Multi-Instantons and Multi-Cuts

TL;DR

The paper develops a comprehensive framework for multi-instanton amplitudes in multi-cut matrix models, showing that instantons arise from shifting filling fractions and can be resummed via a theta-function representation. It derives explicit one- and two-cut results, including a natural regularization for the one-cut limit obtained by degenerating a two-cut background, and validates the formalism through tests in the cubic matrix model and in the 2d gravity (Painlevé I) regime. A key finding is that the instanton gas is ultra-dilute due to eigenvalue repulsion, with amplitudes scaling as $Z^{(\ell)} \sim g_s^{\ell^2/2}$, and the results provide regularized, back-reacted amplitudes for multiple ZZ-branes. The work connects nonperturbative matrix-model phenomena to topological strings and minimal string theories, offering a solid basis for extensions to more cuts and deeper links to continuum Liouville theory.

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.