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August 21, 2006


For somewhat obscure reasons, I’ve been looking, recently, at a recent paper by Aharony, Clark and Karch. They consider the following problem. Say we have two CFT’s, C 1C_1, and C 2C_2. each of which has an AdS dual description. Each CFT has its own conserved, traceless, stress tensor. The AdS dual of C 1×C 2C_1\times C_2 is the disjoint union of the corresponding AdS spaces. (Aharony et al use the phrase “direct sum” instead of “disjoint union”, but one learns to make allowances …) The graviton, on each component of this disjoint union, couples in the usual way to the stress tensor of the corresponding boundary theory.

But now suppose we deform the theory by choosing a pair of relevant operators, O 1O_1, O 2O_2, (with scaling dimensions Δ 1\Delta_1 and Δ 2=dΔ 1\Delta_2=d-\Delta_1) in the two theories and adding

ΔS=ϵd dxO 1(x)O 2(x) \Delta S = \epsilon \int \mathrm{d}^d x O_1(x)O_2(x) For definiteness, let’s normalize the O iO_i so that O(0)O(x)=1|x| 2Δ \langle O(0)O(x) \rangle = \frac{1}{|x|^{2\Delta}}

(More generally, it’s interesting to contemplate a marginal or relevant perturbation of the erstwhile decoupled theory.)

What is the AdS dual of the deformed theory?

Obviously, by turning on ΔS\Delta S, we are correlating the boundary conditions of the fields on the two formerly disjoint AdS spaces. So we should think of these two spaces as glued together at their common boundary. In the boundary field theory, we no longer have two seaparately conserved stress tensors. The total stress tensor, T=T 1+T 2 T = T_1 + T_2 is conserved, but T˜μ ν =c 2T μν (1)c 1T μν (2)+ϵ(c 1Δ 2c 2Δ 1)dη μνO 1O 2 \tensor{\tilde{T}}{_\mu_\nu} = c_2 T^{(1)}_{\mu\nu} - c_1 T^{(2)}_{\mu\nu} + \epsilon \tfrac{(c_1 \Delta_2-c_2\Delta_1)}{d}\eta_{\mu\nu} O_1 O_2 is not. Here cc, defined by T μν(0)T ρσ(x)=c|x| 4(η μρη νσ+η μση νρ2dη μνη ρσ) \langle T_{\mu\nu}(0) T_{\rho\sigma}(x) \rangle = \frac{c}{|x|^4} \left(\eta_{\mu\rho}\eta_{\nu\sigma} + \eta_{\mu\sigma}\eta_{\nu\rho} -\tfrac{2}{d} \eta_{\mu\nu}\eta_{\rho\sigma}\right) is one measure of the central charge of the conformal field theory. Instead of being conserved, to leading order in ϵ\epsilon, μT˜μ ν =K ν=ϵ(c 1+c 2)d(Δ 2( νO 1)O 2Δ 1O 1( νO 2)) \partial^\mu \tensor{\tilde{T}}{_\mu_\nu} = K_\nu = \epsilon\tfrac{(c_1+c_2)}{d} \left(\Delta_2 (\partial_\nu O_1)O_2 - \Delta_1 O_1(\partial_\nu O_2)\right)

Correspondingly, the modes of the bulk graviton which couple to T˜μ ν \tensor{\tilde{T}}{_\mu_\nu} must pick up a mass via the Higgs mechanism (swallowing the bulk vector field which couples to K νK_\nu). The graviton mass is related, in the standard way, to the anomalous dimension of T˜μ ν \tensor{\tilde{T}}{_\mu_\nu}, m grav 2=d(Δ T˜μ ν d)=ϵ 2(1c 1+1c 2)Δ 1Δ 2d(d+2)(d1) m^2_{\text{grav}} = d (\Delta_{\tensor{\tilde{T}}{_\mu_\nu}} -d) = \epsilon^2 \left(\frac{1}{c_1}+\frac{1}{c_2}\right)\frac{\Delta_1 \Delta_2 d}{(d+2)(d-1)}

Rather prettily, a bulk calculation of the effect of the deformed boundary conditions yields an identical expression for the mass of these graviton modes. Note that it is a feature of AdS space, unlike (say) Minkowski space, that the graviton can smoothly acquire a mass.

The authors then turn to some speculations as to which theories have AdS duals (either as a single AdS space, or a set of disjoint spaces glued along their boundaries). I’m not particularly happy with their proposal. Indeed, I think that any QFT which is controlled by a nontrivial UV fixed point should have some sort of AdS dual. This description is, however, unlikely to be a useful one unless there’s some sort of large-NN limit which makes the bulk theory semiclassical.

