The Sugawara generators at arbitrary level

TL;DR

This paper constructs explicit representations of Sugawara generators at arbitrary levels using a vertex operator approach, revealing new transcendental and non-local features, with implications for hyperbolic Kac--Moody algebras.

Contribution

It provides a novel explicit construction of Sugawara generators at any level employing physical vertex operators, extending beyond the level-1 quadratic form.

Findings

Explicit representation of Sugawara generators at arbitrary level.

Introduction of transcendental functions and non-localities in the operators.

Application to higher-level root spaces of hyperbolic Kac--Moody algebras.

Abstract

We construct an explicit representation of the Sugawara generators for arbitrary level in terms of the homogeneous Heisenberg subalgebra, which generalizes the well-known expression at level 1. This is achieved by employing a physical vertex operator realization of the affine algebra at arbitrary level, in contrast to the Frenkel--Kac--Segal construction which uses unphysical oscillators and is restricted to level 1. At higher level, the new operators are transcendental functions of DDF ``oscillators'' unlike the quadratic expressions for the level-1 generators. An essential new feature of our construction is the appearance, beyond level 1, of new types of poles in the operator product expansions in addition to the ones at coincident points, which entail (controllable) non-localities in our formulas. We demonstrate the utility of the new formalism by explicitly working out some higher-level examples. Our results have important implications for the problem of constructing explicit representations for higher-level root spaces of hyperbolic Kac--Moody algebras, and $E_{10}$ in particular.