Equivalence of toral Chern-Simons and Reshetikhin-Turaev theories

Abstract

We prove a natural isomorphism between toral Chern-Simons theory with gauge group $\mathbb{T}=\mathfrak{t}/\Lambda \cong U(1{)}^n$ and the Reshetikhin-Turaev theory associated with the finite quadratic module determined by an even, integral, nondegenerate symmetric bilinear form $K:\Lambda \times \Lambda \to \mathbb{Z}.$ More precisely, let $G_K={\Lambda }^{\ast }/K\Lambda$ 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.