Where there is life there is a pattern, and where there is a pattern there is mathematics. Once that germ of rationality and order exists to turn a chaos into a cosmos, then so does mathematics. There could not be a non-mathematical Universe containing living observers ~ John D. Barrow
It is due time to prove quantum AdS/CFT holographic renormalization, as it is a necessary condition for the consistency of the Sasaki-Einstein Dp-brane ‘elimination’ of spacetime and hence, by GR, gravity. One can only do that by a topological Fukaya embedding of the holomorphic renormalization group generators on the Calabi-Yau conic tip of Dp-branes‘ p+1 dimensional worldspaces. Let us have some fun. To begin with, one must couple massive scalar fields to gravity – then, the bulk on-shell action is:
![\[S = \int {{d^{p + 1}}} x\;d{\phi _{si}}\sqrt g \left( { - \frac{1}{{2{\pi ^2}}}{R_{{\rm{icci}}}} + \frac{1}{2}{g^{\mu \nu }}{{\not \partial }_\mu }\varphi \,{{\not \partial }_\nu }\varphi + V(\varphi ) + ...} \right)\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-05ea017b1317e919dcabd43147048ffa_l3.png "Rendered by QuickLaTeX.com")
with 
the Calabi-Yau 2-D conic string variable and the dots representing contributions from the gauge fermionic and anti-symmetric tensors, and:
![\[{k^2} = 8\pi {G_{d + 1}} \sim {G_{d + 1}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-c6419bb4ba0fb165eb333ff00b87288b_l3.png "Rendered by QuickLaTeX.com")
with 
being the Dp-brane’s p+1 dimensional worldspace Newtonian constant, and:
![\[V(\varphi ) = \frac{\Lambda }{{{k^2}}} + \frac{1}{2}{m^2}{\varphi ^2} + b = \frac{\Lambda }{{{k^2}}} + \frac{1}{2}{m^2} + b\,{\varphi ^3} + ...\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-25036c7caec3fa17cc137f750150e4b5_l3.png "Rendered by QuickLaTeX.com")
with 
the cosmological constant, 
the integral measure of the gauge group generators, and:
![\[{m^2} = \left( {\Delta - d} \right)\Delta \]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-bf5d6b20d4c55f59f349560c208495aa_l3.png "Rendered by QuickLaTeX.com")
holding, with 
the dual conformal operator. Then we have:
![\[\left\{ {\begin{array}{*{20}{c}}{{G^{SE}}_{\mu \nu } = {k^2}{{\widetilde T}_{\mu \nu }}(\varphi )d\,\Omega {{({\phi _{si}})}^2}\,}\\{{{\not \bigcirc }_g}\varphi = \not \partial V/\not \partial \varphi d{\phi _{si}}}\end{array}} \right.\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-2a23a623d801466bfdad4ab22933757e_l3.png "Rendered by QuickLaTeX.com")
where:
![\[{G^{SE}}_{\mu \nu } = R_{\mu \nu }^{{e^{ - {\phi _{si}}}}} - {g_{\mu \nu }}{R_{{\rm{icci}}}}/2\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-0bb729795d1d56e847c4fdc956dce5a8_l3.png "Rendered by QuickLaTeX.com")
is the Sasaki-Einstein ‘AdS’ tensor and the super-covariant Laplacian is:
![\[{\not \bigcirc _g}\varphi = \frac{1}{{\sqrt g }}{\not \partial _\mu }{\phi _{si}}\not \partial \left( {\sqrt g {g^{\mu \nu }}{{\not \partial }_\nu }\varphi } \right)\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-60755a1764e8972ab28f346c773146e8_l3.png "Rendered by QuickLaTeX.com")
