For certain problems, taking the Borel sum of an intuitively chosen divergent series solution will often produce a holomorphic solution. When does this happen, and what is special about the holomorphic solutions obtained in this way? We will present work in progress on these questions, focusing on two kinds of problems: linear ODEs and integrals over Lefschetz thimbles.
Day II
The second day’s lectures are dedicated to a family of examples: the Airy-Lucas functions introduced by Charbonnier et al. (arXiv:2203.16523). These functions satisfy linear ODEs that generalize the Airy equation. They can also, like the Airy function, be expressed as thimble integrals. We will explain, from both perspectives, why these solutions can be obtained by Borel summation.
We will conclude by describing general classes of linear ODEs and 1d thimble integrals that can be analyzed in the same way.
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Aaron Fenyes & Veronica Fantini
