).

As we have demonstrated in the previous section, we can re-group these
multipliers at the right-hand side of the $U(1)$ gauge symmetry integral
identity. By using the same methodology that was used to re-arrange the
parameters of the $SU(2)$ gauge symmetry integral identity we can write the
parameters of the $U(1)$ gauge symmetry integral identity as follows

$\displaystyle{a}_{1},{b}_{1}$ $\displaystyle=-s\pm w\>,$
$\displaystyle{a}_{2}$ $\displaystyle=s+t+y+\omega\rho\>,$ (48)
$\displaystyle{b}_{2}$ $\displaystyle=s+t-y+\omega(1-\rho)\>,$
$\displaystyle{a}_{3},{b}_{3}$ $\displaystyle=-t\pm x\>,$
$\displaystyle{u}_{1},{v}_{1}$ $\displaystyle=-q\pm k\>,$ $\displaystyle
u_{2}$ $\displaystyle=q+p+l+r\sigma\>,$ $\displaystyle{v}_{2}$
$\displaystyle=q+p-l+r(1-\sigma)\>,$ $\displaystyle{u}_{3},{v}_{3}$
$\displaystyle=-p\pm m\>.$

###  4.2 Star-star relation

We will now be discussing Bailey pairs generated from an initial explicit
pair. Noting that $M(t,p)_{z,m;x,j}$ is an integral-sum operator acting on a
sequence of functions $f_{j}(x)$, the relation $\eqref{definition}$ suggests
to start with $\alpha(x,j;t,p)=\delta_{jn}\delta(x-u)$ where $n\in\mathbb{Z}$,
$u\in\mathbb{C}$ are new parameters.

Then, $\beta(z,m;t,p)$ of the following form

$\displaystyle\beta(z,m;t,p)$
$\displaystyle=M(t,p)_{z,m;x,j}\delta_{jn}\delta(x-u)$ (49)
$\displaystyle:=M(t,p;z,m;u,n)\;,$

forms a Bailey pair with $\alpha(x,j;t,p)$. From here, we generate new pairs
with the Bailey lemma,

$\displaystyle\alpha(x,k;t+s;p+q)$
$\displaystyle=D(s,q;y,l;x,k)\alpha(x,k;t,p)\>,$ (50)
$\displaystyle\beta(x,k;t+s;p+q)$
$\displaystyle=D(-t,-p;y,l;x,k)M(s,q)_{x,k;z,m}D(s+t,p+q;y,l;z,m)\beta(z,m;t,p)\;.$
(51)

The relation $\eqref{definition}$ does not give us a particularly interesting
result as it yields the star-triangle relation, which we have used to prove
the Bailey lemma

$\displaystyle M$
$\displaystyle(s,q)_{w,k;z,m}D(s+t,p+q;y,l;z,m)M(t,p;z,m;u,n)$ (52)
$\displaystyle=D(t,p;y,l;w,k)M(s+t,p+q;w,k;u,n)D(s,q;y,l,u,n)\;.$

An immediate consequence of $\eqref{second bailey relation}$ is the functions
$\tilde{\alpha}(z,m;s,q)$ and $\tilde{\beta}(w,k;s,q)$ defined by

$\displaystyle\tilde{\alpha}(z,m;s,q)$
$\displaystyle=D(s+t,p+q;y,l;z,m)M(t,p;z,m;u,n)\>,$ (53)
$\displaystyle\tilde{\beta}(w,k;s,q)$
$\displaystyle=D(t,p;y,l;w,k)M(s+t,p+q;w,k;u,n)D(s,q;y,l,u,n)\;,$ (54)

form a Bailey pair with respect to parameters $s\in\mathbb{C}$,
$q\in\mathbb{Z}$. Applying the lemma once again, we find

$\displaystyle\tilde{\alpha}^{\prime}(z,m;s+c,q+d)=$ $\displaystyle
D(c,d;a,b;z,m)D(s+t,p+q;y,l;z,m)M(t,p;z,m;u,n)\>,$ (55)
$\displaystyle\begin{split}\tilde{\beta}^{\prime}(x,j;s+c,q+d)=&D(-s,-q;a,b;x,j)M(c,d)_{x,j;w,k}D(s+c,q+d;a,b;w,k)\\\
&\times D(t,p;y,l;w,k)M(s+t,p+q;w,k;u,n)D(s,q;y,l,u,n)\;,\end{split}$ (56)

where $a,c\in\mathbb{C}$ and $b,d\in\mathbb{Z}$ are arbitrary. The relation

$\tilde{\beta}^{\prime}(x,j;s+c,q+d)=M(s+c,q+d)_{x,j;z,m}\tilde{\alpha}^{\prime}(z,m;s+c,q+d)\;,$
(57)

yields a non-trivial integral identity

$\displaystyle
M(c,d)_{x,j;w,k}D(s+c,q+d;a,b;w,k)D(t,p;y,l;w,k)M(s+t,p+q;w,k;u,n)$ (58)
$\displaystyle=D(-s,-q;y,l,u,n)D(s,q;a,b;x,j)$ $\displaystyle\text{ }\times
M(s+c,q+d)_{x,j;z,m}D(c,d;a,b;z,m)D(s+t,p+q;y,l;z,m)M(t,p;z,m;u,n)\;,$

which can be recognized as the star-star relation.

