Membranes at quantum criticality

We propose a quantum theory of membranes designed such that the ground-state wavefunction of the membrane with compact spatial topology Sigma(h) reproduces the partition function of the bosonic string on worldsheet Sigma(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.