Resurgence and Lefschetz thimble in 3d N=2 supersymmetric Chern-Simons matter theories

TL;DR

This paper explores the resurgent structure of exact supersymmetric observables in 3d N=2 Chern-Simons matter theories, revealing how perturbative and non-perturbative effects cancel ambiguities and relate to Lefschetz thimbles and complex SUSY solutions.

Contribution

It demonstrates the resurgent nature of SUSY observables, decomposes localization formulas into Lefschetz thimbles, and links non-perturbative effects to complexified SUSY solutions.

Findings

Exact results form a transseries with perturbative and non-perturbative parts.

Stokes phenomena lead to Borel ambiguities canceled by non-perturbative sectors.

Lefschetz thimble decomposition relates to resurgent transseries and complex SUSY solutions.

Abstract

We study a certain class of supersymmetric (SUSY) observables in 3d $\mathcal{N}=2$ SUSY Chern-Simons (CS) matter theories and investigate how their exact results are related to the perturbative series with respect to coupling constants given by inverse CS levels. We show that the observables have nontrivial resurgent structures by expressing the exact results as a full transseries consisting of perturbative and non-perturbative parts. As real mass parameters are varied, we encounter Stokes phenomena at an infinite number of points, where the perturbative series becomes non-Borel-summable due to singularities on the positive real axis of the Borel plane. We also investigate the Stokes phenomena when the phase of the coupling constant is varied. For these cases, we find that the Borel ambiguities in the perturbative sector are canceled by those in nonperturbative sectors and end up with an unambiguous result which agrees with the exact result even on the Stokes lines. We also decompose the Coulomb branch localization formula, which is an integral representation for the exact results, into Lefschetz thimble contributions and study how they are related to the resurgent transseries. We interpret the non-perturbative effects appearing in the transseries as contributions of complexified SUSY solutions which formally satisfy the SUSY conditions but are not on the original path integral contour.