nLab lifting gerbe

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 $G$-principal bundle or more generally of any $G$-principal infinity-bundle, through a shifted central extension

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

of $G$, for some abelian group $A$.

Definition

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

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

in group cohomology.

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

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

Examples

Lifting 1-gerbes

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

For $G$ an ordinary topological or Lie group and

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

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

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

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

$$P \times_X P \to G$$

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 $\hat G \to G$ to get $L$ in

$$\array{ L &\to& \hat G \\ \downarrow && \downarrow \\ P \times_X P &\to& G } \,.$$

This $L$ 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, $U(\mathcal{H})$ its unitary group and $P U(\mathcal{H})$ its projective unitary group. Then the extension

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

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

See projective unitary group for details.

To contrast, in (Roberts 2021), finite dimensional lifting gerbes are very often those gerbes with torsion Dixmier-Douady class. This is in particular true if the fundamental group of the base is finite, or the Lie group $G$ is compact.

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

• Alan Carey, Peter Bouwknegt, Varghese Mathai, Michael Murray and Danny Stevenson, K-theory of bundle gerbes and twisted K-theory , Commun Math Phys, 228 (2002) 17-49 (arXiv)

• David Roberts, Many finite-dimensional lifting bundle gerbes are torsion (2021), arXiv:2104.07936.