Exploring Non-Unitary Euclidean CFTs: What, Why, & How

[atyy]

What is a non-unitary CFT?

Why are Euclidean CFTs allowed to be non-unitary?

I assume the opposite of Euclidean is Lorentzian? Why are those not allowed to be unitary?

These questions are from listening to Hartman's talk that mitchell porter recommended:
https://www.physicsforums.com/showpost.php?p=3387701&postcount=15.

 

[mitchell porter]

What is a non-unitary CFT?

Why are Euclidean CFTs allowed to be non-unitary?

I assume the opposite of Euclidean is Lorentzian? Why are those not allowed to be unitary?

Non-unitary means it's... not unitary. But that doesn't mean it's useless. Open quantum systems have a non-unitary evolution. Non-unitary CFTs get applied in condensed matter physics to the "Yang-Lee edge singularity" and other http://arxiv.org/abs/1012.1080" .

Lorentzian versus Euclidean is just the usual change of signature and it's not related.

 

[Thomas Larsson]

It's been ages since I thought about these things, so some might be misremembered, but here goes.

Any sensible spin model, like the Ising or q-state Potts model, is unitary. Sensible here means that that the Boltzmann weights are local and positive (so the energy is real). More formally it is encoded in the Osterwalder-Schrader axioms, which are the euclidean version of the Wightmann axioms (or possibly vice-versa).

However, there are interesting geometrical models in statphys which can be described as limits of spin models. E.g. percolation is the q -> 1 limit of the q-state Potts model and self-avoiding walks the N -> 0 limit of the O(N) model; both correspond to the non-unitary c -> 0 limit of CFT. In these geometrical models unitary is not necessary, because there is no local definition of percolation of SAWs.

 

[mitchell porter]

I was just reading the previous paper by http://arxiv.org/abs/0910.4587" , and found an arresting concept at the end: a cosmological horizon (as in de Sitter space) corresponds to a thermal state in the dual Euclidean CFT. In AdS/CFT, black holes in the bulk correspond to thermal states on the boundary; it's the holographic explanation of black-hole thermodynamics, and it's why a quark-gluon plasma (a localized thermal state) is dual to an AdS black hole. So it's retrospectively obvious that in dS/CFT, a cosmological horizon might be dual to a thermal state in the CFT, too. Hartman and Anninos figured this out by considering a black hole in de Sitter space, so big that its horizon coincides with the cosmological horizon. The two types of horizon become one!

 

[ordered_chaos]

a cosmological horizon (as in de Sitter space) corresponds to a thermal state in the dual Euclidean CFT.

I'm a layman, but it sounds like your saying outside the cosmological horizon of our universe, it's like a thermal state similar to a black hole, so our universe is on the boundary of a holographic screen.. If I'm completely off, I'll just back to lurking and I won't interrupt the adults who are talking, sorry :)

 

[mitchell porter]

I'm a layman, but it sounds like your saying outside the cosmological horizon of our universe, it's like a thermal state similar to a black hole, so our universe is on the boundary of a holographic screen.. If I'm completely off, I'll just back to lurking and I won't interrupt the adults who are talking, sorry :)

We need to distinguish between holography where the dual theory is "at infinity" in some sense, holography as it pertains to event horizons and cosmological horizons in the "interior", and finally holography as it pertains to a generic surface in space.

In AdS/CFT, you can visualize the bulk space as the interior of a cylinder, and the boundary as the surface of the cylinder. Time goes up or along the cylinder, not around it. (http://www.achtphasen.net/media/users/achtphasen/pwsup5_11-03.jpg" ) Things happening inside the cylinder are equivalent to things happening on the surface of the cylinder.

Since a cylinder has a 2-dimensional surface, and one direction is time, we only have one direction left for space, so in that drawing, the boundary space is just a circle, and the holographically equivalent "bulk space" is the disk region inside the circle. Incidentally, as measured within the disk, it is infinitely far to the boundary (in the picture this is represented by the way the shapes shrink towards the edge).