####  4.2.1  $SU(2)$ gauge symmetry

Now we need to construct this identity as an integral form. For constructing
bailey pairs’ operators, we will just write the same form of the operators
that were used for $SU(2)$ gauge symmetry. However, we have one more operator

$\displaystyle M(t,p;z,m;u,n)=$
$\displaystyle\frac{\Delta_{n}^{u}}{C(t,p)}\gamma^{(2)}(-i(-t+z\pm
u)-i\omega_{1}(m-p\pm n);-i\omega_{1}r,-i\omega)$ (59)
$\displaystyle\times\gamma^{(2)}(-i(-t+z\pm u)-i\omega_{2}(r-(m-p\pm
n));-i\omega_{2}r,-i\omega)$ $\displaystyle\times\gamma^{(2)}(-i(-t-z\pm
u)-i\omega_{1}(-m-p\pm n);-i\omega_{1}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(-t-z\pm u)-i\omega_{r}(r-(-m-p\pm
n));-i\omega_{2}r,-i\omega)\>,$

where

$\displaystyle\Delta_{n}^{u}=\frac{\epsilon(n)}{2r\sqrt{-\omega_{1}\omega_{2}}}\frac{1}{\gamma^{(2)}(\pm
2iu\pm 2i\omega_{1}n;-i\omega_{1}r,-i\omega)\gamma^{(2)}(\pm
2iu-i\omega_{2}(r\pm 2n);-i\omega_{2}r,-i\omega)}\>,$ (60)

and $C(t,p)$ is still (42).

Then one can write down the parameters as follows:

$\displaystyle{a}_{1,2}$ $\displaystyle=-c\pm x\>,$ $\displaystyle a_{3}$
$\displaystyle=s+c+a+\omega\rho\>,$ $\displaystyle a_{4}$
$\displaystyle=s+c-a+\omega(1-\rho)\>,$ (61) $\displaystyle{a}_{5}$
$\displaystyle=t+y+\omega\rho\>,$ $\displaystyle{a}_{6}$
$\displaystyle=t-y+\omega(1-\rho)\>,$ $\displaystyle{a}_{7,8}$
$\displaystyle=-(s+t)\pm u\>,$ $\displaystyle{u}_{1,2}$ $\displaystyle=-d\pm
j\>,$ $\displaystyle u_{3}$ $\displaystyle=q+d+b+r\sigma\>,$ $\displaystyle
u_{4}$ $\displaystyle=q+d-b+r(1-\sigma)\>,$ $\displaystyle{u}_{5}$
$\displaystyle=p+l+r\sigma\>,$ $\displaystyle{u}_{6}$
$\displaystyle=p-l+r(1-\sigma)\>,$ $\displaystyle u_{7,8}$
$\displaystyle=-(p+q)\pm n\>.$

For the right-hand side of the integral identity, we can write the simple
equality between the parameters as follows,

$\displaystyle\begin{aligned}
\tilde{a}_{i}&=a_{i}-s,&\tilde{u}_{i}&=u_{i}-q,&\text{if}\;\;\;i=1,2,3,4\>,\\\
\tilde{a}_{i}&=a_{i}+s,&\tilde{u}_{i}&=u_{i}+q,&\text{if}\;\;\;i=5,6,7,8\>.\end{aligned}$
(62)

####  4.2.2 $U(1)$ gauge symmetry

For this part, all we need to do is just re-use the operators that were
designed to reach the Bailey pair reconstruction of star-triangle form for the
$U(1)$ gauge symmetry.

Again, we need the following operator for the construction

$\displaystyle M(t,p;z,m;x,j)=$
$\displaystyle\frac{1}{C(t,p)}\frac{\epsilon(n)}{r\sqrt{-\omega_{1}\omega_{2}}}$
(63)
$\displaystyle\times\gamma^{(2)}(-i(-t+z+x)-i\omega_{1}(m-p+j);-i\omega_{1}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(-t+z+x)-i\omega_{2}(r-(m-p+j));-i\omega_{2}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(-t-z-x)-i\omega_{1}(-m-p-j);-i\omega_{1}r,-i\omega)$
$\displaystyle\times\gamma^{(2)}(-i(-t-z-x)-i\omega_{r}(r-(-m-p-j));-i\omega_{2}r,-i\omega)\>,$

where $C(t,p)$ still lives as in (42).

Then one can write down the parameters as follows:

$\displaystyle{a}_{1},{b}_{1}$ $\displaystyle=-c\pm x\>,$ $\displaystyle
a_{2}$ $\displaystyle=s+c+a+\omega\rho\>,$ $\displaystyle b_{2}$
$\displaystyle=s+c-a+\omega(1-\rho)\>,$ (64) $\displaystyle{a}_{3}$
$\displaystyle=t+y+\omega\rho\>,$ $\displaystyle{b}_{3}$
$\displaystyle=t-y+\omega(1-\rho)\>,$ $\displaystyle{a}_{4},{b}_{4}$
$\displaystyle=-(s+t)\pm u\>,$ $\displaystyle{u}_{1},{v}_{1}$
$\displaystyle=-d\pm j\>,$ $\displaystyle u_{2}$
$\displaystyle=q+d+b+r\sigma\>,$ $\displaystyle v_{2}$
$\displaystyle=q+d-b+r(1-\sigma)\>,$ $\displaystyle{u}_{3}$
$\displaystyle=p+l+r\sigma\>,$ $\displaystyle{v}_{3}$
$\displaystyle=p-l+r(1-\sigma)\>,$ $\displaystyle{u}_{4},{v}_{4}$
$\displaystyle=-(p+q)\pm n\>,$

For the right-hand side of the integral identity, we can write the simple
equality between the parameters as follows:

$\displaystyle\begin{aligned}
\tilde{a}_{i}&=a_{i}-s,&\tilde{b}_{i}&=b_{i}-s,&\tilde{u}_{i}&=u_{i}-q,&\tilde{v}_{i}&=v_{i}-q,&\text{if}\;\;\;i=1,2\>,\\\
\tilde{a}_{i}&=a_{i}+s,&\tilde{b}_{i}&=b_{i}+s,&\tilde{u}_{i}&=u_{i}+q,&\tilde{v}_{i}&=v_{i}+q,&\text{if}\;\;\;i=3,4\>.\end{aligned}$
(65)

###  4.3 Pentagon identity

We will now consider a different definition of Bailey pairs and its relation
to a pentagon identity on $U(1)$. The discussion will be a slight
generalization of Kashaev:2012cz  (see also Gahramanov:2021pgu ) with discrete
parameters.

