Journal of High Energy Physics

, 2015:93 | Cite as

Wilson surface observables from equivariant cohomology

  • Anton Alekseev
  • Olga ChekeresEmail author
  • Pavel Mnev
Open Access
Regular Article - Theoretical Physics


Wilson lines in gauge theories admit several path integral descriptions. The first one (due to Alekseev-Faddeev-Shatashvili) uses path integrals over coadjoint orbits. The second one (due to Diakonov-Petrov) replaces a 1-dimensional path integral with a 2-dimensional topological σ-model. We show that this σ-model is defined by the equivariant extension of the Kirillov symplectic form on the coadjoint orbit. This allows to define the corresponding observable on arbitrary 2-dimensional surfaces, including closed surfaces. We give a new path integral presentation of Wilson lines in terms of Poisson σ-models, and we test this presentation in the framework of the 2-dimensional Yang-Mills theory. On a closed surface, our Wilson surface observable turns out to be nontrivial for G non-simply connected (and trivial for G simply connected), in particular we study in detail the cases G=U(1) and G=SO(3).


Wilson ’t Hooft and Polyakov loops Differential and Algebraic Geometry Sigma Models Gauge Symmetry 


Open Access

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Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GenevaGenève 4Switzerland
  2. 2.Max Planck Institute for MathematicsBonnGermany
  3. 3.St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of SciencesSt. PetersburgRussia

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