By having a time-reversal symmetry I mean that there is a local anti-unitary symmetry (representing the non-trivial element of $Z_2$) of the state-sum construction (or, if you want, of the associated Hamiltonian). In other words there is a codimension-1 anti-unitary defect, or yet in other words there is a local basis in which all tensors involved in the state-sum have real entries.

Such a symmetry often exists, as for example in the case of the group $Z_2$. However I see no reason why such a symmetry should be there in general, and it seems to me that it actually might not exist for $Z_3$ with one of the non-trivial group cocycles.

For a theory with time-reversal symmetry all invariants associated to oriented $3$-manifolds should be real. Are there manifolds to which the non-trivial $Z_3$ (or some other) Dijkgraaf-Witten theory associates a non-real number? (By construction the invariant is real on manifolds with reflection symmetry, so one would have to test oriented 3-manifolds without reflection symmetry. Guess those exist?)

The motivation why I'm asking is that in physics, models like Dijkgraaf-Witten are called "non-chiral" because they allow gapped boundaries, but on the other hand, people refer to models as "non-chiral" if they have a time-reversal symmetry. I feel that those two notions of "non-chiral" have a large overlap but are not exactly equivalent.

This post imported from StackExchange MathOverflow at 2019-07-25 17:58 (UTC), posted by SE-user Andi Bauer