######  Definition 4.2.

The functions $\alpha(x,n;t,p)$ and $\beta(x,n;t,p)$ with variables
$x\in\mathbb{C}$ and $n\in\mathbb{Z}$ are said to form a pentagon Bailey pair
with respect to parameters $t\in\mathbb{C}$ and $p\in\mathbb{Z}$ if the
following relation is satisfied

$\beta(x,n;t,p)=\sum_{m}\int_{-\infty}^{\infty}\mathrm{d}z\;\mathcal{B}(t+x-z,p+n-m;t-x+z,p-n+m)\alpha(z,m;t,p)\;,$
(66)

where $\mathcal{B}(z_{1},u_{1};z_{2},u_{2})$ is defined as in
$\eqref{pentagon_b}$.

######  Theorem 4.3.

Suppose $\alpha(z,m;t,p)$ and $\beta(x,n;t,p)$ form a pentagon Bailey pair
with respect to $t\in\mathbb{C}$ and $p\in\mathbb{Z}$. Then, the sequences of
functions $\alpha^{\prime}(z,m;t+s,p+q)$ and $\beta^{\prime}(w,k;t+s,p+q)$,
$w\in\mathbb{C}$, $k\in\mathbb{Z}$, defined by

$\displaystyle\alpha^{\prime}(z,m,t+s,p+q)=$
$\displaystyle\mathcal{B}(z+t+y,m+p+l;2s,2q)\alpha(z,m;t,p)\>,$ (67)
$\displaystyle\beta^{\prime}(w,k;t+s,p+q)=$
$\displaystyle\sum_{n}\int_{-\infty}^{\infty}\mathrm{d}x\;\mathcal{B}(s+w-x,q+k-n;y+x,l+n)$
(68)
$\displaystyle\times\mathcal{B}(s+2t+y+w,q+2p+l+k;s-w+x,q-k+n)\beta(x,n;t,p)\>,$

form a Bailey pair with respect to $t+s$ and $p+q$, where $y,s\in\mathbb{C}$,
$l,q\in\mathbb{Z}$.

###### Proof.

We need to show the relation defining $\beta^{\prime}(w,k;t+s,p+q)$ and
$\alpha^{\prime}(z,m,t+s,p+q)$ is satisfied,

$\beta^{\prime}(w,k;t+s,p+q)=\sum_{m}\int_{-\infty}^{\infty}\mathrm{d}z\;\mathcal{B}(t+s+w-z,p+q+k-m;t+s-w+z,p+q-k+m)\alpha^{\prime}(z,m;t+s,p+q)\;.$
(69)

Substituting $\eqref{pentagon alpha}$ and $\eqref{pentagon beta}$ for
$\alpha^{\prime}(z,m,t+s,p+q)$ and $\beta^{\prime}(w,k;t+s,p+q)$, we arrive at

$\displaystyle\sum_{n}\int_{-\infty}^{\infty}$
$\displaystyle\mathrm{d}x\;\mathcal{B}(s+w-x,p+k-n;y+x,l+n)\mathcal{B}(s+2t+y+w,q+2p+l+k;s-w+x,q-k+n)$
(70) $\displaystyle\times\sum_{m}$
$\displaystyle\int_{-\infty}^{\infty}\mathrm{d}z\;\mathcal{B}(t+x-z,p+n-m;t-x+z,q-n+m)\alpha(z,m;t,p)$
$\displaystyle=$
$\displaystyle\sum_{m}\int_{-\infty}^{\infty}\mathrm{d}z\;\mathcal{B}(t+s+w-z,p+q+k-m;t+s-w+z,p+q-k+m)$
$\displaystyle\times\mathcal{B}(z+t+y,m+p+l;2s,2q)\alpha(z,m;t,p)\;.$

Rearrangement of some terms yields

$\displaystyle\sum_{m}\int_{-\infty}^{\infty}$
$\displaystyle\mathrm{d}z\;\alpha(z,m;t,p)\sum_{n}\int_{-\infty}^{\infty}\mathrm{d}x\;\mathcal{B}(s+w-x,p+k-n;y+x,l+n)$
(71) $\displaystyle\times\mathcal{B}(s+2t+y+w,q+2p+l+k;s-w+x,q-k+n)$
$\displaystyle\times\mathcal{B}(t+x-z,p+n-m;t-x+z,q-n+m)$
$\displaystyle=\sum_{m}\int_{-\infty}^{\infty}$
$\displaystyle\mathrm{d}z\;\alpha(z,m;t,p)\mathcal{B}(t+s+w-z,p+q+k-m;t+s-w+z,p+q-k+m)$
$\displaystyle\times\mathcal{B}(z+t+y,m+p+l;2s,2q)\;.$

With the choice of $\mathcal{B}(z_{1},u_{1};z_{2},u_{2})$ in
$\eqref{pentagon_b}$, we see that $\eqref{pentagon integral}$ is indeed
satisfied with the following parametrization,

$\displaystyle\begin{aligned} a_{1}&=s+w,&b_{1}&=y,&u_{1}&=q+k,&v_{1}&=l\>,\\\
a_{2}&=\omega-(2s+2t+y),&b_{2}&=s-w,&u_{2}&=-(2q+2p+l),&v_{2}&=q-k\>,\\\
a_{3}&=t+z,&b_{3}&=t-z,&u_{3}&=p+m,&v_{3}&=p-m\>.\end{aligned}$ (72)

and we arrive at the pentagon relation $\eqref{pentagon}$. Hence,
$\beta^{\prime}(w,k;t+s,p+q)$ and $\alpha^{\prime}(z,m,t+s,p+q)$ satisfy the
Bailey lemma. ∎

