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  Does Dijkgraaf-Witten theory have a time-reversal symmetry?

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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
asked Jun 22, 2019 in Theoretical Physics by Andi Bauer (95 points) [ no revision ]
retagged Jul 25, 2019
Most voted comments show all comments
@MarcelBischoff - Sure, the entries of the trivial cocycle are 0 and 1 which are real. I specifycally mean twisted models (for e.g. $Z_3$). And right, there the two non-trivial cocycles are related by complex conjugation. So an equivalent question is: Do the two non-trivially twisted $Z_3$ theories belong to different topological phases?

This post imported from StackExchange MathOverflow at 2019-07-25 17:58 (UTC), posted by SE-user Andi Bauer
They have different T-matrix, so the answer is yes! Explicitly, the T-matrix $T=(T_{x,y})_{0\leq x,y\leq8}$ is given by $T_{x,y} = \delta_{x,y}\exp(\pm2\pi ix^2/9)$, respectively.

This post imported from StackExchange MathOverflow at 2019-07-25 17:58 (UTC), posted by SE-user Marcel Bischoff
@MarcelBischoff - I see that this is a sufficient condition, but why is it necessary? Can't there be other symmetries than group automorphisms?

This post imported from StackExchange MathOverflow at 2019-07-25 17:58 (UTC), posted by SE-user Andi Bauer
Ok I see if the T-matrix is different then they are in different phases so they can't have time-reversal symmetry. Cool, thanks!

This post imported from StackExchange MathOverflow at 2019-07-25 17:58 (UTC), posted by SE-user Andi Bauer
Let us continue this discussion in chat.

This post imported from StackExchange MathOverflow at 2019-07-25 17:58 (UTC), posted by SE-user Marcel Bischoff
Most recent comments show all comments
@KevinWalker Time-reversal symmetry is usually defined to be a local anti-unitary symmetry without orientation-reversal. In physics though one usually assumes that models (like Dijkgraaf-Witten) are unitary which means that there already automatically is a anti-unitary symmetry associated to orientation-reversal. Combined with the local anti-unitary time-reversal symmetry this yields a unitary symmetry associated with orientation reversal. So time-reversal symmetry and a unitary symmetry associated to orientation-reversal are the same for a unitary theory.

This post imported from StackExchange MathOverflow at 2019-07-25 17:58 (UTC), posted by SE-user Andi Bauer
Doesn't have a Reshetikhin-Turaev TFT $RT(\mathcal C)$ associated with a modular tensor category $\mathcal C$ time-reversal symmetry iff there is a braided equivalence $\mathcal C\to\mathcal C^\mathrm{rev}$? In this case, the two twisted $\mathbb Z/3\mathbb Z$-DW theories are each others time-reversal and thus it cannot be true.

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

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