Schreiber
∞-Lie algebroid valued differential forms

differential cohomology in an (∞,1)-topos -- survey

structures in an (∞,1)-topos

Examples

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Applications

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∞-Lie theory

∞-Lie groupoids and -algebroids

∞-Chern-Weil theory

invariant polynomial
differential cohomology in an (∞,1)-topos
- ∞-Lie algebroid valued differential forms
  - curvature
  - integration
- Cartan-Ehresmann ∞-connection

symplectic ∞-geometry

symplectic ∞-Lie algebroid

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Idea
Definition
- Curvature 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 : Π_{\inf} (X) \to a

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

Remark

In low degrees a morphism out of $Π^{\inf} (X)$ 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

Ω^{•} (X) \leftarrow CE (a) : ω

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 $CE (𝔞)$ . 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 $𝔞 : = Lie (A)$ 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

tra : Π (X) \to A

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

\begin{matrix} Π_{\inf} (X) & \overset{Lie (tra)}{\to} & 𝔞 \\ ↓ & ↓ \\ Π (X) & \overset{tra}{\to} & A \end{matrix} .

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

Curvature forms

For every ∞-Lie algebroid $𝔞$ there is its infinitesimal path $∞$ -groupoid

T 𝔞 : = Π_{\inf} (𝔞) .

For $𝔞$ 0-truncated, this is the ordinary tangent Lie algebroid of $𝔞$ . More generally, it is the $∞$ -Lie algebroid whose Chevalley-Eilenberg algebra is the Weil algebra of $𝔞$ .

Flat differential forms with values in $T 𝔞$ are arbitrary differential forms with values in $𝔞$ : the extra dimension of $T 𝔞$ absorbs the components of the failure of the flatness condition. These are the curvatures.

This way for a given $𝔞$ -valued differential form datum

v : Π_{\inf} (X) \to T 𝔞

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

\begin{matrix} T X & \overset{v}{\to} & T 𝔞 & 𝔞 - valued differential forms \\ ↓^{Id} & ↓ \\ T X & \overset{P (c)}{\to} & T b^{n} ℝ & curvature characteristic forms \end{matrix}

which dually corresponds to the dg-algebra composite

\begin{matrix} Ω^{•} (X) & \overset{ω}{\leftarrow} & W (𝔞) \\ ↑^{Id} & ↑ \\ Ω^{•} (X) & \overset{P (v)}{\leftarrow} & inv (𝔞) \end{matrix} .

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

Revised on May 7, 2010 07:38:05 by Urs Schreiber (87.212.203.135)

Schreiber ∞-Lie algebroid valued differential forms

Examples

Applications

∞-Lie groupoids and -algebroids

∞-Chern-Weil theory

symplectic ∞-geometry

Contents

Idea

Definition

Definition

Remark

Remark

Definition (differential forms associated to finite parallel transport)

Curvature forms

Schreiber
∞-Lie algebroid valued differential forms