Conformally equivariant quantization for spinning particles

TL;DR

This paper establishes the existence and explicit construction of conformally equivariant quantization for spinor differential operators on conformally flat spin manifolds, linking symmetries of the Dirac operator to conformal Killing tensors.

Contribution

It proves existence and uniqueness of conformally equivariant quantization in the spin setting and provides explicit formulas, classifying conformal supercharges and higher symmetries.

Findings

Explicit formula for first-order conformally equivariant quantization.

Classification of conformal supercharges via conformal Killing tensors.

Higher symmetries of the Dirac operator derived from quantized supercharges.

Abstract

This work takes place over a conformally flat spin manifold (M,g). 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, endowed with a symplectic form depending on the metric g. 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 with mixed symmetry, generated by the totally symmetric and skew-symmetric ones. Higher symmetries of the Dirac operator are obtained by quantization of the conformal supercharges.