††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

# Notes on the lens integral pentagon identity

H. Kubra Baga, Osman Ergeca, and Ilmar Gahramanova,b,c<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We obtain the lens integral pentagon identity for three-dimensional mirror
dual theories in terms of hyperbolic hypergeometric functions via reduction of
equality for $S_{b}^{3}/\mathbb{Z}_{r}$ $\mathcal{N}=2$ supersymmetric
partition functions of a certain supersymmetric IR duality.

###### Keywords:

hyperbolic hypergeometric function, pentagon identity, supersymmetric duality

##  1 Introduction

Pentagon relations have plenty of important applications in the theory of
integrable lattice models of statistical mechanics, topological field
theories, knot invariants, etc Kashaev:1996rz ; Kashaev:2014rea ;
Bozkurt:2018xno ; Bozkurt:2020gyy ; Allman:2017xma ; Gahramanov:2014ona ;
Gahramanov:2013rda ; Gahramanov:2016wxi ; Gahramanov:2022jxz ;
Gahramanov:2021pgu ; Dede:2022ofo ; Kashaev:2012cz ; colazzo2020set . From an
algebraic perspective, pentagon relations are related to the Heisenberg double
kashaev1997heisenberg ; militaru2004heisenberg ; aghaei2019heisenberg . In
this paper, we obtain an integral pentagon relation arising from
supersymmetric gauge theory computations. In order to obtain new relation, we
consider a certain three-dimensional $\mathcal{N}=2$ supersymmetric duality on
lens space $S_{b}^{3}/\mathbb{Z}_{r}$. By taking several limits of the
partition functions for supersymmetric theories we end up with a new pentagon
identity for hyperbolic hypergeometric functions.

##  2 Notations

In this paper we use the so-called hyperbolic gamma function which has the
following integral representation:

$\gamma^{(2)}(y;\omega_{1},\omega_{2})=\exp{\bigg{[}-\int_{0}^{\infty}\bigg{(}\frac{\sinh{(2y-\omega_{1}-\omega_{2})}x}{2\sinh{(\omega_{1}x)}\sinh{(\omega_{2}x)}}-\frac{2y-\omega_{1}-\omega_{2}}{2\omega_{1}\omega_{2}x}\bigg{)}\frac{dx}{x}\bigg{]}},$
(1)

The $\gamma^{(2)}(y;\omega_{1},\omega_{2})$ function has following
asymptotics:

$\begin{split}\lim_{y\to\infty}e^{\frac{i\pi}{2}B_{2,2}(y;\omega_{1},\omega_{2})}\gamma^{(2)}(y;\omega_{1},\omega_{2})=1,\arg{\omega_{1}}<\arg{y}<\arg{\omega_{2}}+\pi,\\\
\lim_{y\to\infty}e^{-\frac{i\pi}{2}B_{2,2}(y;\omega_{1},\omega_{2})}\gamma^{(2)}(y;\omega_{1},\omega_{2})=1,\arg{\omega_{1}}-\pi<\arg{y}<\arg{\omega_{2}},\end{split}$
(2)

where the second order Bernoulli polynomial $B_{2,2}(y;\omega_{1},\omega_{2})$
is defined as follows

$B_{2,2}(y;\omega_{1},\omega_{2})=\frac{y^{2}}{\omega_{1}\omega_{2}}-\frac{y}{\omega_{1}}-\frac{y}{\omega_{2}}+\frac{1}{6}\bigg{(}\frac{\omega_{1}}{\omega_{2}}+\frac{\omega_{2}}{\omega_{1}}\bigg{)}+\frac{1}{2}.$
(3)

##  3 Lens partition function

In Imamura:2012rq ; Imamura:2013qxa , the lens partition function was derived
via the supersymmetric localization technique, and in Benini:2011nc ;
Yamazaki:2013fva ; Eren:2019ibl ; Gahramanov:2016ilb , authors used the
dimensional reduction of the four-dimensional lens superconformal index. Here
we briefly outline some basic ingredients of $\mathcal{N}=2$
$S_{b}^{3}/\mathbb{Z}_{r}$ supersymmetric partition function. We mostly follow
the notations of Catak:2021coz ; Bozkurt:2020gyy . A reader interested in
details is encouraged to look at Imamura:2012rq ; Imamura:2013qxa ;
Nieri:2015yia ; Gahramanov:2016ilb .

