Abstract
We study the actions of local conformal vector fields \({X \in {\rm conf}(M,g)}\) on the spinor bundle of (M, g) and on its classical counterpart: the supercotangent bundle \({\mathcal{M}}\) of (M, g). We first deal with the classical framework and determine the Hamiltonian lift of conf (M, g) to \({\mathcal{M}}\) . We then perform the geometric quantization of the supercotangent bundle of (M, g), which constructs the spinor bundle as the quantum representation space. The Kosmann Lie derivative of spinors is obtained by quantization of the comoment map.
The quantum and classical actions of conf (M, g) turn, respectively, the space of differential operators acting on spinor densities and the space of their symbols into conf (M, g)-modules. They are filtered and admit a common associated graded module. In the conformally flat case, the latter helps us determine the conformal invariants of both conf (M, g)-modules, in particular the conformally odd powers of the Dirac operator.
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Aldaya, V., Guerrero, J., Marmo, G.: Quantization on a Lie group: higher-order polarizations. In: Symmetries in science, X (Bregenz, 1997). New York: Plenum, 1998, pp. 1–36
Barducci A., Casalbuoni R., Lusanna L.: Classical spinning particles interacting with external gravitational fields. Nucl. Phys. B 124, 521–538 (1977)
Berezin F.A., Marinov M.S.: Particle spin dynamics as the grassmann variant of classical mechanics. Ann. of Phys. 104, 336–362 (1977)
Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Grundlehren Text Editions. Berlin: Springer-Verlag, 2004. (Corrected reprint of the 1992 original)
Blattner, R.J.: Quantization and representation theory. In: Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972). Providence, RI: Amer. Math. Soc., 1973, pp. 147–165
Bordemann, M.: The deformation quantization of certain super-Poisson brackets and BRST cohomology. In Conférence Moshé Flato 1999, Vol. II (Dijon), Volume 22 of Math. Phys. Stud. Dordrecht: Kluwer Acad. Publ., 2000, pp. 45–68
Bourguignon J.-P., Gauduchon P.: Spineurs, opérateurs de Dirac et variations de métriques. Commun. Math. Phys. 144(3), 581–599 (1992)
Branson T.P.: Sharp inequalities, the functional determinant, and the complementary series. Trans. Amer. Math. Soc. 347(10), 3671–3742 (1995)
Branson T.P.: Second order conformal covariants. Proc. Amer. Math. Soc. 126(4), 1031–1042 (1998)
Cap A., Gover A.R., Soucek V.: Conformally invariant operators via curved casimirs: Examples. Pure Appl. Math. Q. 6(3), 693–714 (2010)
DeWitt, B.: Supermanifolds. Cambridge Monographs on Mathematical Physics. Second edn. Cambridge: Cambridge University Press, 1992
Duval C., Lecomte P.B.A, Ovsienko V.Yu.: Conformally equivariant quantization: existence and uniqueness. Ann. Inst. Fourier (Grenoble) 49(6), 1999–2029 (1999)
Eastwood M.G.: Higher symmetries of the Laplacian. Ann. of Math. (2) 161(3), 1645–1665 (2005)
Eastwood, M.G., Leistner, T.: Higher symmetries of the square of the Laplacian. In Symmetries and overdetermined systems of partial differential equations, Volume 144 of IMA Vol. Math. Appl. New York: Springer, 2008, pp. 319–338
Eastwood, M.G., Rice, J.W.: Conformally invariant differential operators on Minkowski space and their curved analogues. Commun. Math. Phys. 109(2), 207–228 (1987). Erratum Commun. Math. Phys. 144(1), 213 (1992)
Eelbode D., Souček V.: Conformally invariant powers of the Dirac operator in Clifford analysis. Math. Methods Appl. Sci. 33(13), 1558–1570 (2010)
El Gradechi A.M., Nieto L.M.: Supercoherent states, super-Kähler geometry and geometric quantization. Commun. Math. Phys. 175(3), 521–563 (1996)
Ferrara S., Lledó M.A.: Some aspects of deformations of supersymmetric field theories. J. High Energy Phys. 05, 008 (2000)
Getzler E.: Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem. Commun. Math. Phys. 92(2), 163–178 (1983)
Godina M., Matteucci P.: The Lie derivative of spinor fields: theory and applications. Int. J. Geom. Methods Mod. Phys. 2(2), 159–188 (2005)
Gover A.R., Peterson L.J.: Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus. Commun. Math. Phys. 235(2), 339–378 (2003)
Graham C.R., Jenne R., Mason L.J., Sparling G.A.J.: Conformally invariant powers of the Laplacian. I. Existence. J. London Math. Soc. (2) 46(3), 557–565 (1992)
Hitchin N.: Harmonic spinors. Adv. in Math. 14, 1–55 (1974)
Khudaverdian O.M.: Geometry of superspace with even and odd brackets. J. Math. Phys. 32(7), 1934–1937 (1991)
Kosmann Y.: Dérivées de Lie des spineurs. Ann. Mat. Pura Appl. 91(4), 317–395 (1972)