##  5 Conclusions

We have studied novel Bailey pairs constructed from the $3d$ $\mathcal{N}=2$
dual supersymmetric gauge theories on the lens space
$S_{b}^{3}/\mathbb{Z}_{r}$. The equality of the supersymmetric partition
functions of the dual theories on the lens space leads to non-trivial
hyperbolic hypergeometric integral identities. The integral identities have
previously been discussed in terms of lattice spin models (Ising-like and IRF-
type) in statistical mechanics and pentagon identity as a $2-3$ Pachner move
Bozkurt:2020gyy ; Catak:2021coz ; Mullahasanoglu:2021xyf ; Dede:2022ofo . In
this work, we have constructed Bailey pairs that generate these integral
identities. These Bailey pair constructions allow us to study the vertex-type
integrable models Gahramanov:2015cva , knot invariants Kashaev:2012cz ,
supersymmetric quiver gauge theories Brunner:2017lhb , etc. One can use the
Bailey pair construction to generate integral identities for supersymmetric
dualities. One possible future direction is to examine other supersymmetric IR
dualities in this context.

In the context of integrable lattice spin models, we construct the Boltzmann
weights with two types of spectral parameters121212In the literature there are
models with two rapidity parameters, see, e.g. Khachatryan:2012wy ;
Karakhanyan:2013pv ., and these discrete and continuous types of parameters
are preserved in Bailey pair constructions. It would be interesting to see the
implications of this result.

One can obtain the rational beta integral identities, namely the equality of
supersymmetric gauge partition functions on $S^{2}$ by limiting $r\to\infty$
in the integral identities for dualities on $S^{3}_{b}/\mathbb{Z}_{r}$, see
Eren:2019ibl ; Sarkissian:2020pwa . In this work, we constructed Bailey pairs
for the integral identities with the balancing conditions
$\sum_{i=1}^{6}u_{i}=r$ and $\sum_{i=1}^{3}u_{i}+v_{i}=r$. Since the limit
$r\to\infty$ is problematic for the balancing conditions, it would be
interesting to analyze the limiting procedure in our cases.

## Acknowledgments

The authors are grateful to Erdal Catak for the valuable discussions. Ilmar
Gahramanov and Mustafa Mullahasanoglu are supported by the 1002-TUBITAK Quick
Support Program under grant number 121F413. The work of Ilmar Gahramanov
(Sec.4.1) is supported by the Russian Science Foundation grant number
22-72-10122. Ilmar Gahramanov is also partially supported by the Bogazici
University Research Fund under grant number 20B03SUP3.

##  Appendix A Constructing the Bailey pair for the star-triangle relation

Let us construct the governing equation (30) for the star-triangle relation of
the operators and the required identity for the Bailey pairs. We take the
following steps. First, we replace the definitions of the operators on the
left-hand side for a particular model. Then, the specific change of variables
makes the integral part of the specific identity. Before obtaining the right
side of (30), the integral identity allows us to calculate one of the
integrals on the left. Finally, we call back the old variables to write the
right side in proper operator form. One thing to be careful of is the spin-
independent functions in the integral operator. We will mention it again when
it appears in the calculation.

Let’s use the following shorthand notations in the calculations

$\displaystyle\gamma_{h}(z,y;\omega_{1},\omega_{2})=\gamma^{(2)}(-iz-i\omega_{1}y;-i\omega_{1}r,-i\omega)\gamma^{(2)}(-iz-i\omega_{2}(r-y);-i\omega_{2}r,-i\omega)\>,$
(73)

where $\omega=\omega_{1}+\omega_{2}$ and

$\displaystyle\gamma_{h}(\pm z,\pm
y;\omega_{1},\omega_{2})=\gamma_{h}(z,y;\omega_{1},\omega_{2})\gamma_{h}(-z,-y;\omega_{1},\omega_{2})\>,$
(74)

Recall that $[d_{j}x]$ is defined in (41).

Let us explicitly show how to equate the star-triangle equation (30). If one
replaces the definitions of the operators, then obtains the following at the
left-hand side

$\displaystyle M(s,q)_{w,k;z,m}$ $\displaystyle
D(s+t,q+p;y,l;z,m)M(t,p)_{z,m;x,j}=$ (75)
$\displaystyle\frac{1}{C(s,q)C(t,p)}$
$\displaystyle\sum_{j=0}^{[r/2]}\epsilon(j)\int_{-\infty}^{\infty}\frac{[d_{j}x]}{2r\sqrt{-\omega_{1}\omega_{2}}}\sum_{m=0}^{[r/2]}\epsilon(m)\int_{-\infty}^{\infty}\frac{[d_{m}z]}{2r\sqrt{-\omega_{1}\omega_{2}}}$
$\displaystyle\times\gamma_{h}($ $\displaystyle-s+w\pm z,+k-q\pm
m;\omega_{1},\omega_{2})\>\gamma_{h}(-s-w\pm z,-k-q\pm
m;\omega_{1},\omega_{2})$ $\displaystyle\times\gamma_{h}($ $\displaystyle
s+t+y\pm z+\omega\rho,q+p\pm m+r\sigma+l;\omega_{1},\omega_{2})$
$\displaystyle\times\gamma_{h}($ $\displaystyle s+t-y\pm
z+\omega(1-\rho),q+p\pm m+r(1-\sigma)-l;\omega_{1},\omega_{2})$
$\displaystyle\times\gamma_{h}($ $\displaystyle-t+z\pm x,m-p\pm
j);\omega_{1},\omega_{2})\>\gamma_{h}(-t-z\pm x,-m-p\pm
j);\omega_{1},\omega_{2})\>,$

where $C(t,p)=\gamma_{h}(-2t,-2p;\omega_{1},\omega_{2})$ in new notations as
in (42).

