The high point of Friday’s string cosmology workshop here in New York was Witten’s lecture on his new ideas about 2+1 dimensional quantum gravity. I’ll try and reproduce here what I understood from the lecture, but this (2+1 d quantum gravity) is not a subject I’ve ever followed closely, so my understanding of the topic is very limited. It does seem clear to me though that Witten has come up with a striking new idea about this subject, linking together some very beautiful mathematics and physics. He has yet to write a paper on the subject, but presumably there will be one appearing relatively soon. I also suspect this is what he’ll be talking about at Strings 2007.
Witten began by stating his motivation: to study fully quantum black holes in an exactly solvable toy model. There’s no exactly solvable model in 3+1d, and 1+1d is too simple, so that leaves 2+1d. Assuming 2+1d, for positive cosmological constant Λ he is suspicious that the theory is non-perturbatively unstable and one can’t get precise observables, for Λ=0 one doesn’t have black holes, so that leaves negative Λ, here the vacuum solution is anti-deSitter space, AdS3.
Quantum gravity in AdS3 is related to 2d conformal field theory. There have been studies of AdS3/CFT2 as a lower dimensional version of string/gauge duality, but here he uses not string theory on AdS3, but a quantum field theory. In a question afterwards, someone asked about string theory, and Witten just noted that perhaps what he had to say could be embedded in string theory, and that the recent Green et. al. paper showing that one can’t get pure supergravity by taking a limit of string theory did not apply in 3d. If one wants to interpret this new work in light of the the LQG/string theory wars, it’s worth noting that the technique used here, reexpressing gravity in terms of gauge theory variables and hoping to quantize in these variables instead of using strings, is one of the central ideas in the LQG program for quantizing 3+1d gravity. Witten was careful to point out though that there was no 3+1d analog of what he was doing, claiming that one can’t covariantly express gravity in terms of gauge theory in 3+1d (he said that LQG does this non-covariantly).
For negative Λ the theory has so-called BTZ black hole solutions, discovered by Banados, Teitelboim and Zanelli back in 1992, and it is for the quantum theory of these black holes that Witten is trying to find an exact solution. The technique he uses is one that goes back to the 80s, that of re-expressing the theory in terms of SO(2,1) (or its double cover SL(2,R)) gauge theory, where the action becomes the Chern-Simons action. More precisely, the Einstein-Hilbert action
$$I_{EH}=\frac{1}{16\pi G}\int d^3x\sqrt{g}(R +2/l^2)$$
(here the cosmological constant is $\Lambda=-1/l^2$) gets rewritten as an SO(2,2)=SO(2,1)XSO(2,1) gauge theory with connection
$$A= \begin{pmatrix}\omega & e \\ -e & 0 \end{pmatrix}$$
where &\omega; is a 3X3 matrix (the spin-connection), e is the 3d vielbein, and the gauge theory action is the Chern-Simons action
$$I=\frac{k^\prime}{4\pi}\int Tr(A\wedge dA+\frac{2}{3}A\wedge A\wedge A)$$
with $k^\prime=\frac{l}{4G}$ (that 4 may not be quite right…).
Witten wants to exploit the relation between this kind of topological QFT and 2d conformal field theory that he first investigated in several contexts (including one that won him a Fields medal) back in the late eighties. He notes that in this context the existence of left and right Virasoro symmetries with central charges $c_L=c_R=\frac{3l}{2G}$ was first discovered by Brown and Henneaux back in 1986, and he refers to this discovery as the first evidence of an AdS/CFT correspondence. If one really does have a CFT description, one expects that the central charges can’t vary continuously, but that 2+1d gravity will only make sense for certain values of $l/G$, but Witten notes that there is no rigorous way to find the right values one will get upon quantization.
He then goes on to make a “guess”, adding to the action a multiple of the Chern-Simons invariant of the spin connection
$$I^\prime=\frac{k}{4\pi}\int Tr(\omega\wedge d\omega + \frac{2}{3}\omega\wedge\omega\wedge\omega)$$
Now the theory depends on two parameters: $l/G$ and an integer k.
