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

SIGMA 7 (2011), 034, 12 pages      arXiv:1010.0516      https://doi.org/10.3842/SIGMA.2011.034

Natural and Projectively Invariant Quantizations on Supermanifolds

Thomas Leuther and Fabian Radoux
Institute of Mathematics, Grande Traverse 12, B-4000 Liège, Belgium

Received October 05, 2010, in final form March 23, 2011; Published online March 31, 2011

Abstract
The existence of a natural and projectively invariant quantization in the sense of P. Lecomte [Progr. Theoret. Phys. Suppl. (2001), no. 144, 125-132] was proved by M. Bordemann [math.DG/0208171], using the framework of Thomas-Whitehead connections. We extend the problem to the context of supermanifolds and adapt M. Bordemann's method in order to solve it. The obtained quantization appears as the natural globalization of the pgl(n+1|m)-equivariant quantization on Rn|m constructed by P. Mathonet and F. Radoux in [arXiv:1003.3320]. Our quantization is also a prolongation to arbitrary degree symbols of the projectively invariant quantization constructed by J. George in [arXiv:0909.5419] for symbols of degree two.

Key words: supergeometry; differential operators; projective invariance; quantization maps.

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