Musings

2+1 gravity can be recast as a Chern-Simons theory. For negative cosmological constant, the gauge group is $SO(2,1)\times SO(2,1)$, or some covering thereof, usually assumed to be the 2-fold covering, $SO(2,2)$. This was the basis of Witten’s old work on 3D gravity.

But there are two reasons to revisit the subject. One is the discovery of the BTZ blackhole. The idea that 2+1 gravity is a topological field theory seems to conflict with the large degeneracy of states of the blackhole.

The other is AdS/CFT. We now know that, with negative cosmological constant, the observables of quantum gravity are the correlation functions of a boundary CFT.

For $d\geq 4$, quantum gravity with $\Lambda=0$ has the S-matrix as an observable, but there is no S-matrix for gravitational theories in $d=3$. For $\Lambda\gt 0$, no one knows what the observables of quantum gravity are (if they exist at all), so it is unclear what it would mean to “solve” the theory. Hence, Witten restricts himself to the $\Lambda\lt 0$ case.

The action for pure gravity is $$I = \frac{1}{16\pi G} \int d^3 x \sqrt{-g} (R +2/\ell^2)$$ To this, we can add a gravitational Chern-Simons term, $$I' = \frac{k'}{4\pi} \int Tr(\omega d \omega + 2/3 \omega^3)$$ $I+I'$ can be written as a sum of Chern-Simons invariants for $A_{L,R}= \omega \pm e$. If the gravitational Chern-Simons coefficient, $k'=0$, then $k_L=k_R=\ell/16G$

Brown and Henneaux showed that, in 2+1 dimensions, the boundary CFT has $\text{vir}\times\overline{vir}$ symmetry, with $(c_L,c_R) = (24 k_L, 24 k_R)$ where $k_{L,R}$ are the Chern-Simons levels. These are quantized. If we are dealing with $SO(2,1)\times SO(2,1)$, then $k_{L,R}\in\mathbb{Z}$. For $SO(2,2)$, one finds $$\k_L \in \frac{1}{4}\mathbb{Z},\qquad k_L-k_R\in\mathbb{Z}$$ and, more generally, as we go to multiple covers, the quantization condition becomes weaker and weaker, going away entirely for the universal cover.

Something seductive happens for $c\in 24 \mathbb{N}$. Namely, it is precisely for these values of $c$ that there exist holomorphic conformal field theories. For Witten, this is reason enough to decide to focus on $SO(2,1)\times SO(2,1)$. Assuming holomorphic factorization holds allows one to say quite a lot.

The ground state energy $=-c/24 = -k$. In addition to the ground state, the boundary theory has its Virasoro descendents. Even in pure gravity, this can’t be the whole story, as there are the BTZ blackholes, which must correspond to positive energy primary states. In pure gravity, we don’t expect any other primary states.

The holomorphic partition function, then, takes the form

where $$Z_0(q) = q^{-k} \prod_{n=2}^\infty \frac{1}{1-q^n}$$ is the module generated by the ground state and its descendents.

Hoehn proved that, under these assumptions, $Z(q)$ uniquely determined: $$Z(q) = \sum_{r=0}^k a_r J(q)^r$$ where $J(q)= q^{-1} +196884q+\dots$ is the modular-invariant $J$-function. The absence of primary fields with $0\lt h\lt k+1$ allows us to solve for the $a_r$. Then the coefficient of $q^1$ in (1), which gives the degeneracy of the lowest-mass BTZ blackholes is determined.

The simplest case is $k=1$, where we simply have $Z(q)=J(q)$. One of the states at level-1 is the descendent of the ground state. The other 196883 states are primaries. $\log(196883) = 12.194$, which gives not so bad agreement with Bekenstein-Hawking entropy. And $k=1$ was the worst case scenario, with the largest (Planckian) cosmological constant. For large $k$, the cosmological constant is smaller, and the agreement with Bekenstein-Hawking becomes better.

For $k=1$, this theory is the well-known Monster module, and the BTZ blackholes form representations of the Monster group. For higher $k$, we don’t know the CFT’s explictily. But, since $Z(q)$ is a linear combination of powers of $J(q)$, one can show that all the primaries form (reducible) representations of the Monster.

But there are various puzzling features of this proposal.

1. Why, on earth, should the CFT be holomorphically-factorized? Indeed, this poses a puzzle, as there are clearly states the form of a BTZ blackhole for the left-movers and (a descendent of) the ground state for the right-movers. These are very crazy-looking states, and Witten has to explain them away by mumbling words about the quantum theory. But the problem persists for large $k$, where the gravity theory becomes classical.
2. Even if we don’t assume holomorphic factorization, we still demand that the pure gravity theory (if it exists) should be described by a CFT with a gap in its spectrum. That is, there should be no primary fields with conformal weights in the interval $0\lt h \lt k+1$. That’s a very weird property, and particularly hard to satisfy in the large-$k$ limit.

The simplest resolution may be that the pure gravity theory (for large $k$, where one has some right to contend that we really are quantizing something that is recognizably pure gravity) doesn’t exist.

But the main insight, emphasized at several points by Witten in his talk, is that the gauge theory approach is “wrong.” We would not, for instance, ever see the BTZ blackhole from the gauge theory. Conversely, the gauge theory includes configurations that probably shouldn’t be included in the gravity theory. Regardless of whether one can resolve the puzzles of the pure gravity theory in 2+1 dimensions, the boundary CFT is a much more appropriate tool for analysing the physics.