Kostant, B.: Quantization and unitary representations. I. Prequantization. In: Lectures in modern analysis and applications, III, Lecture Notes in Math., Vol. 170. Berlin: Springer, 1970, pp. 87–208
Kostant, B.: Symplectic spinors. In: Symposia Mathematica, Vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973). London: Academic Press, 1974, pp. 139–152
Kostant, B.: Graded manifolds, graded Lie theory, and prequantization. In: Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975). Lecture Notes in Math., Vol. 570. Berlin: Springer, 1977, pp. 177–306
Lawson, H.B., Michelsohn, M.-L.: Spin geometry, Volume 38 of Princeton Mathematical Series. Princeton, NJ: Princeton University Press, 1989
Leĭtes D.A.: New Lie superalgebras, and mechanics. Dokl. Akad. Nauk. SSSR 236(4), 804–807 (1977)
Leĭtes, D.A.: Introduction to the theory of supermanifolds. Uspekhi Mat. Nauk. 35(1(211)), 3–57, 255, (1980)
Mathonet P., Radoux F.: On natural and conformally equivariant quantizations. J. Lond. Math. Soc., II. Ser. 80(1), 256–272 (2009)
Michel, J.-Ph.: Quantification conformément équivariante des fibrés supercotangents. PhD thesis, Université Aix-Marseille II, 2009. available at http://tel.archives-ouvertes.fr/tel-00425576_v1, 2009
Michel, J.-Ph.: Conformally equivariant quantization - a complete classification. SIGMA, 8, Paper 022 (2012)
Musson, I.M., Pinczon, G., Ushirobira, R.: Hochschild cohomology and deformations of Clifford-Weyl algebras. Sigma, 5, Paper 028 (2009)
Nurowski P., Trautman A.: Robinson manifolds as the Lorentzian analogs of Hermite manifolds. Diff. Geom. Appl. 17(2–3), 175–195 (2002)
Ovsienko V.Yu., Redou P.: Generalized transvectants-Rankin-Cohen brackets. Lett. Math. Phys. 63(1), 19–28 (2003)
Palese, M., Winterroth, E.: Noether identities in Einstein-Dirac theory and the Lie derivative of spinor fields. In: Differential geometry and its applications. Hackensack, NJ: World Sci. Publ., 2008, pp. 643–653
Paneitz, S.M.: A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary). SIGMA, 4, Paper 036 (2008)
Papapetrou A.: Spinning test-particles in general relativity. I. Proc. Roy. Soc. London. Ser. A. 209, 248–258 (1951)
Radoux F.: An explicit formula for the natural and conformally invariant quantization. Lett. Math. Phys. 89(3), 249–263 (2009)
Ravndal F.: Supersymmetric Dirac particles in external fields. Phys. Rev. D (3) 21(10), 2823–2832 (1980)
Rothstein, M.: The structure of supersymplectic supermanifolds. In Differential geometric methods in theoretical physics (Rapallo, 1990), Volume 375 of Lecture Notes in Phys. Berlin: Springer, 1991, pp. 331–343
Roytenberg, D.: On the structure of graded symplectic supermanifolds and Courant algebroids. In: Quantization, Poisson brackets and beyond (Manchester, 2001), Volume 315 of Contemp. Math. Providence, RI: Amer. Math. Soc., 2002, pp. 169–185
Silhan, J.: Conformally invariant quantization - towards complete classification. http://arxiv.org/abs/0903.4798v1 [math.DG], 2009
Silhan J., Silhan J., Silhan J.: Higher symmetries of the conformal powers of the laplacian on conformally flat manifolds. J. Math. Phys. 53(3), 26 (2012)
Souriau, J.-M.: Structure des systèmes dynamiques. Maitrises de mathématiques. Paris: Dunod, 1970 (© 1969)
Sparling, G.A.J., Holland, J.E.: Conformally invariant powers of the ambient Dirac operator. http://arxiv.org/abs/math/0112033v2 [math.DG], 2001
Trautman A.: Connections and the Dirac operator on spinor bundles. J. Geom. Phys. 58, 238–252 (2008)
Tuynman G.M.: Geometric quantization of the BRST charge. Commun. Math. Phys. 150(2), 237–265 (1992)
Tuynman, G.M.: Supermanifolds and supergroups, Volume 570 of Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers, 2004
Voronov, F.F.: Quantization on supermanifolds and an analytic proof of the Atiyah-Singer index theorem. In: Current problems in mathematics. Newest results, Vol. 38 (Russian), Itogi Nauki i Tekhniki, 186. Moscow: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., 1990, pp. 3–118. Translated in J. Soviet Math. 64(4), 993–1069 (1993)
Weyl, H.: The classical groups. Their invariants and representations. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press, 1997
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Communicated by N. A. Nekrasov
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Michel, JP. Conformal Geometry of the Supercotangent and Spinor Bundles. Commun. Math. Phys. 312, 303–336 (2012). https://doi.org/10.1007/s00220-012-1475-2
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DOI: https://doi.org/10.1007/s00220-012-1475-2