Torsional Newton–Cartan geometry and the Schrödinger algebra

Abstract

We show that by gauging the Schrödinger algebra with critical exponent z and imposing suitable curvature constraints, that make diffeomorphisms equivalent to time and space translations, one obtains a geometric structure known as (twistless) torsional Newton–Cartan geometry (TTNC). This is a version of torsional Newton–Cartan geometry (TNC) in which the timelike vielbein must be hypersurface orthogonal. For z = 2 this version of TTNC geometry is very closely related to the one appearing in holographic duals of z = 2 Lifshitz space-times based on Einstein gravity coupled to massive vector fields in the bulk. For there is however an extra degree of freedom b0 that does not appear in the holographic setup. We show that the result of the gauging procedure can be extended to include a Stückelberg scalar χ that shifts under the particle number generator of the Schrödinger algebra, as well as an extra special conformal symmetry that allows one to gauge away b0. The resulting version of TTNC geometry is the one that appears in the holographic setup. This shows that Schrödinger symmetries play a crucial role in holography for Lifshitz space-times and that in fact the entire boundary geometry is dictated by local Schrödinger invariance. Finally we show how to extend the formalism to generic TNC geometries by relaxing the hypersurface orthogonality condition for the timelike vielbein .