GAP calculation

gap> G :=SpaceGroupBBNWZ(“Fd-3m”);
SpaceGroupOnRightBBNWZ( 3, 7, 5, 2, 4 )
gap> GroupCohomology(G,2);
[ 2, 2 ]
gap> GroupCohomology(G,2,2);
[ 2, 2, 2, 2, 2 ]
gap> GroupCohomology(G,3);
[ 2, 2, 2 ]
gap> GroupCohomology(G,3,2);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
gap> GroupCohomology(G,4);
[ 2, 2, 2, 2, 2, 12 ]

gap> GroupCohomology(G,4,2);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]

gap> GroupCohomology(G,5);
[ 2, 2, 2, 2, 2 ]
gap> GroupCohomology(G,5,2);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]

gap> G :=SpaceGroupBBNWZ(“Fd-3c”);
SpaceGroupOnRightBBNWZ( 3, 7, 5, 2, 3 )
gap> GroupCohomology(G,2);
[ 2, 2 ]
gap> GroupCohomology(G,2,2);
[ 2, 2, 2, 2 ]
gap> GroupCohomology(G,3);
[ 2, 2 ]
gap> GroupCohomology(G,3,2);
[ 2, 2, 2, 2, 2 ]
gap> GroupCohomology(G,4);
[ 2, 2, 12 ]
gap> GroupCohomology(G,4,2);
[ 2, 2, 2, 2 ]

gap> G := SpaceGroupBBNWZ(“R-3m”);
SpaceGroupOnRightBBNWZ( 3, 5, 5, 1, 1 )
gap> GroupCohomology(G,2,2);
[ 2, 2, 2, 2, 2, 2 ]
gap> GroupCohomology(G,2);
[ 2, 2, 2 ]
gap> G := SpaceGroupBBNWZ(“P4132”);
SpaceGroupOnRightBBNWZ( 3, 7, 3, 1, 2 )
gap> GroupCohomology(G,2,2);
[ 2 ]
gap> GroupCohomology(G,2);
[ 2 ]
gap> G := SpaceGroupBBNWZ(“P4332”);
SpaceGroupOnRightBBNWZ( 3, 7, 3, 1, 2 )
gap> GroupCohomology(G,2,2);
[ 2 ]
gap> GroupCohomology(G,2);
[ 2 ]
gap> G := SpaceGroupBBNWZ(“Fm-3m”);
SpaceGroupOnRightBBNWZ( 3, 7, 5, 2, 1 )
gap> GroupCohomology(G,2,2);
[ 2, 2, 2, 2, 2, 2 ]
gap> GroupCohomology(G,2);
[ 2, 2 ]

gap> G := SpaceGroupBBNWZ(“F-43m”);
SpaceGroupOnRightBBNWZ( 3, 7, 4, 2, 1 )
gap> GroupCohomology(G,2,2);
[ 2, 2, 2, 2 ]
gap> GroupCohomology(G,2);
[ 2 ]

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Good book on field theory

It is Fields and Rings by Kaplansky!

it seems to be a small pamphlet but is concisely and very well composed. Only about 70 pages on Field Theory but almost all major results are there. What’s more, the exercises there are so good that I hav seen many people recommending them, including our Prof. Jacob who puts quite a few in his problem sets. The book is also what Prof. Jacob covered in his first eight week lectures.

I think the Ring Theory part, which is unusually put after Field Theory in the book, is also excellent lecture note. Anyway,  Kaplansky is a big guy in Algebra!

 

There may be other good books on Field Theory, such as Dummit’s and Rotman’s.

 

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)

 

 

Some Latex Skills

I learnt something when

Define function ( use of [2] as “2 arguments”, #1 as “the first argument”),

use [T1]{fontenc} for inputing letters with accents (i.e. french words)

fontawesome for internet icons

fontspec to specify font for each use: use \newfontfamily{][]{}[] to define font for each context, where font size, font type (bold, italic, ..), letter spacing, etc. We can use \scalebox (from graphicx) to horizontally stretch letters

use tikz to plot: draw; node for plots with text; (to plot a rectangular missing one side, must plot three sides separately)

The importantce of fonts; fonts convention for resume