On the relationship between quantum cosmology and the dimensions of the superstring

We examine the possibility of constraining the dimensions M+1 and N of the physical space-time and the internal space, respectively, of the D(=M+N+1)-dimensional theory L=√(-g)(1/2κ-2R+1/2Φ;AΦ;BgAB), which is the field-theory limit of both the bosonic string and the superstring when higher-order terms κ2φ-4/(D-2)R2 and κ2φ-8/(D-2)H2ABC can be ignored, where φ=exp(1/2√(D-2)κΦ) is the dilaton. The Wheeler-De Witt equation for the wave function of the Universe Ψ is derived for the topology R x SM x RN, where the space RN is to approximate a (Ricci-flat) Kaehler space. By requiring that Ψ remain finite at the origin (zero (D-1)-volume) and assuming that the eigenvalue of the dilaton energy-momentum tensor vanishes, we show that the metric initially possesses the ''no-scale'' form ds2=-n2dt2+a2(t)dx2+a-2(t)dy2, provided that (M-N)2=M+N. This condition includes not only the cases D=26 (M=10, N=15) and D=10 (M=3, N=6), for which the dilaton beta function βφ vanishes (to lowest order in α') in the bosonic string and the superstring, respectively, but also the case D=5 (M=3, N=1) corresponding to the original Kaluza-Klein theory. Since M=3 for the physical space, then the superstring is allowed, while the bosonic string appears to be ruled out. The analysis was motivated by a preprint of Enqvist, Mohanty and Nanopoulos. (orig.)