# 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)