A chiral Borel-Weil-Bott theorem

TL;DR

This paper computes the cohomology of modules over twisted chiral differential operators on flag manifolds, applying it to affine Lie algebra representations and elliptic genera, using derived functor techniques.

Contribution

It introduces a novel approach to compute cohomology and characters of modules via a derived functor interpretation of Drinfeld-Sokolov reduction.

Findings

Character formulas for $G$-integrable modules at critical level

Elliptic genus of the flag manifold computed

Cohomology of twisted chiral modules explicitly determined

Abstract

We compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold. This is applied to (1) finding the character of $G$-integrable irreducible highest weight modules over the affine Lie algebra at the critical level, and (2) computing a certain elliptic genus of the flag manifold. The main tool is a result that interprets the Drinfeld-Sokolov reduction as a derived functor.