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AdS/CFT Holography, SUSY, And Renormalization via Gibbons-Hawking-Teichmüller Analysis

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:

    \[S = \int {{d^{p + 1}}} x\;d{\phi _{si}}\sqrt g \left( { - \frac{1}{{2{\pi ^2}}}{R_{{\rm{icci}}}} + \frac{1}{2}{g^{\mu \nu }}{{\not \partial }_\mu }\varphi \,{{\not \partial }_\nu }\varphi + V(\varphi ) + ...} \right)\]

with {\phi _{si}} the Calabi-Yau 2-D conic string variable and the dots representing contributions from the gauge fermionic and anti-symmetric tensors, and:

    \[{k^2} = 8\pi {G_{d + 1}} \sim {G_{d + 1}}\]

with {G_{d + 1}} being the Dp-brane’s p+1 dimensional worldspace Newtonian constant, and:

    \[V(\varphi ) = \frac{\Lambda }{{{k^2}}} + \frac{1}{2}{m^2}{\varphi ^2} + b = \frac{\Lambda }{{{k^2}}} + \frac{1}{2}{m^2} + b\,{\varphi ^3} + ...\]

with \Lambda the cosmological constant, b the integral measure of the gauge group generators, and:

    \[{m^2} = \left( {\Delta - d} \right)\Delta \]

holding, with \Delta the dual conformal operator. Then we have:

    \[\left\{ {\begin{array}{*{20}{c}}{{G^{SE}}_{\mu \nu } = {k^2}{{\widetilde T}_{\mu \nu }}(\varphi )d\,\Omega {{({\phi _{si}})}^2}\,}\\{{{\not \bigcirc }_g}\varphi = \not \partial V/\not \partial \varphi d{\phi _{si}}}\end{array}} \right.\]

where:

    \[{G^{SE}}_{\mu \nu } = R_{\mu \nu }^{{e^{ - {\phi _{si}}}}} - {g_{\mu \nu }}{R_{{\rm{icci}}}}/2\]

is the Sasaki-Einstein ‘AdS’ tensor and the super-covariant Laplacian is:

    \[{\not \bigcirc _g}\varphi = \frac{1}{{\sqrt g }}{\not \partial _\mu }{\phi _{si}}\not \partial \left( {\sqrt g {g^{\mu \nu }}{{\not \partial }_\nu }\varphi } \right)\]

and {\widetilde T_{\mu \nu }}(\varphi ) is the stress-energy tensor. Now, to renormalize Ad{S_5} \times E_S^5 holography, one must first get an asymptotic solution with Dirichlet data:

    \[d{s^2} = \frac{{d{z^2}}}{{{z^2}}}\not \partial {\phi _{si}} + \frac{1}{{{z^2}}}{\varphi _{ij}}(z,x)d{x^i}d{x^j}d\,\Omega {({\phi _{si}})^2}\]

with:

    \[{\varphi _{ij}}(z,x) = {z^{\left( {d - \Delta } \right)\exp \,( - {\phi _{si}})}}{\phi _{si}}(z,x)\]

and:

    \[\begin{array}{c}{g_{ij}}(z,x) = {g_{(o)ij}} + \not \partial \phi _{si}^2{z^2}{g_{(2)ij}} + ... + \not \partial \phi _{si}^d\left( {{g_{(d)ij}} + {\rm{log}}\,{{\rm{z}}^2}\not \partial _{si}^{d + 1}{h_{(d)ij}} + ...{\phi _{si}}(z,x) = {\phi _{(0)}} + {z^2}\not \partial \phi _{si}^2 + ...} \right) \cdot \\{z^{2(\Delta - d)}}\not \partial _{si}^{p + 1}\left( {{\phi _{(2\Delta - d)}} + \log {z^2}{\psi _{(2\Delta - d)}}} \right) + ...\end{array}\]

