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

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.