Generalized Ablowitz-Ladik hierarchy in topological string theory

Abstract

This paper addresses the issue of integrable structures in topological string theory on generalized conifolds. Open string amplitudes of this theory can be expressed as the matrix elements of an operator on the Fock space of 2D charged free fermion fields. The generating function of these amplitudes with respect to the product of two independent Schur functions becomes a tau function of the 2D Toda hierarchy. The associated Lax operators turn out to have a particular factorized form. This factorized form of the Lax operators characterizes a generalization of the Ablowitz-Ladik hierarchy embedded in the 2D Toda hierarchy. The generalized Ablowitz-Ladik hierarchy is thus identified as a fundamental integrable structure of topological string theory on the generalized conifolds.