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Localization of gauge theory on a four-sphere and supersymmetric Wilson loops

Vasily Pestun

TL;DR

The paper establishes an exact localization framework for four-dimensional supersymmetric gauge theories on S^4, proving that supersymmetric Wilson loops in N=4 SYM reduce to Gaussian matrix-model expectations, and generalizes this to N=2 and N=2* with full one-loop and instanton data. The authors formulate a combined Q-complex from gauge-fixing and supersymmetry, compute the 1-loop determinant via the index theory of transversally elliptic operators, and show that in the N=4 case these determinants cancel while instanton contributions vanish, yielding an exact Gaussian model. For N=2* the mass deformation introduces a nontrivial 1-loop measure and Nekrasov instanton factors, producing a richer matrix-model description that captures both perturbative and nonperturbative effects. These results provide exact, nonperturbative tests of gauge/string dualities and offer a powerful tool for computing Wilson-loop observables in diverse supersymmetric theories on curved backgrounds.

Abstract

We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the N=4 supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure N=2 and the N=2* supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional N=2 superconformal gauge theory is treated similarly.

Localization of gauge theory on a four-sphere and supersymmetric Wilson loops

TL;DR

The paper establishes an exact localization framework for four-dimensional supersymmetric gauge theories on S^4, proving that supersymmetric Wilson loops in N=4 SYM reduce to Gaussian matrix-model expectations, and generalizes this to N=2 and N=2* with full one-loop and instanton data. The authors formulate a combined Q-complex from gauge-fixing and supersymmetry, compute the 1-loop determinant via the index theory of transversally elliptic operators, and show that in the N=4 case these determinants cancel while instanton contributions vanish, yielding an exact Gaussian model. For N=2* the mass deformation introduces a nontrivial 1-loop measure and Nekrasov instanton factors, producing a richer matrix-model description that captures both perturbative and nonperturbative effects. These results provide exact, nonperturbative tests of gauge/string dualities and offer a powerful tool for computing Wilson-loop observables in diverse supersymmetric theories on curved backgrounds.

Abstract

We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the N=4 supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure N=2 and the N=2* supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional N=2 superconformal gauge theory is treated similarly.

Paper Structure

This paper contains 14 sections, 269 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fields, action and symmetries
  3. Localization
  4. Determinant factor
  5. Gauge-fixing complex
  6. Supersymmetry complex
  7. The combined $Q$-complex
  8. Computation of the determinant using the index theory of transversally elliptic operators
  9. Example
  10. Instanton corrections
  11. Clifford algebra
  12. Conformal killing spinors on $S^4$
  13. Off-shell supersymmetry
  14. Index of transversally elliptic operators

Figures (1)

  • Figure 1: Wilson loop on the equator of $S^4$