nLab Dixmier-Douady class

Idea

A bundle gerbe or circle 2-bundle has a unique characteristic class in integral cohomology in degree 3, the higher analog of the Chern class of a circle group-principal bundle (or complex line bundle): this is called the Dixmier-Douady class of the bundle gerbe.

Definition

In the literature one find a universal Dixmier-Douady class defined for different entities, notably for projective unitary-principal bundles and for $U(1)$-bundle gerbes, as well as for C-star algebra constructions related to these. All these notions are equivalent in one sense, namely in bare homotopy theory, but differ in other sense, namely in geometric homotopy theory.

In bare homotopy-type theory

The classifying space of the circle 2-group $\mathbf{B}U(1)$ is an Eilenberg-MacLane space $B \mathbf{B} U(1) \simeq B^3 \mathbb{Z} \simeq K(\mathbb{Z}, 3)$. The bare Dixmier-Douday class is the universal characteristic class

$$DD : B B U(1) \stackrel{\simeq}{\to} K(\mathbb{Z}, 3)$$

exhibited by this equivalence. Hence if we identify $B B U(1)$ with $K(\mathbb{Z}, 3)$, then the DD-class is the identity on this space.

This is directly analogous to how the first Chern class is, as a universal characteristic class, the identity on $K(\mathbb{Z},2) \simeq B U(1)$.

This means conversely that the equivalence class of a $U(1)$-bundle gerbe/circle 2-bundle is entirely characterized by its Dixmier-Douady class.

In smooth homotopy-type theory

The circle 2-group $\mathbf{B}U(1)$ naturally carries a smooth structure, hence is naturally regarded not just as an ∞-group in ∞Grpd, but as a smooth ∞-group in $\mathbf{H} \coloneqq$ Smooth∞Grpd.

For each $n$, the central extension of Lie groups

$$U(1) \to U(n) \to PU(n)$$

that exhibits the unitary group as a circle group-extension of the projective unitary group induces the corresponding morphism of smooth moduli stacks

$$\mathbf{B} U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n)$$

in $\mathbf{H}$.

This is part of a long fiber sequence in $\mathbf{H}$ which continues to the right by a connecting homomorphism $\mathbf{dd}_n$

$$\mathbf{B} U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n) \stackrel{\mathbf{dd}_n}{\to} \mathbf{B}^2 U(1)$$

in $\mathbf{H}$. Here the last morphism is presented in simplicial presheaves by the zig-zag/∞-anafunctor of sheaves of crossed modules

$$\array{ [U(1) \to U(n)] &\to& [U(1) \to 1] \\ {}^{\mathllap{\simeq}}\downarrow \\ PU(n) } \,.$$

To get rid of the dependence on the rank $n$ – to stabilize the rank – we may form the directed colimit of smooth moduli stacks

$$\mathbf{B}U \coloneqq \underset{\rightarrow_n}{\lim} \mathbf{B} U(n)$$

$$\mathbf{B} PU \coloneqq \underset{\rightarrow_n}{\lim} \mathbf{B} PU(n) \,.$$

On these we have the smooth universal class

$$\mathbf{dd} : \mathbf{B} PU \to \mathbf{B}^2 U(1) \,.$$

Since the (∞,1)-topos Smooth∞Grpd has universal colimits, it follows that there is a fiber sequence

$$\array{ \mathbf{B}U &\to& \mathbf{B} PU \\ && \downarrow^{\mathbf{dd}} \\ && \mathbf{B}^2 U(1) }$$

exhibiting the moduli stack of smooth stable unitary bundles as the homotopy fiber of $\mathbf{dd}$.

References

• Jacques Dixmier, Adrien Douady, Champs continus d’espaces hilbertiens et de $C^*$-algebres, Bull. Soc. Math. France 91 (1963) 227-284 [numdam:10.24033/bsmf.1596, pdf]

• Jacques Dixmier, Les $C^*$–algébres et leurs représentations, Gauthier–Villars (1969)