Lifshitz Phase of Quantum Gravity: The Infrared Fixed Point of the (2+1)D Unimodular Hořava-Chern-Simons Model

We consider a nonrelativistic gravity theory in (2+1) dimensions with $w_\infty$ gauge symmetry that we call unimodular Hořava-Chern-Simons gravity. The theory has three dimensionless couplings, one of which is the Chern-Simons level $κ$. At large $|κ|$, the theory flows from an asymptotically free UV fixed point to an IR fixed point with anisotropic scale invariance characterized by a dynamical critical exponent $z=2-\tfrac32|κ|^{-1}$. We speculate that unimodular Hořava-Chern-Simons gravity undergoes a phase transition from this scale-invariant phase to a confining phase when $κ$ drops below a critical value.