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Inspired by Matthew Ando’s talk at the Conference on twisted cohomology that I am currently attending, I finally typed up a note on
I have now added what I think is a (simple) formal proof that a -structure on some induces a -fiber -bundle on equipped with a map to which restricts on each fiver to . (I still need an argument that this G-bundle is indeed G-principal).
This is at
Leaving the WZW-terminology aside, this is completely general and formal and has nothing specifically to do with the special situation.
I am using the following general abstract statements, which I hope I am not mixed up about:
dependent sum preserves fiber products (because limits over cospan diagram in the slice are computed as limits over the corresponding co-cone diagrams down in the base and the inclusion of a cospan diagram into its co-cone diagram is final)
by the same argument dependent sum preserves effective epimorphisms .
Maybe some abstractly-minded reader can help me see why the argument at the above link also shows -principality (if it does).
Isn’t there a universal version of this over BG?
Over . That’s what I am trying to construct.
So far I have explicit constructions in local models and I have a general formal argument that lifts from -cocycles to cocycles induced -fiber -bundles with an -connection on the total space that restricts on each fiber to . What I am still lacking is a general formal argument that this -fiber bundle is the -principal bundle classified by the underlying -cocycle.
Prodded by discussion with Eric Sharpe, I have added to the entry pointers to the original articles by Jim Gates, such as Gates-Siegel 88.
Then I have replaced some of the discussion in the entry by pointers to the meanwhile more comprehensive note cwzw. Eventually I’ll go back and turn this into Lab material. For the moment I have just expanded the Idea-section a little bit, mentioning more of the story for the heterotic string.
Much more needs to be done here.
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