Lie algebra cohomology of the positive part of twisted affine Lie algebras

TL;DR

This paper extends Teleman's Lie algebra cohomology vanishing results to twisted affine Lie algebras, crucial for the Verlinde formula in conformal field theory, by establishing a Nakano Identity in this setting.

Contribution

It generalizes cohomology vanishing results to twisted affine Lie algebras with automorphisms, enabling broader applications in conformal blocks and representation theory.

Findings

Proves a cohomology vanishing theorem for twisted affine Lie algebras.

Establishes a Nakano Identity in the twisted setting.

Extends Teleman's results to automorphism-twisted cases.

Abstract

The explicit Verlinde formula for the dimension of conformal blocks, attached to a marked projective curve $\Sigma$, a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ and integrable highest weight modules of a fixed central charge of the corresponding affine Lie algebra $\hat{L}(\mathfrak{g})$ attached to the marked points, requires (among several other important ingredients) a Lie algebra cohomology vanishing result due to C. Teleman for the positive part $\hat{L}^+(\mathfrak{g})$ with coefficients in the tensor product of an integrable highest weight module with copies of finite dimensional evaluation modules. The aim of this paper is to extend this result of Teleman to a twisted setting where $\mathfrak{g}$ is endowed with a special automorphism $\sigma$ and the curve $\Sigma$ is endowed with the action of $\sigma$. In this general setting, the affine Lie algebra gets replaced by twisted affine Lie algebras. The crucial ingredient (as in Teleman) is to prove a certain Nakano Identity.