and 
is the stress-energy tensor. Now, to renormalize 
holography, one must first get an asymptotic solution with Dirichlet data:
![\[d{s^2} = \frac{{d{z^2}}}{{{z^2}}}\not \partial {\phi _{si}} + \frac{1}{{{z^2}}}{\varphi _{ij}}(z,x)d{x^i}d{x^j}d\,\Omega {({\phi _{si}})^2}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-bcfa2036349ede6418afcce4ecc9a87d_l3.png "Rendered by QuickLaTeX.com")
with:
![\[{\varphi _{ij}}(z,x) = {z^{\left( {d - \Delta } \right)\exp \,( - {\phi _{si}})}}{\phi _{si}}(z,x)\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-8e013a42f4d83f29bb00f4cef79d0801_l3.png "Rendered by QuickLaTeX.com")
and:
![\[\begin{array}{c}{g_{ij}}(z,x) = {g_{(o)ij}} + \not \partial \phi _{si}^2{z^2}{g_{(2)ij}} + ... + \not \partial \phi _{si}^d\left( {{g_{(d)ij}} + {\rm{log}}\,{{\rm{z}}^2}\not \partial _{si}^{d + 1}{h_{(d)ij}} + ...{\phi _{si}}(z,x) = {\phi _{(0)}} + {z^2}\not \partial \phi _{si}^2 + ...} \right) \cdot \\{z^{2(\Delta - d)}}\not \partial _{si}^{p + 1}\left( {{\phi _{(2\Delta - d)}} + \log {z^2}{\psi _{(2\Delta - d)}}} \right) + ...\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-d08a64a4781ccf94295a9b155e65bf34_l3.png "Rendered by QuickLaTeX.com")
Now, generally, one gets the following canonical solution:
![\[{g_{(2)ij}}\left[ {{g_{(0)}}} \right] = \frac{1}{{d - 2}}\left( {{R_{{\rm{icci}}}}\left[ {{g_{(0)}} - {{\not \partial }_{ij}}\phi _{si}^{\Delta + 1} - \frac{1}{{2(d - 1)}}{R_{{\rm{icci}}}}\left[ {{g_{(0)}}} \right]{g_{(0)ij}}} \right]} \right)d\,\Omega {({\phi _{si}})^{d - 1}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-444c45eb6d2926d5c654a03cbffe0716_l3.png "Rendered by QuickLaTeX.com")
Now one introduces a Teichmüller radial cut-off 
,
, to derive the SuperGravity action with the power-law expansion:
![\[\begin{array}{c}{S_{{\rm{reg}}}}\left[ {{g_{(0)}},{\phi _{si(0)}};\varepsilon } \right] = \frac{1}{{2\pi {k^2}}}\int {{d^4}} x\not \partial \phi _{si}^{p + 4}x\sqrt {{g_{(0)}}} \cdot \\\left( {\frac{{{g_{(0)}}}}{{{\varepsilon ^d}}} + \frac{{{g_{(1)}}}}{{{\varepsilon ^{d + 1}}}} + ... + {g_{(d)}}{\rm{log}}\,{\varepsilon ^2} + \vartheta ({\varepsilon ^{(0)}})d\,\Omega {{({\phi _{si}})}^{p + 4}}} \right)\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-7de061c5be1aaf12d39ffa9c5240c78f_l3.png "Rendered by QuickLaTeX.com")
and the coefficients 
are the conformal boundary anomaly of the cohomology group corresponding to the Sasaki-Einstein manifold. To move any further and derive pure gravity in 4-D, we need to factor in the Calabi-Yau rotational metric on 
, whose solution is:
