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October 13, 2006

Bundle Gerbes: General Idea and Definition

Posted by Urs Schreiber

I was asked to edit the Wikipedia entry on bundle gerbes.

For one, it does not actually say what a bundle gerbe is. Another thing is that it falsely attributes the insight that the Wess-Zumino term in the WZW-model arises as a special case of parallel 2-transport to us. That should be corrected.

I’ll develop my contribution here, before copying it to Wikipedia.

General.

A bundle gerbe is a categorification of a transition function of a fiber bundle. For that reason, it is sometimes also addressed as a transition bundle.

Despite its name, a bundle gerbe is not a gerbe. A gerbe is rather a categorification of the sheaf of sections of a fiber bundle.

A fiber bundle can be characterized (up to isomorphism) in several different ways, for instance as

  • a fibration,
  • a sheaf of sections,
  • a transition function,
  • an Atiyah groupoid extension.
If it is a principal U(1)U(1)-bundle or a complex line bundle, it can also be characterized as
  • an element in second integral cohomology.

Each of these descriptions has a categorification. They go, respectively, by the name

  • a 2-bundle,
  • a gerbe,
  • a bundle gerbe,
  • an Atiyah 2-groupoid extension,
  • an element in third integral cohomology.

The degree to which these have been studied and understood differs. In as far as they have been understood, one finds essentially the expected equivalences.

In particular, bundle gerbes are equivalent to a certain sub(-2-)category of gerbes, and are classified by third integral cohomology.

This cohomological classification has, historically, been one of the main motivations for the development of gerbes and bundle gerbes. For that reason, bundle gerbes are sometimes addressed as a “geometric realization of third integral cohomology”, in the same sense that complex line bundles are a “geometric realization of second integral cohomology”.

Of course this classification by integral cohomology applies only to what should be more precisely called complex line bundle gerbes or U(1)U(1)-principal bundle gerbes, which were introduced by Murray. Other notions of bundle gerbes can be defined, and in particular nonabelian principal bundle gerbes have been studied by Aschieri-Cantini–Jurčo.

It turns out that line bundle gerbes can alternatively be regarded as central extensions of groupoids. This is not to be confused with the (2-)groupoid extension mentioned above.


Definition.

A bundle gerbe over a space XX is

  • a regular epimorphism Y π X \array{ Y \\ \pi \downarrow \\ X }
  • a fiber bundle B p Y [2] \array{ B \\ p \downarrow \;\; \\ Y^{[2]} } over the fiber product Y [2]Y^{[2]} of YY with itself, i.e. L p Y [2] Y π X, \array{ L \\ p \downarrow \;\; \\ Y^{[2]} &\stackrel{\rightarrow}{\rightarrow}& Y \\ && \pi \downarrow \;\; \\ && X } \,, together with a notion of product \otimes of fibers of BB;
  • on Y [3]Y^{[3]} an isomorphism μ:π 12 *Bπ 23 *Bπ 13 *B, \mu : \pi_{12}^*B \;\otimes\; \pi_{23}^*B \;\stackrel{\sim}{\to}\; \pi_{13}^* B \,, which satisfies an associativity relation on Y [4]Y^{[4]}. Here π 12,π 23,π 13\pi_{12}, \pi_{23}, \pi_{13} are the three obvious maps Y [3]Y [2]. Y^{[3]} \stackrel{\stackrel{\rightarrow}{\rightarrow}}{\rightarrow} Y^{[2]} \,.

For line bundle gerbes, XX is taken to be a smooth space, π:YX\pi : Y \to X a surjective submersion, BB a smooth line bundle and μ\mu a smooth isomorphism. This case was introduced by Murray in dg-ga/9407015.

For (nonabelian) GG-principal bundle gerbes, one chooses BB to be a GG-principal bibundle, which is a bundle that is GG-principal both for a left and a right GG action, which mutually commute. In other words, the fibers of BB are GG-bitorsors. This allows to define the product as p 12 *Bp 23 *B:=p 12 *B× Gp 23 *B. p_{12}^* B \;\otimes\; p_{23}^* B \;:=\; p_{12}^* B \;\times_G\; p_{23}^* B \,. This case was introduced by Aschieri, Cantini & Jurčo in hep-th/0312154.

For complex line bundle gerbes the above definition can be understood as obtained from the definition of a transition function of a line bundle by replacing the monoid of complex numbers by the monoidal category of 1-dimensional complex vector spaces; and by replacing equations between numbers by coherent (here: associative) isomorphisms of vector spaces.

Such a replacement is known as categorification.

Analogously, for principal bundle gerbes the above definition can be understood as obtained from the definition of a transition function of a principal GG-bundle by replacing the group GG by the 2-group GBiTor G\mathrm{BiTor} of GG-bitorsors, and by replacing equations between group elements by coherent (here: associative) isomorphisms of bitorsors.


Interpretation in terms of categorified transition functions.

In order to see this more explicitly, consider the special case where Y=UY = \mathbf{U} is a good covering of XX by open contractible sets {U i}\{U_i\} Y:= iU i, Y := \sqcup_i U_i \,, with Y X \array{ Y \\ \downarrow \\ X } being the obvious map which embeds a point in U iXU_i \subset X into XX. Then Y [2]Y^{[2]} is the disjoint union of double intersections of these open sets Y [2]= i,jU iU j. Y^{[2]} = \sqcup_{i,j} U_i \cap U_j \,.

The transition function of a local trivialization with respect to YY of a complex line bundle on XX is nothing but a complex function g:Y [2]. g : Y^{[2]} \to \mathbb{C} \,. satisfying g ijg jk=g ik g_{ij} g_{jk} = g_{ik} on each triple intersection U iU jU kU_i \cap U_j \cap U_k, where g ijg_{ij} denotes the restriction of gg to to U iU jY [2]U_i \cap U_j \subset Y^{[2]}.

This is equivalent to saying that p 12 *gp 23 *g=p 13 *g. p_{12}^* g \, p_{23}^* g = p_{13}^* g \,. Replacing this equation of (functions with values in) complex numbers by a coherent isomorphism of complex 1-dimensional vector spaces leads to the definition of a line bundle gerbe.


Interpretation in terms of groupoid extensions.

To any morphism YX Y \to X in a category with pullbacks, we may associate the groupoid Y :=Y [2]Y Y^\bullet := Y^{[2]} \stackrel{\to}{\to} Y whose object of objects is YY, whose object of morphisms is Y [2]Y^{[2]} and whose composition law is given by the unique vertical morphsim in Y [3] Y [2] Y [2] Y Y Y [2]. \array{ & & & Y^{[3]} \\ & & \swarrow &&\searrow \\ & Y^{[2]} & &&& Y^{[2]} \\ &\swarrow& &&& \searrow \\ Y &&& \downarrow &&& Y \\ & \nwarrow &&&& \nearrow \\ &&& Y^{[2]} } \,.

For instance, for π:YX\pi : Y \to X a surjective submersion, the objects of Y Y^\bullet are the points of YY, and there is a unique morphism y 1y 2y_1 \to y_2 whenever π(y 1)=π(y 2)\pi(y_1) = \pi(y_2).

The above definition of a bundle gerbe can be understood as an enrichment of this groupoid in the sense of enriched categories. For line bundle gerbes the enrichment is over 1DVect1D\mathrm{Vect}, for principal bundle gerbes the enrichment is over GBiTorG\mathrm{BiTor}.

For line bundle gerbes this is often expressed as saying that a line bundle gerbe is a U(1)U(1)-central extension of Y Y^\bullet.

Posted at October 13, 2006 10:17 AM UTC

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