Geometric quantization of Dirac manifolds

We extend a prequantization procedure to Dirac manifolds by using singular distributions obtained from 2-cocycles associated with Dirac structures. Given a Dirac manifold (M, D), we describe a prequantization formula in terms of a Lie algebroid connection and show that it is a representation of a Poisson algebra consisting of admissible functions on (M, D) on the space of global sections of a hermitian line bundle over M if and only if the curvature derived from the Lie algebroid connection is represented by a skew-symmetric operation which is naturally defined for (M, D). Moreover, we describe a necessary and sufficient condition for the prequantization formula to be the representation in terms of a Lie algebroid cohomology. We introduce the notion of a polarization for (M, D) and construct a representation of a subalgebra of admissible functions. Lastly, we discuss procedures for quantization in two cases: where M is compact and where M is not compact. Published by AIP Publishing.