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groupoid of Lie-algebra valued forms

Idea

For $G$ a Lie group, its delooping $BG$ is a smooth groupoid, which we may think of as a groupoid internal to smooth spaces.

According to the theory of models for ∞-stack (∞,1)-toposes this is to be thought of as modeling the corresponding smooth ∞-stack $G\mathrm{Bund}(-)$ of smooth $G$-principal bundles obtained after ∞-stackification.

The cohomology with coefficients in $BG$ is degree 1 smooth nonabelian cohomology.

Let $g:=\mathrm{Lie}(G)$ be the Lie algebra of $G$.

The groupoid of $g$-valued differential forms , denoted here $\overline{B}G$, is the refinement of $BG$ in differential nonabelian cohomology.

Definition

Definition

For $G$ a Lie group the groupoid of $\mathrm{Lie}(G)$-valued differential forms is as a groupoid internal to smooth spaces, the sheaf of groupoids

$$\overline{B}G:=[P_1(-),BG]$$\bar \mathbf{B}G := [P_1(-), \mathbf{B}G]

that to a smooth test space $U\in \mathrm{Diff}$ assigns the functor category $[P_1(U),BG]$ of smooth functors (functors internal to smooth spaces) from the path groupoid $P_1(U)$ of $U$ to the one-object delooping groupoid $BG$.

Theorem

The groupoid $\overline{B}G$ is canonically equivalent to the smooth groupoid where

• objects are smooth $g$-valued 1-forms $A\in {\Omega }^1(U,g)$;

• morphisms $h:A\to A\prime$ are given by smooth $G$-valued functions $h\in C^{\mathrm{\infty }}(U,G)$ such that

  $$A\prime ={\mathrm{Ad}}_h(A)-h^{*}\overline{\theta }$$A' = Ad_h(A) - h^* \bar \theta

Here $\overline{\theta }$ is the right invariant canonical $g$-valued 1-form on $G$. A more sloppy but common way to write this is $A\prime ={\mathrm{Ad}}_h(A)+hd{h}^{-1}$.

Differential nonabelian cohomology

Theorem

The cohomology with coefficients in $\overline{B}G$ classifies $G$-principal bundles connection on a bundle with connection.

More is true: there is a natural canonical equivalence of groupoids

$$H_{\mathrm{Diff}}(X,\overline{B}G)\simeq G{\mathrm{Bund}}_{\nabla }(X).$$\mathbf{H}_{Diff}(X, \bar \mathbf{B}G) \simeq G Bund_\nabla(X) \,.

Remarks

• There is the obvious projection

  $$\overline{B}G\to BG.$$\bar \mathbf{B}G \to \mathbf{B}G \,.

• Lifting a $G$-cocycle through this projection to a differential $G$-cocycle means equipping it with a connection.

For $G=U(n)$ these differential cocycles model the Yang-Mills field in physics.

• For $G=U(1)$ the sheaf $\overline{B}U(1)(-)$ coincides with the the Deligne complex in degree 2, $\overline{B}U(1)\simeq \mathbb{Z}(2{)}_D^{\mathrm{\infty }}$, as described there.

  • This corresponds to the electromagnetic field.

References

The idea goes back to John Baez. For a detailed history see the discussion at Differential Nonabelian Cohomology.

The details are in

• U. Schreiber, Konrad Waldorf, Parallel Transport and Functors (arXiv)

The definition in terms of differential forms is def 4.6 there. The equivalence to $[P_1(-),BG]$ is proposition 4.7.

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