Getting from (3D gravity + local degrees of freedom) to 4D gravity

[mitchell porter]

Haelfix pointed out the paper http://arxiv.org/abs/1105.4733" , 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" , 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" ). 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.

 

[mitchell porter]

One more idea which offers something concrete to investigate, and which would bring together recent work. The achievement of Maloney et al has been to calculate a partition function for "gravity plus something local" in 3 dimensions. The objective is to adapt their technique to 4 dimensions.

Meanwhile, one spinoff of Maldacena's conformal gravity paper may be to give new life to the twistor string, which contains super-Yang-Mills coupled to conformal gravity. So, if we're looking for a d=4 version of "gravity plus something", why not use the twistor string? There are actually connections between d=4 Yang-Mills and d=3 Chern-Simons, so it's not a totally arbitrary thought.

 

[mitchell porter]

After https://www.physicsforums.com/showthread.php?t=503785" I understand this paper rather better. Some of what I said above was naive or wrong. For example, they didn't manage to completely evaluate the partition function, they just proved that it is finite. However, I also now have a new wave of ideas regarding how to extend the paper. :-)

"A Black Hole Farey Tail" studies string theory on the background AdS3 x S^3 x K3. The corresponding CFT2 has modular symmetry, and this is explained by a "Rademacher expansion" of the AdS partition function in terms of a series of distinct geometries. The counterpart of this in "A de Sitter Farey Tail" is the amendment of the Euclidean path integral for quantum gravity in dS3 to include, not just S^3, but the lens spaces S^3/Z_p, which are also positively curved 3-geometries which define analytic continuations of dS3.

In the real world we are interested in quantum gravity on dS4, and probably that means dS4 x M where M is some compact space. So one avenue of investigation would be to return to the paper from 2000, and see if its techniques can be combined with those of Maloney et al in order to describe quantum gravity on dS3 x S^3 x K3. The "lens space technique" implies we should be looking at a Euclidean path integral dominated by S^3 x S^3 x K3 but with contributions coming from L(p,q) x S^3 x K3 (the L()s are the lens spaces).

On the other hand, retaining the CFT perspective should be problematic because de Sitter space doesn't preserve supersymmetries. I already mentioned the idea (possibly originating with Polyakov) that dS/CFT might be obtained from AdS/CFT by using the same CFT but at imaginary values of the coupling. We also have Tom Banks's idea that de Sitter space might provide a mechanism for susy breaking (something about virtual gravitinos interacting with degrees of freedom on the cosmological horizon). I wonder if some combination of these would provide a way to directly move from AdS x M to dS x M.

We also have the appearance of modular symmetry in the CFT. I am struck by the fact that Gaiotto's strongly coupled "Tn" theories (https://www.physicsforums.com/showthread.php?t=419450&page=41") arise, as I understand it, from the 6-dimensional (2,0) theory compactified on 2-dimensional tori which correspond to subgroups of the modular group (I think the torus corresponds to a fundamental region in the upper half-plane, with opposite edges identified). Could these provide the 4-dimensional generalization of the lens spaces? That is, in trying to define a set of analytic continuations of the causal patch in dS4, apart from S^4, perhaps we should look at a set of Gaiotto theories, and these would allow us to have a counterpart of the Rademacher expansion. (Then again, in M-theory, the (2,0) theory is dual to AdS7 x S^4, so maybe this is really relevant for dS5, not dS4?)

There's also a paper by Witten, http://arxiv.org/abs/1001.2933" , in which it is observed that "the gradient flow equations relevant to analytic continuation of three-dimensional Chern-Simons theory actually have four-dimensional symmetry". This paper is all about analytic continuation to imaginary values of the coupling, as mentioned above, and it proceeds through steepest-descent methods, which are also relevant to the evaluation of the Hartle-Hawking partition function appearing in Maloney et al. In the final section of his paper, Witten goes further and says, not only is the gradient flow described by a form of d=4 N=4 super-Yang-Mills, but it may be best understood at the 5- or 6-dimensional level, in terms of the (2,0) theory.

Returning to the topic of modular symmetry for the moment, I've noticed two-dimensional fundamental domains showing up in the description of http://arxiv.org/abs/1105.2299" (see Figure 2, page 28), where the CFT is a Chern-Simons theory. I wonder if this offers another way to introduce modular symmetry to the dS4 case, and whether Chern-Simons on the boundary would map to Yang-Mills in the bulk in a way akin to Witten's investigation. (So here I'm trying to extend Maloney et al's approach to Yang-Mills coupled to gravity in four dimensions, as promised.)

Something else to bear in mind is that modular symmetry is really a 2-dimensional phenomenon, and that the higher-dimensional generalization may require more general "automorphic symmetries", such as appear in the Langlands program. And of course, the d=6 to d=4 reduction studied by Gaiotto, Witten and others is supposed to explain "geometric Langlands duality", which in physics terms is an S-duality between a gauge theory with gauge group G and another with gauge group ^G, the Langlands dual of G.

Well, all this is slightly insane, and no doubt riddled with errors, but I figure it's better to get it out there and then see if any of it makes sense. (I also haven't forgotten about https://www.physicsforums.com/showthread.php?t=502733"... Maldacena's boundary condition, which turns conformal gravity into Einstein gravity, involves an "S-brane", which sounds Euclidean, could it be another angle on Euclidean dS/CFT? And Caron-Huot achieves manifest dual superconformal symmetry for his amplitudes by reaching into the fifth, AdS dimension, and using Weinberg's 6-dimensional notation (page 15). The other significance of Caron-Huot's paper is that it begins to generalize and systematize the use of the Goncharov symbol technique - which is related to volumes of hyperbolic spaces, especially in odd dimensions; but I had better stop again.)