Abstract
We consider the rarefied elliptic beta integral in various limiting forms. In particular, we obtain an integral identity for parafermionic hyperbolic gamma functions which describes the star-triangle relation for parafermionic Liouville theory.
Highlights
JHEP10(2018)097 that the corresponding relations are implied by the generalization of considerations of [17] to supersymmetric hyperbolic gamma functions found in [15]
It was called in [33] the rarefied elliptic gamma function due to its special product type representation, using which we introduce parafermionic hyperbolic gamma function as a particular limit
The present work can be considered as a complement to [8], where the transition from 4d theories to 3d ones was reached by degenerating elliptic hypergeometric integrals to hyperbolic integrals, — we add to such a connection a relation to the parafermionic LFT
Summary
The standard elliptic gamma function Γ(z; p, q) can be defined as an infinite product:. The lens space elliptic gamma function is defined as a product of two standard elliptic gamma functions with different bases [3]. As shown in [33], the function (2.2) can be written as a special product of the standard elliptic gamma functions with bases pr and qr. Which yields Γ(1)(z, m; p, q) = Γ(z; p, q) It is this object that was called the rarefied elliptic gamma function. It is shown in [33] that if parameters ta, na satisfy the constraints |ta| < 1 and the balancing condition ta = pq , na = −3 , = 0, 1,.
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