Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 111, 11 pages      arXiv:0912.5190

Second-Order Conformally Equivariant Quantization in Dimension 1|2

Najla Mellouli
Institut Camille Jordan, UMR 5208 du CNRS, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France

Received September 22, 2009, in final form December 13, 2009; Published online December 28, 2009

This paper is the next step of an ambitious program to develop conformally equivariant quantization on supermanifolds. This problem was considered so far in (super)dimensions 1 and 1|1. We will show that the case of several odd variables is much more difficult. We consider the supercircle S1|2 equipped with the standard contact structure. The conformal Lie superalgebra K(2) of contact vector fields on S1|2 contains the Lie superalgebra osp(2|2). We study the spaces of linear differential operators on the spaces of weighted densities as modules over osp(2|2). We prove that, in the non-resonant case, the spaces of second order differential operators are isomorphic to the corresponding spaces of symbols as osp(2|2)-modules. We also prove that the conformal equivariant quantization map is unique and calculate its explicit formula.

Key words: equivariant quantization; conformal superalgebra.

pdf (225 kb)   ps (164 kb)   tex (13 kb)


  1. Cohen P., Manin Yu., Zagier D., Automorphic pseudodifferential operators, in Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl., Vol. 26, Birkhäuser Boston, Boston, MA, 1997, 17-47.
  2. Conley C., Conformal symbols and the action of contact vector fields over the superline, J. Reine Angew. Math. 633 (2009), 115-163, arXiv:0712.1780.
  3. Duval C., Lecomte P., Ovsienko V., Conformally equivariant quantization: existence and uniqueness, Ann. Inst. Fourier (Grenoble) 49 (1999), 1999-2029, math.DG/9902032.
  4. Fregier Y., Mathonet P., Poncin N., Decomposition of symmetric tensor fields in the presence of a flat contact projective structure, J. Nonlinear Math. Phys. 15 (2008), 252-269, math.DG/0703922.
  5. Gargoubi H., Mellouli N., Ovsienko V., Differential operators on supercircle: conformally equivariant quantization and symbol calculus, Lett. Math. Phys. 79 (2007), 51-65, math-ph/0610059.
  6. Grozman P., Leites D., Shchepochkina I., Lie superalgebras of string theories, Acta Math. Vietnam. 26 (2001), 27-63, hep-th/9702120.
  7. Grozman P., Leites D., Shchepochkina I., Invariant operators on supermanifolds and standard models, in Multiple Facets of Quantization and Supersymmetry, Editors M. Olshanetski and A. Vainstein, World Sci. Publ., River Edge, NJ, 2002, 508-555, math.RT/0202193.
  8. Lecomte P.B.A., Ovsienko V.Yu., Projectively invariant symbol calculus, Lett. Math. Phys. 49 (1999), 173-196, math.DG/9809061.
  9. Leites D., Supermanifold theory, Petrozavodsk, 1983 (in Russian).
  10. Leites D., Kochetkov Yu., Weintrob A., New invariant differential operators on supermanifolds and pseudo-(co)homology, in General Topology and Applications (Staten Island, NY, 1989), Lecture Notes in Pure and Appl. Math., Vol. 134, Dekker, New York, 1991, 217-238.
  11. Michel J.-P., Duval C., On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative, Int. Math. Res. Not. IMRN 2008 (2008), no. 14, Art. ID rnn054, 47 pages, arXiv:0710.1544.
  12. Ovsienko V., Vector fields in the presence of a contact structure, Enseign. Math. (2) 52 (2006), 215-229, math.DG/0511499.
  13. Ovsienko V.Yu., Ovsienko O.D., Chekanov Yu.V., Classification of contact-projective structures on the supercircle, Russian Math. Surveys 44 (1989), no. 3, 212-213.
  14. Shchepochkina I.M., How to realize Lie algebras by vector fields, Theoret. and Math. Phys. 147 (2006), 821-838, math.RT/0509472.

Previous article   Contents of Volume 5 (2009)