Multi-Instantons and Multi-Cuts

TL;DR

This paper explores multi-instanton configurations in multi-cut matrix models, providing explicit formulas, analyzing their properties, and applying the results to quantum gravity and Painleve I solutions.

Contribution

It introduces explicit multi-instanton amplitude formulas for multi-cut matrix models and applies them to quantum gravity and Painleve I, advancing understanding of non-perturbative effects.

Findings

Multi-instanton amplitudes are explicitly derived for two-cut and one-cut matrix models.

The instanton gas is shown to be ultra-dilute due to eigenvalue repulsion.

Results are validated through applications to 2D quantum gravity and Painleve I equation.

Abstract

We discuss various aspects of multi-instanton configurations in generic multi-cut matrix models. Explicit formulae are presented in the two-cut case and, in particular, we obtain general formulae for multi-instanton amplitudes in the one-cut matrix model case as a degeneration of the two-cut case. These formulae show that the instanton gas is ultra-dilute, due to the repulsion among the matrix model eigenvalues. We exemplify and test our general results in the cubic matrix model, where multi-instanton amplitudes can be also computed with orthogonal polynomials. As an application, we derive general expressions for multi-instanton contributions in two-dimensional quantum gravity, verifying them by computing the instanton corrections to the string equation. The resulting amplitudes can be interpreted as regularized partition functions for multiple ZZ-branes, which take into full account their back-reaction on the target geometry. Finally, we also derive structural properties of the trans-series solution to the Painleve I equation.