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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.
So surely there’s also a notion of fibered Chern-Simons which has the same relation with fibered WZW as that of CS/WZW? Is this supposed to be related to the M2 brane worldvolume theory?
For the example of the heterotic current sector, yes.
But for the GS-functionals I doubt it, because these super Lie algebra cocycles are not transgressive (as per eq. 180, 181, p. 40 here), I think. (I feel like I knew this for a fact once, but now I am just saying it from memory. Certainly no transgression is known.)
This was actually the point in my evolution where I changed my mind from “Oh, everything is fundamentally Chern-Simons theory or its bounday theory.” To “Oh, no, yet more fundamentally everything is fundamentally a WZW model.” Because there are WZW-type theories which do not come from Chern-Simons type theories.
What does one make of that situation where cocycles are not transgressive? What is the meaning of the obstruction, and when can one add other cocycles so as to make the combination transgressive, in the spirit of a GS mechanism (though of course one cannot call this an “anomaly”)?
These are good questions to ask.
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