Wilson surface observables from equivariant cohomology

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

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).

A preprint version of the article is available at ArXiv.

References

  1. [1]

    A. Alekseev, L.D. Faddeev and S.L. Shatashvili, Quantization of symplectic orbits of compact Lie groups by means of the functional integral, J. Geom. Phys. 5 (1988) 391 [INSPIRE].

    MATH  MathSciNet  Article  ADS  Google Scholar 

  2. [2]

    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].

  3. [3]

    C.L. Kane and E.J. Mele, Z 2 topological order and the quantum spin Hall effect, Phys. Rev. Lett. 95 (2005) 146802 [cond-mat/0506581] [INSPIRE].

    Article  ADS  Google Scholar 

  4. [4]

    S. Ryu, C. Mudry, H. Obuse and A. Furusaki, Z 2 topological term, the global anomaly, and the two-dimensional symplectic symmetry class of Anderson localization, Phys. Rev. Lett. 99 (2007) 116601 [cond-mat/0702529].

    Article  ADS  Google Scholar 

  5. [5]

    D. Carpentier, P. Delplace, M. Fruchart, K. Gawedzki and C. Tauber, Construction and properties of a topological index for periodically driven time-reversal invariant 2D crystals, Nucl. Phys. B 896 (2015) 779 [arXiv:1503.04157] [INSPIRE].

    Article  Google Scholar 

  6. [6]

    D. Carpentier, P. Delplace, M. Fruchart and K. Gawedzki, Topological index for periodically driven time-reversal invariant 2D systems, Phys. Rev. Lett. 114 (2015) 106806 [arXiv:1407.7747].

    Article  ADS  Google Scholar 

  7. [7]

    A. Kapustin, Bosonic topological insulators and paramagnets: a view from cobordisms, arXiv:1404.6659 [INSPIRE].

  8. [8]

    O.J. Ganor, Six-dimensional tensionless strings in the large-N limit, Nucl. Phys. B 489 (1997) 95 [hep-th/9605201] [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  9. [9]

    B. Chen, W. He, J.-B. Wu and L. Zhang, M 5-branes and Wilson surfaces, JHEP 08 (2007) 067 [arXiv:0707.3978] [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  10. [10]

    I. Chepelev, Non-Abelian Wilson surfaces, JHEP 02 (2002) 013 [hep-th/0111018] [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  11. [11]

    A.S. Cattaneo and C.A. Rossi, Wilson surfaces and higher dimensional knot invariants, Commun. Math. Phys. 256 (2005) 513 [math-ph/0210037] [INSPIRE].

    MATH  MathSciNet  Article  ADS  Google Scholar 

  12. [12]

    D.S. Freed, Classical Chern-Simons theory. Part 1, Adv. Math. 113 (1995) 237 [hep-th/9206021] [INSPIRE].

    MATH  MathSciNet  Article  Google Scholar 

  13. [13]

    D.S. Freed, Classical Chern-Simons theory, part 2, Houston J. Math. 28 (2002) 293.

    MATH  MathSciNet  Google Scholar 

  14. [14]

    K.G. Wilson, Confinement of quarks, Phys. Rev. D 10 (1974) 2445 [INSPIRE].

    ADS  Google Scholar 

  15. [15]

    R. Giles, The reconstruction of gauge potentials from Wilson loops, Phys. Rev. D 24 (1981) 2160 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  16. [16]

    E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].

    MATH  Article  ADS  Google Scholar 

  17. [17]

    A.A. Kirillov, Lectures on the orbit method, Graduate Studies in Mathematics 64, America Mathematical Society, Providence RI U.S.A. (2004).

  18. [18]

    D.P. Zhelobenko, Compact Lie groups and their representations, Translations of Mathematical Monographs 40, American Mathematical Society, Providence RI U.S.A. (1978).

  19. [19]

    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).

  20. [20]

    A.P. Balachandran, S. Borchardt and A. Stern, Lagrangian and Hamiltonian descriptions of Yang-Mills particles, Phys. Rev. D 17 (1978) 3247 [INSPIRE].

    ADS  Google Scholar 

  21. [21]

    H.B. Nielsen and D. Rohrlich, A path integral to quantize spin, Nucl. Phys. B 299 (1988) 471 [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  22. [22]

    D. Diakonov and V. Yu. Petrov, A formula for the Wilson loop, Phys. Lett. B 224 (1989) 131 [INSPIRE].

    Article  ADS  Google Scholar 

  23. [23]

    S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  24. [24]

    C. Beasley, Localization for Wilson loops in Chern-Simons theory, Adv. Theor. Math. Phys. 17 (2013) 1 [arXiv:0911.2687] [INSPIRE].

    MATH  MathSciNet  Article  Google Scholar 

  25. [25]

    M.F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1 [INSPIRE].

    MATH  MathSciNet  Article  Google Scholar 

  26. [26]

    V.W. Guillemin and S. Sternberg, Supersymmetry and equivariant de Rham theory, Springer, Berlin Heidelberg Germany (1991).

    Google Scholar 

  27. [27]

    N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, Grundlehren Text Editions, Springer, Berlin Heidelberg Germany (2004).

