Talk by Dominic

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 H^{n+1}(G,U(1)) [Chen et. al. 2011]

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

0) How do you gauge internal symmetry?

  1. How do you”gauge”spatial symmetry”?
  2. Classification of phases (U(1)) of phases)

If we have a U(1) symmetry \vec{A},

S_{SPT} \rightarrow S_{SPT}[\vec{A}]\xrightarrow{integrate out SPT} \frac{n}{2\pi} \int \varepsilon_{\mu\nu\lambda} A_\mu \partial_\nu A_\lambda

Hall conductance, integer

H^3(U(1),U(1))=\mathbb{Z}

S_{SPT}\xrightarrow{gauge}S_{SPT}[A]\xrightarrow{integrate out SPT} Topological term, gauge field

[Dijgraaf Witten,1991] H^{d+1}(G,U(1))

What is a gauge field: it is a connection on a principal G-bundle. A physical picture of this: branch cuts, across which a particle state |\psi\rangle becomes g|\psi\rangle, where g is element of the group. (In case of \mathbb{R}^n, 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. G=1, X=\mathbb{R}^2, M = \mathbb{R}^2, f(z) = z^2,

 

Gauge field: M\rightarrow BG, Classifying space: BG=EG/G

“Crystalline gauge field” M\rightarrow X//G=(X\times EG)/G

H^{d+1}(BG,U(1))=H^{d+1}(G,U(1)), H^{d+1}(X//G,U(1)): equivariant cohomology (crystalline SPTs)

 

 

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