The Complete Spectrum of the $W_N$ String

TL;DR

This paper derives the complete physical spectrum of the $W_N$ string for any $N$, revealing a Virasoro-like structure with unique features and providing evidence for unitarity across all $W_N$ strings.

Contribution

It provides the first comprehensive analysis of the $W_N$ string spectrum for arbitrary $N$, including the effects of $W_N$ constraints and the structure of physical states.

Findings

Spectrum includes massless states of the Virasoro string.

$W_N$ constraints reduce the degrees of freedom.

Evidence suggests all $W_N$ strings are unitary.

Abstract

We obtain the complete physical spectrum of the $W_N$ string, for arbitrary $N$. The $W_N$ constraints freeze $N-2$ coordinates, while the remaining coordinates appear in the currents only {\it via} their energy-momentum tensor. The spectrum is then effectively described by a set of ordinary Virasoro-like string theories, but with a non-critical value for the central charge and a discrete set of non-standard values for the spin-2 intercepts. In particular, the physical spectrum of the $W_N$ string includes the usual massless states of the Virasoro string. By looking at the norms of low-lying states, we find strong indications that all the $W_N$ strings are unitary.