How to resum perturbative series in 3d N=2 Chern-Simons matter theories

TL;DR

This paper investigates the Borel summability of perturbative series in 3d N=2 Chern-Simons matter theories, showing that resummation along specific complex axes yields results matching exact solutions.

Contribution

It demonstrates that perturbative series are Borel summable along certain imaginary axes, providing a reliable resummation method for these theories.

Findings

Perturbative series are non-Borel summable along positive real axis.

Series are Borel summable along negative or positive imaginary axes depending on Chern-Simons levels.

Borel resummations along these axes match exact results.

Abstract

Continuing the work arXiv:1603.06207, we study perturbative series in general 3d $\mathcal{N}=2$ supersymmetric Chern-Simons matter theory with $U(1)_R$ symmetry, which is given by a power series expansion of inverse Chern-Simons levels. We find that the perturbative series are usually non-Borel summable along positive real axis for various observables. Alternatively we prove that the perturbative series are always Borel summable along negative (positive) imaginary axis for positive (negative) Chern-Simons levels. It turns out that the Borel resummations along this direction are the same as exact results and therefore correct ways of resumming the perturbative series.