An Affine String Vertex Operator Construction at Arbitrary Level

TL;DR

This paper introduces a new affine vertex operator construction at any level using a compactified chiral bosonic string, providing explicit physical representations of affine modules and insights into hyperbolic Kac-Moody algebras.

Contribution

It presents a novel affine vertex operator construction based on a compactified string, enabling explicit physical representations of affine modules at arbitrary levels.

Findings

Explicit representations of affine highest weight modules using DDF string states

Unified treatment of multiple affine representations within a single framework

New interpretation of affine Weyl group as Lorentz boosts

Abstract

An affine vertex operator construction at arbitrary level is presented which is based on a completely compactified chiral bosonic string whose momentum lattice is taken to be the (Minkowskian) affine weight lattice. This construction is manifestly physical in the sense of string theory, i.e., the vertex operators are functions of DDF ``oscillators'' and the Lorentz generators, both of which commute with the Virasoro constraints. We therefore obtain explicit representations of affine highest weight modules in terms of physical (DDF) string states. This opens new perspectives on the representation theory of affine Kac-Moody algebras, especially in view of the simultaneous treatment of infinitely many affine highest weight representations of arbitrary level within a single state space as required for the study of hyperbolic Kac-Moody algebras. A novel interpretation of the affine Weyl group as the ``dimensional null reduction'' of the corresponding hyperbolic Weyl group is given, which follows upon re-expression of the affine Weyl translations as Lorentz boosts.