By making the identification $(x,y)\sim(e^{\frac{2\pi i}{r}}x,e^{\frac{2\pi
i}{r}}y)$ one can obtain the lens space $S_{b}^{3}/\mathbb{Z}_{r}$ from the
squashed three sphere

$S_{b}^{3}=\\{(x,y)\in\mathbb{C}^{2},b^{2}|x|^{2}+b^{-2}|y|^{2}=1\\}\;.$ (4)

The supersymmetric partition function on this manifold can be reduced to the
following matrix model via supersymmetric localization

$Z=\sum_{m}\int\frac{1}{|W|}\prod_{j}^{rankG}\frac{dz_{j}}{2\pi
ir}Z_{\text{classical}}[z,m]Z_{\text{one-loop}}[z,m]\;,$ (5)

where the sum is over the holonomies
$m=\frac{r}{2\pi}\int_{C}A_{\mu}dx^{\mu}$, $C$ is the non-trivial cycle on
$S^{3}_{b}/\mathbb{Z}_{r}$ and $A_{\mu}$ is the gauge field, and the integral
is over the Cartan subalgebra of the gauge group and $z_{j}$ variables stand
for the gauge fugacities. The prefactor $|W|$ corresponds to the order of the
gauge group, which is broken by the holonomy into a product of $r$ subgroups.

The classical term $Z_{\text{classical}}$ includes the contributions coming
from the classical action of the Chern-Simons term and Fayet-Iliopoulos term.

The one-loop contribution of chiral multiplets is given in terms of hyperbolic
hypergeometric function. A chiral multiplet in the fundamental representation
of the gauge group contributes as

$\begin{split}Z_{\rm
chiral}=\prod_{\rho}\prod_{\phi}&e^{\frac{i\pi}{2r}((\rho(m)+\phi(n))(r-\rho(m)-\phi(n))-(r-1)(\rho(m)+\phi(n))^{2})}\\\
&\times\gamma^{2}(i(\rho(z)+\phi(\Phi))+\omega_{1}(\rho(m)+\phi(n))+(\omega_{1}+\omega_{2})/2;\omega_{1}r,\omega_{1}+\omega_{2})\\\
&\times\gamma^{2}(iz+\omega_{2}(r-(\rho(m)-\phi(n))+(\omega_{1}+\omega_{2})/2;\omega_{2}r,\omega_{1}+\omega_{2}).\end{split}$
(6)

where $\rho$ and $\phi$ weights of the representation of the gauge group, and
the flavor group, respectively. Note that the Weyl weight of a chiral multiple
is absorbed into the flavor fugacity. The one-loop contribution of the vector
multiplet in our case is trivial since we consider only $U(1)$ gauge theories.

The partition function is completely determined by the group-theoretical data
of the theory. Therefore once we know the group-theoretical data of a theory
on $S_{b}^{3}/\mathbb{Z}_{r}$, we can write down the partition function in
terms of hyperbolic hypergeometric integral.

##  4 Supersymmetric duality

Our starting point is the following supersymmetric three-dimensional
$\mathcal{N}=2$ IR duality:

  * •

Theory A: Theory with $U(1)$ gauge symmetry and $SU(3)_{L}\times SU(3)_{R}$
flavor group, having three fundamental chiral multiples and three anti-
fundamental chiral multiplets and no Chern-Simons term.

  * •

Theory B: Theory without gauge degrees of freedom having the same global
symmetries. There are nine “mesons”, transforming in the fundamental
representation of the flavor group $SU(3)_{L}\times SU(3)_{R}$.

