Finite quadratic modules classify extended Abelian Chern-Simons theories, pointed Abelian Reshetikhin-Turaev TQFTs, and pointed modular tensor categories.
Equivalence of Extended $U(1)$ Chern-Simons and Reshetikhin-Turaev TQFTs
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abstract
We establish the equivalence between $U(1)$ Chern-Simons and Reshetikhin-Turaev TQFTs associated with finite quadratic modules. For gauge group $U(1)$ and even level $k$, we prove that the corresponding Chern-Simons TQFT is naturally isomorphic to the Reshetikhin-Turaev TQFT determined by the pointed modular category $C(\mathbb Z_k,q_k)$. The equivalence holds both for closed $3$-manifolds and for bordisms with boundary, so that the two constructions define naturally isomorphic extended $(2+1)$-dimensional TQFTs. In particular, the finite quadratic module $(\mathbb Z_k,q_k)$ completely determines the $U(1)$ Chern-Simons theory.
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math.QA 2years
2026 2representative citing papers
Toral Chern-Simons theory with gauge group U(1)^n is naturally isomorphic to the Reshetikhin-Turaev theory from the associated finite quadratic module.
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Classification of Extended Abelian Chern-Simons Theories
Finite quadratic modules classify extended Abelian Chern-Simons theories, pointed Abelian Reshetikhin-Turaev TQFTs, and pointed modular tensor categories.
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Equivalence of toral Chern-Simons and Reshetikhin-Turaev theories
Toral Chern-Simons theory with gauge group U(1)^n is naturally isomorphic to the Reshetikhin-Turaev theory from the associated finite quadratic module.