# AdS to dS uplifting and its opposite

So as I understand it, localized structures in AdS can wick rotated to dS, the boundary has to be complixified as can be seen here. Also, uplifting is another technique that can be used to move from AdS to dS. (Thanks to Mitchell Porter for this suggestion) My question is that just as an exercise, is it possible to move from dS to AdS, I don't know the name for such a process, it is in some sense "moving down" from dS to AdS. Take the following example, suppose I have a mapping of a scalar field from boundary to bulk in AdS, now that can be uplifted to dS, it won't hold make much sense anymore since AdS allows for SUSY however dS doesn't but let's say for a moment that it can be done. Now consider the opposite case, we have a scalar field mapping in dS, can that be brought down to AdS? I think it can be, and this should be actually easier since AdS allows for SUSY which should make the mapping a little easier. The lack of a concrete example for dS/CFT makes it (about the mapping which we considered in our hypothetical case for dS) difficult to ask this question but does it seem plausible that if such a mapping were to be constructed, it can be brought down to AdS? Thanks.

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There is something I would like to say about the approach one might take to the question I posed, wick rotation works both ways I believe, so going from AdS to dS and back might be possible, if someone could comment on that as well, I'd very much appreciate it. – dhillonv10 Aug 2 '11 at 5:18

## 1 Answer

Well it just so happens that Mitchell answered this question in a private email, I'm posting the relevant part here for the greater good.

The Wick rotation in this case transforms a local patch of dS space into AdS space, or vice versa.

Think of the difference between concave and convex, a valley and a hill. Then you should imagine that we have some formula appropriate for a valley - e.g. you might have a line-of-sight device like surveyors use, and you need to adjust the angle of the view finder, in a way that depends on the curvature of the ground. We can turn it into the corresponding formula for the hill by replacing, somewhere in the formula, a positive number with a negative number - because the curvature now has the opposite sign. This is what's actually going on in these Wick rotations, although the details are somewhat more complicated.

Returning to the analogy, turning dS space as a whole into AdS space is like turning the whole valley into a hill.

Here is a picture for the analogy. Scroll down to the concave meniscus vs convex meniscus. http://www.tutorvista.com/content/physics/physics-iii/solids-and-fluids/shape-meniscus.php In the concave meniscus, the vectors normal to the surface (labeled R) diverge; in the convex meniscus, they converge.

In the previous paragraph, you can see two formulas, basically "Force1 minus Force2" and "Force2 minus Force1", which correspond to the two situations. You can get one formula by multiplying the other by -1.

So here is the analogy. If you start with the force formula for a concave meniscus, and want to get the force formula for a convex meniscus, you perform the transformation "multiply by minus 1". That's analogous to: start with the quantum propagator for a scalar field in AdS space, and get the propagator in dS space by the transformation "perform a Wick rotation".

But if you actually want to turn a concave meniscus into a convex meniscus, you're not just "multiplying the meniscus by minus 1". What you would have to do is increase the intermolecular tension everywhere, until it changes the curvature.

In the same way, the uplift from AdS space to dS space involves changing the local curvature everywhere, by having extra fields whose energy density acts as an extra source of curvature (this is in the "Landscaping" paper by Silverstein and Polchinski).

So my first point is that uplifting isn't Wick rotation. It actually means that you add something to your model, which will make the space curve differently. What the Wick rotation allows you to do, is to take a formula appropriate for one type of space, and modify it so it applies to the other type of space - but these are formulas which describe what happens in a small patch of the space, they aren't describing its total structure.

So far, I've only been talking about physics in the bulk (dS or AdS). The existence of a boundary CFT, and the details of how it relates to the bulk physics, wasn't even discussed yet. Remember that half of what David Lowe writes about is just a mapping from the center of the bulk to the edge of the bulk, that is, it's something entirely within bulk physics. For example, it's about how the wavefunction of a particle changes as it travels from the interior to the edge. Then, at the edge of the bulk, is the truly holographic mapping; or at least, the edge of the bulk is the place where a point in the bulk maps to a single point on the boundary. (Points deep in the interior of the bulk correspond to some sort of sum over whole regions on the boundary.) The truly holographic mapping is this change in variables - from bulk variables to boundary variables.

So a big part of what we should be interested in is, what happens to the holographic change in variables if we go from AdS to dS. But there might be an easier case to study first: what happens to this change in variables if the curvature of AdS space changes (this would usually be expressed in terms of the "radius" of the AdS space, the distance from center of bulk to edge of bulk).

If you look up the notes of Thomas Hartman's talk from Strings 2011, he actually says how the boundary theory changes for Vasiliev's higher-spin gravity, as you go from AdS to dS. In AdS space, the boundary theory is a vector field theory with an O(N) symmetry, in dS space the boundary theory is now a scalar field theory with a Sp(N) symmetry (N is the number of vector fields or number of scalar fields). So going from AdS to dS requires the set of fields to change rather radically - though apparently there's still enough of a similarity that he and Anninos were able to deduce or guess that Sp(N) symmetry would be the dS counterpart. (Let me also say in passing that they have only solved the problem - if they have solved the problem; no paper yet... - for a particular case of AdS/CFT. When the bulk theory is string theory rather than Vasiliev's higher-spin gravity, the boundary theory is a supersymmetric field theory that Hartman doesn't know how to transform appropriately; he says this in questions after his talk.)

If you see Hartman's notes, I think you'll also find that he talks about the cosmological constant in the bulk space going from negative to positive. This just refers to the curvature, and the difference between AdS and dS. But you can also talk about the difference between AdS with strong curvature and AdS with weak curvature. This also requires a change in the boundary theory, but not such a big change.

One reason AdS/CFT is popular is that it is a 'strong-weak' duality. When quantum theories have strong "couplings" - when the interactions are strong - they become difficult or impossible to calculate with, because the usual approach of successive approximations - perturbation theory - doesn't work. The whole idea of perturbation theory is that each correction is smaller and less important than the last, so in order to approximate the actual result, you just keep adding smaller and smaller corrections. But when the interaction is strong, the corrections get bigger as you go, which means that the whole starting point was wrong.

In AdS/CFT, typically, when the boundary theory is strongly coupled, the bulk theory is weakly coupled, and vice versa. So if you have a strongly coupled field theory, it may have an AdS dual gravity description which is weak and therefore calculable. Or if you have a gravity theory in a strong coupling state, it may be dual to a weakly coupled boundary field theory.

If the curvature of the AdS space gets bigger, that should mean that the gravitational coupling is getting stronger, but in turn that means that the boundary coupling is getting weaker. And conversely, if the AdS space is getting flatter - more weakly curved, and approaching the transition to dS space - that should mean the boundary theory is getting strongly coupled. And even independently of AdS/CFT, it's well known that strongly coupled theories often have a dual description in terms of new variables which will be more easy to calculate with. So possibly this is an aspect of what happens on the boundary as you go from AdS to dS.

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