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Lectures on non-perturbative effects in large \(N\) gauge theories, matrix models and strings. (English) Zbl 1338.81335

Summary: 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 consider two applications of these techniques in string theory.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T16 Nonperturbative methods of renormalization applied to problems in quantum field theory
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
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