Abstract
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 N=2 SUSY Chern-Simons matter theories on a sphere. First we construct an infinite number of CSS in the general 3D 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.
Highlights
In supersymmetric (SUSY) field theory, there exist configurations that formally satisfy SUSY conditions but are not on the original path integral contour
While most attention has been paid to SUSY solutions that take values on the original path integral contour, there exist configurations that formally satisfy SUSY conditions but are not on the original contour
While saddle points along the original contour always contribute to the path integral, complex saddles may or may not contribute since this is determined by the details on the steepest descents associated with them
Summary
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. First we construct an infinite number of CSS in the general 3D N 1⁄4 2 SUSY theory with a Lagrangian where an adjoint scalar in the vector multiplet takes a complex value and matter fields are nontrivial We compare their actions with Borel transformations of perturbative expansions by inverse Chern-Simons levels for the observables and see agreement with our conjecture.
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