The Instanton Hunter's Guide to Supersymmetric SU(N) Gauge Theories

TL;DR

This paper provides a comprehensive analysis of multi-instanton solutions in supersymmetric SU(N) gauge theories, offering new formulas and confirming previous results for low-energy dynamics without relying on duality.

Contribution

It extends ADHM multi-instanton constructions to N=1 and N=2 SUSY theories, deriving explicit formulas for instanton contributions to the prepotential in SU(N) models.

Findings

Derived a closed-form formula for the k-instanton contribution to the prepotential.

Confirmed previous instanton calculations for certain flavor numbers.

Obtained new results for specific flavor cases where existing solutions are unclear.

Abstract

We present a compendium of results for ADHM multi-instantons in SU(N) SUSY gauge theories, followed by applications to N=2 supersymmetric models. Extending recent SU(2) work, and treating the N=1 and N=2 cases in parallel, we construct: (i) the ADHM supermultiplet, (ii) the multi-instanton action, and (iii) the collective coordinate integration measure. Specializing to N=2, we then give a closed formula for F_k, the k-instanton contribution to the prepotential, as a finite-dimensional collective coordinate integral. This amounts to a weak-coupling solution, in quadratures, of the low-energy dynamics of N=2 SQCD, without appeal to duality. As an application, we calculate F_1 for all SU(N) and any number of flavors N_F; for N_F<2N-2 and N_F=2N-1 we confirm previous instanton calculations and agree with the proposed hyper-elliptic curve solutions. For N_F=2N-2 and N_F=2N with N>3 we obtain new results, which in the latter case we do not understand how to reconcile with the curves.