Supersymmetric Solutions and Borel Singularities for $\mathcal{N}=2$ Supersymmetric Chern-Simons Theories

In supersymmetric (SUSY) field theory, there exist configurations that formally satisfy SUSY conditions but are not on the original path integral contour. We refer to such configurations as complexified supersymmetric solutions (CSS). In this Letter we discuss that CSS provide important information on the large order behavior of the weak coupling perturbative series in SUSY field theories. We conjecture that CSS with a bosonic (fermionic) free parameter give poles (zeros) of the 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 a sphere. First we construct an infinite number of CSS in the general 3D $\mathcal{N}=2$ SUSY theory with a Lagrangian where an adjoint scalar in the 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.