Schreiber infinity-Chern-Simons functionals

This page is where Chris Rogers and myself are developing a text on $\infty$-Chern-Simons theory

Abstract

Familiar notions from Chern-Weil theory for Lie algebras, such as invariant polynomials, have natural generalizations to $L_\infty$-algebras and more generally to $\infty$-Lie algebroids. One aspect of the resulting $\infty$-Chern-Weil theory is a notion of $\infty$-Chern-Simons elements for every invariant polynomial on an $\infty$-Lie algebroid: the elements in the Weil algebra that witness the transgression to a cocycle. We discuss how these elements induce action functionals on spaces of $\infty$-Lie algebroid-valued connections that generalize the standard Chern-Simons theory action functional. Examples include higher Chern-Simons theories, supergravity theories, also BF-theory coupled to topological Yang-Mills theory as well as all action functionals of AKSZ-theory type induced from symplectic $\infty$-Lie algebroids, such as that of the Poissson $\sigma$-model for the topological string, and the Courant $\sigma$-model for the topological membrane.

Survey

for the moment see ∞-Chern-Weil theory introduction

$\infty$-Lie algebroids

for the moment see ∞-Lie algebroid

$\infty$-Chern-Simons elements

for the moment see Chern-Simons element

$\infty$-Chern-Simons action functionals

see for the moment connection on an ∞-bundle.

∞-Chern-Simons theory