Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-13T05:58:53.732Z Has data issue: false hasContentIssue false
  • English
  • Français

Deformation of Dirac operators along orbits and quantization of noncompact Hamiltonian torus manifolds

Part of: $K$-theory and operator algebras Symplectic geometry, contact geometry Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  09 March 2021

Hajime Fujita
Hajime Fujita*
Affiliation:
Department of Mathematical and Physical Sciences, Japan Women’s University, Tokyo, Japan
*
e-mail: fujitah@fc.jwu.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a formulation of a deformation of Dirac operator along orbits of a group action on a possibly noncompact manifold to get an equivariant index and a K-homology cycle representing the index. We apply this framework to noncompact Hamiltonian torus manifolds to define geometric quantization from the viewpoint of index theory. We give two applications. The first one is a proof of a [Q,R]=0 type theorem, which can be regarded as a proof of the Vergne conjecture for abelian case. The other is a Danilov-type formula for toric case in the noncompact setting, which is a localization phenomenon of geometric quantization in the noncompact setting. The proofs are based on the localization of index to lattice points.

Keywords

Equivariant indexgeometric quantizationlocalization

MSC classification

Primary: 19K56: Index theory
Secondary: 53D50: Geometric quantization 57S25: Groups acting on specific manifolds 58J22: Exotic index theories
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 4 , August 2022 , pp. 1062 - 1092
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author is partly supported by Grant-in-Aid for Scientific Research (C) 18K03288.

References

Andersen, J. E., Geometric quantization of symplectic manifolds with respect to reducible non-negative polarizations. Comm. Math. Phys. 183(1997), no. 2, 401–421.CrossRefGoogle Scholar
Anghel, N., An abstract index theorem on noncompact Riemannian manifolds. Houst. J. Math. 19(1993), no. 2, 223–237.Google Scholar
Atiyah, M. F., Elliptic operators and compact groups. Lecture Notes in Mathematics, 401, Springer-Verlag, Berlin, Germany, 1974.CrossRefGoogle Scholar
Atiyah, M. F. and Singer, I. M., The index of elliptic operators, I. Ann. Math. (2) 87(1968), 484–530.CrossRefGoogle Scholar
Blackadar, B., K-theory for operator algebras. 2nd ed., Mathematical Sciences Research Institute Publications, 5, Cambridge University Press,Cambridge, MA, 1998.Google Scholar
Braverman, M., Index theorem for equivariant Dirac operators on noncompact manifolds . K-Theory 27(2002), no. 1, 61–101.CrossRefGoogle Scholar
Fujita, H., S 1-equivariant local index and transverse index for non-compact symplectic manifolds . Math. Res. Lett. 23(2016), no. 5, 1351–1367.CrossRefGoogle Scholar
Fujita, H., A Danilov-type formula for toric origami manifolds via localization of index. Osaka J. Math. 55(2018), no. 4, 619–645.Google Scholar
Fujita, H., Furuta, M., and Yoshida, T., Torus fibrations and localization of index I—polarization and acyclic fibrations. J. Math. Sci. Univ. Tokyo 17(2010), no. 1, 1–26.Google Scholar
Fujita, H., Furuta, M., and Yoshida, T., Torus fibrations and localization of index II: local index for acyclic compatible system. Comm. Math. Phys. 326(2014), no. 3, 585–633.CrossRefGoogle Scholar
Fujita, H., Furuta, M., and Yoshida, T., Torus fibrations and localization of index III: equivariant version and its applications. Commun. Math. Phys. 327(2014), no. 3, 665–689.CrossRefGoogle Scholar
Higson, N. and Roe, J., Analytic K-homology. Oxford Mathematical Monographs, Oxford University Press and Oxford Science Publications, Oxford, 2000.Google Scholar
Hochs, P. and Song, Y., An equivariant index for proper actions I. J. Funct. Anal. 272(2017), no. 2, 661–704.CrossRefGoogle Scholar
Kasparov, G., Elliptic and transversally elliptic index theory from the viewpoint of KK-theory. J. Noncommut. Geom. 10(2016), no. 4, 1303–1378.CrossRefGoogle Scholar
Kubota, Y., The joint spectral flow and localization of the indices of elliptic operators. Ann. K-Theory 1(2016), no. 1, 43–83.CrossRefGoogle Scholar
Loizides, Y., Meinrenken, E., and Song, Y., Spinor modules for Hamiltonian loop group spaces. J. Symplectic Geom. 18(2020), no. 3, 889–937.CrossRefGoogle Scholar
Loizides, Y., Rodsphon, R., and Song, Y., A KK-theoretic perspective on deformed Dirac operators. Adv. Math. 380(2021), 107604.CrossRefGoogle Scholar
Loizides, Y. and Song, Y., Quantization of Hamiltonian loop group spaces. Math. Ann. 374(2019), nos. 1–2, 681–722.CrossRefGoogle Scholar
Loizides, Y. and Song, Y., Norm-square localization and the quantization of Hamiltonian loop group spaces. J. Funct. Anal. 278(2020), no. 9, 108445, 45p.CrossRefGoogle Scholar
Ma, X. and Zhang, W., Geometric quantization for proper moment maps: the Vergne conjecture. Acta Math. 212(2014), no. 1, 11–57.CrossRefGoogle Scholar
Nicolaescu, L. I., On the space of Fredholm operators. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 53(2007), no. 2, 209–227.Google Scholar
Paradan, P.-E., Localization of the Riemann–Roch character. J. Funct. Anal. 187(2001), no. 2, 442–509.CrossRefGoogle Scholar
Takata, D., An infinite-dimensional index theorem and the Higson–Kasparov–Trout algebra. Preprint, 2018. arxiv:1811.06811 Google Scholar
Takata, D., An analytic LT-equivariant index and noncommutative geometry. J. Noncommut. Geom. 13(2019), no. 2, 553–586.CrossRefGoogle Scholar
Takata, D., LT-equivariant index from the viewpoint of KK-theory. A global analysis on the infinite-dimensional Heisenberg group. J. Geom. Phys. 150(2020), 103591.CrossRefGoogle Scholar
Tian, Y. and Zhang, W., An analytic proof of the geometric quantization conjecture of Guillemin–Sternberg. Invent. Math. 132(1998), no. 2, 229–259.CrossRefGoogle Scholar
Witten, E., Supersymmetry and Morse theory. J. Differ. Geom. 17(1982/1983), no. 4, 661–692.Google Scholar
Yoshida, T., Adiabatic limits, theta functions, and geometric quantization. Preprint, 2019. arxiv:1904.04076 Google Scholar