The possibility that there may exist, at certain points in the moduli space of such theories, multiple conserved spin-2 currents, does not seem — in any robust way — to be correlated with the number of factors in the gauge group1.

Still, it’s a very interesting question to contemplate , and I would try to phrase the answer in terms of the topology of the space of boundary fields.

1 where one defines nn to be the maximum value for which one can decompose the gauge group G=G 1×G 2×...×G n G = G_1\times G_2\times ... \times G_n

into factors, such that no field is charged under more than one factor, G iG_i. Posted by distler at August 21, 2006 2:50 AM

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Re: Coupled

You wrote:

“Indeed, I think that any QFT which is controlled by a nontrivial UV fixed point should have some sort of AdS dual.”

I’m inclined to go further and say that even asymptotically free theories have some sort of AdS dual (albeit with a nontrivial dilaton profile at the boundary). But in any case, I’ve had some confusion about theories with duals consisting of multiple AdS spaces joined at the boundary for a while, and perhaps I can get some new insight on it here.

I started wondering about this after working on mocking up a “topcolor-like” model with two AdS spaces joined at the boundary (hep-ph/0505001). This was of course not the dual of any particular known field theory, just a Randall-Sundrum-like model meant to represent what the low energy effective theory of some such model might look like. It turned out to have a lot of the properties one might expect from a holographic dual of two different strongly interacting theories with the same global symmetries, coupled by a weak gauging of the diagonal subgroup of the product of the symmetries. (Also since there was a UV cutoff at finite scale they are coupled by gravity, but for our purposes that was not particularly important.)

In one sense it’s quite natural that in such a model, turning off the weak gauging should decouple the two sectors and so they should live in independent AdS spaces. On the other hand, some of my basic intuition about holography runs counter to this.

Specifically, I think of holography largely in terms of the renormalization group: the “z” coordinate of AdS corresponds roughly to the inverse of the renormalization scale (at least asymptotically), and the profiles of various fields in AdS relate to RG flows in the field theory.

Now, if you take a theory made of two independent sectors and you give them some coupling, it seems to me that you shouldn’t be able to decouple the RG flows; operators in the two sectors generically will mix, and the matrix of anomalous dimensions can have some nontrivial scale dependence. This suggests to me that the dual should have only one bulk, although it may be a reasonable approximation in some cases that the fields split into sectors that do not communicate strongly in the bulk. On the other hand, the dual picture of AdS spaces joined only by boundary conditions suggests that the RG flows of the two sectors are connected in only a very loose way. And it seems that duals constructed in this manner do correctly capture field theory properties, as Aharony have nicely shown (and also Kiritsis in hep-th/0608088).

Is there a nice way to reconcile these multiple-AdS dual pictures with the intuition of holography as RG flow?

Posted by: Matt Reece on August 21, 2006 9:04 PM | Permalink | Reply to this

Re: Coupled

Now, if you take a theory made of two independent sectors and you give them some coupling, it seems to me that you shouldn’t be able to decouple the RG flows; operators in the two sectors generically will mix, and the matrix of anomalous dimensions can have some nontrivial scale dependence. This suggests to me that the dual should have only one bulk …

Well, this is certainly the heart of the confusion (which I share).

We have long known of examples where turning on VEVs for fields on the boundary leads to a theory where the IR physics breaks up into non-communicating sectors (each with its own conserved stress tensor).

But, of course, that only happens asymptotically in the IR; the throats of these multi-throat solutions only really decouple in the respective near-horizon limits.

If one made a relevant (as opposed to marginal, as in the present papers) deformation of the decoupled Lagrangian of the boundary theory, the picture one would like to see is the inverted version of the above: a geometry which, in the UV looks like the disjoint union of two AdS spaces, but which is actually a single, connected space. The RG flow which mixes the operators of the two theories is the zz-evolution which interpolates between the “two-throated” UV region and the IR.

In the case of a marginal deformation (as considered in these papers), there’s no RG flow. The Hilbert space of the deformed CFT is organized into representations of the total conformal algebra. To the extent that one can construct the correlation functions in conformal perturbation theory, the result fits nicely with this picture of disjoint spaces joined at their boundary (as you can see from the explicit calculations of Aharony et al.)

and also Kiritsis in hep-th/0608088

Yeah, I hadn’t really gotten around to digesting his paper when I wrote this post. But that’s what the comments section is for :-).

Posted by: Jacques Distler on August 21, 2006 10:07 PM | Permalink | PGP Sig | Reply to this

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