What happens at a point in the disk interior is equivalent to a sum over a region on the boundary (you may be able to see what I mean in my attachment https://www.physicsforums.com/showthread.php?t=500688&page=4#49"). So the existence of a black hole at a certain point in the disk corresponds to a thermal state (a plasma) in the corresponding boundary region. Since the boundary is just a circle, in the boundary description you would just have particles or waves moving around the circle, or back and forth on the circle. So a "plasma" on a segment of the circle could be visualized as lots of points moving back and forth, colliding with each other, absorbing and emitting each other. This activity corresponds, in the string theory description of the black hole in the bulk, to all the D-branes (and strings connecting them) which are inside the black hole, exchanging energy between each other, while in a state of thermodynamic equilibrium, i.e. branes and strings are all at about the same energy - this is where the black hole entropy comes from. And all those "parts" of the black hole correspond to particular "components" of the plasma on the boundary.

Ordinary objects on the disk (not black holes) would also have their counterparts on the boundary; not a plasma region, but particular combinations of the boundary fields.

So to sum up, any configuration of objects on the disk - in the interior of the circle - maps to a configuration of objects on the boundary - at spatial infinity, as measured by the metric where "shapes shrink as they approach the boundary". Really, drawing it as a circle of finite radius is just a way of squeezing an infinite space into a picture.

This is all for anti de Sitter space, which is a peculiar hyperbolic space. There ought to be another version of holography, dS/CFT, for de Sitter space, which is a bit more like the universe we see. You can think of de Sitter space as the surface of a sphere which starts infinitely large in the infinite past, then shrinks to a particular size, and then expands infinitely into the future. In this case, the bulk space is just the surface of the sphere, and the boundary is the sphere in the infinite past and infinite future, so that is where the holographically dual description has to live. In this case, events during the history of the universe - events that happen somewhen between the infinite past and the infinite future - will have a dual description in terms of a configuration on the "infinite spheres" in the infinite past and/or the infinite future. i.e. if we compare this with AdS/CFT, in AdS/CFT time in both descriptions is the same, it's normal, the only peculiarity is that particles moving back and forth on the perimeter of the circle systematically encode events happening in the interior of the circle. In dS/CFT, time is the extra holographic dimension, and everything in the history of the universe will instead have an image in a static description at past/future infinity. (Once again, something in the "interior" - here, something in the history of the universe - corresponds to a region on the boundary - a region on the surface of those spheres at infinity. Technically that would be the past and future light cones of the event or object, extrapolated to infinity.)

As I mentioned, in AdS space, it is actually infinitely far to the "edge of space". But it is possible to just consider a finite-sized region of AdS - just a part of the disk. This corresponds to introducing an "energy cutoff" in the boundary theory. Basically, you don't have to worry about what's happening on the boundary below a certain scale of spatial resolution, since details that small will correspond to objects in AdS outside the part of it that you're focused on. Spatial scale corresponds to energy level in a quantum theory - the processes which only show up at short distances have to be high-energy ones. So this is why a cutoff in spatial detail for the boundary theory corresponds also to a cutoff in maximum energy of interest.

Now let us switch to black holes for a moment again. A black hole is a special type of object, which is found in the spatial "interior" - not at infinity - and it turns out that there is a type of duality for black holes too, the "Kerr/CFT correspondence". A Kerr black hole is a spinning black hole, and it seems (I haven't studied it) that it can be described by a CFT (same type of field theory as in these other correspondences; C = "conformal") with one space dimension and one time dimension, where the space dimension in the CFT corresponds to the black hole's rotation - I think to the direction of its axis of rotation. So in Kerr/CFT, if I have understood it correctly, all the space directions away from the axis of rotation have to emerge holographically. In the CFT, you just have things moving back and forth along a line, and that is equivalent to processes happening around the Kerr black hole (e.g. something orbits it, or falls in).

