Integrable Structures in String Field Theory

TL;DR

This paper demonstrates that the Neumann coefficients in string field theory satisfy integrable hierarchy equations, suggesting a deep mathematical structure underlying string interactions and proposing a connection to matrix models.

Contribution

It provides a simple proof linking string field theory coefficients to integrable hierarchies and conjectures a broader underlying mathematical framework.

Findings

Neumann coefficients satisfy Hirota equations for dispersionless KP and Toda hierarchies

Surface state coefficients obey dispersionless KP hierarchy

Three-string vertex coefficients obey dispersionless Toda hierarchy

Abstract

We give a simple proof that the Neumann coefficients of surface states in Witten's SFT satisfy the Hirota equations for dispersionless KP hierarchy. In a similar way we show that the Neumann coefficients for the three string vertex in the same theory obey the Hirota equations of the dispersionless Toda Lattice hierarchy. We conjecture that the full (dispersive) Toda Lattice hierachy and, even more attractively a two--matrix model, may underlie open SFT.