Super{geometric Quantization Stage 1 | Prequantization. Let M Be a Poisson Manifold with the Poisson Bracket (1.1)

Let K be the complex line bundle where the Kostant-Souriau geometric quantization operators are deened. We discuss possible prolongations of these operators to the linear superspace of the K-valued diierential forms, such that the Poisson bracket is represented by the supercommutator of the corresponding operators. We also discuss the possibility to obtain such super-geometric quantizations by (anti)Hermitian operators on a Hilbert superspace. We apply our general considerations to KK ahler manifolds and to cotangent bundles of Riemannian manifolds. In diierential geometry, the problem of geometric quantization is a two stage problem which can be stated in the following terms (e.g., 10], 9]). Find linear representations of the Lie algebra (1.1) on the space ?(K) of cross sections of a complex line bundle K over M by diierential operators of order one and symbol equal to the Hamiltonian vector eld X P f. Stage 2 | Quantization. Restrict prequantization in such a way as to obtain irreducible anti-Hermitian 1 representations of a subalgebra of C 1 (M) with bracket (1.1) on a Hilbert space derived from ?(K). In this paper, we deene the problem of super-geometric quantization as the problem of prolonging the representations mentioned above to linear and Hilbert superspaces. 1 The fact that we use anti-Hermitian operators here is just a technicality. If these operators are multiplied by a purely imaginary constant they become Hermitian operators.