There was a discussion of what "AdS/LQG" could be, two months ago. Today, while puzzling over the appearance of Macdowell-Mansouri gravity in Dmitri Polyakov's construction of AdS4 Vasiliev gravity from string vertex operators, I had the following thoughts. A string theory can be expressed as a theory with an infinite number of fields of arbitrarily high spin; these are the modes of the string. Vasiliev's "higher spin" gravity theory also has this property, the difference being that the high-spin modes are massive in string theory and massless in Vasiliev gravity. Vasiliev gravity also shows up in a type of AdS duality. First of all, I am wondering if both theories can be expressed in terms of spin foams, in which there are "two-complexes" (the transition processes which change the spin-net) of arbitrary complexity, which can be grouped into levels corresponding to the various high-spin fields (or perhaps, more plausibly, corresponding to combinations of the high-spin fields). What I'm thinking is that perhaps every UV-completion of quantum gravity is indeed a string theory, but that it will also have a spin-foam formulation. This hypothesis would be more plausible if we could show that the only way for a spin foam to produce a classical limit is to encode a string theory solution in this way. Next, Vasiliev gravity so far is only defined for AdS space. So if it can be expressed as a spin foam, what corresponds to the boundary theory? Any answer to this question might also tell us about the "AdS/LQG" representation of stringy AdS/CFT. In AdS5/CFT4, the boundary theory is a Yang-Mills gauge theory, but in AdS4/CFT3, the boundary theory is a Chern-Simons theory. These aren't disconnected facts; I read somewhere that these Chern-Simons theories are like a dimensional reduction of a Yang-Mills theory in which the kinetic term disappears. Now let us think about dS4/CFT3 for a moment. Although no concrete examples of dS/CFT are known, one of the standard ideas for how it works is that a gravitational theory in a dS space is dual to a Euclidean CFT in a space of one less dimension, which corresponds to the past infinity of the dS space. Sometimes there are thought to be two dual CFTs for dS, the other one at future infinity, with the cosmological evolution in between somehow corresponding to an RG flow between the CFT in the past and the CFT in the future. In any case, what I want to suggest is that the Hilbert space of states which is constructed in loop quantum gravity (loop states, spin-network states being two alternative bases) is the Hilbert space of the Euclidean CFT3 on the boundary of a dS4, and that dS4 time evolution should arise somehow from an RG flow on that Euclidean CFT; and that this is the appropriate way to think about dynamics in LQG. In other words: in AdS/LQG, the boundary theory is a (2+1)-dimensional Chern-Simons theory, and in dS/LQG, the boundary theory is a "timeless" 3-dimensional gauge theory which is just the Euclidean version of the Chern-Simons theory. In both cases, it should be possible to express the bulk theory as a spin foam encoding RG flow in the boundary theory, though in the AdS case it will be "radial" flow rather than timelike.
Mitchell, no direct bearing I fear but a proposal I wanted to bring up in the context of possible dualities. I'll forget it if it does not connect with the discussion in this thread. http://arxiv.org/abs/1105.0938 The gravity/CFT correspondence Henrique Gomes, Sean Gryb, Tim Koslowski, Flavio Mercati 5 pages, 1 figure (Submitted on 4 May 2011) "We prove a general correspondence between classical gravity in 3+1 dimensions and a pair of classical conformal field theories in 3 dimensions (the generalization to higher dimensions is straightforward). The proof relies on a novel formulation of general relativity called shape dynamics that, despite having different local symmetries, leads to classical trajectories identical to those of general relativity in a particular gauge. The key difference is that general relativity's refoliation invariance is traded for volume-preserving three-dimensional conformal invariance, i.e., local spatial Weyl invariance. It is precisely this symmetry that allows us to establish the general correspondence while resolving exactly the local degrees of freedom, a feat that is not possible in general relativity, without a derivative expansion, due to non-linearity." Gomes, Gryb and Koslowski each presented papers at the recent Loops conference (May 2011) which had to do with this formulation of General Relativity
Let's also mention Laurent Friedel on "Reconstructing AdS/CFT", and I'll throw in "holographic cohomology" too.
