  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.
