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

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