Classification of Extended Abelian Chern-Simons Theories

TL;DR

This paper classifies extended Abelian Chern-Simons theories in 2+1 dimensions using finite quadratic modules derived from lattices, establishing a complete correspondence between these modules and the theories.

Contribution

It proves that finite quadratic modules uniquely determine extended Abelian Chern-Simons theories and that all such modules are realizable from lattices.

Findings

Finite quadratic modules classify extended Abelian Chern-Simons theories.

Every finite quadratic module corresponds to an even integral nondegenerate lattice.

Classification extends to pointed Abelian Reshetikhin-Turaev TQFTs and modular tensor categories.

Abstract

We classify extended Abelian Chern-Simons theories with gauge group $U(1)^n$ as extended $(2+1)$-dimensional topological quantum field theories. For an even integral nondegenerate lattice $(\Lambda,K)$, let $(G_K,q_K)$ denote its discriminant quadratic module. We prove that the associated theory is determined, up to symmetric monoidal natural isomorphism, by this finite quadratic module, and that every finite quadratic module is realized as the discriminant quadratic module of an even integral nondegenerate lattice. It follows that finite quadratic modules classify extended Abelian Chern-Simons theories, pointed Abelian Reshetikhin-Turaev TQFTs, and pointed modular tensor categories.