Classification of abelian spin Chern-Simons theories

TL;DR

This paper provides a new classification scheme for abelian spin Chern-Simons theories, linking classical lattice data to quantum invariants and implications for fractional quantum Hall states.

Contribution

It introduces a novel classification of quantum spin Chern-Simons theories based on invariants derived from classical data, extending classical lattice classification.

Findings

Quantum theories are classified by invariants on spin 3-manifolds.

Two theories are equivalent if they have the same modular group projective representation.

Quantum theory is fully determined by three invariants from classical data.

Abstract

We derive a simple classification of quantum spin Chern-Simons theories with gauge group T=U(1)^N. While the classical Chern-Simons theories are classified by an integral lattice the quantum theories are classified differently. Two quantum theories are equivalent if they have the same invariants on 3-manifolds with spin structure, or equivalently if they lead to equivalent projective representations of the modular group. We prove the quantum theory is completely determined by three invariants which can be constructed from the data in the classical action. We comment on implications for the classification of fractional quantum Hall fluids.