### I know that two and two make four – and should be glad to prove it too if I could – though I must say if by any sort of process I could convert 2 and 2 into five it would give me much greater pleasure ~ George Gordon Byron

Great news for me personally – after nearly 20 years of researching M-Theory: see bottom link on M-Theory and my last formula in this post involving supersymmetry. I ended my last post by showing how Peccei-Quinn invariance leads to the compactification smoothness required for M-theory to isomorphically embed an ‘Einsteinian-Minkowski‘ 4-D space-time in a Calabi-Yau fourfold in a way necessitated by quantum gravity. Let me address here the Kaluza-Klein reduction of string theory on Calabi-Yau fourfolds. Keep your eyes on

and

to be explained below, throughout, to really appreciate the ‘M’agic of M-theory.

My starting point is the low-energy effective D=10 action in the string-frame

with

the D=10 dilaton being the field strength of the anti-symmetric tensor and  the field strength of the  vector  and  the field strength of the  form  and by convention:

Also, and key, the term proportional to

which is analytically related to fourfold-dimensional reduction to the higher derivative term of M-theory

with

imposes a consistency condition on compactification, and the absence of a  tadpole requires hence

to be solvable on manifolds isomorphic to , with  the number of string-fittings. With no loss of generality, let me focus on the conditions

Now, realize, the spectrum of the 2-D theory is determined by super-deformations of the Calabi-Yau metric. So, for the D=10 metric, my ansatz

and since the vectors contain no physical degree of freedom in D=2, then, in light of the lack of 1-forms on  does not contribute any D=2 massless modes: so the anti-symmetric tensors   expand in terms of  forms

which leads to  real scalar field  while  contributes  complex scalars

The  moduli reside in the twisted chiral multiplets where all others are not. Therefore, by twistor-algebra, the dimensional reduction of

gives us, by Teichmüller-integration

with

being crucial for smoothness and

for  integrability.

In such a derivation, the key are the 2 identities

Now, both can be separated via a Kähler potential. Hence, define

and

with  and  and letting the fields  denote

one can see that

becomes

with the conformal gauge being

and this results, after noting that , in the Minkowski-Sylvester space-time Lagrange interpolation continuity condition required of M-theory and only M-theory can thus meet if gravity is to be globally quantized at both scales: the cosmological and the Planck ones, with

and since the moduli space factorizes into chiral and twisted chiral multiplets, which is to say, it is Kähler, we get a fourfolding of M-theory in a way that leads to a finite theory of quantum gravity at the cosmological and Planck scales, as evidenced by

And here is Physicists Sebastian A.R. Ellis, Gordon L. Kane, and Bob Zheng’s paradigm-shifting revolutionary paper on supersymmetry and hence by U-Duality, via string-theory, M-theory:

LHC and Predictions from Constrained Compactified M-Theory

By U-duality, and fibration, one can generalize the above reduction and compactification arguments. What more can one ask of a theory.