One can rewrite equation (75) under the given re-parametrization

$\displaystyle{a}_{1,2}$ $\displaystyle=-s\pm w,$ $\displaystyle{a}_{3}$
$\displaystyle=s+t+y+\omega\rho\>,$ (76) $\displaystyle{a}_{4}$
$\displaystyle=s+t-y+\omega(1-\rho),$ $\displaystyle{a}_{5,6}$
$\displaystyle=-t\pm x\>,$ $\displaystyle{u}_{1,2}$ $\displaystyle=-q\pm k,$
$\displaystyle u_{3}$ $\displaystyle=q+p+l+r\sigma\>,$ $\displaystyle{u}_{4}$
$\displaystyle=q+p-l+r(1-\sigma),$ $\displaystyle{u}_{5,6}$
$\displaystyle=-p\pm m\>.$

Then, (75) becomes

$\displaystyle=\sum_{j=0}^{[r/2]}\epsilon(j)\int_{-\infty}^{\infty}$
$\displaystyle\frac{[d_{j}x]}{2r\sqrt{-\omega_{1}\omega_{2}}}\sum_{m=0}^{[r/2]}\epsilon(m)\int_{-\infty}^{\infty}\frac{[d_{m}z]}{2r\sqrt{-\omega_{1}\omega_{2}}}$
(77) $\displaystyle\times$
$\displaystyle\frac{\prod_{i=1}^{6}\gamma_{h}(a_{i}\pm z,u_{i}\pm
m;\omega_{1},\omega_{2})}{\gamma_{h}(a_{1}+a_{2},u_{1}+u_{2};\omega_{1},\omega_{2})\gamma_{h}(a_{5}+a_{6},u_{5}+u_{6};\omega_{1},\omega_{2})}\;.$

One can integrate with respect to $z$ and sum on $m$ by using the integral
identity (1) and the result is

$\displaystyle=\sum_{j=0}^{[r/2]}\epsilon(j)$
$\displaystyle\int_{-\infty}^{\infty}\frac{\prod_{1\leq i<j\leq
6}\gamma_{h}(a_{i}+a_{j},u_{i}+u_{j};\omega_{1},\omega_{2})}{\gamma_{h}(a_{1}+a_{2},u_{1}+u_{2};\omega_{1},\omega_{2})\gamma_{h}(a_{5}+a_{6},u_{5}+u_{6};\omega_{1},\omega_{2})}\frac{[d_{m}x]}{2r\sqrt{-\omega_{1}\omega_{2}}}\;.$
(78)

If one replaces the parameters (76) with their values found before, it is easy
to see that (78) turns into

$\displaystyle=D(t,p;y,l;w,k)M(s+t,q+p)_{w,k;x,j}D(s,q;y,l;x,j)\>.$ (79)

This is the right-hand side of the star-triangle relation (30).

