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arXiv · 2211.10702

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

Xiaoyue Sun·Junya Yagi
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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.

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Xiaoyue Sun, Junya Yagi. 2022-11-19. Cluster transformations, the tetrahedron equation and three-dimensional gauge theories. https://doi.org/10.4310/atmp.2023.v27.n4.a2

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