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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.10702 |
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| _version_ | 1866911910218170368 |
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| author | Sun, Xiaoyue Yagi, Junya |
| author_facet | Sun, Xiaoyue Yagi, Junya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_10702 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cluster transformations, the tetrahedron equation and three-dimensional gauge theories Sun, Xiaoyue Yagi, Junya Mathematical Physics High Energy Physics - Theory 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. |
| title | Cluster transformations, the tetrahedron equation and three-dimensional gauge theories |
| topic | Mathematical Physics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2211.10702 |