Classes of bundles
Examples and Applications
Special and general types
A bundle gerbe is a special model for the total space Lie groupoid of a -principal 2-bundle for the circle 2-group.
More generally, for a more general Lie 2-group (often taken to be the automorphism 2-group of a Lie group ), a nonabelian bundle gerbe for is a model for the total space groupoid of a -principal 2-bundle.
The definition of bundle gerbe is not in fact a special case (nor a generalization) of the definition of gerbe, even though there are equivalences relating both concepts.
A bundle gerbe’* over a smooth manifold is
a surjective submersion
together with a -principal bundle
over the fiber product of with itself, i.e.
of -bundles on
such that this satisfies the evident associativity condition on .
Here are the three maps
in the Cech nerve of .
In a nonabelian bundle gerbe the bundle is generalized to a bibundle.
A bundle gerbe may be understood as a specific model for the total space Lie groupoid of a principal 2-bundle.
We first describe this Lie groupoid in
and then describe how this is the total space of a principal 2-bundle in
As a groupoid extension
Give a surjective submersion , write
for the corresponding Cech groupoid. Notice that this is a resolution of the smooth manifold itself, in that the canonical projection is a weak equivalence (see infinity-Lie groupoid for details)
The data of a bundle gerbe induces a Lie groupoid which is a -extension of , exhibiting a fiber sequence
This Lie groupoid is the groupoid whose space of morphisms is the total space of the -bundle
with composition given by the composite
As the total space of a principal 2-bundle
We discuss how a bundle gerbe, regarded as a groupoid, is the total space of a -principal 2-bundles.
Recall from the discussion at principal infinity-bundle that the total 2-bundle space classified by a cocycle is simply the homotopy fiber of that cocycle. This we compute now.
(For more along these lines see infinity-Chern-Weil theory introduction. For the analogous nonabelian case see also nonabelian bundle gerbe.)
The Lie groupoid defined by a bundle gerbe is in ∞LieGrpd the (∞,1)-pullback
of a cocycle .
In fact a somewhat stronger statement is true, as shown in the following proof.
We can assume without restriction that the bundle in the data of the bundle gerbe is actually the trivial -bundle by refining, if necessary, the surjective submersion by a good open cover. In that case we may identify with a -valued function
which in turn we may identify with a smooth 2-anafunctor
From here on the computation is a special case of the general theory of groupoid cohomology and the extensions classified by it.
Then recall from universal principal infinity-bundle that we model the -pullbacks that defines principal -bundles in terms of ordinary pullbacks of the universal -principal 2-bundle .
We may model all this in the case at hand in terms of strict 2-groupoips. Then using an evident cartoon-notation we have
and is the 2-groupoid whose morphisms are diagrams
in with composition given by horizontal pasting
and 2-morphisms are paper-cup diagrams
So is the Lie 2-groupoid with a single object, with worth of 1-morphisms and unique 2-morphism between these.
From this we read of that
is indeed a pullback square (in the category of simplicial presheaves over CartSp). The morphisms of the pullback Lie groupoid are pairs of diagrams
hence form a trivial -bundle over the morphisms of , and the 2-morphims are pairs consisting of 2-morphisms
in and paper-cup diagrams of the form
in , which exhibits indeed the composition operation in .
Equivariant bundle gerbes over the point
For a group extension by an abelian group classified by a 2-cocycle in group cohomology, which we may think of as a 2-functopr , the corresponding fiber sequence
exhibits as the bundle gerbe over (in equivariant cohomology of the point, if you wish) with Dixmier-Douady class .
Tautological bunde gerbe
Let be a simply connected smooth manifold and a degree 3 differential form with integral periods.
We may think of this a cocycle in ∞-Lie algebroid cohomology
By a slight variant of Lie integration of oo-Lie algebroid cocycles we obtain from this a bundle gerbe on by the following construction
pick any point ;
let be the based smooth path space of ;
let be the -bundle which over an element in – which is a loop in assigns the -torsor whose elements are equivalence class of pairs , where is a surface cobounding the loop and where , and where the equivalence relation is so that for any 3-ball cobounding two such surfaces and we have that is equivalent to the difference of the labels differs by the integral of the 3-form
the composition operation is loop-wise the evident operation that on loops removes from a figure-8 the inner bit and whch is group multiplication of the labels.
This produces a bundle gerbe whose class in has as its image in de Rham cohomology.
For applications in string theory see also
The notion of bundle gerbe as such was introduced in
Early texts also include
(notice that the title here suppresses one “e” intentionally).
A general picture of bundle -gerbes (with connection) as circle (n+1)-bundles with connection classified by Deligne cohomology is in
- Pawel Gajer, Geometry of Deligne cohomology Invent. Math., 127(1):155–207 (1997) (arXiv)