![\[\begin{array}{c}ds_{\scriptstyle10\atop\scriptstyle}^2 = H_3^{1/2}\left( { - \left[ { - \frac{{1 - r_H^4}}{{{r^4}\Delta }}} \right]d{t^2} + dx_1^2 + dx_2^2 + dx_3^2d\,\Omega {{({\phi _{si}})}^2}} \right) + \\H_3^{1/2}\left[ {\frac{{\Delta d{r^2}}}{{{{\not H}_1}{{\not H}_2}{{\not H}_3} - r_H^4/{r^4}}} - \frac{{2r_H^4\cos \,h{\beta _3}}}{{{r^4}{H_3}\Delta }}dt\left( {\sum\limits_{i = 1}^3 {{l_i}\mu _i^2d{\phi _{si}}} } \right) + {r^2}\sum\limits_{1 = 1}^3 {{{\not H}_i}\left( {d\mu _i^2d\phi _{si}^2} \right)} } \right] + \\\left[ {{{\left( {{r^2}\sum\limits_{i = 1}^3 {{l_i}\mu _i^2d{\phi _{si}}} } \right)}^2}} \right]d\Omega (\phi _{si}^2)\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-fd2b75bcd78cac170a57a8ddb2a2b75d_l3.png "Rendered by QuickLaTeX.com")
with:
![\[\Delta = {\not H_1}{\not H_2}{\not H_3}\sum\limits_{1 = 1}^3 {\frac{{\mu _i^2}}{{{{\not H}_i}}}} \]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-722f962fd2890720fa2c2858167befc0_l3.png "Rendered by QuickLaTeX.com")
holding, and:
![\[{H_3} = 1 + \frac{{r_H^4}}{{{r^4}}}\frac{{\sin \,{h^2}{\beta _3}}}{\Delta } = 1 + \frac{{{\alpha _3}r_3^4}}{{\Delta {r^4}}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-0a6a89fe16726235f533f2b047bdff33_l3.png "Rendered by QuickLaTeX.com")
with:
![\[{\not H_i} = 1 + \frac{{l_i^2}}{{{r^2}}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-0e01b331f246d55f3ca54820c980702c_l3.png "Rendered by QuickLaTeX.com")
for
![\[i = 1,2,3\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-c8af51820e6876110b1b1a2c9cc959cb_l3.png "Rendered by QuickLaTeX.com")
and:
![\[\left\{ {\begin{array}{*{20}{c}}{{\mu _1} = \sin \theta }\\{{\mu _2} = \cos \theta \sin \psi }\\{{\mu _3} = \cos \theta \cos \psi }\end{array}} \right.\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-6c52142a2b918d4776d01fdeb5176d0d_l3.png "Rendered by QuickLaTeX.com")
with 
and 
the Sasaki-Einstein Gaussian angles on
and 
respectively. By 4-dimensional coset reduction, 
, we get, for pure 4-D gravity, the term:
![{g_{(4)}} = \frac{1}{2}\left[ {{\rm{Tr}}\left( {g_{(0)}^{ - 1}g_{(2)}^2 - {{\not \partial }_{\mu \nu }}\phi _{si}^4} \right)} \right]d\,\Omega (\phi _{si}^4)](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-ad0c0a00bcfc67d018873cba4e9c8cc9_l3.png "Rendered by QuickLaTeX.com")
To derive an on-shell action, one must substract all the infinities and have the regulator vanish – hence, eliminating the divergences and ensuring that the Ward identities are satisfied, since the existence of covariant counterterms lets us work on the hyper-surface 
; one then gets, for d = 4 gravity:
![\[{S_{{\rm{reg}}}} = \frac{3}{{{k^2}}}\int_{z = \varepsilon } {{d^4}} \not \partial \phi _{si}^4x\sqrt \gamma \left[ {1 + \frac{1}{{12}}{R_{{\rm{icci}}}}\left[ \gamma \right] - \frac{1}{{48}}\left( {{R_{{\rm{icci}}}}{{\left[ \gamma \right]}^k}{R_{{\rm{icci}}}}{{\left[ \gamma \right]}_k} - \frac{1}{3}{R_{{\rm{icci}}}}{{\left[ \gamma \right]}^2}} \right)} \right] + \log {\varepsilon ^2} + ...\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-39d0757437e4af9dfa175c7b2a6b7c09_l3.png "Rendered by QuickLaTeX.com")
Thus, solving leads to the gauge covariant form of the regulated action. By differentiating the renormalized action, we get:
