Resurgence

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Description

Thursday 09.00-11.00

Asymptotic, divergent power series are ubiquitous in mathematical physics and pure mathematics. In quantum mechanics, quantum field theory, and string theory, the quantization process typically introduces a small parameter, allowing relevant quantities to be expressed as power series in that parameter. The coefficients of these series are often determined recursively, sometimes through Feynman diagrams, and tend to exhibit factorial growth, leading to the divergence of the series.

Resurgence is the systematic mathematical study of such divergent series, particularly those arising in quantization. The goal is to use Borel-Laplace summation to reconstruct an analytic function in some domain, whose asymptotic expansion matches the original formal power series. In general, there are multiple analytic functions with overlapping domains of validity, differing on their overlaps by Stokes factors.

This course will explore both the formal mathematical tools developed in the resurgence program and their applications in geometry and physics. A key geometric example arises in enumerative geometry: the Gromov-Witten invariants of Calabi-Yau manifolds are encoded in asymptotic series, while the Stokes factors capture another set of invariants the Donaldson-Thomas invariants. In physics, asymptotic series typically emerge from perturbative expansions, while Stokes factors encode non-perturbative, exponentially suppressed effects. These effects play a crucial role not only in exact computations but also in the conceptual understanding of physical theories across their full parameter space and
dualities.