# 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
retagged Jul 25, 2019
this is a really great question. A related question, which I am going pose with absolutely no explanation: Is the trace of the T-matrix for the Drinfeld center of a unitary fusion category always real? I just spent the past hour computing and it seems like the answer might be yes....

This post imported from StackExchange MathOverflow at 2019-07-25 17:58 (UTC), posted by SE-user Daniel Barter
I can write a more detailed answer later today, but another perspective on time-reversal symmetry is that $Z$ admits a time-reversal symmetry if it can be defined on unoriented manifolds. Dijkgraaf-Witten theory requires integrating a cohomology class associated to the principal bundle, which is where the orientation appears; in some cases, one can use mod 2 cohomology, or orientation-twisted cohomology, to define the theory on unoriented manifolds. These theories have been constructed by Young.

This post imported from StackExchange MathOverflow at 2019-07-25 17:58 (UTC), posted by SE-user Arun Debray
@ArunDebray - I would say that if $Z$ admits a unitary symmetry associated with orientation-reversal, then it can be defined on unoriented manifolds. People usual stipulate that time-reversal symmetries are anti-unitary.

This post imported from StackExchange MathOverflow at 2019-07-25 17:58 (UTC), posted by SE-user Kevin Walker
@DanielBarter Well the trace of the T-matrix is nothing but the invariant associated to a $3$-manifold, namely $T_2\times I$ (with $T_2$ the $2$-torus and $I$ the interval) where we glue the two boundary components in a Dehn-twisted manner. Don't know much about $3$-manifolds, does this one not have a reflection symmetry?

This post imported from StackExchange MathOverflow at 2019-07-25 17:58 (UTC), posted by SE-user Andi Bauer
@ArunDebray Yes, from my state-sum perspective I also get to the conclusion that models with time-reversal symmetry can be extended to unoriented manifolds in a non-trivial way. This is because in a unitary state-sum model we take a tensor or its (entry-wise) complex conjugate depending on the chirality of the simplices relative to an orientation. If all tensors are real there's no dependence on orientation, so we can define the model on arbitrary (unoriented) manifolds.

This post imported from StackExchange MathOverflow at 2019-07-25 17:58 (UTC), posted by SE-user Andi Bauer
@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
I would say it is true for $[\omega]\in H^3(G,U(1))$ if and only if there is an automorphism $\alpha$ of $G$, such that $\alpha^\ast[\omega]=[\omega^{-1}]$.

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

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