On higher order Sugawara operators

TL;DR

This paper introduces a vertex algebra approach to construct higher Sugawara operators for affine Kac-Moody algebras at the critical level, providing explicit formulas and eigenvalues in Wakimoto modules.

Contribution

It offers a new vertex algebra-based method to explicitly construct higher Sugawara operators and compute their eigenvalues, advancing understanding of affine algebra representations.

Findings

Explicit formulas for higher Sugawara operators derived

Eigenvalues in Wakimoto modules calculated

Connects vertex algebra theory with integrable models

Abstract

The higher Sugawara operators acting on the Verma modules over the affine Kac-Moody algebra at the critical level are related to the higher Hamiltonians of the Gaudin model due to work of Feigin, Frenkel and Reshetikhin. An explicit construction of the higher Hamiltonians in the case of gl_n was given recently by Talalaev. We propose a new approach to these results from the viewpoint of the vertex algebra theory by proving directly the formulas for the higher order Sugawara operators. The eigenvalues of the operators in the Wakimoto modules of critical level are also calculated.