Conformally Equivariant Quantization for Spinning Particles

Abstract

Abstract. This work takes place over a conformally flat spin manifold (M, ě). We prove existence and uniqueness of the conformally equivariant quantization valued in spinor differential operators, and provide an explicit formula for it when restricted to first order operators. The Poisson algebra of symbols is realized as a space of functions on the supercotangent bundle M = T * M ⊕ ΠT M , endowed with a symplectic form depending on the metric ě. It admits two different actions of the conformal Lie algebra: one tensorial and one Hamiltonian. They are intertwined by the uniquely defined conformally equivariant superization, for which an explicit formula is given. This map allows us to classify all the conformal supercharges of the spinning particle in terms of conformal Killing tensors, which are symmetric, skew-symmetric or with mixed symmetry. Higher symmetries of the Dirac operator are obtained by quantization of the conformal supercharges.