Geodesic
Deformation quantization of framed presymplectic manifolds
Teoretičeskaâ i matematičeskaâ fizika, Tome 204 (2020) no. 2, pp. 280-296 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the problem of deformation quantization of presymplectic manifolds in the framework of the Fedosov method. A class of special presymplectic manifolds is distinguished for which such a quantization can always be constructed. We show that in the general case, the obstructions to quantization can be identified with some special elements of the third cohomology group of a differential ideal associated with the presymplectic structure.
Keywords: presymplectic manifold
Mots-clés : deformation quantization, Fedosov quantization. 
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N. D. Gorev; B. M. Elfimov; A. A. Sharapov. Deformation quantization of framed presymplectic manifolds. Teoretičeskaâ i matematičeskaâ fizika, Tome 204 (2020) no. 2, pp. 280-296. http://geodesic.mathdoc.fr/item/TMF_2020_204_2_a7/

[1] B. V. Fedosov, “A simple geometrical construction of deformation quantization”, J. Differ. Geom., 40:2 (1994), 213–238 | DOI | MR | Zbl

[2] B. V. Fedosov, Deformation Quantization and Index Theory, Mathematical Topics, 9, Akademie, Berlin, 1996 | MR

[3] M. A. Grigoriev, S. L. Lyakhovich, “Fedosov deformation quantization as a BRST theory”, Commun. Math. Phys., 218:2 (2001), 437–457 | DOI | MR

[4] I. A. Batalin, M. A. Grigorev, S. L. Lyakhovich, “$\ast$-Umnozhenie dlya sistem so svyazyami vtorogo roda iz BRST-teorii”, TMF, 128:3 (2001), 324–360 | DOI | DOI | MR | Zbl

[5] K. Bering, “Three natural generalizations of Fedosov quantization”, SIGMA, 5 (2009), 036, 21 pp. | DOI | MR | Zbl

[6] M. Bordemann, N. Neumaier, S. Waldmann, “Homogeneous Fedosov star products on cotangent bundles I: Weyl and standard ordering with differential operator representation”, Commun. Math. Phys., 198:2 (1998), 363–396, arXiv: q-alg/9707030 | DOI | MR

[7] A. V. Karabegov, M. Schlichenmaier, “Almost-Kähler deformation quantization”, Lett. Math. Phys., 57:2 (2001), 135–148 | DOI | MR

[8] V. A. Dolgushev, A. P. Isaev, S. L. Lyakhovich, A. A. Sharapov, “On the Fedosov deformation quantization beyond the regular Poisson manifolds”, Nucl. Phys. B, 645:3 (2002), 457–476 | DOI | MR

[9] S. L. Lyakhovich, A. A. Sharapov, “BRST quantization of quasi-symplectic manifolds and beyond”, J. Math. Phys., 47:4 (2006), 043508, 26 pp., arXiv: hep-th/0312075 | DOI | MR

[10] M. J. Gotay, J. Śniatycki, “On the quantization of presymplectic dynamical systems via coisotropic imbedding”, Commun. Math. Phys., 82:3 (1981), 377–389 | DOI | MR

[11] I. Vaisman, “Geometric quantization on presymplectic manifolds”, Monatsh. Math., 96:4 (1983), 293–310 | DOI | MR

[12] C. Crnković, E. Witten, “Covariant description of canonical formalism in geometrical theories”, Three Hundred Years of Gravitation, eds. S. W. Hawking, W. Israel, Cambridge Univ. Press, Cambridge, 1987, 676–684 | MR

[13] G. J. Zuckerman, “Action principles and global geometry”, Mathematical Aspects of String Theory (University of California, San Diego July 21 – August 1, 1986), Advanced Series in Mathematical Physics, 1, ed. S. T. Yau, World Sci., Singapore, 1987, 259–284 | DOI | MR | Zbl

[14] T. J. Bridges, P. E. Hydon, J. K. Lawson, “Multisymplectic structures and the variational bicomplex”, Math. Proc. Cambridge Philos. Soc., 148:1 (2010), 159–178 | DOI | MR

[15] A. A. Sharapov, “Variational tricomplex, global symmetries and conservation laws of gauge systems”, SIGMA, 12 (2016), 098, 24 pp., arXiv: 1607.01626 | DOI | MR

[16] A. A. Sharapov, “On presymplectic structures for massless higher-spin fields”, Eur. Phys. J. C, 76:6 (2016), 305, 16 pp. | DOI

[17] Y. Agaoka, “On a generalization of Cartan's lemma”, J. Algebra, 127:2 (1989), 470–507 | DOI | MR

[18] D. B. Fuks, “Kogomologii beskonechnomernykh algebr Li i kharakteristicheskie klassy sloenii”, Itogi nauki i tekhn. Ser. Sovrem. probl. matem., 10 (1978), 179–285 | DOI | MR | Zbl

[19] P. Tondeur, Geometry of Foliations, Monographs in Mathematics, 90, Birkhäuser, Basel, 1997 | DOI | MR

[20] A. G. Elashvili, “Frobeniusovy algebry Li”, Funkts. analiz i ego pril., 16:4 (1982), 94–95 | DOI | MR | Zbl