Resurgence of large order relations
Coenraad Marinissen, Alexander van Spaendonck, Marcel Vonk
TL;DR
The paper develops a comprehensive framework connecting perturbative and nonperturbative physics through resurgence, showing that large-order relations themselves form a resurgent transseries on a Borel cylinder. By introducing the Stirling transform, it connects the large-order transseries to the original transseries and demonstrates that their Borel transforms exhibit a structured network of singularities with preserved residue data, controlled by Stokes automorphisms. It then proves and tests the existence of exact large-order relations, meaning perturbative coefficients can be recovered by resumming the large-order transseries, with Stokes phenomena playing a crucial role when extending to complex values of the index. The quartic partition function serves as a concrete testbed, where explicit perturbative and instanton sectors, their Borel transforms, and the associated Stokes data validate the theory and reveal subtle contributions from infinity that may or may not affect exactness. Overall, the work advances a geometric and algebraic understanding of nonperturbative physics, enabling precise, contour-dependent resummations and potential extensions to broader classes of transseries and multi-Stokes phenomena.
Abstract
One of the main applications of resurgence in physics is the decoding of nonperturbative effects through large order relations. These relations connect perturbative asymptotic expansions of observables to expansions around other saddle points. Together, this data is unified in transseries that describe the nonperturbative structure. It is known that large order relations themselves also take the form of transseries. We study these large order transseries, uncover an interesting underlying geometry that we call the `Borel cylinder', and show that large order transseries in turn are resurgent -- that is: their nonperturbative sectors `know about each other' through Borel residues that are essentially equal to those of the original transseries. We show that with an appropriate resummation prescription, large order relations are often exact: they can be used to exactly compute perturbative coefficients -- not just their large order growth. Finally, we argue that Stokes phenomenon plays an important role for large order relations, for example if we want to extend the discrete index of the perturbative coefficients to arbitrary complex values.