Darboux Coordinates and Instanton Corrections in Projective Superspace

TL;DR

This paper derives Darboux coordinates for hyperk"ahler manifolds using projective superspace, and applies these results to analyze instanton corrections in the Coulomb branch of 4D and 5D supersymmetric gauge theories.

Contribution

It provides a new derivation of Darboux coordinates in hyperk"ahler geometry and applies this to study instanton corrections in supersymmetric gauge theories.

Findings

Derived explicit Darboux coordinates in hyperk"ahler manifolds.

Reproduced known results for 4D N=2 SYM Coulomb branch.

Explicitly computed electric corrections in 5D N=1 SYM.

Abstract

By demanding consistency of the Legendre transform construction of hyperk\"ahler metrics in projective superspace, we derive the expression for the Darboux coordinates on the hyperk\"ahler manifold. We apply these results to study the Coulomb branch moduli space of 4D, N=2 super-Yang-Mills theory (SYM) on R^3 x S^1, recovering the results by GMN. We also apply this method to study the electric corrections to the moduli space of 5D, N=1 SYM on R^3 x T^2 and give the Darboux coordinates explicitly.