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Conhecimento das teorias não perturbativas e seus efeitos.

1.  Asymptotic series and exponentially small corrections. Formal power series and trans-series in ODEs. Classical asymptotics and Stokes phenomenon. Beyond classical asymptotics: Borel transform. Borel transform and Borel resummation.   2. Trans series and resurgence in Quantum Mechanics. Double well potential:  WKB method and quantization conditions. 3. Non-perturbative effects in Quantum Field Theory. Renormalons in integrable field theories.  Chern-Simons theory. 1/N expansion: large-N instants in Matrix Quantum Mechanics.  4. Non-perturbative effects in large N matrix models: Large N instantons, large-order behaviour and spectral curve. One-cut and multi-cut solutions. Classical asymptotics and Stokes phenomena.

[1] Marcos Mariño, Lectures on non-perturbative effects in large N gauge theories, matrix models and strings     https://arxiv.org/pdf/1206.6272.pdf [2] A. Voros, “The return of the quartic oscillator. The complex WKB method,” Ann. Inst. H. Poincar ́e A 39, 211 (1983).  [3] E. Witten, “Instantons, The Quark Model, And The 1/N Expansion,” Nucl. Phys. B 149, 285 (1979).  [4] J. Zinn–Justin, Quantum field theory and critical phenomena, Oxford University Press, Oxford, 2002.  [5] [1] S. Coleman. The uses of instantons. volume the whys of subnuclear physics, Editor A. Zichichi, 1979.