Fukaya-Category And Sasaki-Einstein AdS/CFT Holographic Renormalization Of Quantum Gravity

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 Ad{S_5} \times E_S^5 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 {\phi _{si}} be the Calabi-Yau 2-D conic string variable and:

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

with {G_{d + p}} being the D-p-brane’s p+1 dimensional worldspace Newtonian constant, with \gamma the Dirichlet data:

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


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

I then deduced the renormalized Hamiltonian action:

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


    \[{g_i} = \frac{1}{2}\left[ {{\rm{Tr}}\,{\rm{(g}}_{(0)}^{ - 1}\,g_{(2)}^2) - {{\not \partial }_{\mu \nu }}{\phi _{si}}^i} \right]d\,\Omega ({\phi _{si}}^i)\]

with k the Gaussian curvature of the Gibbons-Hawkins boundary term of the Calabi-Yau conic tip of E_S^5 and M being the entropic interior of the Ad{S_5} \times E_S^5 d+1 Riemannian conformally compact manifold \widetilde M, and \partial M its boundary. Hence, one gets, on \partial E_S^5:

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

with {L_m} the matter field Lagrangian density and {S_m} transforms as:

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

with \widetilde T the stress-energy tensor and:

    \[{G_{\mu \nu }} = {k^2}\widetilde {{T_{\mu \nu }}}d\,\Omega {({\phi _{si}})^2}\]

the renormalization functional of the total holographic renormalization group. Now, let \Psi _{U\left| {_{INST}} \right.}^{wf}(eh) 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 Ad{S_5} \times E_S^5. Let ^F\Psi _{U\left| {_{INST}} \right.}^{wf}(eh) denote its Fourier transform. Now, look at the Hodge equation:

    \[{\int_{\widetilde M} {\left\langle {{d^E}\Psi _{U\left| {_{INST}} \right.}^{wf}(eh){,^F}\Psi _{U\left| {_{INST}} \right.}^{wf}(eh} \right\rangle } _{k + 1}}d\,\Omega {({\phi _{si}})^{k + 1}} = {\int_{\partial M} {\left\langle {\Psi _{U\left| {_{INST}} \right.}^{wf}(eh),\delta {\,^F}{\Psi _{U\left| {_{ti}} \right.}}} \right\rangle } _k}d\,\Omega {({\phi _{si}})^k}\]

with {\Psi _{U\left| {_{ti}} \right.}} \equiv {\Psi ^{wf}}_{U\left| {_{ti}} \right.} the wavefunction of the universe at t = i, and ^F{\Psi ^{wf}}_{U\left| {_{ti}} \right.} its Fourier transform. Since:

    \[{\Psi ^{wf}}_{U\left| {_{ti}} \right.} = \int\limits_{ - \infty }^\infty {^F{\Psi ^{wf}}_{U\left| {_{ti}} \right.}} (\xi ){e^{2\pi i\xi t}}d\xi \]

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

    \[ds_{1,4}^2 = \left[ {{e^{2A(\xi )}}\left( { - d{t^2} + dx_1^2 + dx_3^2} \right) + d{\tau ^2}} \right]d\,\Omega {({\phi _{si}}(u))^2}\]


    \[u = \frac{{{l_s}}}{{\alpha '}}{e^{\tau /{l_s}}}\]

which preserves Poincaré invariance and:

    \[A(\xi ) \to \tau /{l_s}\quad \quad {\rm{as}}\quad \quad \tau \to \infty \]

where A(\tau ) is the parametrization function of the interpolating region between branes on the Calabi-Yau conic tip of Ad{S_5} \times E_S^5, and hence, the supergravity action can now be derived as:

    \[{S_{SuGra}} = \frac{1}{{2\pi {G_5}}}\int {\int_{\partial M} {{d^5}} } x\sqrt { - G} \left[ {{R_{icci}} - 2{{\not \partial }_\mu }{\varphi _{ij}}{{\not \partial }^\mu }{\varphi _{ij}}} \right]d\,\Omega {({\phi _{si}})^{e(V(\varphi ))}}\]

with V(\varphi ) the Ad{S_5} \times E_S^5 metastable false vacuum potential. Now, inserting ds_{1,4}^2 in the supergravity action above gives us an energy functional:

    \[{\varepsilon _i} = \frac{1}{{16\pi {G_5}}}\int\limits_{ - \infty }^\infty {d\tau {e^{A(\tau )}}} \left[ {2{{\widetilde \varphi }^2}{ - ^F}{\Psi ^{wf}}_{U\left| {_{INST}} \right.} - 12\widetilde A{{(\tau )}^2} + V(\varphi )} \right]d\,\Omega {({\phi _{si}})^i}\]

and by the Hodge equality, its actional measure is:

    \[V(\varphi ) = \frac{4}{{{l_i}}}\left[ {\frac{1}{2}\left( {\frac{{\not \partial {W^2}}}{{\not \partial {\varphi _{ij}}}}} \right) - \frac{4}{3}{W^2}d\,\Omega {{({\phi _{si}})}^2}} \right]\]

Substituting V{(\varphi )^2} in {\varepsilon _i}, we get:

    \[\begin{array}{c}{\varepsilon _i} = \frac{1}{{16\pi {G_5}}}\int\limits_{ - \infty }^\infty {d\tau {e^{A(\tau )}}} \left\{ {2{{\widetilde \varphi }^2}d{\,^F}{\Psi ^{wf}}_{U\left| {_{INST}} \right.} - 12{{\widetilde A}^2} + \frac{4}{{l_i^2}}{{\left( {\frac{{\not \partial W}}{{\not \partial \varphi }}} \right)}^2} - \frac{{16}}{{3l_i^2}}{W^2}} \right\} = \\\frac{1}{{16\pi {G_5}}}\int\limits_{ - \infty }^\infty {d\tau {e^{A(\tau )}}} \left\{ {2\left( {\widetilde \varphi \pm d{\Psi ^{wf}}_{U\left| {_{INST}} \right.} \mp \frac{{4\not \partial W}}{{l_i^2\not \partial {\varphi _{ij}}}}} \right) - 12{{\left( {\widetilde A(\tau ) \pm \frac{2}{{3{l_i}}}W} \right)}^2} \mp \frac{4}{{{l_i}}}\widetilde \varphi \frac{{\not \partial W}}{{\not \partial {\varphi _{ij}}}} \pm \frac{{16}}{{{l_i}}}A(\tau )W} \right\}\end{array}\]

Hence, holographic renormalization analytically is achieved for 2 reasons – one: the integral \int\limits_{ - \infty }^\infty {...} ranges over all ‘times’, and thus {\Psi ^{wf}}_{U\left| {_{INST}} \right.}(eh) does satisfy the Wheeler-DeWitt equation, and since it topologically ‘lives‘ on the folliated bundle of Ad{S_5} \times E_S^5, 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:

    \[\frac{d}{{dt}}\left( {\underbrace {\int_0^{\delta {f_K}} {\frac{{\frac{d}{{d{t^ \circ }}}{\Psi ^{wf}}_{U\left| {_{INST}} \right.}(eh)}}{{\alpha ({t^ \circ })}}} }_{Creation}\,\, + \underbrace {\int_{{\delta _f}}^{{f_K}} {\frac{{\frac{d}{{d{t^ \circ }}}{\Psi ^{wf}}_{U\left| {_{INST}} \right.}{{(eh)}^{2\pi i\xi t}}}}{{\alpha ({t^ \circ })}}} }_{QuantumGravity}} \right)d\,\Omega {({\phi _{si}})^2}dt\]

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.