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Comparing Instanton Contributions with Exact Results in N=2 Supersymmetric Scale Invariant Theories

Philip C. Argyres, Sophie Pelland

TL;DR

The paper addresses the challenge of validating exact low-energy results for scale-invariant ${\cal N}=2$ SU$(N)$ theories with $2N$ flavors against instanton calculations. It develops a systematic framework of holomorphic, symmetry-consistent nonperturbative redefinitions between the parameters and vevs of the two descriptions, enabling precise cross-checks. The authors demonstrate equivalence among different proposed Seiberg-Witten curves, and perform an explicit one-instanton check at a special vacuum, deriving matching constants $C_0$ and $C_1$ that ensure agreement for all $N$. They also analyze the one-loop threshold and electric-magnetic duality ambiguities, showing that with appropriate choices these two approaches yield the same low-energy couplings, thereby strengthening the reliability of both exact and instanton methods in ${\cal N}=2$ gauge theories.

Abstract

We discuss the general issues and ambiguities involved in matching the exact results for the low energy effective action of scale invariant N=2 supersymmetric QCD to those obtained by instanton methods. We resolve the reported disagreements and verify agreement between an infinite series of coefficients in the low energy effective actions calculated in the two approaches. In particular, we show that the exact low-energy effective couplings for SU(N) for all N with 2N fundamental hypermultiplets agree at a special vacuum on the Coulomb branch where a large unbroken discrete global symmetry makes the matching of parameters relatively straightforward.

Comparing Instanton Contributions with Exact Results in N=2 Supersymmetric Scale Invariant Theories

TL;DR

The paper addresses the challenge of validating exact low-energy results for scale-invariant SU theories with flavors against instanton calculations. It develops a systematic framework of holomorphic, symmetry-consistent nonperturbative redefinitions between the parameters and vevs of the two descriptions, enabling precise cross-checks. The authors demonstrate equivalence among different proposed Seiberg-Witten curves, and perform an explicit one-instanton check at a special vacuum, deriving matching constants and that ensure agreement for all . They also analyze the one-loop threshold and electric-magnetic duality ambiguities, showing that with appropriate choices these two approaches yield the same low-energy couplings, thereby strengthening the reliability of both exact and instanton methods in gauge theories.

Abstract

We discuss the general issues and ambiguities involved in matching the exact results for the low energy effective action of scale invariant N=2 supersymmetric QCD to those obtained by instanton methods. We resolve the reported disagreements and verify agreement between an infinite series of coefficients in the low energy effective actions calculated in the two approaches. In particular, we show that the exact low-energy effective couplings for SU(N) for all N with 2N fundamental hypermultiplets agree at a special vacuum on the Coulomb branch where a large unbroken discrete global symmetry makes the matching of parameters relatively straightforward.

Paper Structure

This paper contains 9 sections, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Comparing curves and instantons
  3. Comparing curves and instantons in SU(2) with massive adjoint matter
  4. Comparing different curves for SU(N) with 2N flavors
  5. Comparing curves and instantons in SU(N) with 2N flavors
  6. Curve computation
  7. The electric-magnetic duality ambiguity
  8. One-loop threshold computation
  9. One-instanton contribution

Figures (1)

  • Figure 1: Representation of one sheet of the two-sheeted covering of $\Sigma$ given in (\ref{['curve']}) with a (non-canonical) basis $\{{\widetilde{\alpha}}_j,{\widetilde{\beta}}_k\}$ of cycles in homology. The dark lines represent the $N$ cuts on the $x$-plane. Only half of the ${\widetilde{\beta}}_k$ cycles are visible, the other half lying on the second sheet. The dotted lines represent the $a$ and $b$ contours used in Appendix A.