Picard-Lefschetz theory and alien calculus: a case study

TL;DR

This paper compares Picard--Lefschetz theory and resurgence in three exponential integral models, analyzing thimbles, Stokes phenomena, and alien calculus to establish a detailed correspondence.

Contribution

It provides explicit examples connecting thimble wall-crossing with alien calculus in finite-dimensional integrals.

Findings

Computed Lefschetz thimbles and trajectories for the models.

Analyzed Borel singularities and alien operators to recover Stokes coefficients.

Established a dictionary linking thimble wall-crossing and alien calculus.

Abstract

We compare Picard--Lefschetz theory and resurgence in three basic one-dimensional exponential integrals: the Airy model, the Bessel model, and the Gamma model. On the Picard--Lefschetz side, we describe the Lefschetz thimbles and compute the connecting trajectories between critical points appearing at Stokes phases. On the resurgent side, we analyze the Borel singularities of the saddle expansions and use alien operators to recover the same Stokes coefficients. These examples serve as explicit finite-dimensional test cases for the dictionary between thimble wall-crossing and alien calculus.