Lectures on non-perturbative effects in large N gauge theories, matrix models and strings

TL;DR

This work provides a coherent framework for non-perturbative effects in large $N$ gauge theories and matrix models by leveraging resurgence and trans-series. It develops a three-step approach—formal trans-series, classical asymptotics with Stokes phenomena, and Borel resummation—and demonstrates how these ideas apply from simple differential equations to quantum mechanics, Chern–Simons theory, and large $N$ matrix models. A central theme is the emergence of large $N$ instantons, whose actions $A(t)$ control the large-genus growth ${ m (2g)!}$ and can trigger phase transitions, with clear realizations in eigenvalue tunneling and spectral-curve formalisms. The treatment connects perturbative series to non-perturbative sectors, clarifies the role of D-branes in string duals, and shows how matrix-model techniques (one-cut and multi-cut solutions, spectral curves, and topological recursion) organize the computation of instanton effects and their impact on all orders in $g_s$ and $N$. This provides a practical toolkit for extracting non-perturbative physics in gauge theories and their string-theory duals.

Abstract

In these lectures I present a review of non-perturbative instanton effects in quantum theories, with a focus on large N gauge theories and matrix models. I first consider the structure of these effects in the case of ordinary differential equations, which provide a model for more complicated theories, and I introduce in a pedagogical way some technology from resurgent analysis, like trans-series and the resurgent version of the Stokes phenomenon. After reviewing instanton effects in quantum mechanics and quantum field theory, I address general aspects of large N instantons and then present a detailed review of non-perturbative effects in matrix models. Finally, I also consider two applications of these techniques in string theory

Figures (28)

• Figure 1: We illustrate the method of optimal truncation for the quartic integral (\ref{['quarticint']}) by plotting the difference (\ref{['diffexact']}) between the integral and the partial sum of order $N$ of its asymptotic expansion, as a function of $N$, for $g=0.02$ (left) and $g=0.05$ (right).
• Figure 2: Three paths which lead to solutions of the Airy equation.
• Figure 3: Saddle-point analysis of the integral (\ref{['lamint']}) for different values of $\kappa$. The red dot on the left is the critical point $u^{\rm L}$, while the black point on the right is the critical point $u^{\rm R}$. The continuous lines represent paths of steepest descent, while the dashed lines are paths of steepest ascent.
• Figure 4: Saddle-point analysis of the Airy function ${\rm Ai}(x)$. Full lines (in red) are Stokes lines, while dashed lines (in blue) are anti-Stokes lines. On the Stokes lines $\kappa =\pm 2\pi/3$, a second saddle appears in the integration contour. This saddle is subdominant when $2\pi/3\le |\kappa|<\pi$ and does not contribute to classical asymptotics. However, at $\kappa=\pi$, the saddle is not subdominant anymore and leads to an oscillatory asymptotics.
• Figure 5: The sectors of the $\kappa$-complex plane of opening $8\pi/5$ where the tritronquée solutions $u_{-2}$, $u_{-1}$, $u_0$, $u_1$, $u_2$ are represented by the formal series $u^{(0)}(\kappa)$. The dots in the remaining sector of opening $2\pi/5$ represent the infinite number of poles that the tritronquée solutions have there.

Theorems & Definitions (23)

• Example 2.1
• Example 2.2
• Example 2.3
• Remark 2.4
• Example 2.5
• Remark 2.6
• Remark 2.7
• Example 2.8
• Example 2.9
• Example 2.10