Resurgence and Holomorphy: From Weak to Strong Coupling

TL;DR

This paper explores how resurgence theory links weak and strong coupling regimes in quantum field theory models, showing that global analytic continuation can fix perturbation failures and predict strong coupling behavior from weak coupling data.

Contribution

It demonstrates that resurgent transseries unify weak and strong coupling analyses, providing a method to derive strong coupling results solely from weak coupling perturbation theory.

Findings

Resurgence fixes perturbation theory failures at all couplings.

Weak coupling monodromy matches strong coupling monodromy.

Holomorphy and Picard-Fuchs analysis support the resurgence framework.

Abstract

We analyze the resurgence properties of finite-dimensional exponential integrals which are prototypes for partition functions in quantum field theories. In these simple examples, we demonstrate that perturbation theory, even at arbitrarily weak coupling, fails as the argument of the coupling constant is varied. It is well-known that perturbation theory also fails at stronger coupling. We show that these two failures are actually intimately related. The formalism of resurgent transseries, which takes into account global analytic continuation properties, fixes both problems, and provides an arbitrarily accurate description of exact result for any value of coupling. This means that strong coupling results can be deduced by using merely weak coupling data. Finally, we give another perspective on resurgence theory by showing that the monodromy properties of the weak coupling results are in precise agreement with the monodromy properties of the strong-coupling expansions, obtained using analysis of the holomorphy structure of Picard-Fuchs equations.