The Chern-Weil-Lecomte characteristic map for $L_{\infty}$-algebras

Author:

Juan Sebastian Herrera-Carmona, Cristian Ortiz

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG)

journal:

date:

2023-06-06 16:00:00

Abstract

In this paper we extend the Chern-Weil-Lecomte characteristic map to the setting of $L_{\infty}$-algebras. In this general framework, characteristic classes of $L_{\infty}$-algebra extensions are defined by means of the Chern-Weil-Lecomte map which takes values in the cohomology of an $L_{\infty}$-algebra with coefficients in a representation up to homotopy. This general set up allows us to recover several known cohomology classes in a unified manner, including: the characteristic class of a Lie 2-algebra, the \v{S}evera class of an exact Courant algebroid and the curvature 3-form of a gerbe with connective structure. We conclude by introducing a Chern-Weil map for principal 2-bundles over Lie groupoids.

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