Mathematische Annalen

, Volume 374, Issue 1–2, pp 681–722 | Cite as

Quantization of Hamiltonian loop group spaces

  • Yiannis LoizidesEmail author
  • Yanli Song


We prove a Fredholm property for spin-c Dirac operators \(\mathsf {D}\) on non-compact manifolds satisfying a certain condition with respect to the action of a semi-direct product group \(K\ltimes \Gamma \), with K compact and \(\Gamma \) discrete. We apply this result to an example coming from the theory of Hamiltonian loop group spaces. In this context we prove that a certain index pairing \([{\mathcal {X}}] \cap [\mathsf {D}]\) yields an element of the formal completion \(R^{-\infty }(T)\) of the representation ring of a maximal torus \(T \subset H\); the resulting element has an additional antisymmetry property under the action of the affine Weyl group, indicating \([{\mathcal {X}}] \cap [\mathsf {D}]\) corresponds to an element of the ring of projective positive energy representations of the loop group.



We especially thank Eckhard Meinrenken for many helpful discussions and encouragement, and for providing feedback on an earlier draft. We also thank Nigel Higson for helpful discussions. Y. Song is supported by NSF grant 1800667.


  1. 1.
    Alekseev, A., Malkin, A., Meinrenken, E.: Lie group valued moment maps. J. Differen. Geom. 48, 445–495 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alekseev, A., Meinrenken, E.: Dirac structures and Dixmier–Douady bundles. Int. Math. Res. Not. 2012, 904–956 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alekseev, A., Meinrenken, E., Woodward, C.: The Verlinde formulas as fixed point formulas. J. Symplectic Geom. 1, 1–46 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anghel, N.: An abstract index theorem on non-compact Riemannian manifolds. Houston J. of Math. 19, 223–237 (1993)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Atiyah, M.F.: Elliptic Operators and Compact Groups, vol. 401. Springer, Berlin, Heildelberg, New York (2006)Google Scholar
  6. 6.
    Baaj, S., Julg, P.: Théorie bivariante de Kasparov et opérateurs non bornés dans les C\(^\ast \)-modules hilbertiens. CR Acad. Sci. Paris Sér. I Math 296(21), 875–878 (1983)zbMATHGoogle Scholar
  7. 7.
    Bär, C., Ballmann, W.: Guide to elliptic boundary value problems for dirac-type operators. In: Ballmann, W., Blohmann, C., Faltings, G., Teichner, P., Zagier, D. (eds) Arbeitstagung Bonn 2013. Progress in Mathematics, vol 319. Birkhuser, Cham (2016)Google Scholar
  8. 8.
    Baum, P., Guentner, E., Willett R.: Exactness and the Kadison-Kaplansky Conjecture. In: Doran, R., Park, E. (eds.) Operator algebras and their applications: a tribute to Richard V. Kadison. Contemporary Mathematics, vol 671. American Mathematical Society, Providence, Rhode Island (2016)Google Scholar
  9. 9.
    Blackadar, B.: K-Theory for Operator Algebras, 2nd edn. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  10. 10.
    Blackadar, B.: Operator algebras. Theory of \(C^\ast \)-algebras and von Neumann algebras. Encyclopedia of Mathematical Sciences, vol. 122. Springer-Verlag, Berlin Heidelberg (2006)Google Scholar
  11. 11.
    Bott, R., Tolman, S., Weitsman, J.: Surjectivity for Hamiltonian loop group spaces. Invent. Math. 155, 225–251 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Braverman, M.: Index theorem for equivariant Dirac operators on noncompact manifolds. K-theory 27(1), 61–101 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bunke, U.: A K-theoretic relative index theorem and Callias-type Dirac operators. Math. Annalen 303(1), 241–279 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chernoff, P.R.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12(4), 401–414 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chernoff, P.R.: Schrödinger and Dirac operators with singular potentials and hyperbolic equations. Pac. J. Math. 72(2), 361–382 (1977)CrossRefzbMATHGoogle Scholar
  16. 16.