## References

  * (1) D. Gang, “Chern-Simons Theory on $L(p,q)$ Lens Spaces and Localization,” J. Korean Phys. Soc. 74 (2019) no. 12, 1119–1128, arXiv:0912.4664 [hep-th].
  * (2) F. Benini, T. Nishioka, and M. Yamazaki, “4d Index to 3d Index and 2d TQFT,” Phys. Rev. D86 (2012) 065015, arXiv:1109.0283 [hep-th].
  * (3) Y. Imamura and D. Yokoyama, “$S^{3}/Z_{n}$ partition function and dualities,” JHEP 11 (2012) 122, arXiv:1208.1404 [hep-th].
  * (4) Y. Imamura, H. Matsuno, and D. Yokoyama, “Factorization of the $S^{3}/\mathbb{Z}_{n}$ partition function,” Phys. Rev. D89 (2014) no. 8, 085003, arXiv:1311.2371 [hep-th].
  * (5) F. Nieri and S. Pasquetti, “Factorisation and holomorphic blocks in 4d,” JHEP 11 (2015) 155, arXiv:1507.00261 [hep-th].
  * (6) L. F. Alday, M. Fluder, and J. Sparks, “The Large N limit of M2-branes on Lens spaces,” JHEP 10 (2012) 057, arXiv:1204.1280 [hep-th].
  * (7) M. Yamazaki, “Four-dimensional superconformal index reloaded,” Theor. Math. Phys. 174 (2013) 154–166. [Teor. Mat. Fiz.174,177(2013)].
  * (8) M. Honda, “How to resum perturbative series in 3d N=2 Chern-Simons matter theories,” Phys. Rev. D 94 (2016) no. 2, 025039, arXiv:1604.08653 [hep-th].
  * (9) A. Nedelin, F. Nieri, and M. Zabzine, “$q$-Virasoro modular double and 3d partition functions,” Commun. Math. Phys. 353 (2017) no. 3, 1059–1102, arXiv:1605.07029 [hep-th].
  * (10) V. P. Spiridonov and G. S. Vartanov, “Elliptic Hypergeometry of Supersymmetric Dualities,” Commun. Math. Phys. 304 (2011) 797–874, arXiv:0910.5944 [hep-th].
  * (11) V. P. Spiridonov and G. S. Vartanov, “Elliptic hypergeometry of supersymmetric dualities ii. orthogonal groups, knots, and vortices,” Commun. Math. Phys. 325 (2014) no. 2, 421–486, arXiv:1107.5788 [hep-th].
  * (12) C. Krattenthaler, V. P. Spiridonov, and G. S. Vartanov, “Superconformal indices of three-dimensional theories related by mirror symmetry,” JHEP 06 (2011) 008, arXiv:1103.4075 [hep-th].
  * (13) I. B. Gahramanov and G. S. Vartanov, “Superconformal indices and partition functions for supersymmetric field theories,” in XVIIth Intern. Cong. Math. Phys. 695-703 (2013).  2013\.  arXiv:1310.8507 [hep-th].
  * (14) F. A. H. Dolan, V. P. Spiridonov, and G. S. Vartanov, “From 4d superconformal indices to 3d partition functions,” Phys. Lett. B704 (2011) 234–241, arXiv:1104.1787 [hep-th].
  * (15) I. Gahramanov, “Mathematical structures behind supersymmetric dualities,” Archivum Math. 51 (2015) 273–286, arXiv:1505.05656 [math-ph].
  * (16) Y. Tachikawa, “On ’categories’ of quantum field theories,” in International Congress of Mathematicians.  12, 2017.  arXiv:1712.09456 [math-ph].
  * (17) V. P. Spiridonov, “Superconformal Indices, Seiberg Dualities and Special Functions,” Phys. Part. Nucl. 51 (2020) no. 4, 508–513, arXiv:1912.11514 [hep-th].
  * (18) I. Gahramanov, “Integrability from supersymmetric duality: a short review,” arXiv:2201.00351 [hep-th].
  * (19) V. P. Spiridonov, “Elliptic beta integrals and solvable models of statistical mechanics,” Contemp. Math. 563 (2012) 181–211, arXiv:1011.3798 [hep-th].
  * (20) M. Yamazaki, “Quivers, YBE and 3-manifolds,” JHEP 05 (2012) 147, arXiv:1203.5784 [hep-th].
  * (21) A. P. Kels, “New solutions of the star–triangle relation with discrete and continuous spin variables,” J. Phys. A48 (2015) no. 43, 435201, arXiv:1504.07074 [math-ph].
  * (22) J. Yagi, “Quiver gauge theories and integrable lattice models,” JHEP 10 (2015) 065, arXiv:1504.04055 [hep-th].
  * (23) I. Gahramanov and V. P. Spiridonov, “The star-triangle relation and 3d superconformal indices,” JHEP 08 (2015) 040, arXiv:1505.00765 [hep-th].
  * (24) I. Gahramanov and A. P. Kels, “The star-triangle relation, lens partition function, and hypergeometric sum/integrals,” JHEP 02 (2017) 040, arXiv:1610.09229 [math-ph].
  * (25) G. Sarkissian and V. P. Spiridonov, “From rarefied elliptic beta integral to parafermionic star-triangle relation,” JHEP 10 (2018) 097, arXiv:1809.00493 [hep-th].
  * (26) E. Eren, I. Gahramanov, S. Jafarzade, and G. Mogol, “Gamma function solutions to the star-triangle equation,” Nucl. Phys. B 963 (2021) 115283, arXiv:1912.12271 [math-ph].
  * (27) J. de-la Cruz-Moreno and H. García-Compeán, “Star-triangle type relations from $2d$ $\mathcal{N}=(0,2)$ $USp(2N)$ dualities,” JHEP 01 (2021) 023, arXiv:2008.02419 [hep-th].
  * (28) D. N. Bozkurt, I. Gahramanov, and M. Mullahasanoglu, “Lens partition function, pentagon identity, and star-triangle relation,” Phys. Rev. D 103 (2021) no. 12, 126013, arXiv:2009.14198 [hep-th].
  * (29) M. Yamazaki and W. Yan, “Integrability from 2d ${\mathcal{N}}=(2,2)$ dualities,” J. Phys. A48 (2015) 394001, arXiv:1504.05540 [hep-th].
  * (30) A. P. Kels and M. Yamazaki, “Elliptic hypergeometric sum/integral transformations and supersymmetric lens index,” SIGMA 14 (2018) 013, arXiv:1704.03159 [math-ph].
  * (31) E. Catak, I. Gahramanov, and M. Mullahasanoglu, “Hyperbolic and trigonometric hypergeometric solutions to the star-star equation,” Eur. Phys. J. C 82 (2022) 789, arXiv:2107.06880 [hep-th].
  * (32) M. Mullahasanoglu and N. Tas, “Lens Partition Functions and Integrability Properties,” arXiv:2112.15161 [hep-th].
  * (33) R. Kashaev, F. Luo, and G. Vartanov, “A TQFT of Turaev–Viro Type on Shaped Triangulations,” Annales Henri Poincare 17 (2016) no. 5, 1109–1143, arXiv:1210.8393 [math.QA].
