Calculating the Prepotential by Localization on the Moduli Space of Instantons

TL;DR

This work develops a localization-based framework to compute nonperturbative instanton contributions to the N=2 SU(N) prepotential by integrating over the instanton moduli space, regularized via a noncommutative deformation. By constructing a nilpotent fermionic symmetry Q and rewriting the instanton action as S = Q Xi + Gamma, the authors localize the centred instanton partition function to critical points of Q Xi and extract explicit one- and two-instanton contributions. The results yield concise expressions for the centred one- and two-instanton partition functions, which, when translated into prepotential contributions, agree with Seiberg-Witten predictions for N_F < 2N and illuminate the role of moduli-space singularities and their resolution in the commutative versus noncommutative theories. The approach scales to higher instanton numbers and leverages ADHM data and the Eguchi-Hanson geometry to provide a tractable, first-principles route to the nonperturbative N=2 prepotential.

Abstract

We describe a new technique for calculating instanton effects in supersymmetric gauge theories applicable on the Higgs or Coulomb branches. In these situations the instantons are constrained and a potential is generated on the instanton moduli space. Due to existence of a nilpotent fermionic symmetry the resulting integral over the instanton moduli space localizes on the critical points of the potential. Using this technology we calculate the one- and two-instanton contributions to the prepotential of SU(N) gauge theory with N=2 supersymmetry and show how the localization approach yields the prediction extracted from the Seiberg-Witten curve. The technique appears to extend to arbitrary instanton number in a tractable way.