On what Roger Penrose calls "Fashion, Faith and Fantasy" in Physics Today

The ADS/CFT conjecture arose as another way of looking at the 'string' derivation of the Bekenstein-Hawking black-hole entropy formula ... This would only be of relevance to cosmological-size black holes ... based on some remarkable agreements between 'entropy calculations' done in different ways, rather than on an actual derivation of the Bekenstein-Hawking expression. ... some of the strongest claims can be discounted altogether (such as string theory having provided a complete consistent theory of quantum gravity) ... The strength of string's theory's case appears to rest on a number of remarkable mathematical relationships between seemingly different 'physical situations' ... Are these relationships 'coincidence', or is there some deeper reason behind them? ... indeed such a reason ... as yet undiscovered, ... still does not reassure us that the string theorists are doing physics." 31.18 p. 926

31.16 The Road to Reality

What is the "holographic conjecture"?

"the states of a quantum field theory defined on some spacetime M can be put in direct 1-1 correspondence with the states of another quantum field theory, where the second quantum field theory is defined on another spacetime E of lower dimension! Often, E is presented as though it were a (timelike) boundary of M, or at least some conformally smooth timelike submanifold of M. However, this is not the case in the usual example ... The holographic principle is ... analogous to a hologram, where a 3-dimensional image is perceived when a (basically) 2-dimensional surface is viewed. The most familiar form of this 'holographic principle', ... is sometimes referred to as .. the ADS/CFT conjecture. Here M is ... a (1 + 9) dimensional product AdS5xS5, where AdS5 is the ('unwrapped') (1 + 4) dimensional anti-de Sitter space ... here there are four space dimensions. The S5 is spacelike 5-sphere whose radius is of cosmological dimension, equal to (-/\')^1/2, where /\' is the (negative) cosmological constant of AdS5." I think that is a typo in the text and it is (-/\')^-1/2 because [/\] = 1/AREA = CURVATURE

Note that dark energy is /\' > 0 corresponding to a (1 + 4) de Sitter space without an S5 at all. Why the S5? As Izzi Rabi said of the neutrino "Who ordered that?" So the observational fact is that /\' > 0 although the theory here has /\' < 0.

"The smaller space E is to be the 4-dimensional ... conformal infinity of AdS5. We note that E, being 4-dimensional, is certainly not the boundary of M ... since M = AdS5xS5 is 10 dimensional. Instead the boundary of M can be thought of (but not conformally) as ExS5. The Malcadena conjecture supposes that string-theory on AdSxS5 is ... equivalent to a ... supersymmetric Yang-Mills theory on E. Here there is no chance of appealing to the type of 'quantum energy' argument (in 31.10) for explaining away the gross discrepancy between the functional freedom of an ordinary field on M, namely oo^Moo^9 and an ordinary field on E, namely oo^Eoo^3. Since the extra dimensions of M are in no way 'small' - being of cosmological scale - the flood of additional degrees of freedom, from the fields' dependence on the S5 part of M, would spoil any possibility of an agreement between the two field theories. The same would apply to ordinary QFTs on M and E, since one-particle states are themselves described simply by 'ordinary fields' (26.2). The only chance of the holographic principle being actually true for these spaces is for the QFTs under consideration to be far from 'ordinary'." Penrose elaborates on this with more details.
"It is my opinion that the the importance this kind of discrepancy in functional freedom has been profoundly underrated. ... Why is the ADS/CFT conjecture taken so seriously? The support for it seems to come from a correspondence between BPS states on the two sides, that had been noted by Maldacena and from a number of other correspondences ... there are also some additional 'coincidences' that seem to need explaining. .... The ADS/CFT conjecture arose as another way of looking at the 'string' derivation of the Bekenstein-Hawking black-hole entropy formula ... This would only be of relevance to cosmological-size black holes ... based on some remarkable agreements between 'entropy calculations' done in different ways, rather than on an actual derivation of the Bekenstein-Hawking expression. ... some of the strongest claims can be discounted altogether (such as string theory having provided a complete consistent theory of quantum gravity) ... The strength of string's theory's case appears to rest on a number of remarkable mathematical relationships between seemingly different 'physical situations' ... Are these relationships 'coincidence', or is there some deeper reason behind them? ... indeed such a reason ... as yet undiscovered, ... still does not reassure us that the string theorists are doing physics." 31.18 p. 926