Borel summability of perturbative series in 4d N=2 and 5d N=1 theories

TL;DR

This work addresses whether weak-coupling perturbative expansions in 4d $\mathcal{N}=2$ and 5d $\mathcal{N}=1$ gauge theories are Borel summable, motivated by IR renormalon considerations. The authors leverage supersymmetric localization to recast observables as Laplace transforms with kernels derived from $Z_{\rm VdM}$, one-loop, and Nekrasov factors, showing the small-$t$ expansions converge and define a Borel transform without positive-axis singularities. They prove Borel summability for the zero-instanton sector and extend to arbitrary instanton sectors when Nekrasov functions have no real-axis poles, applying to $S^4$ partition functions, Wilson loops, Bremsstrahlung, extremal correlators, and squashed geometries in 4d, and to squashed $S^5$ in 5d. A key finding is a sector-isolation property: Borel resummation in each instanton sector yields the corresponding truncated full result, enabling exact sector-by-sector reconstructions and offering new insights into resurgence in highly symmetric theories.

Abstract

We study weak coupling perturbative series in 4d N=2 and 5d N=1 supersymmetric gauge theories with Lagrangians. We prove that the perturbative series of these theories in zero instanton sector are Borel summable for various observables. Our result for 4d $\mathcal{N}=2$ case supports an expectation from a recent proposal on a semiclassical realization of infrared renormalons in QCD-like theories, where the semiclassical solution does not exist in N=2 theories and the perturbative series are unambiguous, namely Borel summable. We also prove that the perturbative series in arbitrary number of instanton sector are Borel summable for a wide class of theories. It turns out that exact results can be obtained by summing over the Borel resummations in each number of instanton sector.