Wednesday, 3 September 2008

3D Games with Gravity

These days CERN is running a six-week long TH institute program on black holes. In the eve of the LHC, the subject receives a lot of attention in newspapers and on YouTube. Unfortunately, the Special Task Force Report proved that the LHC black holes are not capable of destroying the Earth, which deprives them of much of their charm. Nevertheless, black holes remain interesting because they are intimately related to many important questions in quantum mechanics, string theory, AdS/CFT and psychoanalysis.

The past week was dominated by "Big Issues", like the black hole information paradox. Big questions make me feel small, especially when no answers are provided. So instead, what I found most interesting was the talk of Wei Li about 3D gravity. That was hardly a big issue, but rather a cute piece of mathematical physics which relates here and there to black holes. The subject is currently going through a phase of accelerated expansion, even though it all happens in AdS.

3D gravity attracts some attention because it is simple enough to hope for an exact solution of the quantum theory, and at the same time it is complicated enough to maybe shed some light on 4D quantum gravity. At first sight, 3D is completely uninteresting. In D dimensions, a graviton has D(D-3)/2 degrees of freedom, which gives the usual 2 degrees of freedom in 4D, and zero degrees of freedom in 3D. However, it turns out that things are not that trivial. In the early nineties it was found that the Einstein gravity with a negative cosmological constant has black hole solutions -- the so-called BTZ black holes. Much like their their 4D cousins, 3D black holes possess everything a respectable black hole should possess: a horizon, Hawking temperature and entropy. This means that the quantum theory should contain microstates that make up the black hole and account for the entropy.

The common line of attack on 3D gravity goes via the dual conformal field theory (CFT). 3D gravity is a topological theory with its degrees of freedom living on the boundary of space-time. Long ago, Brown and Henneaux showed that the physical Hilbert space has an action of the Virasoro algebra corresponding to 2D CFT with the central charges $c_L = c_R = 3l/2G$ (G is the 3D Newton constant and 1/l parametrizes the cosmological constant, in case anybody is still reading). 2D CFTs have little mysteries, so identifying the dual CFT means solving the 3D quantum gravity. In particular, the microstates of the CFT explain the black hole entropy.

The quest for CFT is dangerous as one be eaten by the monster (group). To avoid Witten's fate, Wei Li and company took a different path. They studied the deformation of the 3D gravity called topologically massive gravity that, apart from the Einstein-Hilbert term and the cosmological constant, contains the Chern-Simons term $\sim \frac{1}{\mu} Tr (\Gamma d \Gamma + 2/3 \Gamma \Gamma \Gamma)$. Topologically massive gravity inherits all AdS and black hole solutions of Einstein gravity.

Wei Li and company argue that the theory is stable only for one special value of the parameter $\mu$. The Chern-Simons term modifies the central charges as $c_L = 3l/2G(1- 1/\mu l)$, $c_R = 3l/2G(1 + 1/\mu l)$. If $\mu l <> 1$ is also bad. The Chern-Simons term contains 3 derivatives term which changes the counting of degrees of freedom and, as the result, the theory contains a new degree of freedom -- a massive graviton. This excitation becomes a ghost (it has negative energy) for $\mu l > 1$.
Thus, the whole theory is sensible (at most) for $\mu l = 1$. This is a very special point since the dual CFT is chiral: $c_L = 0$, so that the CFT is holomorphic (contains only right-movers). Hence the name chiral gravity. So the conjecture is that the 3D topologically massive gravity is equivalent to 2D holomorphic CFT. It remains to be found which one.

Slides of all the institute talks are apparently beyond the horizon. Here you find the slides of the related talk by Andy Strominger at Strings'08.

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