Using the fact that SO(2,2)=SO(2,1)XSO(2,1), one can rewrite the total action as the sum of two Chern-Simons terms
$$I= \frac{k_L}{4\pi} \int Tr(A_-\wedge dA_-+\frac{2}{3}A_-\wedge A_-\wedge A_-)$$
$$ \ \ + \frac{k_R}{4\pi}\int Tr(A_+\wedge dA_++\frac{2}{3}A_+\wedge A_+\wedge A_+)$$
for connections
$$A_{\pm}=\omega\pm e$$
Now instead of $l/G$ and k we have $k_L,k_R$ and these are quantized if we take the gauge theory seriously. By matching Chern-Simons and gravity the central charges turn out to be
$$ (c_L, c_R)= (24k_L, 24k_R)$$
and holomorphic factorization is possible in the 2d CFT for just these values
Looking at just the holomorphic part, we have a holomorphic CFT with central charge c=24k and ground state energy -c/24=-k (note, now a different k than before…).
The partition function is expected to be ($q=e^{-\beta}$)
$$Z(q)=q^{-k}\Pi_{n=2}^\infty \frac{1}{1-q^n}$$
The first term in the product is the ground state (AdS3), the only primary state, with the other terms Virasoro descendants (excitations of the vacuum from acting with the stress-energy tensor and derivatives).
Witten then goes on to note that this expression is not modular invariant, so one expects other terms in the product, corresponding to other primary states. By an argument I didn’t understand he claimed that these would be of order $q^{1}$, at an energy k+1 above the ground state, and his proposal was that it would be this modular invariant function that would include black hole states.
In these units the minimum black hole mass is M=k, but here one is getting states only at mass M=k+1 and above. This is because the Bekenstein-Hawking entropy of the M=k black hole is 0, so it doesn’t contribute to the partition function.
Witten claimed that this proposal gives degeneracies of states that agree with the Bekenstein-Hawking entropy formula. As an example, for k=1 the partition function is given by the famous J-function
$$J(q)=j(q)-744=q^{-1}+196884q+\ldots$$
and thus for a black hole of mass 2 the number of primaries is 196883 and the entropy is ln(196883)=12.19, which can be compared to the Bekenstein-Hawking semi-classical prediction of 12.57 (one only expects agreement for large k,M).
The number 196883 is famous as the lowest dimension of an irreducible representation of the monster group, and this partition function is famous as having coefficients that give the dimensions of the other irreducibles (“modular moonshine”). There is a conjecture that there is a unique CFT with this partition function. If so, it must be the CFT that has the monster group as automorphism group. It has always seemed odd that this very special CFT didn’t correspond to a particularly special physical system, but if Witten is right, now it has an interpretation in terms of the quantum theory of black holes in 2+1 dimensions.
Anyway, that’s what I was able to understand of what Witten had to say and what he was claiming. Other people have worked on this problem in the past, for a recent review article on this topic by Carlip, see here. Carlip describes the understanding of the problem at the time as “highly incomplete”, and one of the explanations he describes relates the black hole problem to the Liouville theory. A question from the audience after the talk asked about this, and Witten indicated that he thought the Liouville theory explanation did not work.
I’m no expert here, so unclear on the details, why some of these things might be true, and what the implications might be, but this does seem to be a remarkable new idea, involves beautiful mathematics, and seems to provide promising insight into a crucial lower dimensional toy model. I suspect it will draw a lot of attention from theorists in the future.
For this posting, I especially encourage any comments from people more knowledgable than myself who can correct anything I’ve got wrong. I also strongly discourage people who know little about this from contributing comments that will add noise and incorrect information. Bad enough that I’m trying to provide information about something I’m not expert on; if you can help that’s great, but if not, please don’t make it worse…
Update: Lubos has picked up on this, which he describes as having been “leaked”, and gives the usual argument that this must be part of string theory.
