### Maloney on 2+1

We had Alex Maloney visiting us this week, and he gave a lovely talk about his forthcoming paper with Edward Witten on 2+1 gravity with negative cosmological constant.

You’ll recall that Witten’s proposal is that the dual CFT has a partition function of the form

where the central charge $c=24k= 3\ell/2G$, with $\ell$ the radius of AdS_{3}. $\chi_1(q)$ is the partition function of the famous Monster Module. For higher $k$, the first primary state above the ground state ($h=0$) has $h=k+1$. One can systematically write down the $\chi_k(q)$, but it is not known whether they, in fact, correspond to bona fide CFTs. Indeed, Gaberdiel had presented a strong (though not air-tight) argument that they cannot, for sufficiently large $k$ ($k\geq 42$).

If Gaberdiel is correct, then, either the proposal (1) is wrong, or there is no semiclassical regime for 2+1 gravity^{1}. In either case you can stop reading this post. If not, then an interesting question arises. Semiclassically, we expect there to be a Hawking-Page transition between hot AdS space (with inverse temperature $\beta=Im(\tau)$) and the AdS BTZ blackhole. Indeed, as a function of complex $\tau$, you expect a complicated phase structure, given by the fundamental domains for the action of $SL(2,\mathbb{Z})$.

When we Wick rotate to Euclidean time, the boundary is a torus, $\Sigma$, of modular parameter, $\tau$. Semiclassically, we expect a single bulk geometry^{2} to dominate: a handlebody (a solid torus), $M$, whose boundary is $\Sigma$. This involves a choice of cycle, $\gamma= p\alpha +q\beta$, with $(p,q)$ coprime, such that $\gamma$ is contractible in $M$. The usual Hawking-Page transition is the flip between when spatial circle is contractible (hot AdS) and when the Euclidean time-circle is contractible (the blackhole). But, in 2+1 dimensions, the phase structure is much richer.

One thing that might puzzle you, in this regard, is how there can be a phase transition, in light of (1), where the $\chi_k(q)$ are manifestly analytic. The answer is that $\chi_k(q)$ have $k$ zeroes along the phase boundaries and, in the large-$k$ limit, these zeroes become dense, leading to the desired non-analytic behaviour.

In a phrase: Hawking-Page is Lee-Yang!

^{1} There are other troubling aspects to (1). As I emphasized in my previous post, there are states of the CFT, which are the ground state $\overline{h}=0$ on the right, and some primary state (with $h\geq k+1$) on the left. These correspond to super-rotating BTZ blackholes, which have naked singularities, closed timelike curves, and generally look kinda sick in the semiclassical regime ($k\to\infty$, with $h/k$ fixed).

^{2} A slight (but only slight) complication is that they must use *complex* saddle points of the Euclidean action.

## Re: Maloney on 2+1

But what is the new idea here ?

Also in the dual of the usual SUSY AdS_3xS^3xT^4 gravity there is a 2d CFT which should capture the intricate phase structure (maybe the mechanism is the same). It is again roughly classified by SL(2,Z) (actually one should mod out by actions of T from the left).