Integrability of Schwinger-Dyson equations in 2D quantum gravity and c < 1 non-critical string field theory

Abstract

We investigate the integrability of the Schwinger-Dyson equations in c = 1 − 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 ∞ algebra. We include in the Schwinger-Dyson equations a new process of removing from the loop boundaries the operator sc|H(σ) 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 − 6 m(m+1) string theories.