What are you thinking? Pure Chern-Simons theory doesn't have much physical significance for me; but does it have a special significance for LQG? The history of Chern-Simons in AdS4/CFT3 starts with Schwarz 2004 (see the argument on pages 2-3) and continues with the ABJM paper, and I'd also include Aganagic 2009 for a technically dense string-theory approach to going from Yang-Mills to Chern-Simons. I think she starts with an M-theory background (M2-branes in 3 large space dimensions), then identifies that with a Type IIA string background (D2-branes in 3 large space dimensions) by the usual reduction from 11 to 10 dimensions, but the nontrivial fibering of the M-theory circle over the other 7 compact dimensions gives rise to Ramond-Ramond flux in the 10-dimensional (Type IIA) description. The worldvolume theory of D2-branes should be a Yang-Mills theory, but the RR flux adds a Chern-Simons deformation, and then at low energy the kinetic terms of the Yang-Mills theory disappears, leaving only the Chern-Simons terms. Thus, "a stringy origin of M2-brane Chern-Simons theories".
Spin foams are best understood for discretzing and quantizing BF theory, and I wasn't sure how far the correspondence with lattice gauge theories extends. A review by Bahr et al should be useful. There has been work on spin foams for supersymmetric BF theory and BF theory with matter so I'm hopeful. I do wonder whether it'd be easier to start in AdS3/CFT2. Now I remember Lubos and you had a slightly different idea about this before, linking with canonical LQG states, rather than spin foam states. Physics Monkey has commented on the relationship between tensor networks, spin networks, Chern-Simons and AdS/CFT, and between spin networks and ABJM. The current EPRL/FK spin foams are supposed to be linked to spin networks by KKL. Friedel, Krasnov, and Livine have made a strange, but explicit, observation about spin foams and AdS/CFT.
! Doroud and Smolin: "An action principle is presented for Vasiliev's Bosonic higher spin gauge theory in four spacetime dimensions. The action is of the form of a broken topological field theory, and arises by an extension of the MacDowell-Mansouri formulation of general relativity." I wonder if Doroud and Smolin's proposal is related to Boulanger and Sundell's. But since we want the spin foam to be the boundary theory, I guess it's the conjectured O(N) dual that really matters.
No, I'm looking to have the spin foam in the bulk. Is there such a thing as a "non-gravitational 3d spin foam" for the boundary? edit: How about a (2+1)-dimensional string-net model? Vasiliev gravity in AdS4 is dual to either free or critical "O(N) vector model" in 2+1 dimensions, depending on boundary conditions. I can find many papers on an "O(N) spin model", including in three dimensions, but I can't tell if it's related.
OK, I believe I have the beginning of a way to think about this. The objective is to understand AdS4/CFT3 for Vasiliev gravity (or for ABJM if that doesn't work, but I'll stick with Vasiliev for now) in terms of a bulk spin foam model. This raises the question, how do we think about the boundary? One problem is, how do we even think about the boundary of an "infinite spin network", as would apparently be required to represent a spacelike slice through AdS4? So I propose to work in the opposite direction: Start with a spin network which terminates at points on the boundary. (This is analogous to the bulk-to-boundary correlators in AdS/CFT.) Possibly we should think of the line segments which "end on the boundary" as actually being infinitely long. The other trick is to think about the interior in a somehow renormalized way - there will be "blobs" in the spin network corresponding to sums over whole classes of spin networks. Spin foam models already give us a way to think about the timelike dynamics of changing spin networks, and I'm sure it can be extended to spin networks such as I just described. Now how about the boundary? It's (2+1)-dimensional, so a spacelike slice of the boundary is a plane or sphere with points moving on it, which are the endpoints of the spin network in the bulk. Points can presumably split or join... That just sounds like particles. So maybe it's this simple: the bulk spin networks simply end on particles in the O(N) field theory. A simple-minded beginning, now to see if it makes sense when confronted with the actual details!
I see. I had misunderstood you. There are "non-gravitational" spin foams that are just lattice gauge theory. Even the "gravitational" versions have the same Hilbert space as lattice gauge theory. I think the O(N) model is also known as the classical Heisenberg model or the quantum rotor model. In a certain limit, the classical Heisenberg model looks like Klebanov and Polyakov's starting point. A quantum rotor model can give rise to string nets. McGreevy's lecture notes say the Klebanov and Polyakov cojecture is that the O(N) model does not produce classical gravity in the large N limit. It seems the MacDowell-Mansouri thing goes right back to Vasiliev's himself. MacDowell-Mansouri gravity has been considered for LQG in the context of BF theory coupled to matter. I guess the question is what is the spin foam for Doroud and Smolin's action? I could't understand a word of any reviews of Macdowell-Mansouri or Vasiliev, except for Randono's and Bekaert, Boulanger and Sundell's.
