Formal geometric quantization

Paradan, Paul-Émile

doi:10.5802/aif.2429

Paradan, Paul-Émile ¹

¹ Université Montpellier II Institut de Mathématiques et de Modélisation de Montpellier (I3M) Place Eugène Bataillon 34095 MONTPELLIER (France)

Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 199-238.

Abstract (Original language)
Résumé

Let $K$ be a compact Lie group acting in a Hamiltonian way on a symplectic manifold $(M, Ω)$ which is pre-quantized by a Kostant-Souriau line bundle. We suppose here that the moment map $Φ$ is proper so that the reduced space $M_{μ} : = Φ^{- 1} (K \cdot μ) / K$ is compact for all $μ$ . Then, we can define the “formal geometric quantization” of $M$ as

𝒬_{K}^{- \infty} (M) : = \sum_{μ \in \hat{K}} 𝒬 (M_{μ}) V_{μ}^{K} .

The aim of this article is to study the functorial properties of the assignment $(M, K) \to 𝒬_{K}^{- \infty} (M)$ .

Considérons l’action hamiltonienne d’un groupe de Lie compact $K$ sur une variété symplectique $(M, Ω)$ préquantifiée par un fibré en droites de Kostant-Souriau. On suppose que l’application moment $Φ$ est propre, ainsi les réductions symplectiques $M_{μ} : = Φ^{- 1} (K \cdot μ) / K$ sont compactes pour tout $μ$ . On peut alors définir la quantification formelle de $M$ comme

𝒬_{K}^{- \infty} (M) : = \sum_{μ \in \hat{K}} 𝒬 (M_{μ}) V_{μ}^{K} .

Le but de ce travail est l’étude de certaines propriétés fonctorielles de l’application $(M, K) \to 𝒬_{K}^{- \infty} (M)$ .

MR Zbl

DOI: 10.5802/aif.2429

Classification: 58F06, 57S15, 19L47, 19L10
Keywords: Geometric quantization, moment map, symplectic reduction, index, transversally elliptic
Mots-clés : quantification géométrique, application moment, réduction symplectique, indice, transversalement elliptique

Author's affiliations:

Paradan, Paul-Émile ¹

¹ Université Montpellier II Institut de Mathématiques et de Modélisation de Montpellier (I3M) Place Eugène Bataillon 34095 MONTPELLIER (France)

@article{AIF_2009__59_1_199_0,
     author = {Paradan, Paul-\'Emile},
     title = {Formal geometric quantization},
     journal = {Annales de l'Institut Fourier},
     pages = {199--238},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     doi = {10.5802/aif.2429},
     mrnumber = {2514864},
     zbl = {1163.53056},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2429/}
}

TY  - JOUR
AU  - Paradan, Paul-Émile
TI  - Formal geometric quantization
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 199
EP  - 238
VL  - 59
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2429/
DO  - 10.5802/aif.2429
LA  - en
ID  - AIF_2009__59_1_199_0
ER  -

%0 Journal Article
%A Paradan, Paul-Émile
%T Formal geometric quantization
%J Annales de l'Institut Fourier
%D 2009
%P 199-238
%V 59
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2429/
%R 10.5802/aif.2429
%G en
%F AIF_2009__59_1_199_0

Paradan, Paul-Émile. Formal geometric quantization. Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 199-238. doi : 10.5802/aif.2429. https://aif.centre-mersenne.org/articles/10.5802/aif.2429/

References
Cited by

[1] Atiyah, M. F. Convexity and commuting Hamiltonians, Bull. London Math. Soc., Volume 14 (1982) no. 1, pp. 1-15 | DOI | MR | Zbl

[2] Atiyah, M. F.; Segal, G. B. The index of elliptic operators. II, Ann. of Math. (2), Volume 87 (1968), pp. 531-545 | DOI | MR | Zbl

[3] Atiyah, M. F.; Singer, I. M. The index of elliptic operators. I, Ann. of Math. (2), Volume 87 (1968), pp. 484-530 | DOI | MR | Zbl

[4] Atiyah, M. F.; Singer, I. M. The index of elliptic operators. III, Ann. of Math. (2), Volume 87 (1968), pp. 546-604 | DOI | MR | Zbl

[5] Atiyah, M. F.; Singer, I. M. The index of elliptic operators. IV, Ann. of Math. (2), Volume 93 (1971), pp. 139-149 | DOI | MR | Zbl

[6] Atiyah, Michael Francis Elliptic operators and compact groups, Lecture Notes in Mathematics, Vol. 401, Springer-Verlag, Berlin, 1974 | MR | Zbl

[7] Berline, Nicole; Getzler, Ezra; Vergne, Michèle Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 298, Springer-Verlag, Berlin, 1992 | MR | Zbl

[8] Berline, Nicole; Vergne, Michèle The Chern character of a transversally elliptic symbol and the equivariant index, Invent. Math., Volume 124 (1996) no. 1-3, pp. 11-49 | DOI | MR | Zbl

[9] Berline, Nicole; Vergne, Michèle L’indice équivariant des opérateurs transversalement elliptiques, Invent. Math., Volume 124 (1996) no. 1-3, pp. 51-101 | DOI | MR | Zbl

[10] Brion, Michel Variétés sphériques, Opérations hamiltoniennes et opération de groupes algébriques, Notes de la session de S.M.F., Grenoble, 1997, pp. 1-60

[11] Brion, Michel The behaviour at infinity of the Bruhat decomposition, Comment. Math. Helv., Volume 73 (1998) no. 1, pp. 137-174 | DOI | MR | Zbl

