Membranes at Quantum Criticality

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.