Torsional Newton-Cartan Geometry and the Schr\"odinger Algebra

TL;DR

This paper develops a geometric framework called torsional Newton-Cartan geometry by gauging the Schr"odinger algebra, revealing its significance in holography for Lifshitz space-times and connecting boundary geometry to Schr"odinger symmetry.

Contribution

It extends the gauging of the Schr"odinger algebra to include a St"uckelberg scalar and special conformal symmetry, clarifying the boundary geometry in Lifshitz holography.

Findings

TTNC geometry relates to Lifshitz holography for z=2.

Extra degree of freedom b_0 appears for z≠2 but can be gauged away.

Schr"odinger symmetry dictates the boundary geometry in holographic models.

Abstract

We show that by gauging the Schr\"odinger 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 $\tau_\mu$ 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 $z\neq 2$ there is however an extra degree of freedom $b_0$ that does not appear in the holographic setup. We show that the result of the gauging procedure can be extended to include a St\"uckelberg scalar $\chi$ that shifts under the particle number generator of the Schr\"odinger algebra, as well as an extra special conformal symmetry that allows one to gauge away $b_0$. The resulting version of TTNC geometry is the one that appears in the holographic setup. This shows that Schr\"odinger symmetries play a crucial role in holography for Lifshitz space-times and that in fact the entire boundary geometry is dictated by local Schr\"odinger invariance. Finally we show how to extend the formalism to generic torsional Newton-Cartan geometries by relaxing the hypersurface orthogonality condition for the timelike vielbein $\tau_\mu$.