Think of it: of the infinity of real numbers, those that are most important to mathematics — 0, 1, √2, e and π — are located within less than four units on the number line. A remarkable coincidence? A mere detail in the Creator’s grand design? I let the reader decide.~ Eli Maor, ‘e: The Story of a Number’ (1994)!

The real, not-so-‘publicized’ magic of M-theory lies in the fact that its Sasaki-Einstein renormalization group is analytically finite. This is part 2 of such mathematical inquisition into the holographic renormalization group in the Maldacena duality D-brane context. One can holographically eliminate space-time, and hence, by GR, 4-dimensional gravity, via entropic cohomological bundle analysis on the Sasaki-Einstein space and its De Rahm group. One then shaves, via Occam’s razor and the explanatory completeness and causal closure of quantum field theory, all of space-time and gravity. One needs first to do some holographic renormalization group analysis of the Sasaki-Einstein D-p-brane world-volume. Let be the Calabi-Yau 2-D conic string variable and:

with being the D-p-brane’s p+1 dimensional worldspace Newtonian constant, with the Dirichlet data:

with:

I then deduced the renormalized Hamiltonian action:

with:

with the Gaussian curvature of the Gibbons-Hawkins boundary term of the Calabi-Yau conic tip of and being the entropic interior of the d+1 Riemannian conformally compact manifold , and its boundary. Hence, one gets, on :

with the matter field Lagrangian density and transforms as:

with the stress-energy tensor and:

the renormalization functional of the total holographic renormalization group. Now, let be the Einstein-Hilbert actional wavefuntion of the universe coupled with the instanton, which is a field configuration that is concentrated at a point in time in the worldvolume of the Dirichlet brane of the corresponding string variable, defined on the Hilbert space corresponding to . Let denote its Fourier transform. Now, look at the Hodge equation:

with the wavefunction of the universe at t = i, and its Fourier transform. Since:

with the quantum fluctuational frequency of the string worldsheet, we get the required metric that gauges the graviton and its supersymmetric partner:

with:

which preserves Poincaré invariance and:

where is the parametrization function of the interpolating region between branes on the Calabi-Yau conic tip of , and hence, the supergravity action can now be derived as:

with the metastable false vacuum potential. Now, inserting in the supergravity action above gives us an energy functional:

and by the Hodge equality, its actional measure is:

Substituting in , we get:

Hence, holographic renormalization analytically is achieved for 2 reasons – one: the integral ranges over all ‘times’, and thus does satisfy the Wheeler-DeWitt equation, and since it topologically ‘lives‘ on the folliated bundle of , by holographic elimination, it is a wavefunction describing, vacuously, quantum gravity, and, secondly: its solution describes ‘finite’ renormalization Clifford algebraic variables, and so we have exact-off-hand renormalization. Hence, we can derive the initial singularity ‘creation’ relation in a finite way describing a holographic elimination of space-time, and hence, by GR, gravity, via squaring:

One can hence deduce that space-time and gravity are holographic entropic projections of the Fukaya category of the Sasaki-Einstein AdS/CFT space. To be continued.

THE ART OF DOING MATHEMATICS CONSISTS IN FINDING THAT SPECIAL CASE WHICH CONTAINS ALL THE GERMS…

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