Question

Why is nonorientable surface interesting in condensed matter physics?

Answer 1

especially several papers by Witten and Kapustin, they use extensively the mechanism of putting a theory on a non-orientable manifold, which (to me) magically reproduces many important results in condensed matter physics, e.g. Kitaev 16-fold-way. But the original proof of Kitaev has nothing to do with putting a theory on a non-orientable manifold, and just constructs a direct path from

m>0

m>0

to

m<0

m<0

(heuristically) without closing the gap. My question is, why is non-orientable surface interesting in condensed matter physics a

Answer 2

In many recent papers about topological phases, especially several papers by Witten and Kapustin, they use extensively the mechanism of putting a theory on a non-orientable manifold, which (to me) magically reproduces many important results in condensed matter physics, e.g. Kitaev 16-fold-way. But the original proof of Kitaev has nothing to do with putting a theory on a non-orientable manifold, and just constructs a direct path from m>0m>0 to m<0m<0 (heuristically) without closing the gap. My question is, why is non-orientable surface interesting in condensed matter physics and why can it reproduce many arguments that can be made by more direct methods?