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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeOct 10th 2013
• (edited Oct 10th 2013)

Xiao-Gang Wen has started an entry symmetry protected trivial order.

He mentions ’group super-cohomology theory’ as describing fermionic SPT orders. Given our super-slick account of group cohomology, do we just change the ambient $\infty$-topos to Super $\infty$ Grpd?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 10th 2013

Thanks for highlighting, I would have missed it otherwise.

Yes, if this means (as it seems it does) group cohomology of super-Lie groups, then, yes, this is just mapping spaces in $Super\infty Grpd$.

Right now we are talking about just such super Lie group cohomology at our Super Gerbes meeting. And that’s why right now I have to run and quit reading here. But I’ll try to get back to this later this evening. Thanks again for the heads up.

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeOct 10th 2013
• (edited Oct 10th 2013)

Started a page for Xiao-Gang Wen, who is now at the Perimeter Institute. The description there mentions ’condensed matter’ theory. We could do with an entry on that. Someone at Princeton gives it a go here. So solid-state physics is now seen as a branch of condensed matter physics.

EDIT: Oh, that’s just taken from wikipedia Condensed matter physics.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 10th 2013

We have had a stub solid state physics for some time. I have added more redirects.

• CommentRowNumber5.
• CommentAuthorxgwen
• CommentTimeOct 11th 2013
Group super-cohomology theory is a term that Gu and I invented. It is not the group cohomology for supergroup. It is a "group cohomology" with anticommuting coefficient. The usual group cohomology with U(1) coefficient is denoted as H^d[G,U(1)].
In group super-cohomology we replace U(1) by something that contain anticommuting Grassman numbers. We really want to know is there such kind of group cohomology theory in math (maybe under a different name). I would like to thank Urs for editting the SPT entry.
• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 11th 2013

Thanks for the information. I would expect that also your definition of cohomology with super-geometric coefficients is still given by maps in the higher supergeometric topos.

Could you point me to the precise page of an article where the group super-cohomology in your sense is defined? Thanks!

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeOct 11th 2013

In this paper, we introduce a (special) group super-cohomology theory

See Appendix C on p. 35.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 7th 2015
• (edited Oct 7th 2015)

I have finally begun to cross-linke symmetry protected topological order with higher dimensional WZW model, due to the article

• Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, Xiao-Gang Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013) arXiv:1106.4772; A short version in Science 338, 1604-1606 (2012) pdf

which argues that the bosonic SPT phases are described just by such higher WZW models.

This needs to be expanded on.