I am intrigued by the possibility that the hypothesized duality (of d=3 O(N) vector model and AdS4 bosonic minimal model of Vasiliev gravity) could be used to fix infinitely many otherwise undetermined amplitudes in a corresponding spin foam model. First, the situation: quantum gravity will have "infinitely many couplings" to fix when done perturbatively, or "infinitely many ambiguities" when done non-perturbatively. Vasiliev gravity is no exception (page 8), but "demanding that Vasiliev theory is dual to the free O(N) vector theory should determine [the ambiguities] entirely" (page 80 here). This still doesn't tell us how a Vasiliev-gravity spin foam would look, but it provides leverage: understand how one set of ambiguities corresponds to the other.
Oda wants to start with just pure spinors on the string worldsheet, get the fermions as ghosts, and the spacetime bosons as ghosts of those ghosts. Vasiliev gravity consists of three "master fields" W, S, B, which have a dependence not just on AdS space-time coordinates, but also some auxiliary noncommutative variables. The infinitely many higher spin fields in the bulk show up in the Taylor expansion of W with respect to those extra variables. Giombi and Yin found a gauge for Vasiliev theory in which the dependence on AdS coordinates disappears. Everything just depends on the noncommutative variables. So maybe you could do what Oda did, if you could apply BRST formalism to Vasiliev theory. There are some papers on that, including the case of massive higher spins, which is relevant for string theory.
I'm not sure the latter half of the statement is true. I think it is true if the theory is fundamentally discrete. However, if some sort of UV limit is imposed, then it becomes constrained. The two ideas for imposing a unique limit in spin foams are (i) Dittrich and Bahr's perfect action, which relies on Asymptotic Safety (ii) Rovelli's infinite refinement=summing. Neither is known to exist at the moment. The only known working method is when the theory is topological, as it is in 3D, discussed also more briefly here.
A dS/CFT correspondence for Vasiliev gravity in d=4 de Sitter space has been conjectured by Thomas Hartman and Dionysios Anninos, and the paper will appear soon. It was premiered in a 5-minute talk by Hartman at Strings 2011 (see about one-third of the way through the "Gong Show"). The theory in question is an analytic continuation to negative N of the O(N) model which features in AdS/CFT for Vasiliev gravity.
I was just listening to Witten's talk from Strings 2007, which is about pure gravity in 3 dimensions. Around 3 minutes, he's saying that in any dimension, you can put the spin connection and vierbein into a matrix, and the usual symmetries of general relativity then become gauge transformations of this matrix; but at 3 minutes 30 seconds, he says that only in 3 dimensions can you write the Einstein-Hilbert action in gauge-invariant form (as a Chern-Simons theory, it turns out). What does this imply for the idea of an "AdS4 spin foam"? In this thread I suggested using the AdS/CFT duality for Vasiliev's higher-spin gravity, because at least the boundary theory was simple. But now I'm thinking it might be better to use ABJM or a related theory, in its Euclidean form, because ABJM is a Chern-Simons theory (plus matter). ABJM is a highly supersymmetric (N=6) theory in 2+1 dimensions which is dual to M-theory on an AdS4 background; also to Type IIA string theory on AdS4, if you lose the eleventh dimension. But usually in AdS/CFT one goes to an even simpler classical supergravity limit, so here we would be looking at "Type IIA supergravity" on AdS4. What I'm proposing is to look at the holographic renormalization flow of Wilson loops in Euclidean ABJM. This should correspond to radial "time evolution" in Type IIA supergravity on Euclidean AdS4. There are a few exotic notions here, for example the idea of a "spin foam in the radial direction". But I think this is an idea which could be explored right now... by someone with the necessary expert knowledge.
