The Interacting $W_3$ String

TL;DR

This paper develops a method to compute gauge-invariant scattering amplitudes in the $W_3$ string, revealing the involvement of ghost and matter excitations, and explores duality and connections to the Ising model.

Contribution

It introduces a procedure for calculating gauge-invariant amplitudes in the $W_3$ string and analyzes their properties, including crossing symmetry and relations to the Ising model.

Findings

Non-vanishing amplitudes involve ghost and matter excitations.

Four-point functions exhibit duality similar to Virasoro string amplitudes.

Scattering amplitudes relate to Ising model fusion rules.

Abstract

We present a procedure for computing gauge-invariant scattering amplitudes in the $W_3$ string, and use it to calculate three-point and four-point functions. We show that non-vanishing scattering amplitudes necessarily involve external physical states with excitations of ghosts as well as matter fields. The crossing properties of the four-point functions are studied, and it is shown that the duality of the Virasoro string amplitudes generalises in the $W_3$ string, with different sets of intermediate states being exchanged in different channels. We also exhibit a relation between the scattering amplitudes of the $W_3$ string and the fusion rules of the Ising model.