# New Pentagon Identities Revisited

## TL;DR

This paper introduces a novel gamma function-based solution to the pentagon identity, linking it to three-dimensional index limits, sphere partition functions, and the star-triangle relation in statistical mechanics.

## Contribution

It provides a new gamma function solution to the pentagon identity derived from three-dimensional index limits, connecting mathematical identities with physical models.

## Key findings

- New gamma function solution to the pentagon identity
- Connection between sphere partition functions and star-triangle relation
- Revealed link to strongly coupled Faddeev-Volkov model

## Abstract

We present a new solution to the pentagon identity in terms of gamma function. We obtain this solution by taking the gamma function limit from the pentagon identity related to the three-dimesional index. This limit corresponds to the identification of the sphere partition function of dual theories and being equivalent to the star-triangle relation in statistical mechanics which corresponds to the `strongly coupled' regime of the Faddeev-Volkov model.