Fundamental Membranes and the String Dilaton

We study the quantization of the bosonic sector of supermembrane theory in double dimensional reduction, in order to extract the dependence of the resulting world-sheet action on the string dilaton (which cannot be obtained from a purely kinematic reduction). Our construction relies on a Polyakov-type approach with all six metric components on the world-volume as independent quantum fields, and shows that the correct and unique answer is only obtained if the target-space dimension of the theory is restricted to the critical value ($D=11$ for the supermembrane). As a corollary, our analysis implies that there are no analogs of the non-critical string for (super-)membrane theory.