Quantum Gravity at a Lifshitz Point

TL;DR

The paper introduces a nonrelativistic quantum gravity framework featuring a z=3 Lifshitz fixed point in 3+1 dimensions, aiming to tame UV behavior via anisotropic scaling and a detailed-balance structure tied to the Cotton tensor. The UV theory employs foliation-preserving diffeomorphisms and a K-kinetic term alongside a higher-spatial-derivative potential, then flows to an infrared relativistic regime (z=1) where the effective c, G_N, and Λ emerge from relevant deformations. A key result is the intimate link between detailed balance and three-dimensional topologically massive gravity, with the Cotton tensor playing a central role in the UV action; the theory also accommodates anisotropic Weyl invariance at λ=1/3 and reveals a controlled free-field fixed point. The work further surveys generalizations (z=4, ultralocal limits) and outlines potential holographic interpretations, highlighting how Lorentz invariance becomes an emergent phenomenon and how the framework might address long-standing quantum gravity challenges.

Abstract

We present a candidate quantum field theory of gravity with dynamical critical exponent equal to z=3 in the UV. (As in condensed matter systems, z measures the degree of anisotropy between space and time.) This theory, which at short distances describes interacting nonrelativistic gravitons, is power-counting renormalizable in 3+1 dimensions. When restricted to satisfy the condition of detailed balance, this theory is intimately related to topologically massive gravity in three dimensions, and the geometry of the Cotton tensor. At long distances, this theory flows naturally to the relativistic value z=1, and could therefore serve as a possible candidate for a UV completion of Einstein's general relativity or an infrared modification thereof. The effective speed of light, the Newton constant and the cosmological constant all emerge from relevant deformations of the deeply nonrelativistic z=3 theory at short distances.