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May 8, 2008

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)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

Abstract. We discuss actions of Lie nn-groups and the corresponding action Lie nn-groupoids; discuss actions of Lie nn-algebras (L L_\infty-algebras) and the corresponding action Lie nn-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 nn-algebroid which linearizes the action nn-groupoid of a gauge nn-group acting on the space of fields.

This identifies the ghosts-of-ghosts of degree kk as the cotangents to the space of kk-morphisms of this action nn-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.

Posted at May 8, 2008 11:53 AM UTC

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