A vertex algebra attached to the flag manifold and Lie algebra cohomology

TL;DR

This paper computes the vertex algebra structure on the cohomology of chiral differential operators on flag manifolds, revealing a tensor product structure involving the center and a subalgebra, with applications to bosonization and nonperturbative effects.

Contribution

It explicitly determines the vertex algebra structure on the cohomology of chiral differential operators on flag manifolds, linking it to Lie algebra cohomology and affine Lie algebra modules.

Findings

The vertex algebra is a tensor product of the center and a subalgebra.

The center is isomorphic to the cohomology of the maximal nilpotent Lie algebra.

The cohomology algebra vanishes nonperturbatively for the projective line, confirming Witten's suggestion.

Abstract

Each flag manifold carries a unique algebra of chiral differential operators. Continuing along the lines of arXiv:0903.1281 we compute the vertex algebra structure on the cohomology of this algebra. The answer is: the tensor product of the center and a subalgebra; the center is isomorphic, as a commutative associative algebra, to the cohomology of the corresponding maximal nilpotent Lie algebra; the subalgebra is the vacuum module over the corresponding affine Lie algebra of critical level and 0 central character. We next find the Friedan-Martinec-Shenker-Borisov bosonization of the cohomology algebra in case of the projective line and show that this algebra vanishes nonperturbatively, thus verifying a suggestion by Witten.