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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 $Heis(\mathcal{L}_{WZW})$-structure on some $B$ induces a $G$-fiber $\infty$-bundle on $B$ equipped with a map to $\mathbf{B}^n U(1)_{conn}$ which restricts on each fiver to $\mathcal{L}_{WZW}$. (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 $G$-principality (if it does).
Isn’t there a universal version of this over BG?
Over $\mathbf{B} Heis(\mathcal{L}_{WZW})$. 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 $G$-cocycles to $Heis(\mathcal{L}_{WZW})$ cocycles induced $G$-fiber $\infty$-bundles with an $n$-connection on the total space that restricts on each fiber to $\mathcal{L}_{WZW}$. What I am still lacking is a general formal argument that this $G$-fiber bundle is the $G$-principal bundle classified by the underlying $G$-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 $n$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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