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Thanks.
I think it is really good to give this category-theoretic perspective on what gauge fixing really is, yes. Thanks for starting this entry.
I added an "Idea"-section with some introductory words and also started an "Examples"-section.
Yes, so the anomaly business comes in when we do not start with having an action-function/functor on configuration space, but instead have just a line bundle (with connection) and only know that the action is some flat non-trivial section of it.
By the way, we have stub entries like quantum anomaly and Green-Schwarz mechanism that talk a little about this.
I have meant to give a fully formal nPOV description of this, with the relevant differential cocycles accordingly modeled as functors etc, but haven't really gotten around to doing that.
I did talk about it a bit on the blog, once, though, for instance in Charges and Twisted Bundles, III: Anomalies
There is a very nice general abstract story to be told here. It's good that you are pushing me. Maybe together we can write some decent stuff in some nice entries.
you know, I had been thinking about similar lines but never quite felt that I understood this to the point that it became a general abstract tautology, if you see what I mean.
I think for gauge theories, of the kind described in Freed's article, I'd know how to take what Freed writes and reformulate it in a completely abstract way, such that it boils down to turning a crank on an abstract machinery.
But I am nott sure how to do this with things lilke the Polyakov action.
At some point this made me think the following: I was thinking that maybe we ought to try to think of every physics theory as a gauge theory, thereby realizing it in the realm where we have good control of the abstract whereabouts.
Of course this is not a new idea, but let me make it explicit here: we really want to be talking about the superstring. It's action is, as you know, really a worldsheet supergravity coupled to a bunch of worldsheet "matter" fields (which are of course the sigma-model embedding fields from the target space perspective).
Now there are various attempts to formulate plain gravity as a gauge theory with constraints. They all don't seem entirely satisfactory. But here we are talking supergravity. And I find it striking that for supergravity the situation is the reverse: the formulationns of supergravity that do not formulate it as a gauge theory seem awkward.
I don't know, did you ever have had a look at the original articlle Cremmer-Julia-Scherk on D=11 SUGRA? That whole article, impressive as it is, reads like one big lesson in awkwardness. All of standard supergravity does.
But then, there is the D'Auria-Fre formulation of supergravity and, lo and behold, it is very elegant and manages to show how all the awkward details drop out from turning a nice crank. And: it is crucualliay formuated as a gauge theory -- a higher gauge theory even. (I mean, the original authors did not exactly realize this, but it is evident once one knows what a higher gauge theory looks like).
This is not a proof of anything, of course. But it made me wonder. Maybe we shouldn't be looking too much at the bosonic Polyakov action if we want to come from an abstract point of view. Maybe we should take the RNS worldsheet superstring action before the diffomorphism gauge has been fixed, which is worldsheet supergravity, and then think of that as a gauge theory using standard first order formalism and guidance from D'Auria-Fre.
Maybe then we will see how the conformal Weyl-anomaly fits in exactly into the general absstract picture we had before.
I don't know, but this is something I would like to find time to look into.
Yes.
Well, there are some technical subtleties here and there, but this is certainly the picture, yes.
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