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.

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