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ABCD of instantons

Nikita Nekrasov, Sergey Shadchin

TL;DR

This work extends the Nekrasov Shadchin program SWfromInst to compute the exact prepotentials for N=2 SYM with classical gauge groups SO and Sp, using a five dimensional lift and Omega background to express the instanton sum as contour integrals. By exploiting the ADHM construction and its dual group structure, the authors derive explicit five dimensional and four dimensional representations for the instanton partition function, and extract Seiberg Witten curves and differential that encode the prepotential. They provide detailed Haar measure formulas, contour integral prescriptions, and saddle point analyses, yielding explicit 1-instanton and higher instanton corrections that match known results. The method avoids resolving singularities of instanton moduli spaces and offers a scalable route to SW data for classical groups, with clear avenues for including matter and exploring conformal theories.

Abstract

We solve N=2 supersymmetric Yang-Mills theories for arbitrary classical gauge group, i.e. SU(N), SO(N), Sp(N). In particular, we derive the prepotential of the low-energy effective theory, and the corresponding Seiberg-Witten curves. We manage to do this without resolving singularities of the compactified instanton moduli spaces.

ABCD of instantons

TL;DR

This work extends the Nekrasov Shadchin program SWfromInst to compute the exact prepotentials for N=2 SYM with classical gauge groups SO and Sp, using a five dimensional lift and Omega background to express the instanton sum as contour integrals. By exploiting the ADHM construction and its dual group structure, the authors derive explicit five dimensional and four dimensional representations for the instanton partition function, and extract Seiberg Witten curves and differential that encode the prepotential. They provide detailed Haar measure formulas, contour integral prescriptions, and saddle point analyses, yielding explicit 1-instanton and higher instanton corrections that match known results. The method avoids resolving singularities of instanton moduli spaces and offers a scalable route to SW data for classical groups, with clear avenues for including matter and exploring conformal theories.

Abstract

We solve N=2 supersymmetric Yang-Mills theories for arbitrary classical gauge group, i.e. SU(N), SO(N), Sp(N). In particular, we derive the prepotential of the low-energy effective theory, and the corresponding Seiberg-Witten curves. We manage to do this without resolving singularities of the compactified instanton moduli spaces.

Paper Structure

This paper contains 28 sections, 152 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. $\mathcal{N} = 2$ Super Yang-Mills
  4. The field content and the action
  5. Seiberg-Witten theory
  6. Topological twist
  7. The prepotential
  8. Five dimensional perspective
  9. Tracing over instanton moduli space
  10. A model example
  11. Back to instantons
  12. Four dimensional limit
  13. ADHM construction
  14. $SU(N)$ case
  15. $SO(N)$ case
  16. ...and 13 more sections

Figures (1)

  • Figure 1: Conformal map for $Sp(N)$, $N=1$