# Flipping relation as a reduced star-star relation

## TL;DR

This paper investigates the star-star relation and flipping relation in supersymmetric gauge theories, revealing new solutions and symmetries through hyperbolic gamma functions and specific limits.

## Contribution

It introduces new solutions to the flipping relation and connects these to supersymmetric gauge theory symmetries via hyperbolic gamma functions.

## Key findings

- Derived new solutions to the flipping relation.
- Linked the star-star relation to symmetry transformations.
- Identified limits that produce the flipping symmetry.

## Abstract

In this paper, we consider the lens hyperbolic gamma solution to the star-star relation and the flipping relation from three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories on $S^3_b/\mathbb{Z}_r$. We explore that a certain limit of the star-star relation yields the latter symmetry transformation, which exchanges the edge interactions of two outer spins with a centrally sited spin. Furthermore, we obtain more solutions to the flipping relation in terms of the hyperbolic gamma, basic hypergeometric, and the Euler gamma functions.