Actually, thinking even about AdS/CFT which is commonly said to be emergent space, but not emergent time, if the bulk geometry is pseudo-Riemannian, which has multiple timelike directions at each point, shouldn't time emerge too?

See http://arxiv.org/abs/hep-th/0106113" [Broken]. "In general the dual CFT may be non-unitary". But we want, or we think we want, the bulk theory to be unitary, and since it is defined by RG flow in the "RG space" of the boundary CFT, PM wants "unitary RG flow" in that space. Ultimately it may not be the right way to put it, but I know what he's saying.

A CFT sits a a fixed point(its scale invariant) by definition so presumably once one flows from the fixed point the theory is no-longer a CFT but some perturbation of it in a relevant direction??

That thought was bothering me even as I wrote. I guess that for both conformal and nonconformal boundary theories, the AdS radial dimension encodes behavior at different scales, but for the conformal case, one shouldn't speak of "flow".

Of course it is true that a CFT doesn't itself flow, but it does have a spectrum of operators and controls the flow in its vicinity. In any event, my statement was colloquial and meant for pure amusement.

Unitarity in the RG flow is a strange thing. I would expect it to be connected to operators with imaginary dimension [tex] \Delta [/tex], something not unfamiliar from non-unitary CFTs and other exotica. This way when you flow for an "RG time" T you find expressions like [tex] e^{\Delta T} [/tex] which now look more like unitary evolution if [tex] \Delta = i |\Delta |[/tex]. Hence RG time becomes real physical time!

I think this is well known wild speculation, but it is not much more than that right now.

Thanks guys! Actually, I was thinking of something presumably simpler. Lorentzian spacetimes have "time" or "times" that Riemannian spacetimes don't. In AdS/CFT, what in the boundary theory determines the signature of the bulk geometry?

Papers like http://arxiv.org/abs/0804.3972 and http://arxiv.org/abs/0804.4053 seem to have Galilean boundary theories with Lorentzian bulk, so I presume it isn't the signature of the boundary that determines the bulk signature? (Or is this some weird thing that condensed matter folks do that "real" string theorists wouldn't contemplate?)

Just speaking classically, de Sitter space has a spacelike conformal boundary, anti de Sitter has a timelike conformal boundary, and Minkowski space has a lightlike conformal boundary. (When I say the AdS boundary is timelike, I mean it has a timelike direction - it has spacelike directions too.) Also, of course, the bulk theories all have Lorentzian signature, classically.

I read somewhere that results in AdS can sometimes be analytically continued to results in dS by way of working in Euclidean signature in the AdS boundary theory, but the dS "results" had problems as usual - maybe they were purely formal expressions; I forget the details.