Equivalence of Extended $U(1)$ Chern-Simons and Reshetikhin-Turaev TQFTs

TL;DR

This paper proves the natural isomorphism between $U(1)$ Chern-Simons and Reshetikhin-Turaev TQFTs for finite quadratic modules, establishing their equivalence for closed 3-manifolds and bordisms with boundary.

Contribution

It demonstrates that the $U(1)$ Chern-Simons TQFT is equivalent to the Reshetikhin-Turaev TQFT derived from a specific modular category, clarifying their relationship.

Findings

The equivalence holds for both closed 3-manifolds and bordisms with boundary.

The finite quadratic module $(Z_k, q_k)$ fully determines the $U(1)$ Chern-Simons theory.

The two TQFTs are naturally isomorphic as extended (2+1)-dimensional theories.

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.