Supersymmetric solutions and Borel singularities for N=2 supersymmetric Chern-Simons theories

TL;DR

This paper explores complexified supersymmetric solutions in 3d N=2 SUSY Chern-Simons theories, linking them to Borel singularities and large order behavior of perturbative series, providing a new understanding of non-perturbative effects.

Contribution

It introduces the concept of CSS in 3d N=2 SUSY theories and demonstrates their role in explaining Borel singularities in perturbative expansions.

Findings

CSS correspond to poles and zeroes of Borel transforms.

CSS actions match Borel singularity locations.

CSS explain all Borel singularities in studied theories.

Abstract

In supersymmetric (SUSY) field theory, there exist configurations which formally satisfy SUSY conditions but are not on original path integral contour. We refer to such configurations as complexified supersymmetric solutions (CSS). In this paper we discuss that CSS provide important information on large order behavior of weak coupling perturbative series in SUSY field theories. We conjecture that CSS with a bosonic (fermionic) free parameter give poles (zeroes) of Borel transformation of perturbative series whose locations are uniquely determined by actions of the solutions. We demonstrate this for various SUSY observables in 3d $\mathcal{N}=2$ SUSY Chern-Simons matter theories on sphere. First we construct infinite number of CSS in general 3d $\mathcal{N}=2$ SUSY theory with Lagrangian where adjoint scalar in vector multiplet takes a complex value and matter fields are nontrivial. Then we compare their actions with Borel transformations of perturbative expansions by inverse Chern-Simons levels for the observables and see agreement with our conjecture. It turns out that the CSS explain all the Borel singularities for this case.