nLab
BRST complex

Idea

What is called the “BRST complex” in physics literture is the qDGCA which is the Chevalley-Eilenberg algebra of the $L_{\mathrm{\infty }}$-algebroid which is the differential version in Lie theory if the $\mathrm{\infty }$-groupoid

• whose space of objects is the space of configurations/histories of a given physical system;

• whose morphisms describe the gauge transformations between these configurations/histories;

• whose $k$-morphisms descrive the $k$-fold gauge-of-gauge transformations.

The generators of the BRST complex are called

• in degree 0: fields;

• in degree 1: ghosts;

• in degree 2: ghosts of ghosts;

• etc.

Relation to BV theory

In as far as the BRST complex can be regarded as being the algebra of functions on an $L_{\mathrm{\infty }}$-algebroid, the theory of integration over this $L_{\mathrm{\infty }}$-algebroid is given by the BV formalism.