A (1+1)-dimensional Lifshitz Weyl Anomaly From a Schr$\mathrm{\ddot{o}}$dinger-invariant Non-relativistic Chern-Simons Action

Abstract

The main result of this paper is that the Weyl anomaly of a $z=2$ (1+1)-dimensional Lifshitz effective action can be derived from a (2+1)-dimensional non-relativistic Schr$\mathrm{\ddot{o}}$dinger-invariant Chern-Simons (NRSCS) action which was shown to be equivalent to a specific Weyl-invariant non-projectable Horava-Lifshitz action of gravity. On a manifold with a boundary, we will show that the (1+1)-dimensional Lifshitz Weyl anomaly can be derived from a specific term, the torsional CS (tCS) term, in the NRSCS action built from the gauge fields of the Weyl and special conformal symmetry generators of the centrally-extended Schr$\mathrm{\ddot{o}}$dinger algebra. We also focus on the $z=1$ Lifshitz Weyl anomaly and attempt to elicit its geometric and physical nature, in particular its relationship with the Lorentz anomaly of a (1+1)-dimensional CFT effective actions. We show that it is directly related to the curvature scalar of the dual Lorentz connection, the integral of which is known to be a topological invariant. We also point out that making the anomalous Lifshitz quantum effective action Weyl-invariant amounts to obtaining the equation of motion for a stationary chiral boson which happens to be the spatial-component of the acceleration vector. By putting boundary conditions on the spatial slices, the time dependence of the lapse function in the Arnowitt, Deser and Misner (ADM) decomposition is eliminated and the result is a Rindler metric. We finally discuss several issues related to the (1+1)-dimensional Lifshitz Weyl anomaly regarding edge physics of fractional quantum Hall states and anomaly cancellation by anomaly inflow.