
SIGMA 5 (2009), 111, 11 pages arXiv:0912.5190
https://doi.org/10.3842/SIGMA.2009.111
SecondOrder Conformally Equivariant Quantization in Dimension 12
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
Abstract
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 11. We will
show that the case of several odd variables is much more difficult.
We consider the supercircle S^{12} equipped with the standard
contact structure. The conformal Lie superalgebra K(2)
of contact vector fields on S^{12} contains the Lie superalgebra
osp(22). We study the spaces of linear differential
operators on the spaces of weighted densities as modules over
osp(22). We prove that, in the nonresonant case, the
spaces of second order differential operators are isomorphic to the
corresponding spaces of symbols as osp(22)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)
References
 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, 1747.
 Conley C.,
Conformal symbols and the action of contact vector fields over the superline,
J. Reine Angew. Math. 633 (2009), 115163,
arXiv:0712.1780.
 Duval C., Lecomte P., Ovsienko V.,
Conformally equivariant quantization: existence and uniqueness,
Ann. Inst. Fourier (Grenoble) 49 (1999), 19992029,
math.DG/9902032.
 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), 252269,
math.DG/0703922.
 Gargoubi H., Mellouli N., Ovsienko V.,
Differential operators on supercircle: conformally equivariant quantization and symbol calculus,
Lett. Math. Phys. 79 (2007), 5165,
mathph/0610059.
 Grozman P., Leites D., Shchepochkina I.,
Lie superalgebras of string theories,
Acta Math. Vietnam. 26 (2001), 2763,
hepth/9702120.
 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, 508555,
math.RT/0202193.
 Lecomte P.B.A., Ovsienko V.Yu.,
Projectively invariant symbol calculus,
Lett. Math. Phys. 49 (1999), 173196,
math.DG/9809061.
 Leites D.,
Supermanifold theory, Petrozavodsk, 1983 (in Russian).
 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, 217238.
 Michel J.P., Duval C.,
On the projective geometry of the supercircle: a unified construction of the super crossratio and Schwarzian derivative,
Int. Math. Res. Not. IMRN 2008 (2008), no. 14, Art. ID rnn054, 47 pages,
arXiv:0710.1544.
 Ovsienko V.,
Vector fields in the presence of a contact structure,
Enseign. Math. (2) 52 (2006), 215229,
math.DG/0511499.
 Ovsienko V.Yu., Ovsienko O.D., Chekanov Yu.V.,
Classification of contactprojective structures on the supercircle,
Russian Math. Surveys 44 (1989), no. 3, 212213.
 Shchepochkina I.M.,
How to realize Lie algebras by vector fields,
Theoret. and Math. Phys. 147 (2006), 821838,
math.RT/0509472.

