Sugawara Construction and Casimir Operators for Krichever-Novikov Algebras

TL;DR

This paper extends the Sugawara construction to Krichever-Novikov algebras, linking highest weight representations of affine type to vector field algebra representations and constructing Casimir operators, generalizing classical Virasoro results.

Contribution

It introduces a generalized Sugawara construction for higher genus affine Kac-Moody algebras and constructs Casimir operators for these algebras.

Findings

Representation relations between weights are established.

Casimir operators are explicitly constructed.

The construction is extended to multi-point cases.

Abstract

We show how to obtain from highest weight representations of Krichever-Novikov algebras of affine type (also called higher genus affine Kac-Moody algebras) representations of centrally extended Krichever-Novikov vector field algebras via the Sugawara construction. This generalizes classical results where one obtains representations of the Virasoro algebra. Relations between the weights of the corresponding representations are given and Casimir operators are constructed. In an appendix the Sugawara construction for the multi-point situation is done.