Topological cystalline insulator (Liang Fu, 2011)
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
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.
“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)