A note on perturbation series in supersymmetric gauge theories

TL;DR

The paper investigates perturbation series in several supersymmetric gauge theories using exact localization results. Across ABJM, ${\cal N}=2$ SU(2) with $N_f=4$, and ${\cal N}=2^*$ SU(N) theories, it demonstrates that perturbative expansions are often factorially divergent yet Borel summable, with divergences arising from complex-plane singularities rather than instantons. It provides explicit Borel structures for partition functions and Wilson/'t Hooft loops, analyzes large-$N$ planar limits, and shows how mass deformations induce running couplings and modify strong-/weak-coupling behavior. The work highlights the interpretive role of localization in clarifying perturbative versus nonperturbative content, and suggests a broader framework where observable-specific analytic structure governs summability and ambiguities, with potential implications for resurgent analyses in gauge theories.

Abstract

Exact results in supersymmetric Chern-Simons and N=2 Yang-Mills theories can be used to examine the quantum behavior of observables and the structure of the perturbative series. For the U(2) x U(2) ABJM model, we determine the asymptotic behavior of the perturbative series for the partition function and write it as a Borel transform. Similar results are obtained for N=2 SU(2) super Yang-Mills theory with four fundamental flavors and in N=2* super Yang-Mills theory, for the partition function as well as for the expectation values for Wilson loop and 't Hooft loop operators (in the 0 and 1 instanton sectors). In all examples, one has an alternate perturbation series where the coefficient of the nth term increases as n!, and the perturbation series are Borel summable. We also calculate the expectation value for a Wilson loop operator in the N=2* SU(N) theory at large N in different regimes of the 't Hooft gauge coupling and mass parameter. For large masses, the calculation reproduces the running gauge coupling for the pure N=2 SYM theory.