Partition functions of dual theories agree and as a result one obtains the
following hyperbolic hypergeometric integral identity Sarkissian:2018ppc

$\displaystyle\sum\limits_{y=0}^{[r/2]}\epsilon(y)\int_{-i\infty}^{i\infty}\prod_{i=1}^{3}\gamma^{(2)}(a_{i}-z+\omega_{1}(u_{i}-y);\omega_{1}r,\omega_{1}+\omega_{2})\gamma^{(2)}(a_{i}-z+\omega_{2}(r-(u_{i}-y));\omega_{2}r,\omega_{1}+\omega_{2})$
$\displaystyle\times\gamma^{(2)}(b_{i}+z+\omega_{1}(v_{i}+y);\omega_{1}r,\omega_{1}+\omega_{2})\gamma^{(2)}(b_{i}+z+\omega_{2}(r-(v_{i}+y));\omega_{2}r,\omega_{1}+\omega_{2})\frac{dz}{ir\sqrt{\omega_{1}\omega_{2}}}$
$\displaystyle=\prod\limits_{i,j=1}^{3}\gamma^{(2)}(a_{i}+b_{j}+\omega_{1}(u_{i}+v_{j});\omega_{1}r,\omega_{1}+\omega_{2})\gamma^{(2)}(a_{i}+b_{j}+\omega_{2}(r-(u_{i}+v_{j}));\omega_{2}r,\omega_{1}+\omega_{2})$
(7)

with the balancing conditions $\sum_{i}a_{i}+b_{i}=\omega_{1}+\omega_{2}$ and
$\sum_{i}u_{i}+v_{i}=r$. Here $a_{i}$ are $SU(3)_{L}$ flavor fugacities and
$b_{i}$ are $SU(3)_{R}$ flavor fugacities. The $\epsilon(y)$ function is
defined as $\epsilon(0)=\epsilon(\lfloor\frac{r}{2}\rfloor)=1$ and
$\epsilon(y)=2$ otherwise.

Note that this integral identity can be written as the star-triangle relation
and pentagon identity Bozkurt:2020gyy ; Mullahasanoglu:2021xyf ;
Sarkissian:2018ppc . The Bailey pair for this identity was constructed in
Gahramanov:2022jxz  (see, also Gahramanov:2015cva ). A similar identity was
discussed for the $S^{3}$ sphere partition functions (without summation) in
Spiridonov:2010em ; Kashaev:2012cz  (see also Kels:2013ola ) and for the
$S^{2}\times S^{1}$ partition function in Gahramanov:2013rda ;
Gahramanov:2014ona ; Gahramanov:2016wxi .

Another supersymmetric duality considered in this paper is mirror symmetry.
Three-dimensional mirror symmetry was introduced in Intriligator:1996ex  for
$\mathcal{N}=4$ supersymmetric gauge theories, and in Aharony:1997bx  for
$\mathcal{N}=2$ gauge theories. The simplest example of $\mathcal{N}=2$ mirror
symmetry is the duality between supersymmetric quantum electrodynamics with
one flavor and the free Wess-Zumino theory:

  * •

Theory A: Theory has one flavor consisting of two chiral fields and one vector
multiplet. This theory possesses two U$(1)$ global symmetries: one is the
topological U$(1)_{J}$, and the other is the flavor symmetry U$(1)_{A}$.

  * •

Theory B: The mirror theory, the free Wess-Zumino model (this theory is often
called the XYZ model) is the theory containing three chiral fields interacting
through the trilinear superpotential. This theory has also two U$(1)$ global
symmetries, named U$(1)_{V}$ and U$(1)_{A}$.

In the context of mirror symmetry, one identifies U$(1)_{J}$ and U$(1)_{A}$ of
the SQED theory with U$(1)_{V}$ and U$(1)_{A}$ of the corresponding Wess-
Zumino model. The equality of lens partition functions for mirror dual
theories can be written as follows

