Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux

Abstract

The Eisenhart-Duval lift allows embedding non-relativistic theories into a Lorentzian geometrical setting. In this paper we study the lift from the point of view of the Dirac equation and its hidden symmetries. We show that dimensional reduction of the Dirac equation for the Eisenhart-Duval metric in general gives rise to the non-relativistic Levy-Leblond equation in lower dimension. We study in detail in which specific cases the lower dimensional limit is given by the Dirac equation, with scalar and vector flux, and the relation between lift, reduction and the hidden symmetries of the Dirac equation. While there is a precise correspondence in the case of the lower dimensional massive Dirac equation with no flux, we find that for generic fluxes it is not possible to lift or reduce all solutions and hidden symmetries. As a by-product of this analysis we construct new Lorentzian metrics with special tensors by lifting Killing-Yano and Closed Conformal Killing-Yano tensors and describe the general Conformal Killing-Yano tensor of the Eisenhart-Duval lift metrics in terms of lower dimensional forms. Lastly, we show how dimensionally reducing the higher dimensional operators of the massless Dirac equation that are associated to shared hidden symmetries it is possible to recover hidden symmetry operators for the Dirac equation with flux.