When does a three-dimensional Chern-Simons-Witten theory have a time reversal symmetry? (English) Zbl 1541.81143
Summary: 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.
MSC:
| 81T32 | Matrix models and tensor models for quantum field theory |
| 58J28 | Eta-invariants, Chern-Simons invariants |
| 70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 06F20 | Ordered abelian groups, Riesz groups, ordered linear spaces |
| 05C60 | Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) |
| 11T24 | Other character sums and Gauss sums |
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