Annales Henri Poincaré

, Volume 12, Issue 2, pp 351–395 | Cite as

Gauge Orbit Types for Theories with Gauge Group O(n), SO(n) or Sp(n)

  • Alexander Hertsch
  • Gerd Rudolph
  • Matthias Schmidt
Article
  • 53 Downloads

Abstract

We determine the orbit types of the action of the group of local gauge transformations on the space of connections in a principal bundle with structure group O(n), SO(n) or Sp(n) over a closed, simply connected manifold of dimension 4. On the way we derive a classification of Howe subgroups of SO(n) up to conjugacy.

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References

  1. 1.
    Alvarez-Gaumé L., Ginsparg P.: The topological meaning of nonabelian anomalies. Nucl. Phys. B 243, 449–474 (1984)ADSCrossRefGoogle Scholar
  2. 2.
    Asorey M., Falceto F., López J.L., Luzón G.: Nodes, monopoles, and confinement in 2 + 1-dimensional gauge theories. Phys. Lett. B 345, 125–130 (1995)ADSGoogle Scholar
  3. 3.
    Asorey M.: Maximal non-Abelian gauges and topology of the gauge orbit space. Nucl. Phys. B 551, 399–424 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Atiyah M.F., Singer I.M.: Dirac operators coupled to vector potentials. Proc. Natl. Acad. Sci. USA. 81(No. 8), 2597–2600 (1984)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Borel A.: Topics in the Homology Theory of Fibre Bundles Lecture Notes in Mathematics 36. Springer, Berlin (1967)Google Scholar
  6. 6.
    Bott R., Tu L.W.: Differential Forms in Algebraic Topology. Springer, Berlin (1982)MATHGoogle Scholar
  7. 7.
    Carey A., Mickelsson J., Murray M.: Index theory, gerbes, and Hamiltonian quantization. Commun. Math. Phys. 183, 707–722 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Charzynski SZ., Kijowski J., Rudolph G., Schmidt M.: On the stratified classical configuration space of lattice QCD. J. Geom. Phys. 55, 137–178 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Charzynski SZ., Rudolph G., Schmidt M.: On the topology of the reduced classical configuration space of lattice QCD. J. Geom. Phys. 58, 1607–1623 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Dieudonné J.: On the Automorphisms of the Classical Groups. Memoirs of the American Mathematical Society, 2. American Mathematical Society, Providence (1980)Google Scholar
  11. 11.
    Dold A., Whitney H.: Classification of oriented sphere bundles over a 4-complex. Ann. Math. 69, 667–677 (1959)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Emmrich C., Römer H.: Orbifolds as configuration spaces of systems with gauge symmetries. Commun. Math. Phys. 129, 69–94 (1990)ADSMATHCrossRefGoogle Scholar
  13. 13.
    Fischer E., Rudolph G., Schmidt M.: A lattice gauge model of singular Marsden-Weinstein reduction. Part I. Kinematics. J. Geom. Phys. 57, 1193–1213 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Ford C., Tok T., Wipf A.: Abelian projection on the torus for general gauge groups. Nucl. Phys. B 548, 585–612 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Ford C., Tok T., Wipf A.: SU(N)-gauge theories in Polyakov gauge on the torus. Phys. Lett. B 456, 155–161 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Gribov V.N.: Quantization of non-Abelian gauge theories. Nucl. Phys. B 139, 1–19 (1978)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Heil A., Kersch A., Papadopoulos N.A., Reifenhäuser B., Scheck F.: Anomalies from nonfree action of the gauge group. Ann. Phys. 200, 206–215 (1990)ADSMATHCrossRefGoogle Scholar
  18. 18.
    Huebschmann J., Rudolph G., Schmidt M.: A lattice gauge model for quantum mechanics on a stratified space. Commun. Math. Phys. 286(Nr. 2), 459–494 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    Ireland K., Rosen M.: A Classical Introduction to Modern Number Theory. Springer, Berlin (1990)MATHGoogle Scholar
  20. 20.
    Kobayashi S., Nomizu K.: Foundations of Differential Geometry I. Wiley Classics Library, Wiley, New York (1996)Google Scholar
  21. 21.
    Kondracki, W., Rogulski, J.: On the notion of stratification. Institute of Mathematics, Polish Academy of Sciences, Preprint 281, Warszawa (1983)Google Scholar
  22. 22.
    Kondracki, W., Rogulski, J.: On the stratification of the orbit space for the action of automorphisms on connections. Dissertationes Mathematicae 250, Panstwowe Wydawnictwo Naukowe, Warszawa (1986)Google Scholar
  23. 23.
    Langmann E., Salmhofer M., Kovner A.: Consistent axial-like gauge fixing on hypertori. Mod. Phys. Lett. A 9(31), 2913–2926 (1994)MathSciNetADSMATHCrossRefGoogle Scholar
  24. 24.
    Massey, W.S.: A Basic Course in Algebraic Topology. Graduate Texts in Mathematics 127, Springer (1991)Google Scholar
  25. 25.
    Narasimhan M.S., Ramadas T.R.: Geometry of SU(2) gauge fields. Commun. Math. Phys. 67, 121–136 (1979)MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Pflaum M.: Analytic and geometric study of stratified spaces. In: Lect. Notes Math. 1768. Springer, Berlin (2001)Google Scholar
  27. 27.
    Pontryagin L.: Classification of some skew products. Dokl. Akad. Nauk SSSR 47, 322–325 (1945)MATHGoogle Scholar
  28. 28.
    Rudolph G., Schmidt M., Volobuev I.P.: Classification of gauge orbit types for SU(n)-gauge theories. Math. Phys. Anal. Geom. 5, 201–241 (2002)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Rudolph G., Schmidt M., Volobuev I.P.: Partial ordering of gauge orbit types for SU(n)-gauge theories. J. Geom. Phys. 42, 106–138 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  30. 30.
    Rudolph G., Schmidt M., Volobuev I.P.: On the gauge orbit space stratification: a review. J. Phys. A Math. Gen. 35, R1–R50 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  31. 31.
    Rudolph G., Schmidt M.: On a certain stratification of the gauge orbit space. Rep. Math. Phys. 50, 99–110 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  32. 32.
    Schmidt M.: Classification and partial ordering of reductive Howe dual pairs of classical Lie group. J. Geom. Phys. 29, 283–318 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  33. 33.
    Shabanov S.V.: 2D Yang Mills theories, gauge orbit spaces and the path integral quantization. Commun. Theor. Phys. (Allahabad) 4(1), 1–62 (1995)MathSciNetGoogle Scholar
  34. 34.
    Singer I.M.: Some remarks on the Gribov ambiguity. Commun. Math. Phys. 60, 7–12 (1978)ADSMATHCrossRefGoogle Scholar
  35. 35.
    t‘Hooft G.: On the phase transition towards permanent quark confinement. Nucl. Phys. B 138, 1–25 (1978)MathSciNetADSCrossRefGoogle Scholar
  36. 36.
    Witten E.: An SU(2)-anomaly. Phys. Lett. B 117, 324–328 (1982)MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Wu W.T.: On Pontryagin classes I. Sci. Sin. 3, 353–367 (1954)Google Scholar
  38. 38.
    Wu W.T.: On Pontryagin classes II. Sci. Sin. 4, 455–490 (1955)MATHGoogle Scholar
  39. 39.
    Wu W.T.: On Pontryagin classes III. Acta Math. Sin. 4, 323–346 (1954)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Alexander Hertsch
    • 1
  • Gerd Rudolph
    • 1
  • Matthias Schmidt
    • 1
  1. 1.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany

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