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

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.