So here's the plan. First, here's the original paper on Wilson loops in AdS/CFT. See figure 2, page 3: the expectation value of a Wilson loop in the boundary theory, equals exp(-S), where S is the action of a string worldsheet, stretching in the AdS direction, which has the loop as its boundary. This action is a quantity that can be calculated in the classical supergravity theory. That gives us an idea of what a Wilson loop in the CFT is supposed to mean in the AdS bulk: it's the area of a minimal surface bounded by the loop. Second, here's an example of spin networks actually showing up in AdS/CFT. The paper is by Nathan Berkovits, and Lubos Motl provides the exposition. Lubos wishes to say that spin networks are only meaningful here, since they're being applied to string theory, not to LQG, but LQG readers may wish to tune out those comments and focus on the positive content. I need to explain 't Hooft's planar expansion of gauge theory first. That is the original paper, made available on 't Hooft's own website; see the diagrams on "page 466". The point is that a very complicated Feynman diagram can be associated with a surface. If the Feynman diagram can be drawn on a page without any lines crossing, then the surface is a plane, and it's a "planar diagram". More complicated diagrams map naturally to surfaces with holes and handles. This correspondence applies to AdS/CFT, except that the complicated Feynman diagrams in the boundary theory correspond to topological surfaces in the bulk, which are the string worldsheets in the AdS space. In Berkovits's paper, he reexpresses the string worldsheet in the bulk in terms of Feynman diagrams which (under the interpretation of Lubos) are also spin networks. The vertices of the spin network appear to correspond to holes in the worldsheet, which should mean the appearance of a virtual open string (open string has endpoints, so the worldsheet has a boundary, the rim of the circle). Also, this interpretation only works in a particular limit - I think it is that the AdS radius is zero - and he employs the corresponding topological string theory (a limit of string theory in which the metric has dropped out). That is all a little complicated, so I'll try to summarize. The boundary theory is being considered at weak coupling, so Feynman diagrams are meaningful. In the bulk that corresponds to strong coupling and an AdS radius smaller than the string. Berkovits studies the bulk string theory under these conditions using a topological string theory. He obtains the path integrals for the string worldsheet by summing over networks of Wilson lines on the worldsheet. These networks of Wilson lines are the spin networks. Here I think an interlude for further explanation is needed. So far, we have one application of AdS/CFT, in which a Wilson loop on the boundary is matched with a string worldsheet in the bulk, and another application of AdS/CFT, in which scattering processes on the boundary are matched with sums over spin networks (Wilson lines) in the bulk. How can we match these up? By using a special property of N=4 super-Yang-Mills which is also shared by ABJM: a duality between scattering amplitudes and Wilson loops. In a scattering amplitude, you have particles with incoming and outgoing four-momenta. If they are massless particles, all those four-vectors are lightlike. Because of momentum conservation, their sum equals zero (if you think four-dimensionally). So you can put the four-vectors together in space, head to tail, like high-school vector addition, and they will form a loop. This is the dual Wilson loop! - a polygon of null vectors. The equivalence of the amplitude with the Wilson loop VEV is dual superconformal invariance. So our third ingredient is to use the dual superconformal invariance of the boundary theory to swap the scattering amplitudes in Berkovits's construction for the corresponding Wilson loops. Now we have a mapping between a sum over spin networks in the interior, and the expectation value of a Wilson loop on the boundary. But we are working in ABJM, in order to be able to employ the special features of Chern-Simons theory. So a fourth, even more technical ingredient is the ABJM version of the pure-spinor topological sigma model employed by Berkovits. This can, maybe, hopefully, give us a version of the mapping between "spin networks in the bulk, loop states on the boundary" for AdS4 (Berkovits was working in AdS5). Those are the ingredients I can extract from stringy AdS/CFT. But we're trying to realize "AdS/LQG". Here's where I could use some help. :-) I know that spin networks and loop states offer alternative Hilbert space bases for the d=3 kinematical state space, so the idea is that the Wilson loop on the boundary can be expressed as a sum over spin networks; and spin networks are the boundaries of spin foams. What I want to do is to recover the Berkovits sum-over-spin-networks (assuming that it exists for ABJM) for a spin foam in the bulk. So far we have spin networks in the bulk, rather than a spin foam, but maybe they correspond to saddle points in the spin foam path integral... Obviously, my ultimate objective here is to transpose, as much as possible, the AdS/CFT relations into LQG language, because it may offer clues as to how to make LQG work.
What I said about "radial time evolution" may sound strange. I mean the radial evolution described in hep-th/9912012, "On the Holographic Renormalization Group". You can write equations of motion for supergravity in the bulk, in which, instead of time dependence, you have dependence on the anti de Sitter radial coordinate (the extra dimension, from the boundary perspective). There is a radial Hamiltonian analogous to the ADM Hamiltonian with which LQG began. I have been advocating that AdS/LQG should be pursued in AdS4 (ABJM) because the boundary theory is 3-dimensional, so the familiar loop states could exist there - albeit only in the Euclidean version of the boundary theory. In Lorentzian signature, the boundary theory has 2+1 dimensions, so we are talking about an analytic continuation to 3 spatial dimensions. However, it occurs to me that perhaps one could work in AdS5, for which the boundary has 3+1 dimensions. One could define knotted Wilson loop operators in the boundary theory (which is a gauge theory here, not a Chern-Simons theory), and talk about their time evolution on the boundary. The surprise (from the perspective of ordinary LQG) would be that this equals a five-dimensional theory of gravity. On the LQG side, two papers which might be useful are Mikovic on spin foams in AdS and Kisielowski et al on "spin network Feynman diagrams". In general, the concept is to use the existing quantitative results from AdS/CFT as a guide for the formulation of AdS/LQG.