Now, generally, one gets the following canonical solution:

    \[{g_{(2)ij}}\left[ {{g_{(0)}}} \right] = \frac{1}{{d - 2}}\left( {{R_{{\rm{icci}}}}\left[ {{g_{(0)}} - {{\not \partial }_{ij}}\phi _{si}^{\Delta + 1} - \frac{1}{{2(d - 1)}}{R_{{\rm{icci}}}}\left[ {{g_{(0)}}} \right]{g_{(0)ij}}} \right]} \right)d\,\Omega {({\phi _{si}})^{d - 1}}\]

Now one introduces a Teichmüller radial cut-off z \ge \varepsilon,

\varepsilon > 0, to derive the SuperGravity action with the power-law expansion:

    \[\begin{array}{c}{S_{{\rm{reg}}}}\left[ {{g_{(0)}},{\phi _{si(0)}};\varepsilon } \right] = \frac{1}{{2\pi {k^2}}}\int {{d^4}} x\not \partial \phi _{si}^{p + 4}x\sqrt {{g_{(0)}}} \cdot \\\left( {\frac{{{g_{(0)}}}}{{{\varepsilon ^d}}} + \frac{{{g_{(1)}}}}{{{\varepsilon ^{d + 1}}}} + ... + {g_{(d)}}{\rm{log}}\,{\varepsilon ^2} + \vartheta ({\varepsilon ^{(0)}})d\,\Omega {{({\phi _{si}})}^{p + 4}}} \right)\end{array}\]

and the coefficients {g_{(n)}} 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 Ad{S_5} \times E_s^5, whose solution is:

    \[\begin{array}{c}ds_{\scriptstyle10\atop\scriptstyle}^2 = H_3^{1/2}\left( { - \left[ { - \frac{{1 - r_H^4}}{{{r^4}\Delta }}} \right]d{t^2} + dx_1^2 + dx_2^2 + dx_3^2d\,\Omega {{({\phi _{si}})}^2}} \right) + \\H_3^{1/2}\left[ {\frac{{\Delta d{r^2}}}{{{{\not H}_1}{{\not H}_2}{{\not H}_3} - r_H^4/{r^4}}} - \frac{{2r_H^4\cos \,h{\beta _3}}}{{{r^4}{H_3}\Delta }}dt\left( {\sum\limits_{i = 1}^3 {{l_i}\mu _i^2d{\phi _{si}}} } \right) + {r^2}\sum\limits_{1 = 1}^3 {{{\not H}_i}\left( {d\mu _i^2d\phi _{si}^2} \right)} } \right] + \\\left[ {{{\left( {{r^2}\sum\limits_{i = 1}^3 {{l_i}\mu _i^2d{\phi _{si}}} } \right)}^2}} \right]d\Omega (\phi _{si}^2)\end{array}\]

with:

    \[\Delta = {\not H_1}{\not H_2}{\not H_3}\sum\limits_{1 = 1}^3 {\frac{{\mu _i^2}}{{{{\not H}_i}}}} \]

holding, and:

    \[{H_3} = 1 + \frac{{r_H^4}}{{{r^4}}}\frac{{\sin \,{h^2}{\beta _3}}}{\Delta } = 1 + \frac{{{\alpha _3}r_3^4}}{{\Delta {r^4}}}\]

with:

    \[{\not H_i} = 1 + \frac{{l_i^2}}{{{r^2}}}\]

for

    \[i = 1,2,3\]

and:

    \[\left\{ {\begin{array}{*{20}{c}}{{\mu _1} = \sin \theta }\\{{\mu _2} = \cos \theta \sin \psi }\\{{\mu _3} = \cos \theta \cos \psi }\end{array}} \right.\]

with \theta and \psi the Sasaki-Einstein Gaussian angles on

Ad{S_5} and E_S^5 respectively. By 4-dimensional coset reduction, ds_{10}^2/4, 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)

