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Exact Results for 't Hooft Loops in Gauge Theories on S^4

Jaume Gomis, Takuya Okuda, Vasily Pestun

TL;DR

This work delivers an exact localization computation of supersymmetric 't Hooft loop expectation values in four-dimensional ${ m N}=2$ gauge theories on ${ m S}^4$, extending Pestun's Wilson loop results. The observable factorizes into north/south pole instanton/anti-instanton contributions (Nekrasov functions) and an equator monopole piece, with a crucial monopole screening sum over allowed magnetic charges. The authors establish precise quantitative agreement with Liouville/Toda CFT predictions from AGT, including non-perturbative monopole screening effects, thereby providing a robust realization of S-duality for loop operators. They develop a comprehensive framework combining localization, equivariant index theory, ADHM/Kronheimer monopole moduli spaces, and the Nekrasov machinery, with potential extensions to other disorder operators and to broader duality webs.

Abstract

The path integral of a general N=2 supersymmetric gauge theory on S^4 is exactly evaluated in the presence of a supersymmetric 't Hooft loop operator. The result we find - obtained using localization techniques - captures all perturbative quantum corrections as well as non-perturbative effects due to instantons and monopoles, which are supported at the north pole, south pole and equator of S^4. As a by-product, our gauge theory calculations successfully confirm the predictions made for 't Hooft loops obtained from the calculation of topological defect correlators in Liouville/Toda conformal field theory.

Exact Results for 't Hooft Loops in Gauge Theories on S^4

TL;DR

This work delivers an exact localization computation of supersymmetric 't Hooft loop expectation values in four-dimensional gauge theories on , extending Pestun's Wilson loop results. The observable factorizes into north/south pole instanton/anti-instanton contributions (Nekrasov functions) and an equator monopole piece, with a crucial monopole screening sum over allowed magnetic charges. The authors establish precise quantitative agreement with Liouville/Toda CFT predictions from AGT, including non-perturbative monopole screening effects, thereby providing a robust realization of S-duality for loop operators. They develop a comprehensive framework combining localization, equivariant index theory, ADHM/Kronheimer monopole moduli spaces, and the Nekrasov machinery, with potential extensions to other disorder operators and to broader duality webs.

Abstract

The path integral of a general N=2 supersymmetric gauge theory on S^4 is exactly evaluated in the presence of a supersymmetric 't Hooft loop operator. The result we find - obtained using localization techniques - captures all perturbative quantum corrections as well as non-perturbative effects due to instantons and monopoles, which are supported at the north pole, south pole and equator of S^4. As a by-product, our gauge theory calculations successfully confirm the predictions made for 't Hooft loops obtained from the calculation of topological defect correlators in Liouville/Toda conformal field theory.

Paper Structure

This paper contains 28 sections, 332 equations, 1 figure.

Table of Contents

  1. Introduction
  2. ${\cal N}=2$ Gauge Theories in $S^4$ and Localization
  3. 't Hooft Loop in $S^4$ and Localization Equations
  4. Symmetries and Fields
  5. Localization Equations in $B_3\times S^1$
  6. Completeness of Solutions
  7. Classical Contribution
  8. Instanton Contribution
  9. One-Loop Determinants
  10. Review of the Atiyah-Singer Equivariant Index Theory
  11. North and South Pole Contributions
  12. Equator Contribution
  13. Examples
  14. Non-Perturbative Effects of Monopole Screening
  15. Physical Picture of Monopole Screening
  16. ...and 13 more sections

Figures (1)

  • Figure 1: Instanton, monopole and anti-instanton field configurations