Advances in Theoretical and Mathematical Physics

Volume 27 (2023)

Number 3

Dynamical quantization of contact structures

Pages: 881 – 959

DOI: https://dx.doi.org/10.4310/ATMP.2023.v27.n3.a7

Authors

Roger Casals (Department of Mathematics, University of California, Davis, Calif., U.S.A.)

Gabriel Herczeg (Department of Physics, Brown University, Providence, Rhode Island, U.S.A.)

Andrew Waldron (Department of Mathematics, and the Center for Quantum Mathematics & Physics, University of California, Davis, Calif., U.S.A.)

Abstract

We construct a dynamical quantization for contact manifolds in terms of a flat connection acting on a Hilbert tractor bundle. We show that this contact quantization, which is independent of the choice of contact form, can be obtained by quantizing the Reeb dynamics of an ambient strict contact manifold equivariantly with respect to an ${\mathbb R}^+$-action. The contact quantization further determines a certain contact tractor connection whose parallel sections determine a distinguished choice of Reeb dynamics and their quantization. This relationship relies on tractor constructions from parabolic geometries and mirrors the tight relationship between Einstein metrics and conformal geometries. Finally, we construct in detail the dynamical quantization of the unique tight contact structure on the 3-sphere, where the Holstein-Primakoff transformation makes a surprising appearance.

Published 6 June 2024