Natural and Projectively Invariant Quantizations on Supermanifolds

Details Natural and Projectively Invariant Quantizations on Supermanifolds Natural and Projectively Invariant Quantizations on Supermanifolds

2010-10-04

arxiv.org/abs/1010.0516v2

SIGMA 7:034,2011

Mathematics

Differential Geometry

Scientific paper

10.3842/SIGMA.2011.034

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 $\mathfrak{pgl}({n+1|m})$-equivariant quantization on ${\mathbb{R}}^{n|m}$ 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.

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