223. Starting dS/CFT: Hilbert space

Since I was recently thinking of the dS/CFT correspondence, I find it natural to also start discussing facts and hypotheses related to dS/CFT  and other gauge theory – gravity dualities on the blog.  In what follows, we will mostly discuss 4-dimensional gauge theories.

First of  all, what is the motivation for us to look for gauge theory-gravity dualities? The reason is that a general confining gauge theory at strong couplings  looks effectively like a string theory – flux tubes connecting fermions charged over the gauge group do behave like relativistic strings. In the case of QCD we still have no idea which one in particular is this string theory describing confining asymptotically free QCD at strong coupling, but from 1970s (thanks to ‘t Hooft) we are pretty sure that this string theory is free if the number of colors in the gauge theory is very large (that is, if the gauge group is , should be taken to infinity).

Although it proved to be extremely hard to get any other information about string theory dual to QCD, one important piece of information became available in the end of 1990s -  as it turns out,  supersymmetric YM theory (which is a conformal field theory) is dual to IIB string theory on background. That is, we generally have to expect that if duality between a string theory and a gauge theory exists, string theory should live on a curved background spacetime.

1. AdS/CFT

More precisely, this particular duality works as follows.

Let us first focus on its gravity side. The near horizon geometry of coincident -branes is exactly . Strings attached to these branes are relevant degrees of freedom that we are interested in. As we know, spectrum of the string contains graviton and at low energies we can describe dynamics of the string simply by using Einstein-Hilbert action (with matter fields). More precisely, this is possible when the radius of the curvature of the background is much larger than the string length (in other words, the curvature of the background is sufficiently small), i.e., .

(Note: Well, I am note completely honest here. If you say that , honestly it is only guaranteed that the SUGRA approximation holds for the stringy degrees of freedom, but the effective action is not necessarily the Einstein-Hilbert one. To suppress higher curvature corrections, you need in addition to take the limit .)

On the other hand, if the curvature of the background is large, the only relevant degrees of freedom are the ones that live in the world volume of – brane. The dynamics of these degrees of freedom is governed by gauge theory with supersymmetry, and thus we come to the field theory side of the correspondence. Description of the system in terms of the supersymmetric gauge field theory holds when the curvature of the background is very large, i.e., when .

Therefore, dynamics of the super Yang-Mills theory at strong couplings is equal to the dynamics of gravity + matter degrees of freedom on asymptotically background. Does this fact give us a lot of information? Yes and no.

Yes, because the AdS/CFT duality is established extremely well. For example, has the global – symmetry, and this fact is reflected on the gravity side – extra 5 dimensions of the 10-dimensional space are compactified in the sphere with the group of symmetry . 4-dimensional conformal symmetry of the SYM is , and the symmetry of the space is also .

(Note: By the way, why do we need supersymmetry so much? Bosonic, that is, non-supersymmetric string contains a tachyon in its spectrum, so we cannot expect that the dual gauge theory is stable.)

No, because the duality is a kind of boring. First of all, dynamics of conformal (this is maximal possible amount of supersymmetry in 4 dimensions!) at strong coupling is much more trivial than the dynamics of QCD at strong coupling that we are really interested to learn about. Second, dual description in terms of gravity really works for large and large ‘t Hooft coupling , while planar QCD confining string is the string in the regime of large N and small (even worse, the low energy regime of QCD is the regime with relatively small and small , since does not become terribly larger than 1 in the real life).

So, the second question one really comes to is…

2. What if we deform AdS/CFT

Naturally, we would like to deform the duality and see what happens – does it hold? How much should I deform the theory to break the gauge-gravity duality? For example, we could try to deform it on the CFT side – say, we take a slightly non-conformally invariant theory and try to establish whether it is possible to find its dual gravitational description describing behavior of the theory in the regime of strong coupling (gravity on the background which is slightly deformed ). This is actually the route we will really have to pursuit if we want to construct a gravity dual to QCD. Indeed, the latter is asymptotically free, that is, has negative beta-function while a conformal field theory corresponds to a situation with precisely zero beta-function.

This way however is a hard one, so we can try to deform instead the gravity side of the duality – say, by deforming the asymptotic background. As it turns out, many other CFTs (than SYM) exist which correspond at strong coupling to a gravity with some non-trivial asymptotic behavior. For example, we can try to factor asymptotic AdS space with a discrete group with compact fundamental domain or discuss, for example, . As it seems, both cases correspond to some conformal field theories, but what are the theories exactly is not quite clear yet.

(Note: Some of string theorists (who maybe really like to exxagerate) say that every single CFT that can be found in Nature (and even those that cannot)  has a gravity dual.)

But what if we will do something even more radical – say, take de Sitter space time as a background instead of Anti-de Sitter? 5-dimensional de Sitter space has group – that is, Euclidean conformal group, not really weaker than the conformal group of symmetry of .

Can gravity on an asymptotically de Sitter 5-dimensional background be dual to a Euclidean gauge theory? As it turns out, we come to many obstacles trying to test this idea. First of all …

3. Hilbert space of dS is finite dimensional and de Sitter space has entropy associated with it

Indeed, no single observer can have access to the whole spacetime – its Hubble colume is separated from the rest of the spacetime by the horizon (I discussed the causal structure of dS space in details before, but if you remain dissatisfied after reading my post, I can advise you to take a look at Hawking, Ellis book, Chapter 5). Similar to the case of a black hole emitting Hawking radiation, the observer detects thermal radiation which comes from the horizon (the difference is that thermal radiation is emitted inside the horizon, while in the case of BH – outside) with temperature proportional to the Hubble scale .

Since the region outside is not accessible, we have to average over quantum modes with wavelength larger than the horizon – we are unable to discreminate between them and the constant zero mode. This is a source of entropy associated with de Sitter space. This entropy can be calculated to be

,

where is the area of the de Sitter horizon in exact analogy with Bekenstein-Hawking entropy of BHs.

So, why the Hilbert space of dS is supposed to be finite-dimensional? The reason is that string theory really teaches us to interpret gravitational entropy like any other entropy – as the logarithm of the number of states in the causally connected region. Finite entropy therefore means the finite number of states:

.

4. Dimension of the Hilbert space in perturbation theory

Suppose that we start from a flat spacetime (say, by setting the cosmological constant to zero), where the dimension of de Sitter space is infinite, and then slowly increase the value of the cosmological constant. We then conclude that the dimension of the Hiblert space is infinite for the flat space time, i.e., the case , while for any arbitrarily small positive value of the dimension is infinite. If we go vise versa, from the case of small positive to the case of exactly zero , we have a jump from finite values of the total number of states to infinite .

As we see, one interesting question is how to reconcile the fact that Hilbert space of dS is finite non-perturbatively with the fact that it is infinite perturbatively? I am going to discuss the answer to this question in the next few days.

Update: I have found a terrific book on Amazon which contains almost all TASI lectures related to string theory (in particular, it contains Klebanov’s lectures on AdS/CFT).