W-Strings from Affine Lie Algebras

TL;DR

This paper introduces a new class of $ ext{W}$-strings derived from affine Lie algebra representations using quantum Drinfel'd-Sokolov reduction, leading to explicit constructions of $ ext{W}_3$-string spectra.

Contribution

It presents a novel method to construct $ ext{W}$-strings from affine Lie algebras, including explicit BRST operators and spectrum calculations for $ ext{W}_3$-strings.

Findings

Constructed a BRST operator for $ ext{W}_3$-string using $ ext{sl}_3$ currents.

Reproduced known $ ext{W}_3$-string spectra with free-field realization.

Generalized the construction to any $ ext{W}$-algebra from affine algebras.

Abstract

The most disappointing aspect of $\W$-strings is probably the fact that at least for the known models one does not recover a physical spectrum that differs much from that of ordinary string theory. It is hoped that this is not an intrinsic shortcoming of $\W$-string theory, but rather that it is a consequence of the $\W$-algebra realizations that have been chosen. In this note we point out a whole new class of possible $\W$-strings built from representations of affine Lie algebras via (quantum) Drinfel'd-Sokolov reduction. Explicitly, we construct a BRST operator in terms of $\widehat{sl}_3$ currents which computes the physical spectrum of a $\W_3$-string. As a special case of this construction, if we take a free-field realization of $\widehat{sl}_3$, we recover the 2-scalar $\W_3$-string. These results generalize to any $\W$-algebra which can be obtained via quantum Drinfel'd-Sokolov reduction from an affine algebra.