Equivalence of toral Chern-Simons and Reshetikhin-Turaev theories
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| Format: | Preprint |
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2026
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| _version_ | 1866913063803813888 |
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| author | Galviz, Daniel |
| author_facet | Galviz, Daniel |
| contents | We prove a natural isomorphism between toral Chern-Simons theory with gauge group $\mathbb T=\mathfrak t/Λ\cong U(1)^n$ and the Reshetikhin-Turaev theory associated with the finite quadratic module determined by an even, integral, nondegenerate symmetric bilinear form $K:Λ\timesΛ\to\mathbb Z.$ More precisely, let $G_K=Λ^*/KΛ$ be the discriminant group of $K$, equipped with its induced quadratic form $q_K$, and let $C(G_K,q_K)$ be the corresponding pointed modular category. Using the geometric quantization formulation of toral Chern-Simons theory, we show that the resulting TQFT is naturally isomorphic to the Reshetikhin--Turaev TQFT determined by $C(G_K,q_K)$. The equivalence is established at the level of closed 3-manifold invariants, bordism operators for manifolds with boundary, and the extended $(2+1)$-dimensional structure, yielding a natural isomorphism of extended TQFTs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_01982 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Equivalence of toral Chern-Simons and Reshetikhin-Turaev theories Galviz, Daniel Quantum Algebra Mathematical Physics Geometric Topology We prove a natural isomorphism between toral Chern-Simons theory with gauge group $\mathbb T=\mathfrak t/Λ\cong U(1)^n$ and the Reshetikhin-Turaev theory associated with the finite quadratic module determined by an even, integral, nondegenerate symmetric bilinear form $K:Λ\timesΛ\to\mathbb Z.$ More precisely, let $G_K=Λ^*/KΛ$ be the discriminant group of $K$, equipped with its induced quadratic form $q_K$, and let $C(G_K,q_K)$ be the corresponding pointed modular category. Using the geometric quantization formulation of toral Chern-Simons theory, we show that the resulting TQFT is naturally isomorphic to the Reshetikhin--Turaev TQFT determined by $C(G_K,q_K)$. The equivalence is established at the level of closed 3-manifold invariants, bordism operators for manifolds with boundary, and the extended $(2+1)$-dimensional structure, yielding a natural isomorphism of extended TQFTs. |
| title | Equivalence of toral Chern-Simons and Reshetikhin-Turaev theories |
| topic | Quantum Algebra Mathematical Physics Geometric Topology |
| url | https://arxiv.org/abs/2604.01982 |