Is the first differential Pontryagin class a morphism of stacks?

Question

In Cech Cocycles for Characteristic Classes, Jean-Luc Brylinski and Dennis McLaughlin provide explicit formulas for Cech cocycles for characteristic classes of real and complex vector bundles, and show how to refine these to Cech-Deligne cocycles for differential characteristic classes in differential cohomology.

When the compact Lie group $G$ involved is enough connected, e.g., if one is interested in the second Chern class of a principal $SU(n)$-bundle, the Brylinski-McLaughlin formula drastically simplifies, and it can be shown that in this case it actually gives a morphism of stacks from the classifying stack of principal $G$-bundles with connections to the (higher) stack of principal $U(1)$-$n$-gerbes with connections (here $n+2$ is the degree of the characteristic class involved).

This stacky interpretation is emphatised, e.g., in the follow-up Cech cocycles for differential characteristic classes - An $\infty$-Lie theoretic construction, by Urs Schreiber, Jim Stasheff and myself, where it is obtained (as the title suggests) via integration of $L_\infty$-algebras to higher Lie groups. For this approach, the connectivity of $G$ is essential. For instance one can see that the first fractional differential Pontryagin class $\frac{1}{2}\hat{p}_1$ is a morphism of stacks from the stack of princiapl Spin bundles with connection to the 3-stack of $U(1)$-2-gerbes with connection, and this precisely reproduces Brylinski-McLaughlin construction, the but one cannot see the Brylinski-McLaughlin cocycle for the first differential Pontryagin class $\hat{p}_1$ for princiapal $SO$-bundles with connections via Lie integration: $SO$ is not enough connected to allow this. This pheneomenon is in a sense not surprising: for instance the "identity" morphism from the Lie algebra of $O(2)$ to the Lie algebra of $SO(2)$ cannot be integrated to a morphism of Lie groups from $O(2)$ to $SO(2)$ due to "lack of connectivity" reasons.

Yet the fact that a particular technique fails does not mean that a statement is false. So here is my question: is there a natural interpretation of Brylinski-McLaughlin cocycle for the first differential Pontryagin class $\hat{p}_1$ as a morphism of stacks (from $SO$-bundles with connections to $U(1)$-2-gerbes with connections)? or is there a natural interpretation of Brylinski-McLaughlin cocycle for the second differential Chern class $\hat{c}_2$ as a morphism of stacks from $U$-bundles (not $SU$!) with connections to $U(1)$-2-gerbes with connections)?

My feeling is that once there are no topological obstruction (i.e., once the characteristic classes are defined at the non-very-conected level, as in these cases), the morphism of stacks (which surely exists at the highly connected level) descends from the higher connected cover of $G$ involved to the original $G$. So, for instance since $\frac{1}{2}p_1$ is not an integral class for $SO$-bundles one can not make $\frac{1}{2}\hat{p}_1$ descend from principal Spin-bundles with connections to princiapal $SO$-bundles with connections; but since $p_1$ is an integral class for $SO$-bundles, then it should be possible that $\hat{p}_1$ descends. But all my attemps towards a rigorous proof of this have failed so far, so I've begun thinking that I may be wrong, and that $\hat{p}_1$ can be given no natural interpretation as a morphism of stacks after all.

Any suggestion, reference or criticism concerning this problem is welcome.

Answer 1

Yes, every differential characteristic class is a stack morphism.

The point is that there exist universal differential characteristic classes. These are not easy to describe since they involve a notion of differential cohomology of classifying spaces. One way is to use Urs Schreiber's approach. At least in the case of degree four classes one can alternatively use the theory of multiplicative bundle gerbes.

In this context, I'd use the following working definition:

Definition: A universal degree four differential characteristic class on a Lie group $G$ is a multiplicative $U(1)$-bundle gerbe over $G$ with (multiplicative) connection of curvature $H$.

Here, $H$ is the "canonical" 3-form on $G$, it is defined using a bilinear symmetric linear form on the Lie algebra of $G$ - I think any such form is fine.

Multiplicative bundle gerbes have been introduced in the paper (1); they represent classes in $H^4(BG,\mathbb{Z})$. The notion of a multiplicative connection is subtle, I'd refer to Definition 1.3 of my paper (2). The point is that the "naive" definition is too strict and leaves essentially no space for examples. In particular, while a multiplicative gerbe over $G$ can be seen as a 2-gerbe over $BG$, a multiplicative connection is NOT a connection (in the ordinary sense) on this 2-gerbe.

Example 1: If $G=Spin(n)$, the basic gerbe $\mathcal{G}$ over $G$ carries a canonical connection of curvature $H$ and a canonical multiplicative structure. It is the universal differential half first Pontryagin class, $\frac{1}{2}\widehat{p_1}$. It underlies the definition of string connections I have proposed in my paper (3).

Example 2: If $G = SO(n)$, the bundle gerbe $\mathcal{G}$ descends together with its connection along the projection $Spin(n) \to SO(n)$, but its multiplicative structure does not descend. Instead, only the multiplicative bundle gerbe $\mathcal{G}^2 := \mathcal{G} \times \mathcal{G}$ descends together with its connection and its multiplciative structure. The descended gerbe over $SO(n)$ is the universal differential first Pontryagin class, $\widehat{p_1}$. Descent theory for multiplicative gerbes, together with obstructions is discussed in (4), see Table 1 at the end of the paper.

Now suppose that $\mathcal{G}$ is a universal differential characteristic class, $X$ is a smooth manifold and $P$ is a principal $G$-bundle with connection $A$ over $X$. The Chern-Simons 2-gerbe $\mathbb{CS}_P(\mathcal{G})$ is a $U(1)$-bundle 2-gerbe over $X$, see (1). A connection on $\mathbb{CS}_P(\mathcal{G})$ is constructed from the Chern-Simons 3-form $CS(A)$ and the multiplicative connection on $\mathcal{G}$; here the condition that the curvature of $\mathcal{G}$ is $H$ is important. This construction is described in detail in Section 3.2 of (2). It has the following properties:

Theorem:

• if $\xi_P: M \to BG$ is a classifying map for $P$, then $[\mathbb{CS}_P(\mathcal{G})] = \xi_P^*[\mathcal{G}] \in H^4(M,\mathbb{Z})$.

• a connection-preserving bundle morphism $P_1 \to P_2$ (covering some smooth map $X_1 \to X_2$ between base manifolds) induces a morphism $$ \mathbb{CS}_{P_1}(\mathcal{G}) \to \mathbb{CS}_{P_2}(\mathcal{G}) $$ between bundle 2-gerbes.

The first statement is Theorem 3.2.3 in (2). It means that on the level of the underlying (non-differential) characteristic class the construction is just pullback. The second statement follows by inspection of the definition of $\mathbb{CS}_P(\mathcal{G})$. It means precisely that we have a stack morphism $$ \mathbb{CS}(\mathcal{G}): Bun_G^{\nabla} \to 2\text{-}Grb_{U(1)}^{\nabla}. $$

References:

1. "Bundle gerbes for chern-simons and wess-zumino-witten theories"
2. "Multiplicative bundle gerbes with connection"
3. "String connections and Chern-Simons theory"
4. "Polyakov-Wiegmann formula and multiplicative gerbes"