$\displaystyle\sum\limits_{y=0}^{[r/2]}\epsilon(y)\int_{-i\infty}^{i\infty}$
$\displaystyle e^{\frac{2\pi
i}{\omega_{1}\omega_{2}r}(z+\alpha)(\frac{3\alpha+\beta-\omega}{2})+\frac{i\pi}{r}(y+n_{\alpha})(2n_{\beta}-r)}\gamma^{(2)}(z+\alpha+\omega_{1}(y+n_{\alpha});\omega_{1}r,\omega)$
$\displaystyle\times\gamma^{(2)}(z+\alpha+\omega_{2}(r-y-
n_{\alpha}));\omega_{2}r,\omega)\gamma^{(2)}(-z+\alpha+\omega_{1}(-y+n_{\alpha});\omega_{1}r,\omega)$
$\displaystyle\times\gamma^{(2)}(-z+\alpha+\omega_{2}(r+y-n_{\alpha}));\omega_{2}r,\omega)\frac{dz}{ir\sqrt{\omega_{1}\omega_{2}}}$
$\displaystyle=e^{\frac{2\pi
i}{\omega_{1}\omega_{2}r}(\alpha)(\frac{2\beta-\omega}{2})+\frac{i\pi}{r}(-n_{\alpha})(2n_{\beta}-r)}\gamma^{(2)}(\omega-\alpha-\beta+\omega_{1}(r-n_{\alpha}-n_{\beta});\omega_{1}r,\omega)$
$\displaystyle\times\gamma^{(2)}(\omega-\alpha-\beta+\omega_{2}(n_{\alpha}+n_{\beta});\omega_{2}r,\omega)\gamma^{(2)}(\beta-\alpha+\omega_{1}(n_{\beta}-n_{\alpha});\omega_{1}r,\omega)$
$\displaystyle\times\gamma^{(2)}(\beta-\alpha+\omega_{2}(r-n_{\beta}+n_{\alpha});\omega_{2}r,\omega)$
$\displaystyle\times\gamma^{(2)}(2\alpha+\omega_{1}(2n_{\alpha});\omega_{1}r,\omega)\gamma^{(2)}(2\alpha+\omega_{2}(r-2n_{\alpha});\omega_{2}r,\omega)\;,$
(8)

where $\alpha$, $\beta$ are fugacities for the global symmetries and we use
the shorthand $\omega=\omega_{1}+\omega_{2}$.

##  5 Pentagon identity for mirror symmetry

For a special limit of flavor fugacities, the first duality can be reduced to
the mirror symmetry for $\mathcal{N}=2$ supersymmetric gauge theories. We will
make the breaking of flavor symmetry on the level of partition functions. Here
we follow the procedure described in Sarkissian:2022mlb . First we take the
limit $a_{3}\to\infty$ and of course $b_{3}=\omega-
a_{1}-a_{2}-a_{3}-b_{1}-b_{2}$ with the balancing condition
$\sum_{i=1}^{3}u_{i}+v_{i}=r$. If we collect terms coming from relevant
Bernoulli polynomials which are not used in limit operations on the left-hand
side, the following remains:

$\displaystyle\sum\limits_{y=0}^{[r/2]}\epsilon(y)\int_{-i\infty}^{i\infty}$
$\displaystyle
e^{\frac{i\pi}{2}A}\prod_{i=1}^{2}\gamma^{(2)}(a_{i}-z+\omega_{1}(u_{i}-y);+\omega_{1}r,\omega)\gamma^{(2)}(a_{i}-z)+\omega_{2}(r-(u_{i}-y));\omega_{2}r,\omega)$
$\displaystyle\times\gamma^{(2)}(b_{i}+z+\omega_{1}(v_{i}+y);\omega_{1}r,\omega)\gamma^{(2)}(b_{i}+z+\omega_{2}(r-(v_{i}+y));\omega_{2}r,\omega)\frac{dz}{ir\sqrt{\omega_{1}\omega_{2}}}$
$\displaystyle=\prod\limits_{i,j=1}^{2}\gamma^{(2)}(a_{i}+b_{j}+\omega_{1}(u_{i}+v_{j});\omega_{1}r,\omega)\gamma^{(2)}(a_{i}+b_{j}+\omega_{2}(r-(u_{i}+v_{j}));\omega_{2}r,\omega)$
$\displaystyle\qquad\quad\times\gamma^{(2)}(\omega-
a_{1}-b_{1}-a_{2}-b_{2}+\omega_{1}(r-u_{1}-v_{1}-u_{2}-v_{2});\omega_{1}r,\omega)$
$\displaystyle\qquad\quad\times\gamma^{(2)}(\omega-
a_{1}-b_{1}-a_{2}-b_{2}+\omega_{2}(u_{1}+v_{1}+u_{2}+v_{2});\omega_{2}r,\omega),$
(9)

where the exponential factor is

$\begin{split}A=\frac{1}{\omega_{1}\omega_{2}r}\big{(}2z(a_{1}+b_{1}+a_{2}+b_{2})+(a_{2}+b_{2})(-a_{1}+b_{1})+(a_{1}+b_{1})(-a_{2}+b_{2})\big{)}\\\
+\frac{2y}{r}\big{(}u_{1}+v_{1}+u_{2}+v_{2}\big{)}-\frac{2}{\omega
r}\big{(}\omega_{1}-\omega_{2}\big{)}\big{(}u_{1}u_{2}-v_{1}v_{2}\big{)}.\end{split}$

