https://arxiv.org/abs/1612.00846

Topological cystalline insulator (Liang Fu, 2011)

Interacting SPTs/SETs

Internal symmetry (charge conservation, Z^T_2, etc.)

Bosonic SPTs: Group cohomology classification [Chen et. al. 2011]

Bosonic SETs -crossed braided tensor categories [Barkeshli, et.al,2014]

0) How do you gauge internal symmetry?

- How do you”gauge”spatial symmetry”?
- Classification of phases () of phases)

If we have a U(1) symmetry ,

Hall conductance, integer

Topological term, gauge field

[Dijgraaf Witten,1991]

What is a gauge field: it is a connection on a principal -bundle. A physical picture of this: branch cuts, across which a particle state becomes , where is element of the group. (In case of , the group is GL). Branch cuts always have codim=1.

Examples (pass)

“Topological crystalline liquid”: topological part of it (TQFT) does not care about lattice. (liquid in the sense that it doesn’t care about the lattice)

E.g. , , , ,

Gauge field: , Classifying space:

“Crystalline gauge field”

, : equivariant cohomology (crystalline SPTs)

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