### On Lie oo-Theory

#### Posted by Urs Schreiber

Taking a day off at HIM (will report tomorrow on what Liang Kong has been teaching us about vertex operator algebras from Segal’s CFT axioms, using theorems by Huang), today I am giving a talk in Hamburg in our series on BRST-BV formalism (as you will have guessed), the goal being to illuminate the geometric $\infty$-categorical meaning of the BRST complex regarded as a $L(ie) \infty$-algebroid:

The things I’ll say and draw to the board are those at the beginning of section 1 of

On action Lie $\infty$-groupoids and action Lie $\infty$-algebroids

(pdf)

Abstract.We discuss actions of Lie $n$-groups and the corresponding action Lie $n$-groupoids; discuss actions of Lie $n$-algebras ($L_\infty$-algebras) and the corresponding action Lie $n$-algebroids; and discuss the relation between the two by integration and differentiation.As an example of interest, we discuss the BRST complex that appears in quantum field theory. We describe it as the Chevalley-Eilenberg algebra of the Lie $n$-algebroid which linearizes the action $n$-groupoid of a gauge $n$-group acting on the space of fields.

This identifies the ghosts-of-ghosts of degree $k$ as the cotangents to the space of $k$-morphisms of this action $n$-groupoid.

Several separate aspects of what we say here are essentially “well known” to those who know it well. But a coherent description as attempted here is certainly missing in the literature and deserves to be better known.