The "dS/dS correspondence", and AdS-to-dS uplift
This post is about yet another approach to holography in de Sitter space, which may at least be halfway successful.
The starting point is "The dS/dS correspondence" from 2004. The idea is that physics in dS_n is holographically dual to gravity plus two CFTs in dS_(n-1). This comes from a timelike slicing of de Sitter space - or rather, a timelike slicing of one portion of de Sitter space, the "static patch" - into a product of de Sitter space in one less dimension by a line (the extra space dimension), with a warp factor varying along the line as graphically depicted here. The central dS_(n-1) slice is where the graviton concentrates, just like in Randall-Sundrum, and the two CFTs represent the bulk space on either side.
The authors say that this duality, which they call "semi-holographic" because you still have gravity in the lower-dimensional theory, could in principle be iterated until you just have a quantum mechanical model in (0+1)-dimensional space (i.e., a matrix model, I assume). One might intuitively think of this as follows: We take the n-dimensional hypersphere which is the spacelike cross-section of dS_(n+1), and we collapse it onto its (n-1)-dimensional equatorial hypersphere, then we collapse that hypersphere onto its (n-1)-dimensional equator... and so on down to the 2-sphere of dS3, being reduced to the circle of dS_2, and then that being reduced to - two points in "dS_1"? The endpoint of the process is a little mysterious. It's also unclear to me whether this is a correct understanding of how their sequence of dualities would work, because they are working in the static patch, one of the several different metrics available in de Sitter space, and this particular metric only covers a quarter of the space.
In any case, the paper in 2004 discusses the dS/dS correspondence in a rather hypothetical fashion. Six years later in 2010, in "Micromanaging de Sitter holography", an example has finally been constructed - it's Type IIB string theory on dS3 x (a lens space x T^4) - but as far as I can tell, this space is asymmetrically shaped especially so as to realize the correspondence. The dS3 factor seems to be like a football, with a stack of branes at each tip. So the equator of the football should be where the projection to dS2 occurs, and the brane-stack on either side presumably defines one of the CFT sectors.
The discussion of dS/dS in 2004 doesn't say anything that I can see about an inherently preferred direction in the dS geometry; the warping involved in the slicing looks like it's a result of the coordinate system - that it's a result of working in the "static patch". Probably I'm missing something, that will come clear when I do my remedial studies of de Sitter geometry... But the impression I got, was that the duality should work from within any static-patch coordinate system, suggesting that there might be many equivalent dual theories of the form "gravity plus two conformal sectors" - that is, just one dual theory, but a theory with many equivalent descriptions corresponding to the different "directions" from which one might perform the duality. If this is how it works, it should help with the iteration of the duality too - it won't matter from which direction you reached the next rung down, it'll still just be physics on de Sitter space and you can once again apply the duality from any direction you wish.
That's how I imagine it working, but if the paper from 2010 is any indication, perhaps that's not how it works at all - since they had to employ this special football construction. That's why I call the idea only halfway successful - it may be that the original idea is not completely correct, when put into practice. And by the way, the football geometry is just the beginning of it - there are lots of other branes too, four different types of them, all playing a role. More on this below.
In a recent post, I pondered how to get from some constructions of gravity in dS3 due to Maloney et al, to something comparable in four dimensions. One path was to pursue an algebraic generalization appropriate to four dimensions (thus my remark about Gaiotto theories playing the same role, in d=4, that the lens spaces were playing in d=3). The other path - but these are not necessarily disjoint alternatives - is to turn an AdS construction into a dS construction. It's now dawned on me that, of course, this is what was done in the famous "KKLT" paper which launched the landscape mania in string theory; and that it's about adding new branes and fluxes to the AdS configuration, which will serve as new sources of positive curvature, sufficient to move the net curvature from negative (AdS) to positive (dS).
But it turns out that there are difficulties. Polchinski and Silverstein note the immediate difficulty: in many (most?) known AdS/CFT dualities, the size of the compact (non-AdS) dimensions is comparable to that of the AdS radius. So first they set out to construct new AdS solutions to string theory in which the AdS radius is orders of magnitude larger than the radius of the non-AdS dimensions. The space which features in "Micromanaging de Sitter holography" starts from one of these new AdS solutions, and this is why it's so complicated: first they need an AdS space with a hierarchy of dimensional sizes, then they need to add branes and fluxes which will change the overall curvature from negative to positive.
To see just how complicated the resulting construction is, turn to page 14 of "Micromanaging...". The table indicates the kinds of branes and the dimensions into which they extend. D1-branes and D5-branes (the initial ingredient), "rho-branes" which are degenerations in a fibration of the T^4 torus over the lens space, NS5-branes, D7-branes, and there are orientifold planes too. Also see page 22, where they calculate various quantities (like the radius of curvature) for a specific number of branes of each type. Note that they have a stack of 156 D1-branes! You know, most of these AdS/CFT dualities involve "large N" theories on the field side - and N corresponds to the number of branes in a stack somewhere. So this is a reminder of what large N actually means. :-)
Assuming that everything works as advertised, I have to respect this construction. And it evidently took years to figure out how to make a working model. But the question remains, is this complexity typical of how de Sitter space gets realized in string theory? Even if it is, that's not necessarily an argument against its "correctness". Particle physics in the real world is pretty complicated, and maybe that's due to an inflationary selection effect, whereby inflation only occurs if the local initial conditions strike this fine balance.
Or maybe the complexity is just a feature of this first implementation of the idea, and later ones will be more streamlined.
Now I want to mention a feature of the construction which really does sound cool, or noteworthy. They say that it provides a microscopic derivation of the entropy associated with the cosmological horizon in de Sitter space. The original horizon-with-an-entropy is the event horizon of a black hole, and in the mid-1990s Andrew Strominger and others were able to construct black holes in string theory for which the Bekenstein entropy / Hawking temperature, as calculated from macroscopic thermodynamic considerations, could be matched by a microscopic description of the black hole, as a bound state of branes. True, it was a black hole in five dimensions, and an "extremal" black hole in a highly supersymmetric theory, so it didn't have much in common with astrophysical black holes, but it was a beginning. (In fact, such investigations were a step towards the discovery of AdS/CFT, which also started life as a statement about the near-horizon geometry of black brane stacks.) But the cosmological horizon in de Sitter space is observer-dependent! So I think it's remarkable that they have a microscopic construction which accounts for the associated entropy, and I'm keen to understand how it works. (Right now, I don't understand at all; the relevant calculations come right at the end of the paper, and consist of showing that certain quantities scale appropriately.)
I think that's about all I have to say - except for one last observation. Polchinski and Silverstein, in the paper cited above (on finding new AdS systems suitable for uplift to dS), say that the dual theories may contain Argyres-Douglas fixed points. Well, that's not too far away, conceptually, from saying that they contain Gaiotto theories, I think, because in both cases you're talking about these strongly coupled sectors, possibly with no perturbative expansion, which descend from the M5-brane worldvolume theory. I brought up the Gaiotto theories as a possible 4-dimensional implementation of modular or automorphic invariance in the string-theoretic partition function, to play the same role as the lens spaces in the 3-dimensional path integral of Maloney et al. So I'm moderately encouraged to here P & S talking about something similar, for an entirely different reason.
Final note: Let the reader beware; in the preceding post, I made a number of statements which are really guesses based on an incomplete understanding of what I was reading! Namely, everything where there's a qualifier like "I think" or "I imagine", etc. Of course, I hope to acquire a truly correct understanding in the near future, and will post any corrections here - if someone else doesn't do it first.
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