On the spectrum, no ghost theorem and modular invariance of $W_3$ strings

TL;DR

This paper constructs a spectrum generating algebra for the $W_3$ string, identifies physical states with positive norm, and demonstrates modular invariance through inclusion of Ising model characters, revealing new states in the cohomology.

Contribution

It introduces a spectrum generating algebra for $W_3$ strings and establishes modular invariance by incorporating additional states linked to the Ising model.

Findings

Physical states have positive norm

Partition function involves Ising model characters 0 and 1/16

Modular invariance achieved with states of Ising weight 1/2

Abstract

A spectrum generating algebra is constructed and used to find all the physical states of the $W_3$ string with standard ghost number. These states are shown to have positive norm and their partition function is found to involve the Ising model characters corresponding to the weights 0 and 1/16. The theory is found to be modular invariant if , in addition, one includes states that correspond to the Ising character of weight 1/2. It is shown that these additional states are indeed contained in the cohomology of $Q$.