# Time reversal symmetry for non-orientable manifold

From a recent paper by Kapustin(https://arxiv.org/abs/1406.7329), he argued that for non-orientable manifold with spin structure $Pin^{\pm}$, the corresponding time reversal symmetry $T$ squares to ${\mp}1$. The reason for this, as stated in his paper, is that reflection is related to time reversal symmetry by a Wick rotation. So if his claim is correct, wouldn't that contradict some of the results obtained in a recent paper by Ashvin(http://arxiv.org/abs/1505.04193)? In that paper, the essential argument for a tighter bound of filling is to avoid Kramer's degeneracy since time reversal symmetry $T$ squares to $-1$. However, for non-orientable Bieberbach manifolds, we should not have this degeneracy since we have $Pin^{-}$ spin structure on the manifold and $T$ should instead square to $1$ according to Kapustin.

• Your first sentence is not a full sentence, and I'm not sure what exactly it is saying. Generally, I'm having a hard time understanding your writing. I understand that you may not be a native English speaker, but please have another look at your writing and try to make it clearer what you're trying to say: Pay special attention to writing complete sentences that are grammatically correct.
– Danu
Jun 29 '16 at 21:37
• Looks clear to me now, still, don't know those papers. Jun 29 '16 at 23:00