Stokes Phenomena and Non-perturbative Completion in the Multi-cut Two-matrix Models

TL;DR

This paper explores the role of Stokes multipliers in non-perturbative string theory, analyzing multi-cut two-matrix models and their stability, leading to solutions connected to Painlevé equations and off-shell formulations.

Contribution

It introduces a new analysis of Stokes phenomena in multi-cut matrix models, classifies solutions via Young diagrams, and links non-perturbative stability to Riemann-Hilbert problems.

Findings

Explicit solutions for multi-cut critical points.

D-instanton chemical potentials fixed in 2-cut case.

Connection to Painlevé II and off-shell string formulations.

Abstract

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 Painlev\'e II equation. It is also stressed that the Riemann-Hilbert approach realizes an off-shell background independent formulation of non-critical string theory.