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

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 consider two applications of these techniques in string theory

Quantum geometry of resurgent perturbative/nonperturbative relations

“…Further motivation along these lines comes from the ODE/Integrable Model correspondence [66][67][68][69][70], which provides explicit mappings between monodromy operators in certain Schrödinger systems and Yang-Baxter operators in integrable models. We are also strongly motivated by the geometric relation between supersymmetric gauge theories, matrix models and topological strings [71][72][73][74][75], for which a rich web of resurgent structures has been comprehensively established both analytically and numerically [76][77][78][79][80][81][82][83][84][85][86]. There are surprisingly close parallels between the resurgent structures found in such theories for the partition function (or free energy) as a function of (at least) two parameters, g s and N , and the resurgent structure of the Schrödinger energy eigenvalue u( , N ), as a function of and the perturbative level number N .…”

Deconfinement in $ \mathcal{N}=1 $ super Yang-Mills theory on $ {{\mathbb{R}}^3}\times {{\mathbb{S}}^1} $ via dual-Coulomb gas and “affine” XY-model

“…The functions G(a, b) and Θ(a, A) are the discrete versions of the corresponding continuum expressions. 17 We use the Metropolis algorithm to simulate the grand canonical dual-Coulomb gas (4.1). The details of the algorithm for the Coulomb gas are given in appendix (C).…”

“…A recent example is the argument that they are relevant to studies of the thermal deconfinement phase transition in pure YM theory, via the idea of "continuity". 2 Further, magnetic bions have been shown to be crucial for understanding the 1 The ongoing studies of "resurgence"-a generalization of Borel resummation-in field theories in various dimensions are shedding further light on the role of these molecules and other path integral saddle points, showing a fascinating interplay between perturbative and nonperturbative contributions at weak coupling (this is a currently active area of research, see, e.g., the recent work [17][18][19][20][21] and references therein).…”

Thresholds of large N factorization in CFT4: exploring bulk spacetime in AdS5

“…Once the formal transseries is constructed, there are approaches that can recover the exact full form of the path integral from the series; this is called the theory of resurgence. The lecture notes [31] give a review of transseries and resurgence in QFT and string theory. In our analysis of the threshold, the series we have found has not come from a path integral, but still has exponentially-suppressed corrections.…”