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

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.

29 pages. v2: minor changes, references added, published version

References in corpus (11)

• SUSY Gauge Theories on Squashed Three-Spheres
• The quantum dilogarithm and representations quantum cluster varieties
• Braids, Walls, and Mirrors
• Cluster algebras from dualities of 2d N=(2,2) quiver gauge theories
• Classical and Quantum Dilogarithm Identities
• Elliptic gamma-function and multi-spin solutions of the Yang-Baxter equation
• Labeling Schemes for Tetrahedron Equations and Dualities between Them
• Tetrahedron and 3D reflection equations from quantized algebra of functions
• 3d N=2 Theories from Cluster Algebras
• Superspin chains from superstring theory
• Integrable 3D lattice model in M-theory

Cited by in corpus (2)

• Tetrahedron equation and Schur functions
• State integral models and the tetrahedron equation