Membranes at Quantum Criticality

TL;DR

The paper develops a nonrelativistic gravity framework at a Lifshitz point with $z=2$ in $D+1$ dimensions and specializes to $D=2$ to model membrane worldvolumes, pairing it with Lifshitz matter to reproduce the bosonic string partition function via the ground-state of a corresponding membrane theory. It builds the second-quantized membrane theory as a Bose-Einstein condensate across genera, and carefully analyzes both the first-quantized Lifshitz-matter sector and the gravitational sector, including a detailed balance structure, foliation-preserving diffeomorphisms, and an anisotropic Weyl symmetry that fixes key couplings. The work reveals a two-parameter free-field fixed point ($\gamma$, $\lambda$) at $z=2$, derives the spectrum of excitations, and argues that relevant deformations can drive the theory to $z=1$ in the infrared, reproducing general relativity-like behavior with emergent light. It further explores the coupling to 26 Lifshitz scalars, the algebra of constraints with anisotropic Weyl invariance, and the possibility of realizing a ground state that enforces the string partition function constraints, while outlining avenues for richer dynamics such as Liouville modes and connections to Ricci flow and foliated geometries.

Abstract

We propose a quantum theory of membranes designed such that the ground-state wavefunction of the membrane with compact spatial topology Σ_h reproduces the partition function of the bosonic string on worldsheet Σ_h. The construction involves worldvolume matter at quantum criticality, described in the simplest case by Lifshitz scalars with dynamical critical exponent z=2. This matter system must be coupled to a novel theory of worldvolume gravity, also exhibiting quantum criticality with z=2. We first construct such a nonrelativistic "gravity at a Lifshitz point" with z=2 in D+1 spacetime dimensions, and then specialize to the critical case of D=2 suitable for the membrane worldvolume. We also show that in the second-quantized framework, the string partition function is reproduced if the spacetime ground state takes the form of a Bose-Einstein condensate of membranes in their first-quantized ground states, correlated across all genera.