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Conformal geometry of the supercotangent and spinor bundles. (English) Zbl 1257.53078

Summary: We study the actions of local conformal vector fields \({X \in \operatorname{conf}(M,g)}\) on the spinor bundle of \((M, g)\) and on its classical counterpart: the supercotangent bundle \(\mathcal{M}\) of \((M, g)\). We first deal with the classical framework and determine the Hamiltonian lift of \(\operatorname{conf}(M, g)\) to \({\mathcal{M}}\). We then perform the geometric quantization of the supercotangent bundle of \((M, g)\), which constructs the spinor bundle as the quantum representation space. The Kosmann Lie derivative of spinors is obtained by quantization of the comoment map.
The quantum and classical actions of \(\operatorname{conf}(M, g)\) turn, respectively, the space of differential operators acting on spinor densities and the space of their symbols into \(\operatorname{conf}(M, g)\)-modules. They are filtered and admit a common associated graded module. In the conformally flat case, the latter helps us determine the conformal invariants of both \(\operatorname{conf}(M, g)\)-modules, in particular the conformally odd powers of the Dirac operator.

MSC:

53C27 Spin and Spin\({}^c\) geometry
81R25 Spinor and twistor methods applied to problems in quantum theory
58A50 Supermanifolds and graded manifolds
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