# School on Resurgence

The school consist of four courses of five lectures each all covering various aspects of the theory of resurgence together with applications of this theory in Quantum Field Theory and String Theory.

#### Resurgent Functions: From Linear to Non--Linear ODEs, with Resurgence Monomials

Frederic Fauvet (IRMA)

We shall first introduce the main features of resurgence theory by working on an explicit and elementary example of a linear differential equation with polynomial coefficients (the one satisfied by the "prolate spheroidal wave functions").

We shall then proceed to non--linear ODEs, present the families of alien derivations in some generality and expose the basics on resurgence monomials. These monomials are resurgent functions which are dual to alien derivations; they constitute building blocks to perform a resurgent analysis of a given problem and have a vast scope of applications in mathematics and for questions in theoretical physics which display the phenomenon of resurgence.

#### Introduction to Resurgent Asymptotics and Non-Perturbative Physics

Gerald Dunne (University of Connecticut)

These lectures will give an elementary introduction to resurgent asymptotics as applied to problems in theoretical physics. One of the main motivations is to gain deeper understanding of the mathematical structure of the quantum path integral. I will review methods to extract physical information from asymptotic series, and illustrate these methods with explicit examples from nonlinear differential equations, matrix models, quantum mechanics and quantum field theory.

#### Initiation to Resurgence

David Sauzin (CNRS)

This will be on the foundations of Resurgence Theory, illustrated by many concrete examples. The topics (not necessarily in this order, more or less developed according to the audience's taste) will be:

• Review of the Borel-Laplace summation in complex directions.
• Ecalle's definition of resurgent series in terms of endless continuability in the Borel plane.
• First examples: Euler's series, the Stirling series, asymptotics of the Airy function, other special functions.
• Ecalle's formalism of singularities: minors and majors.
• Ecalle's alien derivations: alien calculus!
• Stability under convolution in the Borel plane, nonlinear analysis with resurgent series.
• Example of the exponential of the Stirling series.
• The topics of the last part will depend on the audience...

#### Resurgence and Transseries in Quantum Field Theory and String Theory

Ricardo Schiappa (IST Lisboa)

In general interacting theories — quantum mechanical, field, gauge, or string theories — perturbation theory is divergent: perturbative expansions have zero radius of convergence and seemingly cannot be summed. Nonperturbatively well-defined results can still be constructed out of perturbation theory by the uses of resurgence and transseries.

Asymptotic series require the use of resurgence and transseries in order for their associated observables to become nonperturbatively well-defined. Resurgent transseries encode the complete large-order asymptotic behaviour of the coefficients from a perturbative expansion, generically in terms of (multi) instanton sectors and for each problem in terms of its Stokes coefficients. By means of two very explicit examples, we plan to introduce the aforementioned resurgent, large-order asymptotics of general perturbative expansions, including discussions of transseries, Stokes phenomena, generalized steepest-descent methods, Borel transforms, nonlinear resonance, and alien calculus.

The program of the lectures will include:

• Introduction
• Resurgent Analysis of a Quartic-Potential Integral
• Lefschetz Thimbles for Linear Problems
• Borel Transforms for Nonlinear Problems
• Physical Resurgence: From Lattices to Virasoro Algebras
• Resurgent Analysis of an Elliptic-Potential Integral