Instantons in Non-Critical strings from the Two-Matrix Model

TL;DR

This paper derives non-perturbative corrections to the two-matrix model's free energy using algebraic curves, linking eigenvalue instantons to geometric cycles and confirming consistency with conformal field theory results.

Contribution

It provides a geometric interpretation of instanton effects in the two-matrix model and demonstrates their agreement with CFT predictions across various backgrounds.

Findings

Instanton corrections relate to vanishing cycles of the algebraic curve.

Agreement with CFT results extends to perturbed backgrounds.

Eigenvalue instantons' interpretation as ZZ-branes remains ambiguous under perturbations.

Abstract

We derive the non-perturbative corrections to the free energy of the two-matrix model in terms of its algebraic curve. The eigenvalue instantons are associated with the vanishing cycles of the curve. For the (p,q) critical points our results agree with the geometrical interpretation of the instanton effects recently discovered in the CFT approach. The form of the instanton corrections implies that the linear relation between the FZZT and ZZ disc amplitudes is a general property of the 2D string theory and holds for any classical background. We find that the agreement with the CFT results holds in presence of infinitesimal perturbations by order operators and observe that the ambiguity in the interpretation of the eigenvalue instantons as ZZ-branes (four different choices for the matter and Liouville boundary conditions lead to the same result) is not lifted by the perturbations. We find similar results to the c=1 string theory in presence of tachyon perturbations.