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Notes on Equivariant Localization

  • Anton Alekseev
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 543)

Abstract

We review the localization formula due to Berline-Vergne and Atiyah- Bott, with applications to the exact stationary phase phenomenon discovered by Duistermaat-Heckman. We explain the Weil model of equivariant cohomology and recall its relation to BRST. We show how to quantize the Weil model, and obtain new localization formulas which, in particular, apply to Hamiltonian spaces with group valued moment maps.

Keywords

Equivariant Form Erential Form Equivariant Cohomology Circle Action Dominant Weight 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AB]
    Atiyah, M., Bott, R. (1984): The moment map and equivariant cohomology. Topology 23 no. 1, 1–28zbMATHCrossRefMathSciNetGoogle Scholar
  2. [AMM]
    Alekseev, A., Malkin, A., Meinrenken E. (1998): Lie group valued moment maps. J. Differential Geom. 48 no. 3, 445–495zbMATHMathSciNetGoogle Scholar
  3. [AM]
    Alekseev, A., Meinrenken, E. (1999): The non-commutative Weil algebra. Preprint math.DG/990352, to be published in Inv. Math.Google Scholar
  4. [AMW1]
    Alekseev, A., Meinrenken, E., Woodward, C. (1999): Duistermaat-Heckman distributions for group-valued moment maps. Preprint math.DG/9903087Google Scholar
  5. [AMW2]
    Alekseev, A., Meinrenken, E., Woodward, C. (1999): Group-valued equivariant localization. Preprint math.DG/9905130Google Scholar
  6. [BV]
    Berline, N., Vergne, M. (1983): Zéro d’un champ de vecteurs et classes caractéristiques équivariantes. Duke Math. J. 50, 539–549zbMATHCrossRefMathSciNetGoogle Scholar
  7. [BGV]
    Berline, N., Getzler, E., Vergne, M. (1992): Heat kernels and Dirac operators. Grundlehren der mathematischen Wissenschaften, vol. 298, Springer-Verlag, Berlin-Heidelberg-New YorkGoogle Scholar
  8. [BT]
    Blau, M., Thompson, G. (1995): Equivariant Kähler Geometry and Localization in G=G Model. Nucl. Phys. B439, 367–394CrossRefADSMathSciNetGoogle Scholar
  9. [C1]
    Cartan, H. (1950): La transgression dans un groupe de Lie et dans un fibré principal. In Colloque de topologie (espaces fibrés), Bruxelles, 73–81Google Scholar
  10. [C2]
    Cartan, H. (1950): Notions d’algèbre différentielle; application aux groupes de Lie et aux variétés où opŕe un groupe de Lie. In Colloque de topologie (espaces fibrès), BruxellesGoogle Scholar
  11. [CMR]
    Cordes, S., Moore, G., Ramgoolam, S. (1995): Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories. Nucl. Phys. Proc. Suppl. 41, 184–244zbMATHMathSciNetGoogle Scholar
  12. [DH]
    Duistermaat, J.J., Heckman, G.J. (1982): One the variation in the cohomology of the symplectic form on the reduced phase space. Inv. Math. 69, 259–268zbMATHCrossRefADSMathSciNetGoogle Scholar
  13. [G]
    Gerasimov, A. (1993): Localization in GWZW and Verlinde formula. Preprint hep-th/9305090Google Scholar
  14. [MNP]
    A., Morozov, A., Niemi, A., Palo, K. (1991): Supersymmetry and loopspace geometry. Phys. Lett. B 271 365–371ADSMathSciNetGoogle Scholar
  15. [JK]
    Jeffrey, L., Kirwan, F. (1995): Localization for Nonabelian Group Actions. Topology 34, 291–327zbMATHCrossRefMathSciNetGoogle Scholar
  16. [K]
    Kalkman, J. (1993): A BRST model applied to symplectic geometry. Ph.D. thesis, Universiteit Utrecht.Google Scholar
  17. [MNS]
    Moore, G., Nekrasov, N., Shatashvili, S. (1997): Integrating over Higgsbranches. Preprint hep-th/9712241Google Scholar
  18. [P]
    Plamenevskaya, O. (1999): A Residue Formula for SU(2)-valued Moment Maps. Preprint math.DG/9906093Google Scholar
  19. [W1]
    Witten, E. (1982): Supersymmetry and Morse theory. J. Differential Geom. 17, 661–692zbMATHMathSciNetGoogle Scholar
  20. [W2]
    Witten, E. (1992): Two-dimensional gauge theory revisited. J. Geom. Phys. 9, 303–368zbMATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Anton Alekseev
    • 1
  1. 1.Institutionen för Teoretisk FysikUppsala UniversitetUppsalaSweden

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