In the latter expression, we used the balancing condition to rewrite
$u_{3}+v_{3}$ in terms of other flavor charges as
$u_{3}+v_{3}=r-u_{1}-u_{2}-v_{1}-v_{2}$.

Now if we take $a_{2}\to\infty$ limit, and define $a_{1}+b_{1}=g_{1}$,
$a_{2}+b_{2}=g_{2}$, $u_{1}+v_{1}=t_{1}$, $u_{2}+v_{2}=t_{2}$ while sending111
Actually in the latter expression we only have the following combinations
$u_{1}+v_{1}$ and $a_{1}+b_{1}$, therefore by sending we do not lose any
information. $b_{1},v_{1}\to 0$, we get:

$\displaystyle\sum\limits_{y=0}^{[r/2]}\int_{-i\infty}^{i\infty}$
$\displaystyle e^{\frac{2\pi
iz}{\omega_{1}\omega_{2}r}(\frac{2g_{1}+g_{2}-\omega}{2})+\frac{i\pi
y}{r}(t_{1}+2t_{2}-r)}\gamma^{(2)}(z+\omega_{1}y;\omega_{1}r,\omega)\gamma^{(2)}(z+\omega_{2}(r-y));\omega_{2}r,\omega)$
$\displaystyle\times\gamma^{(2)}(-z+g_{1}+\omega_{1}(t_{1}-y);\omega_{1}r,\omega)\gamma^{(2)}(-z+g_{1}+\omega_{2}(r-t_{1}+y));\omega_{2}r,\omega)\frac{dz}{ir\sqrt{\omega_{1}\omega_{2}}}$
$\displaystyle=e^{\frac{2\pi
i}{\omega_{1}\omega_{2}r}(\frac{g_{1}}{2})(\frac{2g_{2}+g_{1}-\omega}{2})+\frac{i\pi}{r}(\frac{-t_{1}}{2})(t_{1}+2t_{2}-r)}\gamma^{(2)}(\omega-
g_{1}-g_{2}+\omega_{1}(r-t_{1}-t_{2});\omega_{1}r,\omega)$
$\displaystyle\qquad\times\gamma^{(2)}(\omega-
g_{1}-g_{2}+\omega_{2}(t_{1}+t_{2});\omega_{2}r,\omega)\gamma^{(2)}(g_{2}+\omega_{1}t_{2};\omega_{1}r,\omega)\gamma^{(2)}(g_{2}+\omega_{2}(r-t_{2});\omega_{2}r,\omega)$
$\displaystyle\qquad\times\gamma^{(2)}(g_{1}+\omega_{1}t_{1};\omega_{1}r,\omega)\gamma^{(2)}(g_{1}+\omega_{2}(r-t_{1});\omega_{2}r,\omega)$
(10)

As one can see, the expression (10) has the form of the integral pentagon
relation. The redefinition of the fugacities as $g_{1}/2\to\alpha$, $z\to
z+\alpha$, $g_{2}\to\beta-\alpha$ and $t_{1}\to 2n_{\alpha}$, $y\to
y+n_{\alpha}$, $t_{2}\to n_{\beta}-n_{\alpha}$ gives the integral identity (8)
for mirror symmetry Imamura:2012rq . The case $r=1$ corresponds to the
identity from Sarkissian:2022mlb ; Faddeev:2000if .

##  6 Summary

We report a solution to the pentagon equation obtained via reduction of
equality for $S_{b}^{3}/\mathbb{Z}_{r}$ partition functions of a certain
$\mathcal{N}=2$ supersymmetric duality via the approach described in
Sarkissian:2022mlb .

