Lectures on non-perturbative effects in large N gauge theories, matrix models and strings (1206.6272v2)

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

• The paper presents a comprehensive analysis of non-perturbative effects in large N gauge theories and matrix models using trans-series and instanton methods.
• The paper details how large N expansions reveal key phenomena like confinement, duality, and eigenvalue tunneling beyond conventional perturbative series.
• The paper demonstrates the vital role of matrix models in linking quantum mechanics to string theory via exact solutions and resurgent analysis.

Overview of Non-Perturbative Effects in Large N Gauge Theories, Matrix Models, and Strings

The paper authored by Marcos Mariño provides an extensive review of the non-perturbative effects observed in quantum theories, with a particular concentration on large $N$ gauge theories and matrix models. The discourse is structured to illuminate how non-perturbative effects, which manifest beyond the conventional perturbative series, can be understood and computed in these specific domains, ultimately seeking applications in string theory.

The document canvasses several pivotal topics, delineating the contributions of large $N$ expansions and matrix models to comprehend these effects. The author's exploration begins by establishing a foundation with ordinary differential equations (ODEs), utilizing techniques from resurgent analysis such as trans-series and the Stokes phenomenon to model these phenomena comprehensively. The discussion extends into the field of quantum mechanics, elucidating concepts through double-scaling limits and matrix models, before transitioning to their broader applications in field theories.

Key Components

1. Instanton Effects in Quantum Mechanics: The paper delineates the role of instantons, topologically non-trivial solutions to classical field equations, within quantum mechanics. These are illustrated as necessary to complete the non-convergent perturbative series, highlighting their contribution to understanding quantum phenomena beyond perturbative approximations.
2. Large $N$ Gauge Theories: Large $N$ theories, particularly through their $1/N$ expansion, serve as a bridge to understand phenomena like confinement and duality in quantum chromodynamics (QCD). The paper provides a detailed analysis of how these theories encapsulate non-perturbative phenomena, such as topological transitions and string-theoretic correspondences, in large $N$ limits.
3. Matrix Models: Mariño emphasizes how matrix models form an essential toolkit in analyzing non-perturbative physics. By employing the 't Hooft coupling and exploiting matrix model techniques, the author presents a structured method to grasp various phases and critical phenomena encountered in large $N$ dynamics. The paper also demonstrates how non-perturbative effects manifest as eigenvalue tunneling between different potential wells, leading to distinct large $N$ phases.
4. Trans-series and Exact Solutions: The paper leverages trans-series expansions, which integrate perturbative and non-perturbative series, to describe solutions of quantum mechanical systems accurately. These serve as formal reconciliations of divergent series and allow a more profound understanding of the underlying physics in large $N$ theories.
5. Applications in String Theory: The culmination of the discussion in the application to string theory underscores the broader implications of non-perturbative effects. Mariño's exploration into how large $N$ dualities and matrix models can shed light on string-theoretic non-perturbative effects highlights the potential of these theoretical tools to not only solve longstanding puzzles in string theory but also to draw parallels across various branches of modern physics.

Implications and Future Directions

The implications of these findings are manifold, serving as a conduit to bridge quantum mechanics, field theory, and string theory through comprehensive mathematical frameworks like matrix models and large $N$ techniques. The methodologies proposed enrich theoretical physics, offering pathways to evaluate quantum effects that deviate from classical predictions.

The advancements forecast potential exploratory pathways in AI, where complex systems can be approximated or modeled using similar non-perturbative methodologies. Furthermore, these insights could be pivotal in developing new algorithms that simulate quantum systems or predict phase transitions in complex networks.

In summation, Marcos Mariño's paper is a profound elucidation of non-perturbative effects in the landscape of large $N$ gauge theories, offering substantial insights into how these phenomena govern underlying quantum processes, with implications that resonate across theoretical physics and computational domains.