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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeApr 23rd 2018

I came across mention of homogeneous spaces for supergroups, and since we don’t have an entry for this I’ve started one.

The quotients for those two superspheres are cited in the literature, but I haven’t checked them.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 23rd 2018

Thanks!

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeApr 23rd 2018

Almost all work here seems to be by physicists for particular applications rather than a pure mathematical account, so perhaps only worth adding if the physics is interesting. E.g., how about

• Jaume Gomis, Dmitri Sorokin, Linus Wulff, The complete AdS(4) x CP(3) superspace for the type IIA superstring and D-branes, (arXiv:0811.1566)

which deals with superspaces like the type IIA superspace $OSp(8|4)/SO(7) \times SO(1,3)$?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 23rd 2018
• (edited Apr 23rd 2018)

References like this are about super anti de Sitter spacetime (we should cross-link!). There are very many of such references, since this is what enters the AdS-CFT correspondence.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeApr 23rd 2018

made explicit how super-Minkowski is a super-group quotient, and similarly for super anti de Sitter spacetime

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeApr 30th 2019

Gave the general result for cosets of orthosymplectic groups being superspheres.

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeApr 30th 2019
• (edited Apr 30th 2019)

Tiny formatting change, $OSp(8/4) \mapsto OSp(8\vert 4)$ etc. One needs to use \vert not |.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeApr 30th 2019

Thanks. Let’s add reference for that. And there ought to be an entry supersphere

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeApr 30th 2019

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeApr 30th 2019

In case you’re wondering, I was just idly contemplating whether there might be a super-cohomotopy.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeApr 30th 2019

Sure. I have been wondering myself, and maybe we have talked about it before. One can certainly define it. Whether it’s the right thing to use in M-theory I still don’t see, either way.

• CommentRowNumber12.
• CommentAuthorDavid_Corfield
• CommentTimeApr 30th 2019

There’s a note (p. 13) of arXiv:hep-th/0409257 that only some super cosets should considered superspaces. I’ll add in that reference.

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeApr 30th 2019

Re #11, perhaps if framings of supermanifolds are needed.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeApr 30th 2019

Right, I guess what you have in mind is part of the general statement that when passing to supergeometry, it is natural to do so throughout (systyematic internalization) instead of only for part of the data.

And it is, which makes it very suggestive that the super-spheres should appear. It’s just that beyond this general abstract suggestion, I have not yet any more concrete insight into it.

Incidentally, this same general suggestion also means that it would be more surprising than not if there were no supersymmetry in physics: This is because the phase space of any field theory with fermions is a supermanifold, so that it is more natural than not that also the symmetry group acting on the phase space is a supergroup, instead of a plain group.