    Freed, D.S., Hopkins, M.J., Teleman, C.: Loop groups and twisted K-theory III. Ann. Math. 4(4), 947–1007 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Freed, D.S., Hopkins, M.J., Teleman, C.: Loop groups and twisted K-theory II. J. Am. Math. Soc. 26, 595–644 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Guillemin, V., Sternberg, S.: Geometric quantization and multiplicities of group representations. Invent. Math. 67(3), 515–538 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Higson, N.: A primer on KK-theory. Operator theory: operator algebras and applications 1, 239–283 (1990)MathSciNetGoogle Scholar
  20. 20.
    Higson, N., Roe, J.: Analytic K-Homology. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  21. 21.
    Hochs, P., Song, Y.: Equivariant indices of Spin-c Dirac operators for proper moment maps. Duke Math. J. 166, 1125–1178 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kac, V.: Infinite-Dimensional Lie Algebras. Cambridge University Press, Cambridge (1994)Google Scholar
  23. 23.
    Kasparov, G.: Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91, 147–201 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kucerovsky, D.: The KK-product of unbounded modules. K-theory 11(1), 17–34 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kucerovsky, D.: A short proof of an index theorem. In: Proceedings of the Am. Math. Soc. 3729–3736 (2001)Google Scholar
  26. 26.
    Loizides, Y.: Geometric K-homology and the Freed-Hopkins-Teleman theorem. arXiv:1804.05213
  27. 27.
    Loizides, Y.: Norm-square localization for Hamiltonian LG-spaces. J. Geom. Phys. 114, 420–449 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Loizides, Y., Meinrenken, E., Song, Y.: Spinor modules for Hamiltonian loop group spaces. arXiv:1706.07493
  29. 29.
    Loizides, Y., Song, Y.: Witten deformation for Hamiltonian loop group spaces. arXiv:1810.02347
  30. 30.
    Meinrenken, E.: Symplectic surgery and the Spin-c Dirac operator. Adv. Math. 134, 240–277 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Meinrenken, E.: Twisted K-homology and group-valued moment maps. Int. Math. Res. Not. 2012, 4563–4618 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Meinrenken, E., Woodward, C.: Hamiltonian loop group actions and Verlinde factorization. J. Diff. Geom. 50, 417–469 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mislin, G., Valette, A.: Proper Group Actions and the Baum–Connes Conjecture. Birkhäuser, Basel (2012)zbMATHGoogle Scholar
  34. 34.
    Paradan, P.-E.: Localization of the Riemann–Roch character. J. Fun. Anal. 187, 442–509 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Paradan, P.-E., Vergne, M.: Equivariant Dirac operators and differentiable geometric invariant theory. arXiv:1411.7772
  36. 36.
    Paradan, P.-E., Vergne, Michele: Witten non abelian localization for equivariant K-theory, and the \([Q, R]= 0\) theorem. arXiv:1504.07502
  37. 37.
    Phillips, N.C.: An introduction to crossed product \(C^\ast \)-algebras and minimal dynamics. Lecture notes.
  38. 38.
    Pressley, A., Segal, G.B.: Loop groups. Clarendon Press, Oxford (1986)zbMATHGoogle Scholar
  39. 39.
    Reed, M., Simon, B.: Methods of modern mathematical physics. vol. 4: Analysis of operators. Academic Press, New York-London (1978)Google Scholar
  40. 40.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Functional Analysis, vol. 1. Academic Press, New York (1980)zbMATHGoogle Scholar
  41. 41.
    Shubin, M.A.: Spectral Theory of the Schrödinger Operators on Non-compact Manifolds: Qualitative Results. Spectral Theory and Geometry, pp. 226–283. Cambridge University Press, Cambridge (1999). (Edinburgh, 1998)zbMATHGoogle Scholar
  42. 42.
    Song, Y.: Dirac operators on quasi-Hamiltonian G-spaces. J. Geom. Phys. 106, 70–86 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Takata, D.: An analytic LT-equivariant index and noncommutative geometry. arXiv:1701.06055
  44. 44.
    Takata, D.: LT-equivariant index from the viewpoint of KK-theory. arXiv:1709.06205
  45. 45.
    Tian, Y., Zhang, W.: An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg. Invent. Math. 132, 229–259 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Washington UniversityMissouriUSA

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