Bailey pairs for the q-hypergeometric integral pentagon identity

In this work, we construct new Bailey pairs for the integral pentagon identity in terms of q-hypergeometric functions. The pentagon identity considered here represents the equality of the partition functions of certain three-dimensional supersymmetric dual theories. It can be also interpreted as the star-triangle relation for the Ising-type integrable lattice model.


Introduction
Bailey's lemma [1,2] is a powerful tool to derive hypergeometric identities (ordinary, trigonometric, and elliptic type).In this work, we construct new integral Bailey pairs for the pentagon identity in terms of q-hypergeometric functions.The pentagon identity can be interpreted as a Pachner's 3-2 move for triangulated three-dimensional manifolds.Such identities also play a role in the study of supersymmetric gauge theories, integrable models, knot theory, etc. 1Let q, z ∈ Z with |q| < 1.We define the infinite q-product (z; q) ∞ := ∞ k=0 (1 − zq k ) . (1.1) We also adopt the following convention (a, b; q) ∞ := (a, q) ∞ (b, q) ∞ . (1.2) where the balancing conditions are and T represents the positively oriented unit circle.
We would like to mention that the integral identity represents the supersymmetric duality for three-dimensional N = 2 supersymmetric gauge theories with the flavor symmetry2 SU (3) × SU (3) × U (1).This identity can also be written as the star-triangle relation3 for some integrable model of statistical mechanics.
The proof of the form above is given in [8] for the balancing conditions4 3 i=0 The absolute values can be eliminated by the identity [12] (q and one ends up with the following q-hypergeometric sum/integral identity [6][7][8] m∈Z T

Integral pentagon identity
In [6][7][8] it was shown that the identity (1.3) can be written as an integral pentagon identity where we define the following function as In a general sense, any algebraic relation for operators which can be interpreted as a 2-3 Pachner move of a triangulated three-dimensional manifold is called a pentagon relation [4,5].Note that the integral pentagon identity (2.1) for the N = 2 supersymmetric S 2 × S 1 partition functions is supposed to be related to some topological invariant of corresponding 3-manifold via 3d-3d correspondence [12,13] that connects three-dimensional N = 2 supersymmetric theories and triangulated 3-manifolds.

Bailey pairs
Rogers-Ramanujan type identities are being continuously used in the solution of the integrable models, namely to derive the Yang-Baxter and the pentagon identities.In fact, a well-known example of this usage is conducted during the investigations of the hard hexagon model by Baxter.It turns out that Bailey discovered a systematic way to derive these types of identities [1,2,16,17].As generalized by Andrews [18,19], there exists an iterative scheme to derive infinitely many of these identities if one pair, called a Bailey pair is known.This form the so-called Bailey chain.The induction step of generating the particular Bailey pairs is referred as the Bailey lemma for the chain we consider.
A generalization of the Bailey pairs approach to the integral identities is firstly done by Spiridonov in [20,21].
Accordingly, the generalized version of the Bailey chain is a couple of infinite sequences of holomorphic functions {α n } n≥0 and {β n } n≥0 such that there exists an identity independent of i which connect α n and β n as where F can be an operator which may now include sum or integrals.Here, α n and β n are constructed according to where G and H represent integral-sum operators.
form a Bailey pair with respect to the parameter st.
Proof.We have to show that Hence, by regrouping the terms accordingly, we obtain5 p∈Z m∈Z where we required the sum of the powers of x to vanish, namely Upon renaming the variables as we identify the integral relation (1.3).Also observe that the constraint (3.10) resulted in the balancing condition (1.5).We hence get upon simplification and regrouping of the terms p∈Z α p (y, t)dy(−q

Conclusions
In this work, we constructed new integral Bailey pairs for pentagon identity in terms of q-hypergeometric functions.One can use this construction to obtain new supersymmetric dualities for quiver theories.
We would like to mention that the pentagon identity presented here can also be written as the star-triangle relation for some integrable lattice model of statistical mechanics.It would be interesting to construct the Bailey pairs corresponding to the star-triangle form of the same integral identity.