Schreiber
∞-Lie algebroid valued differential forms

Idea

A collection of differential forms on a space $X$ with values in a ∞-Lie algebroid $𝔞$ is the data given by a choice of parallel transport along infinitesimal paths in $X$ with values in $𝔞$.

It is the image under ∞-Lie differentiation of parallel transport along finite paths in $X$ with values in an ∞-Lie groupoid $A$ integrating $𝔞$.

Definition

Consider a context in which there is a notion of ∞-Lie theory, as described there.

Definition

For $𝔞$ an ∞-Lie algebroid and $X$ some ∞-Lie groupoid, a collection of flat $a$-valued differential forms is a morphism

$$v:{\Pi }^{\inf }(X)\to a$$v : \Pi^{inf}(X) \to a

from the infinitesimal path ∞-groupoid of $X$ to $𝔞$.

Remark

In low degrees a morphism out of ${\Pi }^{\inf }(X)$ is essentially what is known as a Grothendieck connection.

Remark

For $X$ an ordinary manifold and after forming the corresponding morphism of generalized Chevalley-Eilenberg algebras this becomes a morphism

$${\Omega }^{•}(X)\leftarrow \mathrm{CE}(a):\omega$$\Omega^\bullet(X) \leftarrow CE(a) : \omega

of differential graded algebras from the Chevalley-Eilenberg algebra of $a$ to the deRham dg-algebra of differential forms on $X$.

The underlying morphism of algebras produces a collection of differential forms on $X$, one for each generator of $\mathrm{CE}(𝔞)$. The condition that this is a morphism of differential algebras puts constraints on these differential forms. This are the flatness constraints.

Definition (differential forms associated to finite parallel transport)

For $A$ an ∞-Lie groupoid write $𝔞:=\mathrm{Lie}(A)$ for its image under ∞-Lie differentiation. Then, from the discussion there, every flat finite $A$-valued parallel transport or $A$-valued local system on $X$ given by a morphism

$$\mathrm{tra}:\Pi (X)\to A$$tra : \Pi(X) \to A

out of the path ∞-groupoid fits naturally into a diagram

$$\begin{array}{ccc}{\Pi }^{\inf }(X)& \stackrel{\mathrm{Lie}(\mathrm{tra})}{\to }& 𝔞\\ \downarrow & & \downarrow \\ \Pi (X)& \stackrel{\mathrm{tra}}{\to }& A\end{array}.$$\array{ \Pi^{inf}(X) &\stackrel{Lie(tra)}{\to}& \mathfrak{a} \\ \downarrow && \downarrow \\ \Pi(X) &\stackrel{tra}{\to}& A } \,.

The top vertical morphism is the flat $\mathrm{Lie}(A)$-valued differential form data associated with the finite parallel transport $\mathrm{tra}$.

Curvature forms

For every ∞-Lie algebroid $𝔞$ there is ∞-Lie algebroid $\mathrm{cone}(a)$, the cone on $𝔞$, i.e. the pushout

$$\begin{array}{ccc}𝔞& \stackrel{{\delta }_0}{\to }& 𝔞\times {\Delta }^1\\ \downarrow & & \downarrow \\ *& \to & \mathrm{cone}(𝔞)\end{array}$$\array{ \mathfrak{a} &\stackrel{\delta_0}{\to}& \mathfrak{a} \times \Delta^1 \\ \downarrow && \downarrow \\ {*} &\to& cone(\mathfrak{a}) }

The Chevalley-Eilenberg algebra of $\mathrm{cone}(𝔞)$ should be the Weil algeba of $𝔞$:

$$\mathrm{CE}(\mathrm{cone}(𝔞))=W(𝔞).$$CE(cone(\mathfrak{a})) = W(\mathfrak{a}) \,.

Flat differential forms with values in $\mathrm{cone}(𝔞)$ are arbitrary differential forms with values in $𝔞$: the extra dimension of $\mathrm{cone}(𝔞)$ absorbs the components of the failure of the flatness condition. These are the curvatures. More precisely, there is the suspension $\Sigma 𝔞$ of $𝔞$ defined as the colimit

$$\begin{array}{ccccc}& & 𝔞& \to & *\\ & & {\downarrow }^{{\delta }_1}& & \downarrow \\ 𝔞& \stackrel{{\delta }_0}{\to }& 𝔞\times {\Delta }^1\\ \downarrow & & \downarrow & & \downarrow \\ *& \to & \mathrm{cone}(𝔞)& \to & \Sigma 𝔞\end{array}$$\array{ && \mathfrak{a} &\to& {*} \\ && \downarrow^{\delta_1} && \downarrow \\ \mathfrak{a} &\stackrel{\delta_0}{\to}& \mathfrak{a} \times \Delta^1 \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& cone(\mathfrak{a}) &\to& \Sigma \mathfrak{a} }

and the Chevalley-Eilenberg algebra of $\Sigma 𝔞$ should be the algebra of invariant polynomials on $𝔞$. This way for a given $𝔞$-valued differential form datum

$$v:{\Pi }^{\inf }(X)\to \Sigma 𝔞$$v : \Pi^{inf}(X) \to \Sigma \mathfrak{a}

the corresponding curvature characteristic forms are given by the composite $P(v)$

$$\begin{array}{ccccc}{\Pi }^{\inf }(X)& \stackrel{v}{\to }& \Sigma 𝔞& & 𝔞-\mathrm{valued}\mathrm{differential}\mathrm{forms}\\ {\downarrow }^{\mathrm{Id}}& & \downarrow \\ {\Pi }^{\inf }(X)& \stackrel{P(c)}{\to }& \Sigma 𝔞& & \mathrm{curvature}\mathrm{characteristic}\mathrm{forms}\end{array}$$\array{ \Pi^{inf}(X) &\stackrel{v}{\to}& \Sigma \mathfrak{a} && \mathfrak{a}-valued\;differential\;forms \\ \downarrow^{Id} && \downarrow \\ \Pi^{inf}(X) &\stackrel{P(c)}{\to}& \Sigma \mathfrak{a} && curvature\;characteristic\;forms }

which dually corresponds to the dg-algebra composite

$$\begin{array}{ccc}{\Omega }^{•}(X)& \stackrel{\omega }{\leftarrow }& W(𝔞)\\ {\uparrow }^{\mathrm{Id}}& & \uparrow \\ {\Omega }^{•}(X)& \stackrel{P(v)}{\leftarrow }& \mathrm{inv}(𝔞)\end{array}.$$\array{ \Omega^\bullet(X) &\stackrel{\omega}{\leftarrow}& W(\mathfrak{a}) \\ \uparrow^{Id} && \uparrow \\ \Omega^\bullet(X) &\stackrel{P(v)}{\leftarrow}& inv(\mathfrak{a}) } \,.

The refinement of this statement relative to a principal ∞-bundle $P\to X$ yields the notion of Cartan-Ehresmann ∞-connection.