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August 4, 2007

Gauge Tranformations of n-Bundles and (n-1)-Gerbes

Posted by Urs Schreiber

Over on sci.math.research Christoph Wockel today asks:

can anyone help me with finding the appropriate notion of a gauge transformation on a gerbe (abelian or not)? I would be interested in particular in lifting gerbes. I tried to find it several times but did not succeed. My guess would be to define it as a vertical 2-bundle automorphismsm of the corresponding 2-bundle, but I did not find this mentioned anywhere… Accordingly, gauge transformations would build up a 2-group rather than a group.

I’ll give a quick reply here, indicating the basic idea. Upon request I can spell out more details.

Quite generally, “gauge transformation” is really just another word for automorphism.

So, given any object PP in some category – like a GG-bundle, say, in the category of GG-bundles, its “gauge transformations” are simply all morphism

g:PP g : P \to P

from the object to itself (i.e. endomorphisms) which are invertible.

For bundles here, one would want to distinguish, possibly, between the category of bundles whose morphisms are required to fix the base space and that where we don’t impose this restriction. The gauge transformations proper of bundles are the automorphisms in the former version of this category. But it one may feel the need to look at automorphisms in the latter, too, thus getting a notion of gauge transformation which combines the more standard one with diffeomorphisms of the base space. This is useful, for instance, for the study of equivariant bundles, in case there is a group acting by diffeomorphisms on the base space.

The only subtlety as we climb up the dimensional ladder, now, is that the notion of “automorphism” gets weakened to that of an equivalence.

For 2-categories, a morphism g:PPg : P \to P' is called an equivalence, if there is another morphism, h:PPh : P' \to P such that the 1-morphisms hg:PP h\circ g : P \to P and gh:PP g\circ h : P' \to P' are – not necessarily equal but isomorphic, i.e. connected by an invertible 2-morphisms, to the identity 1-morphism on PP and on PP', respectively.

From that, the pattern is clear: in a general nn-category, we say, recursively, that an nn-morphism is an equivalence if it is invertible, and a (k<n)(k \lt n)-morphism is an equivalence if it has an inverse up to a (k+1)(k+1)-equivalence.

That said, all one needs to understand then to understand gauge transformations of nn-bundles and of (n1)(n-1)-gerbes is what the kk-morphisms between these beasts are, in the first place.

This is usually very obvious. Of course the details depende on which of a bunch of equivalent formulations of these objects one is looking at.

In as far, for instance, as an nn-bundle is an nn-category PP with a suitable projection p:PX p : P \to X for XX the discrete nn-category (i.e. no nontrivial morphisms) over the elements of some space XX, then morphisms between these guys are simply the nn-functors between the total nn-spaces g:PP g : P \to P' respecting whichever structure one demands to respect (like smoothness, usually, or like the base space projection, usually).

Conversely, if one rather likes to think of one’s nn-bundle as a fiber-assigning (n+1n+1)-functor XnCat X \to n\mathrm{Cat} which sends each point xx \in to the fiber P xP_x (an nn-category) living over it, then morphisms are simply the morphisms of such nn-functors, respecting the extra structure (smoothness) which is around.

Or, if you like to think of the stacks of sections of these beasts, i.e. of (n1n-1)-gerbes as (n1)(n-1)-stacks on something like the site of open subsets of a fixed space, or on that of all manifolds, say, morphisms are simply the morphisms of these (n1)(n-1)-stacks (hence, since (n1)(n-1) stacks are pseudofunctors, are nothing but morphisms of pseudofunctors).

Similarly, in both nn-bundle perspectives mentioned before, one tends to want to work with local trivializations of the full global thing, i.e. with the descent data of these gadgets. Details may depends on the setup, but the descent data are essentially themselves pseudofunctors, so they have obvious notions of morphisms between them.

Applying this general nonsense to concrete special realizations, like bundle gerbes, then tells one what the morphisms of these are. For instace the funny twist which is introduced for morphisms of bundle gerbes, which was originally missed and therefore later called “stable morphism” of bundle gerbes (unfortunately) is in fact precisely what one gets from the general nonsense once one realizes that a bundle gerbe is really a pseudofunctor.

I, and others here, should be able to provide literature and precise details and formulas for whatever special case is requested.

Posted at August 4, 2007 8:38 AM UTC

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Re: Gauge Tranformations of n-Bundles and (n-1)-Gerbes

Very minor comment:
If this is meant to be defining morphisms so as to define the category, am I right that
these morphisms are over the identity of the base? the other alternative is also viable and of interst in the physics.

Posted by: jim stasheff on August 11, 2007 4:23 PM | Permalink | Reply to this

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