Musings

Starinets gave a wonderful survey of the application of the AdS/CFT correspondence to calculate hydrodynamic properties of the quark-gluon plasma. Transport coefficients are not amenable to lattice gauge theory techniques and weak-coupling perturbative calculations are not of much help. So AdS/CFT, despite the fact that the calculations are done at large-N and strong 't Hooft coupling, in a theory dominated by the $\mathcal{N}=4$ UV fixed point, is one of the few theoretical windows we have. He talked about a panoply of results; here are a couple.

At weak coupling, the ratio of the shear viscosity to the entropy density in the gauge theory is $$\frac{\eta}{s}\sim \frac{1}{g^4\log(1/g^2)}\gg 1$$ But, more or less universally in AdS/CFT, one finds $$\frac{\eta}{s}= \frac{1}{4\pi}$$ at strong 't Hooft coupling. This is conjectured to be a lower-bound. At finite 't Hooft coupling, $\lambda$, one finds corrections, $$\frac{\eta}{s}= \frac{1}{4\pi} + \frac{135\zeta(3)}{32\pi}\frac{1}{(2\lambda)^{3/2}}+\dots$$

Interestingly, measurements of elliptic flow at RHIC favour small values for this ratio, seemingly in good accord with the above prediction.

For the $\mathcal{N}=4$ theory, the sound speed, $$v_s = \left(\frac{\partial P}{\partial \mathcal{E}}\right)^{1/2} = \frac{1}{\sqrt{3}}$$ where $P$ is the pressure, and $\mathcal{E}$ is the volume energy density. This follows from conformal invariance; the tracelessness of the stress tensor implies $\mathcal{E}=3P$. Buchel and Liu studied the AdS dual of the finite-temperature mass-deformed theory (the so-called $\mathcal{N}=2^*$ theory, whose zero-temperature limit was studied by Pilch and Warner) and, using their solution, Benincasa, Buchel and Starinets computed the corrections to the sound-speed, and predictions for the bulk viscosity, in the gauge theory.

On Friday, Cachazo reviewed the near revolution in perturbative QCD calculations that have taken place in the past year. The insights which flowed from Witten’s reformulation of $\mathcal{N}=4$ SYM as a string theory in twistor space continue to pay dividends.

Per Kraus talked about his paper with Finn Larsen, where they provide a robust derivation of the corrections to the Bekenstein-Hawking formula for the blackhole entropy. Their situation isn’t quite the familiar 4D one, the BPS blackhole in 4 dimensions, whose near-horizon geometry is $AdS_2\times S^2$. They assume (as happens, say, in 5 dimensions, when you compactify M-theory on a Calabi-Yau), that the near-horizon geometry of the blackhole is $AdS_3\times S^p$. They then use anomaly arguments to show that the entropy is given by the Cardy formula $$S = 2\pi \left[\sqrt{c_L h_L/6}+\sqrt{c_R h_R/6}\right]$$ where $c_{L,R}$ are the central charges of the CFT on the boundary of $AdS_3$ and $h_{L,R}=\frac{1}{2}(M\mp J)$ are the left- and right-moving momenta of the solution. For $c_L=c_R=c$, this central charge is obtained by extremizing a “central charge function”, involving the on-shell action. When $c_L\neq c_R$, the difference is entirely determined by the anomalies. What’s cool is that their derivation makes no assumptions about extremality, and so is applicable to a much wider class of blackholes.

There were lots of other great talks, and the fact that I didn’t discuss them here should not be construed as an editorial comment on my part.