Constraining Spacetime Dimensions in Quantum Gravity by Scale Invariance and Electric-Magnetic Duality

We consider a low-energy effective theory of p-branequaes in a D-dimensional spacetime, and impose two conditions: (1) the theory is scale-invariant, and (2) the electric-magnetic dual $(D-p-4)$-branes exist and they obey the same type of interactions to the p-branes. (We also assume other natural conditions such as Lorentz invariance but not string theory, supersymmetry, supergravity, and so on.) We then ask what values of p and D are consistent with these conditions. Using simple dimensional analysis, we find that only two solutions are possible: $(p,D)=(2,11)$ and $(p,D)=(2n-1,4n+2)$, ($n=1,2,3,\cdots$). The first solution corresponds to M-theory, and the second solutions at $n=1$ and $n=2$ correspond to self-dual strings in little string theory and D3-branes in type IIB superstring theory, respectively, while the second solutions for $n \ge 3$ are unknown but would be higher spin theories. Thus, quantum gravity (massless spin two theory) satisfying our two conditions would only be superstring theories, and the conditions would be strong enough to characterize superstring theories in quantum gravity.