Advances in Theoretical and Mathematical Physics

Volume 27 (2023)

Number 4

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

Pages: 1065 – 1106

DOI: https://dx.doi.org/10.4310/ATMP.2023.v27.n4.a2

Authors

Xiaoyue Sun (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Junya Yagi (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

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.

Published 6 June 2024