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Haelfix pointed out the paper http://arxiv.org/abs/1105.4733" [Broken], and Witten 2007 (discussed in that thread) expresses doubt that 4D gravity could be exactly solved, precisely because it has local excitations. And yet here Maloney et al have done it in 3 dimensions. Can something about their method be extended to 4 dimensions? For that matter, what is their method, and is it that reliable?

A glance at the paper indicates that their sum over geometries is tractable because they have focused on lens spaces, but they admit that other geometries may also make a prominent contribution.

Also of interest is the conceptual genealogy of the paper: it harks back to a recent work by the same authors, http://arxiv.org/abs/1103.4620" [Broken], which is intriguing in that it starts with a set of relationships known from number theory, and manages to lift them to a set of relationships between partition functions on either side of an AdS/CFT duality. In their earlier paper, Castro, Lashkari and Maloney explicitly say that they have found a de Sitter analogue of the AdS/CFT "fareytail transform", for (2+1)-dimensional de Sitter space. Can it be generalized to (3+1) dimensions?

Their path integral, before they extend it to include lens spaces, is over a 3-sphere (they are working in Euclidean signature). The extension leads to "deviations from the standard thermal behaviour obtained by analytic continuation from the three sphere". Now consider section 5 of http://arxiv.org/abs/1105.5632" [Broken]). Here we find partition functions on the 3-sphere showing up again, but this time we are talking about the (hypothetical) d=3 boundary theory dual to quantum gravity on d=4 de Sitter space. Note that the CFTs which show up in AdS4/CFT3 are Chern-Simons theories, and that the CFTs in dS4/CFT3 may be analytic continuations of these Chern-Simons theories to imaginary coupling, and that the extra, local degree of freedom in topologically massive gravity arises from a Chern-Simons term.

I suppose I am led to wonder: can quantum gravity on dS4 become tractable, by employing something like Maloney et al's method

A glance at the paper indicates that their sum over geometries is tractable because they have focused on lens spaces, but they admit that other geometries may also make a prominent contribution.

Also of interest is the conceptual genealogy of the paper: it harks back to a recent work by the same authors, http://arxiv.org/abs/1103.4620" [Broken], which is intriguing in that it starts with a set of relationships known from number theory, and manages to lift them to a set of relationships between partition functions on either side of an AdS/CFT duality. In their earlier paper, Castro, Lashkari and Maloney explicitly say that they have found a de Sitter analogue of the AdS/CFT "fareytail transform", for (2+1)-dimensional de Sitter space. Can it be generalized to (3+1) dimensions?

Their path integral, before they extend it to include lens spaces, is over a 3-sphere (they are working in Euclidean signature). The extension leads to "deviations from the standard thermal behaviour obtained by analytic continuation from the three sphere". Now consider section 5 of http://arxiv.org/abs/1105.5632" [Broken]). Here we find partition functions on the 3-sphere showing up again, but this time we are talking about the (hypothetical) d=3 boundary theory dual to quantum gravity on d=4 de Sitter space. Note that the CFTs which show up in AdS4/CFT3 are Chern-Simons theories, and that the CFTs in dS4/CFT3 may be analytic continuations of these Chern-Simons theories to imaginary coupling, and that the extra, local degree of freedom in topologically massive gravity arises from a Chern-Simons term.

I suppose I am led to wonder: can quantum gravity on dS4 become tractable, by employing something like Maloney et al's method

*on the boundary*? The problem with this is (1) the boundary theory isn't gravitation + Chern-Simons (which is what Maloney et al are studying) (2) you don't normally have topology changes on the boundary, in (A)dS/CFT. But with respect to the latter point, I want to quote Maldacena (page 16): "if one were to replace the S3 by other manifolds, such as a S1 × S2, or T3, then one would get different answers. In fact, black branes in AdS4 can be viewed as computing such factors, due to the analytic continuation from EAdS to dS." Well, I don't know what that last sentence is referring to, at all. And there's nothing there to countenance a sum over topologies on the boundary, just the observation that one may consider a boundary with a topology other than S^3.
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