When Does A Three-Dimensional Chern-Simons-Witten Theory Have A Time Reversal Symmetry?

TL;DR

This paper characterizes when three-dimensional Chern-Simons theories with torus gauge groups have time-reversal symmetry, linking classical lattice structures to quantum invariants and conjecturing a criterion for non-Abelian cases.

Contribution

It provides a complete classification of time-reversal invariant toral Chern-Simons theories and proposes a conjecture relating quantum symmetries to higher Gauss sums.

Findings

Quantum time-reversal symmetry occurs iff the associated higher Gauss sums are real.

Classical Lagrangians with self-perpendicular embeddings into unimodular lattices define symmetric theories.

Conjecture: Reality of higher Gauss sums is necessary and sufficient for non-Abelian Chern-Simons T-symmetry.

Abstract

In this paper, we completely characterize time-reversal invariant three-dimensional Chern-Simons gauge theories with torus gauge group. At the level of the Lagrangian, toral Chern-Simons theory is defined by an integral lattice, while at the quantum level, it is entirely determined by a quadratic function on a finite Abelian group and an integer mod 24. We find that quantum time-reversally symmetric theories can be defined by classical Lagrangians defined by integral lattices which have self-perpendicular embeddings into a unimodular lattice. We find that the quantum toral Chern-Simons theory admits a time-reversal symmetry iff the higher Gauss sums of the associated modular tensor category are real. We conjecture that the reality of the higher Gauss sums is necessary and sufficient for a general non-Abelian Chern-Simons to admit quantum T-symmetry.