Cluster transformations, the tetrahedron equation and three-dimensional gauge theories

TL;DR

This paper constructs specific cluster transformations related to braid relations, which yield solutions to the tetrahedron equation and connect to partition functions of 3D supersymmetric gauge theories.

Contribution

It introduces three families of quivers where braid relations are realized via mutations, linking cluster algebra, quantum dilogarithm, and 3D gauge theories.

Findings

Derived solutions to the tetrahedron equation from cluster transformations.

Connected these solutions to partition functions of 3D supersymmetric gauge theories.

Demonstrated explicit cluster transformations with specific mutation counts.

Abstract

We define three families of quivers in which the braid relations of the symmetric group $S_n$ are realized by mutations and automorphisms. A sequence of eight braid moves on a reduced word for the longest element of $S_4$ yields three trivial cluster transformations with 8, 32 and 32 mutations. For each of these cluster transformations, a unitary operator representing a single braid move in a quantum mechanical system solves the tetrahedron equation. The solutions thus obtained are constructed from the noncompact quantum dilogarithm and can be identified with the partition functions of three-dimensional $\mathcal{N} = 2$ supersymmetric gauge theories on a squashed three-sphere.