If we imagine a Kerr black hole in an AdS space, it's clear that the "black hole CFT" can't be the whole story - there's the whole rest of the AdS universe. So the CFT dual to an individual black hole has to be just a "sub-CFT" of the full field theory dual to the whole of such an AdS universe. I can't tell you the details, unfortunately; I don't think anyone knows them. But just as part of the full CFT corresponds to a finite region of the AdS interior, the CFT dual to an individual black hole must be part of the sub-CFT for the region containing the black hole. For example, I talked about an energy scale cutoff in the boundary CFT corresponding to a finite size chunk of AdS; if there was a boundary plasma in which the temperatures of everything in the plasma stayed below that cutoff energy, that might correspond to the presence of a black hole in that patch of AdS, and the plasma dynamics should have a description in terms of the Kerr CFT.

As difficult as it can be to figure such things out for AdS/CFT, it's even worse for dS/CFT, where we don't know the boundary CFT (although Thomas Hartman announced one at Strings 2011 a few weeks ago, paper forthcoming; see atyy's post at the start of this thread). Also, because of the expansion of de Sitter space, there is a different type of horizon there, a cosmological horizon. Discussing coordinate systems in de Sitter space really needs a diagram that I don't have to hand, but basically, you can think about being at a point on the sphere, and then there's an opposite point on the sphere that you can never reach, see, or be influenced by, because whether you follow the light cones into the past or into the future, the sphere expands too quickly for you to reach that opposite point. This means that most of the boundary CFT at past or future infinity would be irrelevant for you, and so people try defining dS/CFT on other surfaces in the interior of de Sitter space, such as the limits of your light cones, or your cosmological horizon. These other surfaces would hypothetically have CFTs associated with them that are sub-CFTs of the full CFT at infinity, like in the AdS case. But since there is no working example of dS/CFT, it's all somewhat speculative.

So returning to your comment, from the past/future infinity perspective, which ought to be the more fundamental one, it's not that there's a thermal state on or beyond our cosmic horizon; rather, there should be a thermal state "at timelike infinity", which corresponds to the existence of the cosmological horizon during the history of the universe. But one of the mysteries of dS/CFT is how this should work, because the region beyond the cosmological horizon is indeed analogous to the interior of a black hole's event horizon; but the region beyond our cosmological horizon should just be ordinary space - just like here - and not a mass of tightly gravitationally bound D-branes, such as one should find in the interior of a black hole. A de Sitter universe, at least in string theory, should correspond to an expanding braneworld, so the temperature associated with a cosmological horizon might be somehow spread throughout the braneworld rather than being concentrated in one place, though that still doesn't tell you what the holographic counterpart description at infinity is.

http://pirsa.org/11070007/" which might resemble the truth, but it's really a matrix model, which is almost the thing beyond holography - all the dimensions have lost their ordinary character, and just live on the diagonal of the matrix, and now there are off-diagonal degrees of freedom which look highly nonlocal from the perspective of ordinary space. In a matrix model, a black hole is a filled-in block on the main diagonal, so maybe Banks's matrix theory can explain patches of de Sitter space in some analogous way; but we still don't know what that will be.

 

[atyy]

http://arxiv.org/abs/1108.5735: "The Sp(N) CFT₃ dual to de Sitter space has anticommuting scalar fields and is therefore non-unitaryThis peculiarity does not rule out the duality because in dS/CFT, the CFT is Euclidean and never continued to Lorentzian signature."

Doesn't that sound like the Lorentzian CFTs can't be non-unitary?

 

[mitchell porter]

In AdS/CFT, you have an equivalence between one Lorentzian theory (boundary) and another Lorentzian theory (bulk). Time evolution exists in both descriptions so if one is unitary, so must the other be. But dS/CFT is an equivalence between a Euclidean theory (future boundary) and a Lorentzian theory (bulk). There's no time evolution in the Euclidean theory and so no requirement that it is unitary.

But see http://arxiv.org/abs/0705.4657" . The theory is "pseudo-unitary" and there is more to discover here.