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

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.

Automated summary

In this paper, the time-reversal-invariant three-dimensional Chern-Simons gauge theories with torus gauge group are fully characterized. The toral Chern-Simons theory is defined by an integral lattice at the Lagrangian level, and at the quantum level, it is determined by a quadratic function on a finite Abelian group and an integer mod 24. It is found that quantum time-reversally symmetric theories can be defined by classical Lagrangians defined by integral lattices with self-perpendicular embeddings into a unimodular lattice. It is also found that the quantum toral Chern-Simons theory exhibits time-reversal symmetry if and only if the higher Gauss sums of the associated modular tensor category are real. It is conjectured that the reality of the higher Gauss sums is necessary and sufficient for a general non-Abelian Chern-Simons theory to exhibit quantum T-symmetry.