[12] Brylinski, Jean-Luc Décomposition simpliciale d’un réseau, invariante par un groupe fini d’automorphismes, C. R. Acad. Sci. Paris Sér. A-B, Volume 288 (1979) no. 2, p. A137-A139 | MR | Zbl

[13] Colliot-Thélène, J.-L.; Harari, D.; Skorobogatov, A. N. Compactification équivariante d’un tore (d’après Brylinski et Künnemann), Expo. Math., Volume 23 (2005) no. 2, pp. 161-170 | MR | Zbl

[14] De Concini, C.; Procesi, C. Complete symmetric varieties, Invariant theory (Montecatini, 1982) (Lecture Notes in Math.), Volume 996, Springer, Berlin, 1983, pp. 1-44 | Zbl

[15] De Concini, C.; Procesi, C. Complete symmetric varieties. II. Intersection theory, Algebraic groups and related topics (Kyoto/Nagoya, 1983) (Adv. Stud. Pure Math.), Volume 6, North-Holland, Amsterdam, 1985, pp. 481-513 | Zbl

[16] Duistermaat, J. J. The heat kernel Lefschetz fixed point formula for the spin- $c$ Dirac operator, Progress in Nonlinear Differential Equations and their Applications, 18, Birkhäuser Boston Inc., Boston, MA, 1996 | MR | Zbl

[17] Guillemin, V.; Sternberg, S. Geometric quantization and multiplicities of group representations, Invent. Math., Volume 67 (1982) no. 3, pp. 515-538 | DOI | MR | Zbl

[18] Jeffrey, Lisa C.; Kirwan, Frances C. Localization and the quantization conjecture, Topology, Volume 36 (1997) no. 3, pp. 647-693 | DOI | MR | Zbl

[19] Kempf, George; Ness, Linda The length of vectors in representation spaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978) (Lecture Notes in Math.), Volume 732, Springer, Berlin, 1979, pp. 233-243 | MR | Zbl

[20] Kirwan, Frances Clare Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, 31, Princeton University Press, Princeton, NJ, 1984 | MR | Zbl

[21] Kostant, Bertram Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Springer, Berlin, 1970, p. 87-208. Lecture Notes in Math., Vol. 170 | MR | Zbl

[22] Lerman, Eugene Symplectic cuts, Math. Res. Lett., Volume 2 (1995) no. 3, pp. 247-258 | MR | Zbl

[23] Lerman, Eugene; Meinrenken, Eckhard; Tolman, Sue; Woodward, Chris Nonabelian convexity by symplectic cuts, Topology, Volume 37 (1998) no. 2, pp. 245-259 | DOI | MR | Zbl

[24] Meinrenken, Eckhard On Riemann-Roch formulas for multiplicities, J. Amer. Math. Soc., Volume 9 (1996) no. 2, pp. 373-389 | DOI | MR | Zbl

[25] Meinrenken, Eckhard Symplectic surgery and the ${Spin}^{c}$ -Dirac operator, Adv. Math., Volume 134 (1998) no. 2, pp. 240-277 | DOI | MR | Zbl

[26] Meinrenken, Eckhard; Sjamaar, Reyer Singular reduction and quantization, Topology, Volume 38 (1999) no. 4, pp. 699-762 | DOI | MR | Zbl

[27] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, Springer-Verlag, Berlin, 1994 | MR | Zbl

[28] Oda, Tadao Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 15, Springer-Verlag, Berlin, 1988 (An introduction to the theory of toric varieties, Translated from the Japanese) | MR | Zbl

[29] Paradan, Paul-Emile Localization of the Riemann-Roch character, J. Funct. Anal., Volume 187 (2001) no. 2, pp. 442-509 | DOI | MR | Zbl

[30] Sjamaar, Reyer Symplectic reduction and Riemann-Roch formulas for multiplicities, Bull. Amer. Math. Soc. (N.S.), Volume 33 (1996) no. 3, pp. 327-338 | DOI | MR | Zbl

[31] Tian, Youliang; Zhang, Weiping An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg, Invent. Math., Volume 132 (1998) no. 2, pp. 229-259 | DOI | MR | Zbl

[32] Vergne, Michele Multiplicities formula for geometric quantization. I, II, Duke Math. J., Volume 82 (1996) no. 1, p. 143-179, 181–194 | DOI | MR | Zbl

[33] Vergne, Michèle Quantification géométrique et réduction symplectique, Astérisque (2002) no. 282, pp. 249-278 (Séminaire Bourbaki, Vol. 2000/2001, Exp. No. 888, viii) | Numdam | MR | Zbl

[34] Weitsman, Jonathan Non-abelian symplectic cuts and the geometric quantization of noncompact manifolds, Lett. Math. Phys., Volume 56 (2001) no. 1, pp. 31-40 EuroConférence Moshé Flato 2000, Part I (Dijon) | DOI | MR | Zbl

[35] Witten, Edward Two-dimensional gauge theories revisited, J. Geom. Phys., Volume 9 (1992) no. 4, pp. 303-368 | DOI | MR | Zbl

[36] Woodward, Chris The classification of transversal multiplicity-free group actions, Ann. Global Anal. Geom., Volume 14 (1996) no. 1, pp. 3-42 | DOI | MR | Zbl

Cited by Sources:

ANNALES DE L'INSTITUT FOURIER

ARTICLES TO APPEAR

BROWSE ISSUES

ABOUT THE JOURNAL

EDITORIAL BOARD

SUBMIT A PAPER

SUBSCRIPTION