Notes on the algebraic curves in (p,q) minimal string theory

TL;DR

This paper explores algebraic curves and amplitudes in (p,q) minimal string theory using continuum string field theory, deriving equations and amplitudes for various brane configurations and clarifying their algebraic structure.

Contribution

It derives Schwinger-Dyson equations and algebraic curves for (p,q) minimal string theory directly from W_{1+ abla} constraints, extending previous results to general backgrounds.

Findings

Explicit algebraic curves for disk amplitudes in (p,q) models

Annulus amplitudes for various brane pairs

Equivalence between Douglas equation and KP hierarchy-based string field theory

Abstract

Loop amplitudes in (p,q) minimal string theory are studied in terms of the continuum string field theory based on the free fermion realization of the KP hierarchy. We derive the Schwinger-Dyson equations for FZZT disk amplitudes directly from the W_{1+\infty} constraints in the string field formulation and give explicitly the algebraic curves of disk amplitudes for general backgrounds. We further give annulus amplitudes of FZZT-FZZT, FZZT-ZZ and ZZ-ZZ branes, generalizing our previous D-instanton calculus from the minimal unitary series (p,p+1) to general (p,q) series. We also give a detailed explanation on the equivalence between the Douglas equation and the string field theory based on the KP hierarchy under the W_{1+\infty} constraints.