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Instantons and Supersymmetry

Massimo Bianchi, Stefano Kovacs, Giancarlo Rossi

TL;DR

This review surveys how instantons illuminate nonperturbative effects across globally supersymmetric gauge theories with ${\cal N}=1,2,4$. It develops a unified instanton calculus, emphasizes the role of zero modes and Ward identities, and shows how holomorphy fixes effective actions and exact prepotentials, notably the Seiberg-Witten construction for ${\cal N}=2$ and the Veneziano-Yankielowicz approach for ${\cal N}=1$. It then connects these field-theoretic results to string theory, via open/closed string realizations, D-instantons, and the AdS/CFT correspondence, providing striking nonperturbative checks of dualities and illuminating the nonperturbative structure of supersymmetric vacua, anomalies, and moduli spaces. The discussion extends to topological twists and localization techniques, showing how instanton sums can be computed to arbitrary charge using Nekrasov's framework, and culminates in detailed comparisons with D-instanton amplitudes in AdS5×S5, including the BMN limit. Overall, instanton methods reveal deep structure linking gauge dynamics, supersymmetry, and string dualities, with concrete predictions for condensates, beta functions, effective actions, and holographic correlators.

Abstract

The role of instantons in describing non-perturbative aspects of globally supersymmetric gauge theories is reviewed. The cases of theories with N=1, N=2 and N=4 supersymmetry are discussed. Special attention is devoted to the intriguing relation between instanton solutions in field theory and branes in string theory.

Instantons and Supersymmetry

TL;DR

This review surveys how instantons illuminate nonperturbative effects across globally supersymmetric gauge theories with . It develops a unified instanton calculus, emphasizes the role of zero modes and Ward identities, and shows how holomorphy fixes effective actions and exact prepotentials, notably the Seiberg-Witten construction for and the Veneziano-Yankielowicz approach for . It then connects these field-theoretic results to string theory, via open/closed string realizations, D-instantons, and the AdS/CFT correspondence, providing striking nonperturbative checks of dualities and illuminating the nonperturbative structure of supersymmetric vacua, anomalies, and moduli spaces. The discussion extends to topological twists and localization techniques, showing how instanton sums can be computed to arbitrary charge using Nekrasov's framework, and culminates in detailed comparisons with D-instanton amplitudes in AdS5×S5, including the BMN limit. Overall, instanton methods reveal deep structure linking gauge dynamics, supersymmetry, and string dualities, with concrete predictions for condensates, beta functions, effective actions, and holographic correlators.

Abstract

The role of instantons in describing non-perturbative aspects of globally supersymmetric gauge theories is reviewed. The cases of theories with N=1, N=2 and N=4 supersymmetry are discussed. Special attention is devoted to the intriguing relation between instanton solutions in field theory and branes in string theory.

Paper Structure

This paper contains 77 sections, 564 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalities about instantons
  3. The geometry of instantons
  4. Quantum tunneling
  5. Introducing fermions
  6. Fermionic functional integration
  7. Fermionic zero modes
  8. Putting together fermion and boson contributions
  9. The general case
  10. The supersymmetric case
  11. Chiral and supersymmetric Ward--Takahashi identities
  12. Space-time dependence
  13. Mass and $g$ dependence
  14. a) Mass analyticity
  15. b) Mass dependence
  16. ...and 62 more sections

Figures (5)

  • Figure 1: Contact and exchange contributions to a four-point amplitude in AdS$_5\times S^5$
  • Figure 2: D-instanton induced minimal amplitudes in AdS$_5\times S^5$.
  • Figure 3: Contact and exchange contributions to a four-point amplitude in AdS$_5\times S^5$
  • Figure 4: Contributions to the twenty-point amplitude dual to the correlation function (\ref{['G20LamE']}).
  • Figure 5: Leading D-instanton contribution to a two-point scattering amplitude.