  * (34) R. M. Kashaev, “Beta pentagon relations,” Theor. Math. Phys. 181 (2014) no. 1, 1194–1205, arXiv:1403.1298 [math-ph].
  * (35) R. Kashaev, “Euler’s beta function and pentagon relations,” Acta Mathematica Vietnamica 39 (2014) no. 4, 561–566.
  * (36) I. Gahramanov and H. Rosengren, “A new pentagon identity for the tetrahedron index,” JHEP 11 (2013) 128, arXiv:1309.2195 [hep-th].
  * (37) I. Gahramanov and H. Rosengren, “Integral pentagon relations for 3d superconformal indices,” arXiv:1412.2926 [hep-th]. [Proc. Symp. Pure Math.93,165(2016)].
  * (38) I. Gahramanov and H. Rosengren, “Basic hypergeometry of supersymmetric dualities,” Nucl. Phys. B913 (2016) 747–768, arXiv:1606.08185 [hep-th].
  * (39) D. N. Bozkurt and I. Gahramanov, “Pentagon identities arising in supersymmetric gauge theory computations,” Teor. Mat. Fiz. 198 (2019) no. 2, 215–224, arXiv:1803.00855 [math-ph].
  * (40) S. Jafarzade, “New Pentagon Identities Revisited,” J. Phys. Conf. Ser. 1194 (2019) no. 1, 012054, arXiv:1812.01325 [math-ph].
  * (41) M. Dede, “A comment on the solutions of the generalized Faddeev-Volkov model,” arXiv:2206.14271 [math-ph].
  * (42) V. P. Spiridonov, “A bailey tree for integrals,” Theoretical and mathematical physics 139 (2004) no. 1, 536–541.
  * (43) F. Brünner and V. P. Spiridonov, “4d $\mathcal{N}=1$ quiver gauge theories and the $\mathrm{A_{n}}$ Bailey lemma,” JHEP 03 (2018) 105, arXiv:1712.07018 [hep-th].
  * (44) V. P. Spiridonov, “The rarefied elliptic bailey lemma and the yang–baxter equation,” Journal of Physics A: Mathematical and Theoretical 52 (2019) no. 35, 355201, arXiv:1904.12046 [math-ph].
  * (45) I. Gahramanov and O. E. Kaluc, “Bailey pairs for the q-hypergeometric integral pentagon identity,” arXiv:2111.14793 [math-ph].
  * (46) L. Hadasz, M. Pawelkiewicz, and V. Schomerus, “Self-dual Continuous Series of Representations for $\mathcal{U}_{q}(sl(2))$ and $\mathcal{U}_{q}(osp(1|2))$,” JHEP 10 (2014) 091, arXiv:1305.4596 [hep-th].
  * (47) Y. Fan and T. G. Mertens, “From quantum groups to Liouville and dilaton quantum gravity,” JHEP 05 (2022) 092, arXiv:2109.07770 [hep-th].
  * (48) E. Apresyan, G. Sarkissian, and V. P. Spiridonov, “A parafermionic hypergeometric function and supersymmetric 6j-symbols,” arXiv:2205.10276 [hep-th].
  * (49) I. Gahramanov and S. Jafarzade, “Integrable lattice spin models from supersymmetric dualities,” Phys. Part. Nucl. Lett. 15 (2018) no. 6, 650–667, arXiv:1712.09651 [math-ph].
  * (50) M. Yamazaki, “Integrability As Duality: The Gauge/YBE Correspondence,” Phys. Rept. 859 (2020) 1–20, arXiv:1808.04374 [hep-th].
  * (51) V. V. Bazhanov, V. V. Mangazeev, and S. M. Sergeev, “Faddeev-Volkov solution of the Yang-Baxter equation and discrete conformal symmetry,” Nucl. Phys. B 784 (2007) 234–258, arXiv:hep-th/0703041 [hep-th].
  * (52) V. V. Bazhanov, V. V. Mangazeev, and S. M. Sergeev, “Exact solution of the Faddeev-Volkov model,” Phys. Lett. A372 (2008) 1547–1550, arXiv:0706.3077 [cond-mat.stat-mech].
  * (53) N. Hama, K. Hosomichi, and S. Lee, “SUSY Gauge Theories on Squashed Three-Spheres,” JHEP 05 (2011) 014, arXiv:1102.4716 [hep-th].
  * (54) R. J. Baxter, “Star-triangle and star-star relations in statistical mechanics,” International Journal of Modern Physics B 11 (1997) no. 01n02, 27–37.
  * (55) N. Seiberg, “Electric - magnetic duality in supersymmetric nonAbelian gauge theories,” Nucl. Phys. B435 (1995) 129–146, arXiv:hep-th/9411149 [hep-th].
  * (56) K. A. Intriligator and N. Seiberg, “Mirror symmetry in three-dimensional gauge theories,” Phys. Lett. B 387 (1996) 513–519, arXiv:hep-th/9607207.
  * (57) O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg, and M. J. Strassler, “Aspects of N=2 supersymmetric gauge theories in three-dimensions,” Nucl. Phys. B 499 (1997) 67–99, arXiv:hep-th/9703110.
  * (58) A. Kapustin and B. Willett, “Generalized Superconformal Index for Three Dimensional Field Theories,” arXiv:1106.2484 [hep-th].
  * (59) A. Kapustin, B. Willett, and I. Yaakov, “Nonperturbative Tests of Three-Dimensional Dualities,” JHEP 10 (2010) 013, arXiv:1003.5694 [hep-th].
  * (60) A. Amariti, “Integral identities for 3d dualities with SP(2N) gauge groups,” arXiv:1509.02199 [hep-th].
  * (61) S. Kharchev, D. Lebedev, and M. Semenov-Tian-Shansky, “Unitary representations of U(q) (sl(2, R)), the modular double, and the multiparticle q deformed Toda chains,” Commun. Math. Phys. 225 (2002) 573–609, arXiv:hep-th/0102180 [hep-th].
  * (62) B. Ponsot and J. Teschner, “Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U(q)(sl(2,R)),” Commun. Math. Phys. 224 (2001) 613–655, arXiv:math/0007097 [math-qa].
  * (63) L. D. Faddeev, “Modular double of quantum group,” Math. Phys. Stud. 21 (2000) 149–156, arXiv:math/9912078 [math-qa]. [,149(1999)].
  * (64) L. D. Faddeev, R. M. Kashaev, and A. Yu. Volkov, “Strongly coupled quantum discrete Liouville theory. 1. Algebraic approach and duality,” Commun. Math. Phys. 219 (2001) 199–219, arXiv:hep-th/0006156 [hep-th].
  * (65) A. Y. Volkov, “Noncommutative hypergeometry,” Commun. Math. Phys. 258 (2005) 257–273, arXiv:math/0312084.
  * (66) V. V. Bazhanov and S. M. Sergeev, “A Master solution of the quantum Yang-Baxter equation and classical discrete integrable equations,” Adv. Theor. Math. Phys. 16 (2012) no. 1, 65–95, arXiv:1006.0651 [math-ph].
