A string field theory of (p, q) minimal superstrings is constructed with the freefermion realization of 2-component KP (2cKP) hierarchy, starting from 2-cut ansatz of two-matrix models. Differential operators of 2cKP hierarchy are identified with operators in super Liouville theory, and we obtain algebraic curves for the disk amplitudes of η = −1 FZZT-branes and the partition functions of neutral/charged η = −1 ZZ branes, which correctly reproduce those of type 0B (p, q) minimal superstrings in conformal backgrounds. In the course of study, some subtle points are clarified, including a difference of (p, q) even/odd models and quantization of flux, and we show that the Virasoro constraints naturally incorporate quantized fluxes without ambiguity. We also argue within this string field framework that type 0A minimal superstrings can be obtained by orbifolding the type 0B strings with a Z 2 symmetry existing when special backgrounds are taken. *

The Stokes multipliers in the matrix models are invariants in the string-theory moduli space and related to the D-instanton chemical potentials. They not only represent non-perturbative information but also play an important role in connecting various perturbative string theories in the moduli space. They are a key concept to the non-perturbative completion of string theory and also expected to imply some remnant of strong coupling dynamics in M theory. In this paper, we investigate the non-perturbative completion problem consisting of two constraints on the Stokes multipliers. As the first constraint, Stokes phenomena which realize the multi-cut geometry are studied in the Z k symmetric critical points of the multi-cut two-matrix models. Sequence of solutions to the constraints are obtained in general k-cut critical points. A discrete set of solutions and a continuum set of solutions are explicitly shown, and they can be classified by several constrained configurations of the Young diagram. As the second constraint, we discuss non-perturbative stability of backgrounds in terms of the Riemann-Hilbert problem. In particular, our procedure in the 2-cut (1, 2) case (pure-supergravity case) completely fixes the D-instanton chemical potentials and results in the Hastings-McLeod solution to the Painlevé II equation. It is also stressed that the Riemann-Hilbert approach realizes an off-shell background independent formulation of non-critical string theory.

Loop amplitudes in (p, q) minimal string theory are studied in terms of the continuum string field theory based on the free fermion realization of the KP hierarchy. We derive the Schwinger-Dyson equations for FZZT disk amplitudes directly from the W 1+∞ constraints in the string field formulation and give explicitly the algebraic curves of disk amplitudes for general backgrounds. We further give annulus amplitudes of FZZT-FZZT, FZZT-ZZ and ZZ-ZZ branes, generalizing our previous D-instanton calculus from the minimal unitary series (p, p + 1) to general (p, q) series. We also give a detailed explanation on the equivalence between the Douglas equation and the string field theory based on the KP hierarchy under the W 1+∞ constraints.

Multi-cut critical points and their macroscopic loop amplitudes are studied in the multi-cut two-matrix models, based on an extension of the prescription developed by Daul, Kazakov and Kostov. After identifying possible critical points and potentials in the multi-cut matrix models, we calculate the macroscopic loop amplitudes in the Z_k symmetric background. With a natural large N ansatz for the matrix Lax operators, a sequence of new solutions for the amplitudes in the Z_k symmetric k-cut two-matrix models are obtained, which are realized by the Jacobi polynomials.Comment: 46 pages, 3 figures; v2: 51 pages, 7 figures, notations changed, explanations in Section 2.4 extended, figures for topology of the curves added, Appendix E added, final version to appear in Nucl. Phys.