There are many possible directions for future research. The supersymmetric
partition functions are related to topological invariants of triangulated
three-manifolds via $3-3$ relation Dimofte:2011ju ; Dimofte:2012pd . The
special $r=1$ case of integral identities discussed here is used in
Kashaev:2012cz  for the construction of the Turaev-Viro invariant of three-
manifolds. It would be interesting to construct the invariant of the lens
space suzuki2002turaev . Much work remains to be done in this direction.

It would be interesting to construct a solution to the corresponding Yang-
Baxter equation using the obtained solution to the pentagon equation in the
context of gauge/YBE correspondence Gahramanov:2017ysd ; Yamazaki:2018xbx .

Another possible future direction is obtaining the five-term identity
Alexandrov:2015xir  implied by Kontsevich-Soibelman wall-crossing via the
procedure described in Faddeev:2011tah .

## Acknowledgements

It is pleasure to thank Mustafa Mullahasanoglu for the helpful discussions.
The work of Ilmar Gahramanov is partially supported by TUBITAK grant 220N106,
by the 1002-TUBITAK Quick Support Program under grant number 121F413, and by
the Russian Science Foundation grant number 22-72-10122.

## References

  * (1) R. M. Kashaev and S. M. Sergeev, “On pentagon, ten term, and tetrahedron relations,” arXiv:q-alg/9607032.
  * (2) R. M. Kashaev, “Beta pentagon relations,” Theor. Math. Phys. 181 no. 1, (2014) 1194–1205, arXiv:1403.1298 [math-ph].
  * (3) D. N. Bozkurt and I. Gahramanov, “Pentagon identities arising in supersymmetric gauge theory computations,” Teor. Mat. Fiz. 198 no. 2, (2019) 215–224, arXiv:1803.00855 [math-ph].
  * (4) D. N. Bozkurt, I. Gahramanov, and M. Mullahasanoglu, “Lens partition function, pentagon identity, and star-triangle relation,” Phys. Rev. D 103 no. 12, (2021) 126013, arXiv:2009.14198 [hep-th].
  * (5) J. Allman and R. Rimányi, “Quantum dilogarithm identities for the square product of A-type Dynkin quivers,” Math. Res. Lett. 25 (2018) 1037–1087, arXiv:1702.04766 [math.RT].
  * (6) I. Gahramanov and H. Rosengren, “Integral pentagon relations for 3d superconformal indices,” arXiv:1412.2926 [hep-th]. [Proc. Symp. Pure Math.93,165(2016)].
  * (7) I. Gahramanov and H. Rosengren, “A new pentagon identity for the tetrahedron index,” JHEP 11 (2013) 128, arXiv:1309.2195 [hep-th].
  * (8) I. Gahramanov and H. Rosengren, “Basic hypergeometry of supersymmetric dualities,” Nucl. Phys. B913 (2016) 747–768, arXiv:1606.08185 [hep-th].
  * (9) I. Gahramanov, B. Keskin, D. Kosva, and M. Mullahasanoglu, “On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_{b}^{3}/\mathbb{Z}_{r}$,” arXiv:2210.11455 [hep-th].
  * (10) I. Gahramanov and O. E. Kaluc, “Bailey pairs for the q-hypergeometric integral pentagon identity,” arXiv:2111.14793 [math-ph].
  * (11) M. Dede, “A comment on the solutions of the generalized Faddeev-Volkov model,” arXiv:2206.14271 [math-ph].
  * (12) R. Kashaev, F. Luo, and G. Vartanov, “A TQFT of Turaev–Viro Type on Shaped Triangulations,” Annales Henri Poincare 17 no. 5, (2016) 1109–1143, arXiv:1210.8393 [math.QA].
  * (13) I. Colazzo, E. Jespers, and Ł. Kubat, “Set-theoretic solutions of the pentagon equation,” Communications in Mathematical Physics 380 no. 2, (2020) 1003–1024.
  * (14) R. Kashaev, “Heisenberg double and the pentagon relation,” in St. Petersburg Math. J, Citeseer.  1997\.
  * (15) G. Militaru, “Heisenberg double, pentagon equation, structure and classification of finite-dimensional hopf algebras,” Journal of the London Mathematical Society 69 no. 1, (2004) 44–64.
