Integrability of Schwinger-Dyson Equations in 2D Quantum Gravity and c < 1 Non-critical String Field Theory

TL;DR

This paper studies the integrability of Schwinger-Dyson equations in 2D quantum gravity and non-critical string theories with central charge less than one, revealing a closed algebra structure and deriving a universal string field Hamiltonian.

Contribution

It demonstrates the algebraic structure of continuum Schwinger-Dyson equations and derives a universal string field Hamiltonian for c<1 string theories.

Findings

The continuum Schwinger-Dyson equations form a closed algebra including Virasoro but not W_infinity.

A new process for removing operators from loop boundaries is introduced.

A universal form of the string field Hamiltonian is derived for all c<1 theories.

Abstract

We investigate the integrability of the Schwinger-Dyson equations in $c = 1 - \frac{6}{m(m+1)}$ string field theory which were proposed by Ikehara et al as the continuum limit of the Schwinger-Dyson equations of the matrix chain model. We show the continuum Schwinger-Dyson equations generate a closed algebra. This algebra contains Virasoro algebra but does not coincide with $W_{\infty}$ algebra. We include in the Schwinger-Dyson equations a new process of removing from the loop boundaries the operator ${\cal H}(\sigma)$ which locally changes the spin configuration. We also derive the string field Hamiltonian from the continuum Schwinger-Dyson equations. Its form is universal for all $c = 1 - \frac{6}{m(m+1)}$ string theories.