  * (67) V. P. Spiridonov, “Essays on the theory of elliptic hypergeometric functions,” Russian Mathematical Surveys 63 (2008) no. 3, 405, arXiv:0805.3135 [math.CA].
  * (68) S. N. M. Ruijsenaars, “First order analytic difference equations and integrable quantum systems,” J. Math. Phys. 38 (1997) no. 2, 1069–1146.
  * (69) F. van de Bult et al., “Hyperbolic hypergeometric functions,” Ph.D. Thesis, University of Amsterdam, Amsterdam Netherlands (2007) .
  * (70) J. E. Andersen and R. Kashaev, “Complex Quantum Chern-Simons,” arXiv:1409.1208 [math.QA].
  * (71) L. Faddeev, “Discrete Heisenberg-Weyl group and modular group,” Lett. Math. Phys. 34 (1995) 249–254, arXiv:hep-th/9504111.
  * (72) S. Woronowicz, “Quantum exponential function,” Reviews in Mathematical Physics 12 (2000) no. 06, 873–920.
  * (73) K. Hikami, “Hyperbolic structure arising from a knot invariant,” International Journal of Modern Physics A 16 (2001) no. 19, 3309–3333.
  * (74) K. Hikami, “Generalized volume conjecture and the a-polynomials: The neumann–zagier potential function as a classical limit of the partition function,” Journal of Geometry and Physics 57 (2007) no. 9, 1895–1940.
  * (75) K. Hikami and R. Inoue, “Braiding operator via quantum cluster algebra,” J. Phys A 47 (2014) no. 47, 474006.
  * (76) C.-T. Chan, A. Mironov, A. Morozov, and A. Sleptsov, “Orthogonal Polynomials in Mathematical Physics,” Rev.Math.Phys 30 (2018) 1840005, arXiv:1712.03155 [hep-th].
  * (77) J. Teschner and G. Vartanov, “6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories,” Lett. Math. Phys. 104 (2014) 527–551, arXiv:1202.4698 [hep-th].
  * (78) R. J. Baxter, Exactly Solved Models in Statistical Mechanics.  Academic, London, 1982\.
  * (79) V. P. Spiridonov and G. S. Vartanov, “Superconformal indices for N = 1 theories with multiple duals,” Nucl. Phys. B 824 (2010) 192–216, arXiv:0811.1909 [hep-th].
  * (80) T. Dimofte and D. Gaiotto, “An E7 Surprise,” JHEP 10 (2012) 129, arXiv:1209.1404 [hep-th].
  * (81) V. V. Bazhanov and S. M. Sergeev, “Elliptic gamma-function and multi-spin solutions of the Yang-Baxter equation,” Nucl. Phys. B856 (2012) 475–496, arXiv:1106.5874 [math-ph].
  * (82) V. V. Bazhanov, A. P. Kels, and S. M. Sergeev, “Comment on star-star relations in statistical mechanics and elliptic gamma-function identities,” J. Phys. A46 (2013) 152001, arXiv:1301.5775 [math-ph].
  * (83) U. Pachner, “P.l. homeomorphic manifolds are equivalent by elementary shellings,” European journal of Combinatorics 12 (1991) no. 2, 129–145.
  * (84) T. Dimofte, D. Gaiotto, and S. Gukov, “Gauge Theories Labelled by Three-Manifolds,” Commun. Math. Phys. 325 (2014) 367–419, arXiv:1108.4389 [hep-th].
  * (85) T. Dimofte, D. Gaiotto, and S. Gukov, “3-Manifolds and 3d Indices,” Adv. Theor. Math. Phys. 17 (2013) no. 5, 975–1076, arXiv:1112.5179 [hep-th].
  * (86) W. N. Bailey, “Identities of the rogers-ramanujan type,” Proceedings of the London Mathematical Society s2-50 (1948) no. 1, 1–10.
  * (87) L. Slater, Generalized Hypergeometric Functions.  Cambridge University Press, 1966.
  * (88) D. M. Bressoud, “Some identities for terminating q-series,” Mathematical Proceedings of the Cambridge Philosophical Society 89 (1981) no. 2, 211–223.
  * (89) G. Andrews and D. Bowman, “The Bailey transform and D.B. Sears,” Quaestiones Mathematicae 22 (1999) 19–26.
  * (90) A. Schilling and S. O. Warnaar, “A higher-level bailey lemma,” International Journal of Modern Physics B 11 (1997) no. 01n02, 189–195.
  * (91) G. E. Andrews, “Multiple series Rogers-Ramanujan type identities.,” Pacific Journal of Mathematics 114 (1984) no. 2, 267 – 283.
  * (92) S. C. Milne and G. M. Lilly, “The $A_{\textit{l}}$ and $C_{\textit{l}}$ Bailey transform and lemma,” Bull. Amer. Math. Soc.(NS) 26 (1992) 258–263.
  * (93) G. M. Lilly and S. C. Milne, “The $C_{\textit{l}}$ Bailey transform and Bailey lemma,” Constructive Approximation 9 (1993) no. 4, 473–500.
  * (94) S. C. Milne and G. M. Lilly, “Consequences of the al and cl bailey transform and bailey lemma,” Discrete mathematics 139 (1995) no. 1-3, 319–346.
  * (95) A. Berkovich, B. M. McCoy, and A. Schilling, “N = 2 supersymmetry and bailey pairs,” Physica A: Statistical Mechanics and its Applications 228 (1996) no. 1-4, 33–62.
  * (96) G. E. Andrews and A. Berkovich, “A trinomial analogue of bailey’s lemma and n = 2 superconformal invariance,” Communications in Mathematical Physics 192 (1998) no. 2, 245–260.
  * (97) S. O. Warnaar, “50 years of bailey’s lemma,” in Algebraic Combinatorics and Applications, pp. 333–347.  Springer Berlin Heidelberg, Berlin, Heidelberg, 2001.
  * (98) S. Khachatryan and A. Sedrakyan, “On the solutions of the Yang-Baxter equations with general inhomogeneous eight-vertex $R$-matrix: Relations with Zamolodchikov’s tetrahedral algebra,” J. Statist. Phys. 150 (2013) 130, arXiv:1208.4339 [math-ph].
  * (99) D. Karakhanyan and S. Khachatryan, “New solutions to the $sl_{q}(2)$ -invariant Yang-Baxter equations at roots of unity: Cyclic representations,” Nucl. Phys. B 868 (2013) 328–349, arXiv:1203.6528 [math-ph].
  * (100) G. A. Sarkissian and V. P. Spiridonov, “Rational hypergeometric identities,” arXiv:2012.10265 [math.CA].