On the index theorem for symplectic orbifolds
Annales de l'Institut Fourier, Volume 54 (2004) no. 5, p. 1601-1639

We give an explicit construction of the trace on the algebra of quantum observables on a symplectiv orbifold and propose an index formula.

Nous donnons une construction explicite de la trace sur l’algèbre des observables quantiques sur une orbifolde symplectique et proposons une formule de l’indice.

DOI : https://doi.org/10.5802/aif.2061
Classification:  53D55,  37J10
Keywords: star-product, symmetry group, G-trace, G-index
@article{AIF_2004__54_5_1601_0,
     author = {Fedosov, Boris and Schulze, Bert-Wolfang and Tarkhanov, Nikolai},
     title = {On the index theorem for symplectic orbifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {5},
     year = {2004},
     pages = {1601-1639},
     doi = {10.5802/aif.2061},
     zbl = {1071.53055},
     mrnumber = {2127860},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2004__54_5_1601_0}
}
Fedosov, Boris; Schulze, Bert-Wolfang; Tarkhanov, Nikolai. On the index theorem for symplectic orbifolds. Annales de l'Institut Fourier, Volume 54 (2004) no. 5, pp. 1601-1639. doi : 10.5802/aif.2061. https://aif.centre-mersenne.org/item/AIF_2004__54_5_1601_0/

[1] M. F. Atiyah Elliptic operators and compact groups, Lect. Notes Math, Tome 401, Springer-Verlag, Berlin, 1974 | MR 482866 | Zbl 0297.58009

[2] F. Bayen; M. Flato; C. Fronsdal; A. Lichnerovicz; D. Sternheimer Deformation theory and quantization, Ann. Phys, Tome 111 (1978), pp. 61-151 | Zbl 0377.53025

[3] L. Boutet de Monvel; V. Guillemin The Spectral Theory of Toeplitz Operators, Princeton University Press, Princeton, NJ, 1981 | MR 620794 | Zbl 0469.47021

[4] L. Charles Aspects semi-classiques de la quantification géométrique (2000) (Thèse, Université Paris IX - Dauphine, Paris)

[5] L. Charles Spectral invariants of Toeplitz operators over symplectic two-dimensional orbifolds (2002) (Preprint, Università di Bologna)

[6] J. Duistermaat The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Birkhäuser, Boston et al, 1996 | MR 1365745 | Zbl 0858.58045

[7] F. Faure; B. Zhilinskii Qualitative features of intra-molecular dynamics. What can be learned from symmetry and topology?, Acta Applicandae Mathematicae, Tome 70 (2002), pp. 265-282 | MR 1892384 | Zbl 01747783

[8] B. Fedosov A simple geometrical construction of deformation quantization, J. Differential Geom, Tome 40 (1994), pp. 213-238 | MR 1293654 | Zbl 0812.53034

[9] B. Fedosov Deformation Quantization and Index Theory, Akademie-Verlag, Berlin, 1995 | MR 1389013 | Zbl 0867.58061

[10] B. Fedosov On normal Darboux coordinates, Amer. Math. Soc. Transl, Tome 206 (2002) no. 2, pp. 81-93 | MR 1939488 | Zbl 1032.53065

[11] B. Fedosov On the trace density in deformation quantization, Deformation Quantization. Proceedings of the Meeting of Theoretical Physicist and Mathematicians (Strasbourg, 2001) (2002), pp. 67-83 | Zbl 1014.53056

[12] B. Fedosov On G-trace and G-index in deformation quantization, Lett. Math. Phys, Tome 52 (2000), pp. 29-49 | MR 1800489 | Zbl 0998.53058

[13] T. Kawasaki The index of elliptic operators over $V$-manifolds, Nagoya Math. J, Tome 84 (1981), pp. 135-157 | MR 641150 | Zbl 0437.58020

[14] M. Pflaum On the deformation quantization of symplectic orbispaces (2003) (To appear in Differential Geometry and its Applications) | MR 2013100 | Zbl 1055.53069

[15] M. Vergne Equivariant index formula for orbifolds, Duke Math. J, Tome 82 (1996), pp. 637-652 | MR 1387687 | Zbl 0874.57029