  28. [28]

    E. Meinrenken, Equivariant cohomology and the Cartan model, http://www.math.toronto.edu/mein/research/enc.pdf.

  29. [29]

    N. Ikeda, Two-dimensional gravity and nonlinear gauge theory, Annals Phys. 235 (1994) 435 [hep-th/9312059] [INSPIRE].

    MATH  Article  ADS  Google Scholar 

  30. [30]

    P. Schaller and T. Strobl, Poisson structure induced (topological) field theories, Mod. Phys. Lett. A 9 (1994) 3129 [hep-th/9405110] [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  31. [31]

    P. Schaller and T. Strobl, A brief introduction to Poisson σ-models, Lect. Notes Phys. 469 (1996) 321 [hep-th/9507020] [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  32. [32]

    D. Birmingham, M. Blau, M. Rakowski and G. Thompson, Topological field theory, Phys. Rept. 209 (1991) 129 [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  33. [33]

    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.

  34. [34]

    S.-S. Chern and J. Simons, Characteristic forms and geometric invariants, Annals Math. 99 (1974) 48 [INSPIRE].

    MATH  MathSciNet  Article  Google Scholar 

  35. [35]

    R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].

    MATH  MathSciNet  Article  ADS  Google Scholar 

  36. [36]

    J.W. Milnor and J.D. Stasheff, Characteristic classes, Princeton University Press, Princeton U.S.A. (1974).

    Google Scholar 

  37. [37]

    A.A. Migdal, Recursion equations in gauge theories, Sov. Phys. JETP 42 (1975) 413 [Zh. Eksp. Teor. Fiz. 69 (1975) 810] [INSPIRE].

  38. [38]

    N.E. Bralic, Exact computation of loop averages in two-dimensional Yang-Mills theory, Phys. Rev. D 22 (1980) 3090 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  39. [39]

    V.A. Kazakov and I.K. Kostov, Nonlinear strings in two-dimensional U(∞) gauge theory, Nucl. Phys. B 176 (1980) 199 [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  40. [40]

    V.A. Kazakov and I.K. Kostov, Computation of the Wilson loop functional in two-dimensional U(∞) lattice gauge theory, Phys. Lett. B 105 (1981) 453 [INSPIRE].

    Article  ADS  Google Scholar 

  41. [41]

    V.A. Kazakov, Wilson loop average for an arbitrary contour in two-dimensional U(N ) gauge theory, Nucl. Phys. B 179 (1981) 283 [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  42. [42]

    L. Gross, C. King and A. Sengupta, Two-dimensional Yang-Mills theory via stochastic differential equations, Annals Phys. 194 (1989) 65 [INSPIRE].

    MATH  MathSciNet  Article  ADS  Google Scholar 

  43. [43]

    B.E. Rusakov, Loop averages and partition functions in U(N ) gauge theory on two-dimensional manifolds, Mod. Phys. Lett. A 5 (1990) 693 [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  44. [44]

    D.S. Fine, Quantum Yang-Mills on the two-sphere, Commun. Math. Phys. 134 (1990) 273 [INSPIRE].

    MATH  Article  ADS  Google Scholar 

  45. [45]

    D.S. Fine, Quantum Yang-Mills on a Riemann surface, Commun. Math. Phys. 140 (1991) 321 [INSPIRE].

    MATH  Article  ADS  Google Scholar 

  46. [46]

    M. Blau and G. Thompson, Quantum Yang-Mills theory on arbitrary surfaces, Int. J. Mod. Phys. A 7 (1992) 3781 [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  47. [47]

    E. Witten, On quantum gauge theories in two-dimensions, Commun. Math. Phys. 141 (1991) 153 [INSPIRE].

    MATH  Article  ADS  Google Scholar 

  48. [48]

    S. Cordes, G.W. Moore and S. Ramgoolam, Lectures on 2D Yang-Mills theory, equivariant cohomology and topological field theories, Nucl. Phys. Proc. Suppl. 41 (1995) 184 [hep-th/9411210] [INSPIRE].

    MATH  MathSciNet  Article  ADS  Google Scholar 

  49. [49]

    E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303 [hep-th/9204083] [INSPIRE].

    MATH  MathSciNet  Article  ADS  Google Scholar 

  50. [50]

    N. Steenrod, The topology of fiber bundles, Princeton Mathematical Series 14, Princeton University Press, Princeton U.S.A. (1951).

  51. [51]

    J. Fuchs, C. Schweigert and A. Valentino, A geometric approach to boundaries and surface defects in Dijkgraaf-Witten theories, Commun. Math. Phys. 332 (2014) 981 [arXiv:1307.3632] [INSPIRE].

    MATH  MathSciNet  Article  ADS  Google Scholar 

  52. [52]

    A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge U.K. (2002).

    Google Scholar 

Download references

Open Access

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Olga Chekeres.

Additional information

ArXiv ePrint: 1507.06343

Dedicated to Ludwig Faddeev on the occasion of his 34th anniversary

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alekseev, A., Chekeres, O. & Mnev, P. Wilson surface observables from equivariant cohomology. J. High Energ. Phys. 2015, 93 (2015). https://doi.org/10.1007/JHEP11(2015)093

Download citation

Keywords

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