ohwilleke said:
So, shorter version (assuming that I have understood) - the AsD/CFT correspondence can be established in a way that is in many ways more general and less string theory specific than the conventional proof and that illustrates more clearly why this is the case, at the expense of not being quite as comprehensive in some respects. The real bummer of the paper though (at the end) is that the paper says the math for the case we really care about (real GR) is so ugly that even a mother couldn't love it.
That's an interesting comment! Can you be specific about where in the Conclusions section at the end they say that, in effect? I'd enjoy seeing how you interpret some of these. It's all interesting so I will quote the entire Conclusions section. Looks like 8 bullets. Maybe we can refer to them by number:
======quote
http://arxiv.org/abs/1305.6315 pages 19, 20=====
4
Comments and conclusions
We make several comments on these results
• The reason why a conformal field theory on a d-dimensional manifold with a fixed metric is related to a gravitational theory in one higher dimension now becomes transparent. The correspondence maps the constraint generating d-dimensional spatial diffeomorphisms of the gravitational theory to the conservation law of the energy-momentum tensor of the CFT. The remaining constraint, the Hamiltonian constraint generating many fingered time in d + 1 dimensions is shown, through the correspondence with Shape Dynamics, to be equivalent to the generator of volume-preserving conformal transformations in that theory, plus a single global time-reparametrization constraint (14). The latter is mapped by the correspondence using Wald’s axioms to the trace anomaly of the energy-momentum tensor of the CFT.
Thus, we can now understand why the geometry of the renormalization group of a CFT is given by General Relativity, possibly coupled to matter fields, a diffeomorphism invariant theory in one higher dimension.
• We can conjecture a more general correspondence. Given a CFT we get an expansion of the form (1) with coefficients, c
r. We can define a theory with the gauge symmetries of Shape Dynamics by the corresponding expansion, Eq. (4), where we equate the values of the coefficients, c ̃
r = c
r. This gives us some gravitational theory with a possibly different global Hamiltonian constraint replacing (14). We can reconstruct HSD by its volume expansion. This gives what may be called a generalized Shape Dynamics theory which by construction is matched to the original CFT.
We can then conjecture that by gauge symmetry trading of the VPCT for many fingered time, this can be matched to some spacetime diffeomorphism invariant theory in d + 1 dimensions. This may not be General Relativity, but by construction will have spacetime diffeomorphism invariance. If this matching can be achieved there is a general correspondence between any CFT and some gravitational theory in one higher dimension.
• We have studied only the ⟨0| T
cd |0⟩ but it is possible that the renormalization group flow of other conserved currents can be explained by expanding Wald’s axioms to them, and then by coupling Shape Dynamics to suitable matter fields.
• The correspondence we show is more general than the conjectured AdS/CFT correspondence as stated originally by Maldacena, in that it is not restricted to supersymmetric theories and String Theory and the properties of 10 dimensional supersymmetry algebra play no role in establishing the general correspondence at this level. However supersymmetry certainly plays a role in specific examples of correspondences, such as those involving supersymmetric Yang-Mills theory.
• The correspondence we demonstrate here is however weaker than the original Maldacena conjecture in that we claim only the classical bulk gravitational theory plays a role. Rather then a conjectured equivalence between the Hilbert spaces and observables algebras of a bulk and boundary theory, we demonstrate only an equivalence between the expectation values of operators in the CFT and solutions of the classical field equations of the bulk theory.
• However, some of the evidence used to support the stronger Maldacena conjecture can now be explained by the weaker and more general correspondence we give here. Hence, only results that are not explicable by our weaker correspondence can count as evidence for the stronger Maldacena conjecture.
• In particular, Shape Dynamics does not, so far, directly explain the correspondence between pure General Relativity and N = 4 super-Yang–Mills theory in the N → ∞ limit except as a coincidence of the coefficients in the two expansions. This coincidence is well represented by the work of Skenderis et al, and also from a purely gravitational perspective [3, 7].
• We can also conjecture that the correspondence we have demonstrated here extends to a stronger correspondence between a quantization of Shape Dynamics and the quantum conformal field theory. The evidence we possesses for an extension of the correspondence into the quantum regime is the apparent absence of anomalies (for odd dimensions) of our spatial conformal transformations (see appendix A). In spite of their classical correspondence, the quantization of Shape Dynamics is unlikely to coincide with a quantization of General Relativity due to the very different structure of their constraints.
==endquote==
One hopeful thing I notice here is that although Shape Dynamics and GR agree at a classical level they may lead to
different quantum theories! So we have more chance of being right. SD and GR are equally good gravity theories empirically but we may end up with two inequivalent
quantum gravity theories, and thus a possibility of comparing what they say about the early universe.
For anyone new to the subject. The abbreviation VPCT in bullet #2 stands for "Volume Preserving Conformal Transformation". These are physically meaningless ("gauge") transformations in SD, which allows local conformal scale change as long as the OVERALL volume remains unchanged. Expanding one place means you have to shrink somewhere else.
