Musings

Coupled

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, $C_1$, and $C_2$. each of which has an AdS dual description. Each CFT has its own conserved, traceless, stress tensor. The AdS dual of $C_1\times C_2$ 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, $O_1$, $O_2$, (with scaling dimensions $\Delta_1$ and $\Delta_2=d-\Delta_1$) in the two theories and adding

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

(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 $\Delta S$, 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, $$T = T_1 + T_2$$ is conserved, but $$\tensor{\tilde{T}}{_\mu_\nu} = c_2 T^{(1)}_{\mu\nu} - c_1 T^{(2)}_{\mu\nu} + \epsilon \tfrac{(c_1 \Delta_2-c_2\Delta_1)}{d}\eta_{\mu\nu} O_1 O_2$$ is not. Here $c$, defined by $$\langle T_{\mu\nu}(0) T_{\rho\sigma}(x) \rangle = \frac{c}{|x|^4} \left(\eta_{\mu\rho}\eta_{\nu\sigma} + \eta_{\mu\sigma}\eta_{\nu\rho} -\tfrac{2}{d} \eta_{\mu\nu}\eta_{\rho\sigma}\right)$$ is one measure of the central charge of the conformal field theory. Instead of being conserved, to leading order in $\epsilon$, $$\partial^\mu \tensor{\tilde{T}}{_\mu_\nu} = K_\nu = \epsilon\tfrac{(c_1+c_2)}{d} \left(\Delta_2 (\partial_\nu O_1)O_2 - \Delta_1 O_1(\partial_\nu O_2)\right)$$

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

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-$N$ 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.

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