AdS/CFT Duality, GKP-Witten Relation, and U(1)-Symmetric Holography

All high mathematics serves to do is to beget higher mathematics. ~ Ashim Shanker!

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere. W. S. Anglin, in Mathematics and History

In my last post I showed via AdS/CFT analysis, that gravity is an emergent holographic notion: namely, that one can holographically derive (logically deduce) gravity from conformal field theoretic entropic properties of quantum entanglement, and that such a property is a necessary condition for the ‘bundle’ existence of the gravitonic field. To do so, I had to deform CFT by source-fields via the addition of \int {{d^4}} xJ(x)\vartheta (x), which is a dual AdS theory with a bundle field J and a boundary condition

    \[{J^{\Delta - d + k}} = {J_{CFT}}\]

with \Delta the conformal dimension, a local operator \vartheta and \kappa equals the number of indices of J substracting the contravariant ones to get the AdS/CFT quasi-isomorphic Maldacena correspondence ( = AdS/CFT correspondence), thus the identity

    \[\begin{array}{c}{\left\langle {\Im \{ \exp (\int {{d^4}x{J_{4D}}(X)\vartheta (x))\} } } \right\rangle _{CFT}} = \\{Z_{AdS}}\left[ {\frac{{\lim }}{{boundary}}J{\omega ^{\Delta - d + k}} = {J_{4D}}} \right]\end{array}\]

with the left-hand side being the vacuum expectation value of the time-ordered exponential of the operator over CFT, the right-hand side being the quantum gravity functional with topological-conformal boundary condition, thus leading to holographic emergence, and in a sense, elimination, of gravity. Recall that the Heisenberg Uncertainty Relation holds for energy and time, leading to many anomalies for the Green‘s function of string-propagation:

    \[G({X_{{\sigma _t}(a)}},{X_{{\sigma _t}(b)}}) = \int {{}^sD\exp \left( { - \int_{{\sigma _t}(b)}^{{\sigma _t}(a)} {d{\sigma _t}\int_0^\pi {d{\sigma _t}} } } \right)} \,L\]

with L the Lagragian, due to the fact the {\sigma _t} superpositionality with respect to energy makes Feynman path-summation:

    \[P = \frac{\sum }{{{\rm{Topologies}}}}d\mu {\prod _{\mu {\sigma _s}\,{\sigma _t}}}d{X_\mu }({\sigma _s},{\sigma _t})\]

incoherent since some topologies will degenerate and violate existence conditions for tangent bundles over Minkowski spacetime and some will not correspond to the categorical CFT-manifold, and hence we need to replace the Green’s function with the Källén–Lehmann spectral representation. This is where the GKP-Witten Relation enters with all its glory:

    \[{Z_{CFT}} = {e^{ - S{\,_{GRAVITY}}}}({\phi _i})\]

with background deficit angle \Delta \varphi = 2\pi (1 - n) and the curvature localized on the boundary with an angular deficit:

    \[R = 4\pi (n - 1) \cdot \delta ({\gamma _A}) + ...\,{\rm{regular terms}}\]

with action

    \[\begin{array}{c}{S_A} = \frac{1}{{16\pi {G_N}}}\int {d{x^{d + 2}}} \sqrt g R + \to \\{\rm{Area}}\frac{{({\gamma _A})}}{{4{G_N}}} \cdot (n - 1)\end{array}\]

giving us

    \[\begin{array}{*{20}{c}}{{S_A} = \frac{\partial }{{{\partial _n}}}{\rm{log}}{\mkern 1mu} {\rm{T}}{{\rm{r}}_A}{\mkern 1mu} {\rho ^n} = - \frac{\partial }{{{\partial _n}}}{\rm{log}}\left( {\frac{{{Z_n}}}{{{{\left( {{Z_1}} \right)}^n}}}} \right)}\\\begin{array}{c} = {\rm{Area}}\frac{{\left( {{\gamma _A}} \right)}}{{4{G_N}}} \cdot \\\delta {S_{GRAVITY}} = 0 \to {\gamma _A} = {\rm{ minimal surface!}}\end{array}\end{array}\]

with

    \[{S_{GRAVITY}} = \frac{1}{{16\pi {G_N}}}\int {d{x^{d + 2}}} \sqrt {{g_{\mu \nu }}} R + ... \to \frac{{{\rm{Area(}}{\gamma _A})}}{{4{G_N}}} \cdot (n - 1)\]

hence solving the ‘Ricci/dilaton’ problem I discussed in my last post, since now the holographic formula is

    \[{S_A} = \frac{{\left| {{\gamma _A}} \right|}}{{4G_N^3}} = \frac{c}{3}{\rm{log}}\left( {\frac{{{l_s}}}{a}} \right)\]

with the ‘magical’ expression ({l_s} being the string lenght):

    \[c = \frac{{3R}}{{2G_N^{(3)}}}\]

and with that, the GKP-Witten relation solves the ‘Ricci/dilaton’ problem for the action of supergravity theory.

 

Now let me set up the mathematical context needed to show, in a forthcoming post, that even in M-Theory, or for that matter: any quantum-gravity theory, one cannot coherently quantize gravity in a way that satisfies General Relativistic ‘necessity-criteria’ – as I will show that this would imply, via gravitonic quantum entanglement, the point-‘instantaneous’ collapse of spacetime to a zero-dimensional point like singularity. Not a pretty picture! To do that I have to show that boundary AdS/CFT admits of a ‘local’ symmetry in the bulk theory that is dual to a ‘global’ symmetry corresponding to the boundary and that the (Gubser-Klebanov-Polyakov)-Witten relation deduces the Green correlation functions and that they must have negative Källén–Lehmann spectral representation

    \[\Delta (p) = \int_0^\infty {d{\mu ^2}} \rho ({\mu ^2})\frac{1}{{{p^2} - {\mu ^2} + i\varepsilon }}\]

with \rho ({\mu ^2}) being the gauge-theoretic positive-definite spectral density function.

In the AdS/CFT duality, one must note that the second derivative of the on-shell action principle with respect to the bulk U(1) second-quantized field, must, by unitarity, be identical to the Green function of the U(1) current

    \[\begin{array}{c}1.\quad {\left. {\frac{{{\delta ^2}{S_{CFT}}}}{{\delta {A_\mu }(x)\delta A\nu (x')}}} \right|_{u = 0}}\\ \to {G^{\mu \nu }}(x - x') = - {\left\langle {{T_E}{J^\mu }(x){J^\nu }(x')} \right\rangle _G}\end{array}\]

with {T_E} being the Euclidean time-ordering, and G the Green function.

For equation 1. to be true, the connected Green function {G^{ij}}(x) should provably reduce to the static