Equivalence of toral Chern-Simons and Reshetikhin-Turaev theories

TL;DR

This paper establishes a natural isomorphism between toral Chern-Simons theory and Reshetikhin-Turaev theory, linking geometric quantization with modular tensor categories for certain gauge groups.

Contribution

It proves the equivalence of two topological quantum field theories (TQFTs) associated with toral gauge groups and finite quadratic modules, clarifying their relationship at multiple levels.

Findings

The TQFTs are naturally isomorphic at the level of 3-manifold invariants.

The equivalence extends to bordism operators and extended (2+1)-dimensional structures.

The geometric quantization approach is used to establish the isomorphism.

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 $G_K=\Lambda^*/K\Lambda$ 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.