As you may know from three previous recent posts on research (here, here, and here), I’ve been thinking and calculating a lot in the area of dynamical cosmological constant – concerning mostly (but not entirely) thermodynamics and quantum gravity. Specifically, the cosmological constant becomes the pressure variable in the thermodynamics. I think it is important, and will teach us something about things like gauge/gravity duality, string theory, black holes, and perhaps even cosmology, but I am not sure what yet. I’ve made some suggestions in recent papers, and computed some interesting things along the way.

Anyway, the larger community has not been following this story much, since: (1) It means a break with some powerful and still very fruitful frameworks where the cosmological constant being fixed is a given – like AdS/CFT – and it is not clear what that means yet, so the motivation is not super-strong; and (2) Let’s be honest, there’s no superstar working on it, so it is not going to get anyone any points. So I’ve been trying to shout about it in my little way from the periphery, as I think it might be useful, and since several people have been doing really good and interesting work on this issue for many years and it is worth more people seeing what they’ve been up to.

So imagine my pleasant surprise when I looked on the listing of new papers on the arXiv for today and saw three (!) papers on the subject, moving things forward in various ways. (They all seem to have noticed some of what I’ve written and cited it too, which is awfully kind.) It all looks interesting: There’s the paper by Grumiller et. al., working in two dimensional dilation gravity, the one by Frassino et. al., studying Lovelock gravity, and the one by Dolan, in five dimensions, which seems to be trying to make more explicit an idea I’ve been thinking about concerning getting to grips with having a chemical potential for the number of degrees of freedom in a gauge theory (as measured by some power of N, the rank of the gauge group). The point here (see my holographic heat engines paper) is that pressure – and hence cosmological constant – is connected directly to N, and hence its conjugate, volume, ought to be this chemical potential. What is the best way to think about such an object in a field theory?

[Er…, the random drawing at the top has no connection to the post. I flew back from Newark to LA on Saturday and did my usual (as you may know from some previous posts of mine) practice of flipping through the airline magazine for an interesting face to do a bit of drawing practice with. This one struck me as an interesting face, and so I did some pencil work, followed by ink work, and then threw on a little watercolour. It was a challenge at several points since the plane decided to shake a bit every time I tried to work on crucial bits like eyes and lips… sigh.]

-cvj

In the matrix model language that would involve varying the size of the matrices-and that brings to mind the paper by BrÃ©zin and Zinn-Justin, http://arxiv.org/abs/hep-th/9206035, that might be useful.

Hi Stam,

Yes, indeed… dynamical cosmological constant and varying N do show up together in matrix models of c<=1 strings, and they try to get to a bigger picture of those models by varying N. I've been trying for years to see if some of the matrix model lessons learned in those old days can still instruct us today. Who knows, there might be clues there. I like that they think of the story as a RG flow, since this is rather similar in spirit (at least on the surface) to what I've been thinking about in the context of the holographic interpretation. Cheers, -cvj

Hi Clifford,

One difference, of course, between the matrix models and the case here, is gauge invariance: Varying N, i.e. going from SU(N)->SU(N-1), for instance, means that certain gauge bosons decouple. So a question might be, how to describe their decoupling precisely.

Hi,

Actually, it is the same in both pictures. The bonus here is that on the gravity side gauge invariance is built in (in the sense that while being dual to a gauge theory, it does not care about NxN matrices explicitly), but the decoupling happens nonetheless. That’s why I think holographic RG flow is a nice piece of the story – it shows how to implement such a decoupling gauge invariantly. See the penultimate section of the Holographic Heat Engines paper.

-cvj

Hi,

Thanks, I’m reading the papers and I agree, of course, that the RG flow is an interesting object to study. I’m wondering how does the decoupling of the 2N-1 gauge fields, when going from SU(N) to SU(N-1), becomes visible in the bulk-or doesn’t this make sense?

Upon rereading your paper ðŸ™‚ I think I understand better what’s meant: There must exist operators on the dual field theory side that play the role of pressure and volume on the gravity side. These operators, should, in particular, change N, the rank of the group (for U(N), rank+1 for SU(N)) on the field theory side, thereby describing how the cosmological constant varies on the gravity side. On the field theory side, they would lead to gauge bosons decoupling. The cycles of the engine would be akin to the cycles through phases of the field theory-would this be a correct description?

-Stam

Hi,

Yes. That is closer. Note that the engines are a fun thing you can do, but that is not at all essential to the core business of understanding what p and V mean in field theory. The penultimate section’s discussion about how that all maps to field theory is entirely independent of the heat engine discussion. The fact that there is some regime where it all maps to thermo is nice, and then heat engines would mean something interesting in the field theory, but that’s not essential to understand what p and V mean in field theory. And it is more V that I find curious… I think I understand p and its dynamics…. holographic RG flow tells us what to do. If anything, I think the full picture moves us not just within one field theory, but perhaps on the space of field theories, in the sense that SU(N) and SU(M) for [tex]M\neq N[/tex] are different, and beyond (since I can flow to different operator content, like from N=4 SYM to N=2* SYM, or cascading theories, and so forth… Also…. finite temperature seems to be crucial in the thermo picture, which is not very well explored on the RG flow side.

Cheers,

-cvj

I wonder, whether it would be correct to say that V on the gravity side is related to flux on the field theory side, that would be quantized (hence the dependence on N), since, by adding flux (i.e. Wilson lines), it’s possible to change the rank of the gauge group for the boundary theory. For the volume takes continuous values, but the rank can change by integer values, so the way to do this in the boundary theory is through fluxes. Does this make sense?

-Stam

It may be possible to think of it that way. I don’t know for sure. Do check out Brian Dolan’s paper that I mention in the post. That might help point the way.

-cvj

Indeed-I think what I’m trying to work out is what he’s discussing at the end-to set up a concrete calculation needs some more work…