![\[\begin{array}{c}\left\langle {{T_{ij}}(x)} \right\rangle _s^{CFT} \equiv \frac{2}{{\sqrt {{g_{(0)}}(x)} }}\frac{{\delta {S_{{\rm{renorm}}}}}}{{\delta g_{(0)}^{ij}(x)}}\not \partial \,\Omega {({\phi _{si}})^2} = \\\mathop {\lim }\limits_{\varepsilon \to 0} \left( {\frac{1}{{{\varepsilon ^{d - 2}}}}\frac{2}{{\sqrt {\gamma (\varepsilon ,x)} }}\frac{{\delta {S_{{\rm{sub}}}}}}{{\delta {\gamma ^{ij}}(\varepsilon ,x)}}} \right) = \frac{d}{{2{k^2}}}{g_{{{(d)}^{ij}}}} + {X_{ij}} \cdot \\\left[ {{g_{(0)}},{\phi _{si(0)}}} \right]\left\langle {\vartheta (x)} \right\rangle _s^{Ad{S_5}} \equiv \frac{1}{{\sqrt {{g_{(0)}}(x)} }}\frac{{\delta {S_{{\rm{renorm}}}}}}{{\delta {\phi _{si(0)}}(x)}}\not \partial \,\Omega {({\phi _{si}})^5} = \\\mathop {\lim }\limits_{\varepsilon \to 0} \left( {\frac{1}{{{\varepsilon ^\Delta }}}\frac{1}{{\sqrt {\gamma (\varepsilon ,x)} }}\frac{{\delta {S_{{\rm{sub}}}}}}{{\delta \varphi (\varepsilon ,x)}}} \right) = (d - 2\Delta ){\phi _{si\left( {2\Delta - d} \right)}} + \Upsilon \left[ {{g_{(0)}},{\phi _{si(0)}}} \right]\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-b6bc0b29743ae5dd41837f5c7a53068c_l3.png "Rendered by QuickLaTeX.com")
We are now in a position to derive a renormalized Hamiltonian action:
letting 
be a d+1 Riemannian conformally compact manifold, 
its boundary, and 
its entropic interior, then with 
the corresponding metric on , we get:
![\[{S_{{\rm{Gra}}}}\left[ g \right] = \frac{1}{{2{k^2}}}\left[ {\int_M {{d^{p + 1}}\not \partial \phi _{si}^{p + 1}x\sqrt g {R_{icci}} + \int_{\partial M} {{d^4}d\,\Omega {{({\phi _{si}})}^{ - p + 1}}x\sqrt \gamma 2K} } } \right]\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-58ee9695a963deb2b49eeedf921d7cbd_l3.png "Rendered by QuickLaTeX.com")
with 
the trace of the Gaussian curvature of the boundary. This is equivalent to the Gibbons-Hawking boundary term of the Calabi-Yau conic tip of the Sasaki-Einstein 5-D manifold. Therefore, we can derive, on 
:
![\[{S_m} = \int_{\partial M} {{d^{d + 1}}\not \partial \phi _{si}^{d + 1}} x\sqrt g {L_m}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-3f02c99db8643e996d52e6a2dd0ba72f_l3.png "Rendered by QuickLaTeX.com")
with 
the matter field Lagrangian density, and the action above transforms as:
![\[{\delta _g}{S_m} \equiv \frac{1}{2}\int_{\widetilde M} {{d^{p + 1}}} d\,\Omega (\phi _{si}^{p + 1})x\sqrt {{G_{\mu \nu }}} {\widetilde T_{\mu \nu }}\delta {g^{\mu \nu }}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-27ced4d1a4267654b81740990035c8bb_l3.png "Rendered by QuickLaTeX.com")
with:
![\[{G_{\mu \nu }} = {k^2}{\widetilde T_{\mu \nu }}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-ac634ab8866fcf865878d48104dd0480_l3.png "Rendered by QuickLaTeX.com")
being the crucial finitizing Euler-Lagrange equation of the total renormalization action:
![\[S = {S_{Gra}} + {S_m} + {S_{reg}} + {S_{renorm}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-fdd301bc628b593d18d12caf9b178dd8_l3.png "Rendered by QuickLaTeX.com")
Fait accompli!
FROM THE INTRINSIC EVIDENCE OF HIS CREATION, THE GREAT ARCHITECT OF THE UNIVERSE NOW BEGINS TO APPEAR AS A PURE MATHEMATICIAN. ~ SIR JAMES JEANS!
