I think that every reader who has at least a vague idea what physics is must know that such a method of determining the critical dimension as well as the result are completely absurd. M-theory as we know it just can't work in the total number of 8 dimensions.
The number D=8 mentioned here makes me feel terribly guilty of being quantum gravity crackpot.
First confession: I (yes,it is me, I cannot blame anyone else) regard the target space dimension of string theories as a purely formal notion. The construction of stringy vertex operators in terms of ordered exponentials of bosonic free fields brings in formal space-time dimension equal to that of Cartan algebra of the Kac-Moody algebra defining symmetries of the theory. This stringy dimension is roughly the dimension of space-time as it is determined in string theories. And no it comes: I have used almost 28 years of my life to construct a "theory" in which space-times are 4-D surfaces in 8-D space-time with stringy dimension equal to 10 and cannot but continue arguing that D=8 is the physical dimension.
Second confession: This D=8 imbedding space is fixed and non-dynamical apart from the scaling factors of metric but leaves us without landscape.
Third confession: D=4 for space-time surfaces is critical dimension for D=2 super-conformal closed strings with metric signature (2,2) or (0,4): this algebra actually extends to D=4 superconformal algebra which is the symmetry algebra of my 8-D theory. Very conservatively (even condensed matter physicist would agree), the dynamical spacetime surfaces in my own theory have the same role as 10-D dynamical target space in super string models. Hence I believe that gravity is at classical level four-dimensional phenomenon, not 10- or 11-D one.
Fourth confession: I dare to regard also strings as purely formal objects. In my own "theory" particles correspond to 2-D partonic 2-surfaces, submanifolds of space-time surface. N-point functions of conformal field theory at partonic 2-surfaces give stringy scattering amplitudes using the usual formulas. No fundamental strings in the "theory". The 4-D classical dynamics creates correlations between partons and even N=4 topological string could be physically realistic in this framework as a description of perturbative phase.
I have now confessed all that I can in the limitations posed by the limit the length of posting and I feel immense relief. Am I a quantum-gravity-crackpot? Jury can decide.
Jury can find a short summary about the most recent activities of the accused in the last section of chapter Construction of Quantum Theory.