nLab
lifting gerbe

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

A lifting gerbe is a cocycle/gerbe – or equivalently the principal infinity-bundle that it classifies – arising as the obstruction to the lift of a GG-principal bundle or more generally of any GG-principal infinity-bundle, through a shifted central extension

B n1AG^G \mathbf{B}^{n-1} A \to \hat G \to G

of GG, for some abelian group AA.

Definition

Such an extension is classified (or rather its delooping is) by a cocycle

BGB n+1A \mathbf{B}G \to \mathbf{B}^{n+1} A

in group cohomology.

For XBGX \to \mathbf{B}G the cocycle in nonabelian cohomology classifying a GG-principal bundle or principal infinity-bundle, the corresponding lifting AA-gerbe is the thing classified by the composite cocycle

XBGB nA. X \to \mathbf{B}G \to \mathbf{B}^n A \,.

Examples

Lifting 1-gerbes

In the literature the following simplest case of the general situation is usually considered exclusively:

for GG an ordinary group and

U(1)G^G U(1) \to \hat G \to G

a U(1)U(1)-central group extension, classified by a 2-cocycle BGB 2U(1)\mathbf{B}G \to \mathbf{B}^2 U(1) in ordinary group cohomology, and for XBGX \to \mathbf{B}G the cocycle of a GG-principal bundle, the corresponding lifting gerbe is given by the cocycle

XBGB 2U(1). X \to \mathbf{B}G \to \mathbf{B}^2 U(1) \,.

In the literature this is often discussed in terms of the model for gerbes given by U(1)U(1)-principal bundle gerbes: let PXP \to X be the total space of the bundle classified by XBGX \to \mathbf{B}G. Regard this as the surjective submersion that is part of the data of the bundle gerbe to be constructed. By principality of PP, there is a canonical morphism

P× XPG P \times_X P \to G

that sends two elements in the fiber of the bundle to the unique group element that takes the first to the second. Along this morphism, form the pullback of the extension G^G\hat G \to G to get LL in

L G^ P× XP G. \array{ L &\to& \hat G \\ \downarrow && \downarrow \\ P \times_X P &\to& G } \,.

This LL is the line bundle ingredient in the lifting bundle gerbe. There is canocically a multiplication map that completes the definition.

Proposition

Let \mathcal{H} be a separable infinite-dimensiona Hilbert space, U()U(\mathcal{H}) its unitary group and PU()P U(\mathcal{H}) its projective unitary group. Then the extension

0U(1)U()PU()0 0 \to U(1) \to U(\mathcal{H}) \to P U(\mathcal{H}) \to 0

is universal with the above property: every U(1)U(1)-bundle gerbe/circle 2-bundle is the lifting gerbe of some PU()P U(\mathcal{H})-principal bundle through this extension.

See projective unitary group for details.

Lifting 2-gerbes

The archetypical example of a lifting 2-gerbe is a Chern-Simons gerbe.

References

A review is for instance in section 2.2 of

Revised on January 9, 2013 00:18:06 by Urs Schreiber (89.204.139.156)