  * (16) N. Aghaei and M. Pawelkiewicz, “Heisenberg double and drinfeld double of the quantum superplane,” arXiv preprint arXiv:1909.04565 (2019) .
  * (17) Y. Imamura and D. Yokoyama, “$S^{3}/Z_{n}$ partition function and dualities,” JHEP 11 (2012) 122, arXiv:1208.1404 [hep-th].
  * (18) Y. Imamura, H. Matsuno, and D. Yokoyama, “Factorization of the $S^{3}/\mathbb{Z}_{n}$ partition function,” Phys. Rev. D89 no. 8, (2014) 085003, arXiv:1311.2371 [hep-th].
  * (19) 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].
  * (20) M. Yamazaki, “Four-dimensional superconformal index reloaded,” Theor. Math. Phys. 174 (2013) 154–166. [Teor. Mat. Fiz.174,177(2013)].
  * (21) 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].
  * (22) 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].
  * (23) E. Catak, I. Gahramanov, and M. Mullahasanoglu, “Hyperbolic and trigonometric hypergeometric solutions to the star-star equation,” Eur. Phys. J. C 82 no. 9, (2022) 789, arXiv:2107.06880 [hep-th].
  * (24) F. Nieri and S. Pasquetti, “Factorisation and holomorphic blocks in 4d,” JHEP 11 (2015) 155, arXiv:1507.00261 [hep-th].
  * (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) M. Mullahasanoglu and N. Tas, “Lens Partition Functions and Integrability Properties,” arXiv:2112.15161 [hep-th].
  * (27) I. Gahramanov and V. P. Spiridonov, “The star-triangle relation and 3d superconformal indices,” JHEP 08 (2015) 040, arXiv:1505.00765 [hep-th].
  * (28) V. P. Spiridonov, “Elliptic beta integrals and solvable models of statistical mechanics,” Contemp. Math. 563 (2012) 181–211, arXiv:1011.3798 [hep-th].
  * (29) A. P. Kels, “A new solution of the star-triangle relation,” J. Phys. A47 (2014) 055203, arXiv:1302.3025 [math-ph].
  * (30) K. A. Intriligator and N. Seiberg, “Mirror symmetry in three-dimensional gauge theories,” Phys. Lett. B387 (1996) 513–519, arXiv:hep-th/9607207 [hep-th].
  * (31) O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg, and M. Strassler, “Aspects of N=2 supersymmetric gauge theories in three-dimensions,” Nucl.Phys. B499 (1997) 67–99, arXiv:hep-th/9703110 [hep-th].
  * (32) G. Sarkissian and V. Spiridonov, “Elliptic, hyperbolic, complex gamma functions and QFT in various dimensions,” PoS Regio2021 (2022) 037.
  * (33) L. D. Faddeev, R. M. Kashaev, and A. Y. Volkov, “Strongly coupled quantum discrete Liouville theory. 1. Algebraic approach and duality,” Commun. Math. Phys. 219 (2001) 199–219, arXiv:hep-th/0006156.
  * (34) 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].
  * (35) T. Dimofte and D. Gaiotto, “An E7 Surprise,” JHEP 10 (2012) 129, arXiv:1209.1404 [hep-th].
  * (36) K. Suzuki and M. Wakui, “On the turaev-viro-ocneanu invariant of 3-manifolds derived from the $e_{6}$-subfactor,” Kyushu Journal of Mathematics 56 no. 1, (2002) 59–81.
  * (37) I. Gahramanov and S. Jafarzade, “Integrable lattice spin models from supersymmetric dualities,” Phys. Part. Nucl. Lett. 15 no. 6, (2018) 650–667, arXiv:1712.09651 [math-ph].
  * (38) M. Yamazaki, “Integrability As Duality: The Gauge/YBE Correspondence,” Phys. Rept. 859 (2020) 1–20, arXiv:1808.04374 [hep-th].
  * (39) S. Alexandrov and B. Pioline, “Theta series, wall-crossing and quantum dilogarithm identities,” Lett. Math. Phys. 106 no. 8, (2016) 1037–1066, arXiv:1511.02892 [hep-th].
  * (40) L. D. Faddeev, “Volkov’s Pentagon for the Modular Quantum Dilogarithm,” Funct. Anal. Appl. 45 (2011) 291, arXiv:1201.6464 [math.QA].