### Extremal CFTs

I’ve already written a couple of posts about pure 2+1 gravity with negative cosmological constant. But there have been some odds and ends I wanted to deal with, so perhaps another post is in order. In particular, there are papers by Gaberdiel and Gaiotto, that bear remarking upon.

On general grounds, AdS/CFT duality suggests that these theories should be completely described by writing down a 1+1 dimensional conformal field theory with central charge $c = 24k = 3\ell/2G$, where $\ell$ is the radius of the AdS space. Witten proposed that this CFT is “extremal.” That is, the partition function can be written as

where

The expressions for the $\chi_k(q)$ are polynomials in the modular J-function, $J(q)=q^{-1}+196884q+\dots$, and are uniquely determined. For $k=1$, $\chi_1(q)=J(q)$ is the familiar Monster Module.

What we don’t know is

- whether, for $k\gt 1$, they correspond to the partition functions of actual conformal field theories
- whether these conformal field theories also carry a Monster symmetry
- whether, in fact, pure 2+1 gravity exists as a quantum theory and, if it does, whether the partition function of the dual CFT takes the “extremal” form (1)

The “gap” in the spectrum of conformal primaries in (2) is important. The ground state is the AdS vacuum. The only other states that should be present, in pure gravity, are BTZ blackholes of positive mass. The dictionary between conformal primaries and BTZ blackholes is simple. A spinning BTZ blackhole of mass, $M$, and spin, $J$, corresponds to a primary of conformal weight $(h,\overline{h})$, where $M = \Delta + \overline{\Delta},\qquad J = \Delta - \overline{\Delta}$ and we’ve abbreviated $\Delta= h -c/24 = h-k$. In the semiclassical ($k\to \infty$) limit, the asymptotic density of states accords nicely with the Bekenstein-Hawking entropy $S \sim 4\pi (\sqrt{\Delta} + \sqrt{\overline{\Delta}})$

However, there’s a problem, alluded to in my previous posts. Whereas “most” of the states just described satisfy the “extremal bound” $\frac{|J|}{M} \leq 1$ there’s one class of states which violate it: take the ground state, say, for the right-movers ($\overline{\Delta}=-k$) and a BTZ blackhole, $\Delta\geq 1$, for the left-movers. Classically, these would correspond to over-rotating blackholes, $\frac{J}{M} = \frac{\Delta+k}{\Delta -k}$ and the corresponding classical solutions (which we ought to trust in the $k\to\infty$ limit) are pathological, with naked singularities and closed timelike curves.

Actually, this problem is not peculiar to the extremal partition functions (1); it crops up whenever the ground state, corresponding to the AdS vacuum, is in the same conformal block as the BTZ blackhole. And this seems hard to avoid, given the requirements of modular invariance and the gap in the spectrum.

Gaberdiel’s paper addresses the first point above. He argues that:

- Whenever $\chi_k(q)$ obeys a linear differential equation, this corresponds to the existence of a null vector at a level equal to the order of the differential equation.
- There is an upper bound on the minimal
*order*of the differential equation (i.e., there’s an upper bound on the level at which the first such null vector occurs). - For sufficiently large $k$, this order becomes less than $k+1$, where one can explicitly show that there are no null vectors. (Up to level-$k$, by
*assumption*, we just have a vacuum Virasoro module whose only null vector is $L_{-1}|0\rangle$, since $c\geq 24$.)

The differential equation takes the form

where $D^{(n)} = \left(q \frac{d}{d q} -\frac{1}{6} (n-1) E_2(\tau) \right) \left(q \frac{d}{d q} -\frac{1}{6} (n-2) E_2(\tau) \right) \cdots \left(q \frac{d}{d q} -\frac{1}{6} E_2(\tau) \right) q \frac{d}{d q}$ and $f_r(q)$ is a polynomial in $E_4(\tau)$ and $E_6(\tau)$ of modular weight $2(s-r)$.

Giaotto, in his paper, argues that

- Gaberdiel’s upper bound need not be saturated. The $\chi_k(q)$ could obey a lower order differential equation.
- The differential equation could be “accidental,” unrelated to the existence of a null vector.

The first point, of course, does not affect Gaberdiel’s argument. The second is, however, rather more puzzling. Giaotto’s counterexample, a tensor product of two Monster Modules, isn’t really satisfactory. The latter, in fact, has *more* null vectors than required by Gaberdiel’s argument. But it is true that this is the weak point in Gaberdiel’s argument. Proving that a solution to (3) implies the existence of a null vector, at level $\leq s$, is still a loophole.

But the main point of Gaiotto’s paper is to show that, for $k=2$, where it seems likely that a CFT exists, the theory does not carry a representation of the Monster Group. While it’s tempting to extrapolate that the higher-$k$ theories (if they exist) don’t have Monster symmetry either, Giaotto’s methods are not sufficient to show that.

^{1} The Eisenstein series
$E_{2p}(\tau) = \frac{1}{2\zeta(2p)} \sum_{(m,n)\neq (0,0)} \frac{1}{{(m+n\tau)}^{2p}}$
satisfy
$\begin{aligned}
E_2(\tau) &= 1 - 24 \sum_{n=1}^\infty \frac{n q^n}{1-q^n}\\
E_4(\tau) &= 1 + 240\sum_{n=1}^\infty \frac{n^3 q^n}{1-q^n}\\
E_6(\tau) &= 1 - 504\sum_{n=1}^\infty \frac{n^5 q^n}{1-q^n}
\end{aligned}$
and (for $p\geq 2$) are modular forms of weight $2p$. The ring of modular forms (and hence the possible terms in $f_r(q)$) is freely-generated by $E_4$ and $E_6$.

## Re: Extremal CFTs

Gerald Hoehn has a result very similar to Giaotto’s, but it is phrased in vertex algebra language.