To derive an on-shell action, one must substract all the infinities and have the regulator vanishhence, eliminating the divergences and ensuring that the Ward identities are satisfied, since the existence of covariant counterterms lets us work on the hyper-surface z = \varepsilon; one then gets, for d = 4 gravity:

    \[{S_{{\rm{reg}}}} = \frac{3}{{{k^2}}}\int_{z = \varepsilon } {{d^4}} \not \partial \phi _{si}^4x\sqrt \gamma \left[ {1 + \frac{1}{{12}}{R_{{\rm{icci}}}}\left[ \gamma \right] - \frac{1}{{48}}\left( {{R_{{\rm{icci}}}}{{\left[ \gamma \right]}^k}{R_{{\rm{icci}}}}{{\left[ \gamma \right]}_k} - \frac{1}{3}{R_{{\rm{icci}}}}{{\left[ \gamma \right]}^2}} \right)} \right] + \log {\varepsilon ^2} + ...\]

Thus, solving leads to the gauge covariant form of the regulated action. By differentiating the renormalized action, we get:

    \[\begin{array}{c}\left\langle {{T_{ij}}(x)} \right\rangle _s^{CFT} \equiv \frac{2}{{\sqrt {{g_{(0)}}(x)} }}\frac{{\delta {S_{{\rm{renorm}}}}}}{{\delta g_{(0)}^{ij}(x)}}\not \partial \,\Omega {({\phi _{si}})^2} = \\\mathop {\lim }\limits_{\varepsilon \to 0} \left( {\frac{1}{{{\varepsilon ^{d - 2}}}}\frac{2}{{\sqrt {\gamma (\varepsilon ,x)} }}\frac{{\delta {S_{{\rm{sub}}}}}}{{\delta {\gamma ^{ij}}(\varepsilon ,x)}}} \right) = \frac{d}{{2{k^2}}}{g_{{{(d)}^{ij}}}} + {X_{ij}} \cdot \\\left[ {{g_{(0)}},{\phi _{si(0)}}} \right]\left\langle {\vartheta (x)} \right\rangle _s^{Ad{S_5}} \equiv \frac{1}{{\sqrt {{g_{(0)}}(x)} }}\frac{{\delta {S_{{\rm{renorm}}}}}}{{\delta {\phi _{si(0)}}(x)}}\not \partial \,\Omega {({\phi _{si}})^5} = \\\mathop {\lim }\limits_{\varepsilon \to 0} \left( {\frac{1}{{{\varepsilon ^\Delta }}}\frac{1}{{\sqrt {\gamma (\varepsilon ,x)} }}\frac{{\delta {S_{{\rm{sub}}}}}}{{\delta \varphi (\varepsilon ,x)}}} \right) = (d - 2\Delta ){\phi _{si\left( {2\Delta - d} \right)}} + \Upsilon \left[ {{g_{(0)}},{\phi _{si(0)}}} \right]\end{array}\]

We are now in a position to derive a renormalized Hamiltonian action:

letting \widetilde M be a d+1 Riemannian conformally compact manifold, \partial M its boundary, and M its entropic interior, then with {g_{\mu \nu }} the corresponding metric on \widetilde M, we get:

    \[{S_{{\rm{Gra}}}}\left[ g \right] = \frac{1}{{2{k^2}}}\left[ {\int_M {{d^{p + 1}}\not \partial \phi _{si}^{p + 1}x\sqrt g {R_{icci}} + \int_{\partial M} {{d^4}d\,\Omega {{({\phi _{si}})}^{ - p + 1}}x\sqrt \gamma 2K} } } \right]\]

with K 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 \partial E_S^5:

    \[{S_m} = \int_{\partial M} {{d^{d + 1}}\not \partial \phi _{si}^{d + 1}} x\sqrt g {L_m}\]

with {L_m} the matter field Lagrangian density, and the action above transforms as:

    \[{\delta _g}{S_m} \equiv \frac{1}{2}\int_{\widetilde M} {{d^{p + 1}}} d\,\Omega (\phi _{si}^{p + 1})x\sqrt {{G_{\mu \nu }}} {\widetilde T_{\mu \nu }}\delta {g^{\mu \nu }}\]

with:

    \[{G_{\mu \nu }} = {k^2}{\widetilde T_{\mu \nu }}\]

being the crucial finitizing Euler-Lagrange equation of the total renormalization action:

    \[S = {S_{Gra}} + {S_m} + {S_{reg}} + {S_{renorm}}\]

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!