$W_3$ strings, parafermions and the Ising model

TL;DR

This paper demonstrates that $W_3$ string scattering amplitudes include factors obeying Ising model correlation function equations and reveals the presence of an $N=2$ parafermion theory, linking string theory to statistical models and algebraic structures.

Contribution

It establishes a connection between $W_3$ string amplitudes, Ising model correlation functions, and $N=2$ parafermion theories, highlighting new algebraic structures within the string framework.

Findings

Scattering amplitudes contain Ising model correlation function factors.

$W_3$ string includes an $N=2$ parafermion theory.

Physical states form a representation of the associated $W$-algebra.

Abstract

We show that any covariant scattering amplitude of the $W_3$ string will contain, as part of its integrand, a factor that obeys the differential equations satisfied by an Ising model correlation function. This factor can thus be identified with such a correlation function, in agreement with a previous result of the authors. The $W_3$ string is also shown to contain an $N=2$ parafermion theory, and hence to contain in addition the non-linear infinite-dimensional $W$-algebra corresponding to this parafermion theory. The physical states form a representation of this algebra.