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Journal of High Energy Physics

, 2010:57 | Cite as

Gauge theory loop operators and Liouville theory

  • Nadav Drukker
  • Jaume Gomis
  • Takuya Okuda
  • Jörg Teschner
Open Access
Article

Abstract

We propose a correspondence between loop operators in a family of four dimensional \( \mathcal{N} \) = 2 gauge theories on S 4 — including Wilson, ‘t Hooft and dyonic operators — and Liouville theory loop operators on a Riemann surface. This extends the beautiful relation between the partition function of these \( \mathcal{N} \) = 2 gauge theories and Liouville correlators found by Alday, Gaiotto and Tachikawa. We show that the computation of these Liouville correlators with the insertion of a Liouville loop operator reproduces Pestun’s formula capturing the expectation value of a Wilson loop operator in the corresponding gauge theory. We prove that our definition of Liouville loop operators is invariant under modular transformations, which given our correspondence, implies the conjectured action of S-duality on the gauge theory loop operators. Our computations in Liouville theory make an explicit prediction for the exact expectation value of ’t Hooft and dyonic loop operators in these \( \mathcal{N} \) = 2 gauge theories. The Liouville loop operators are also found to admit a simple geometric interpretation within quantum Teichmüller theory as the quantum operators representing the length of geodesics. We study the algebra of Liouville loop operators and show that it gives evidence for our proposal as well as providing definite predictions for the operator product expansion of loop operators in gauge theory.

Keywords

Supersymmetric gauge theory Field Theories in Lower Dimensions Duality in Gauge Field Theories 

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© The Author(s) 2010

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Nadav Drukker
    • 1
  • Jaume Gomis
    • 2
  • Takuya Okuda
    • 2
  • Jörg Teschner
    • 3
  1. 1.Institut für PhysikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  3. 3.DESY TheoryHamburgGermany

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