Wilson surface observables from equivariant cohomology
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).
KeywordsWilson ’t Hooft and Polyakov loops Differential and Algebraic Geometry Sigma Models Gauge Symmetry
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
- D. Diakonov and V. Petrov, Non-Abelian Stokes theorem and quark-monopole interaction, published version in Nonperturbative approaches to QCD, Proceedings of the Internat. workshop at ECT, Trento Italy July 10-29 1995, D. Diakonov ed., Petersburg Nucl. Phys. Inst., v. Gatchina Russia (1995) [hep-th/9606104] [INSPIRE].
- A.A. Kirillov, Lectures on the orbit method, Graduate Studies in Mathematics 64, America Mathematical Society, Providence RI U.S.A. (2004).Google Scholar
- D.P. Zhelobenko, Compact Lie groups and their representations, Translations of Mathematical Monographs 40, American Mathematical Society, Providence RI U.S.A. (1978).Google Scholar
- R. Bott, The geometry and representation theory of compact Lie groups, in Representation theory of Lie groups, London Mathematical Society Lecture Note Series 34, Cambridge University Press, Cambridge U.K. (1979).Google Scholar
- V.W. Guillemin and S. Sternberg, Supersymmetry and equivariant de Rham theory, Springer, Berlin Heidelberg Germany (1991).Google Scholar
- N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, Grundlehren Text Editions, Springer, Berlin Heidelberg Germany (2004).Google Scholar
- E. Meinrenken, Equivariant cohomology and the Cartan model, http://www.math.toronto.edu/mein/research/enc.pdf.
- J. Cheeger and J. Simons, Differential characters and geometric invariants, in Geometry and topology, Proceedings of the Special Year held at the University of Maryland, College Park 1983-1984, J. Alexander and J. Harer eds., Springer, Berlin Heidelberg Germany Lect. Notes Math. 1167 (1985) 50.Google Scholar
- A.A. Migdal, Recursion equations in gauge theories, Sov. Phys. JETP 42 (1975) 413 [Zh. Eksp. Teor. Fiz. 69 (1975) 810] [INSPIRE].
- N. Steenrod, The topology of fiber bundles, Princeton Mathematical Series 14, Princeton University Press, Princeton U.S.A. (1951).Google Scholar