# $\mathrm{AdS_3}$ bulk with BTZ black holes and particles in AdS/CFT

Consider three-dimensional anti-de Sitter space $\mathrm{AdS_3}$ treated as the $SL(2,\mathbb{R})$ group manifold, thus parametrised by elements $g \in SL(2,\mathbb{R})$. This space has as isometry group $SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$ since we can act on an element on the left or right, that is,

$$g \mapsto h_L g h_R.$$

It is known that in the presence of a BTZ black hole, taking a closed path around the black hole will not result in coming back to the same $g$, but in fact $h_L g h_R$ where,

$$h_{L,R} = \exp \left[ \frac{\pi(r_+ \pm r_-)\sigma_3}{\ell} \right]$$

where $r_{\pm}$ are related to the inner and outer horizon of the BTZ black hole and $\ell$ is the $\mathrm{AdS_3}$ radius. Similarly, in the presence of a particle which gives rise to a cusp in the classical geometry, one experiences a transformation on $g$.

Quantum mechanically, we now consider a system of $N$ particles in this space. The $N$-particle wave function $\psi(z_1,\dots,z_n;t)$ for particles at positions $\{z_i\}$ must be single-valued and possesses so-called braid statistics, due to the fact that the equivalent transformation for $z$ when moving around a particle is given by,

$$z \mapsto \frac{az+b}{cz+d}$$

with $ad-bc=1$. Solving these constraints for $\psi$ is shown to be equivalent to solving the bootstrap equations for the CFT on the boundary of $\mathrm{AdS_3}$, as Verlinde states. My questions are:

1. Can anyone refer me to literature wherein the CFT approach to this is presented, as well as perhaps a direct study of $\psi$ from the bulk side?
2. Solving the boostrap constaints for this CFT essentially defines the CFT; is a Lagrangian description known of a CFT with this corresponding CFT data?
• I didn't quite understand what you were saying about the braid statistics. However, if you are simply treating AdS as a fixed metric space (i.e. with no dynamical gravity), then you can put a massive QFT inside of it, and the boundary correlators of this theory define a boundary conformal theory, which is however doesn't have a local Lagrangian description (because its stress-tensor would be dual to a dynamical graviton field, which is absent in this construction). Feb 7 '17 at 8:23
• I am not exactly sure what happens when you have a BTZ background instead of empty AdS. Perhaps what I am talking about is different from what Verlinde meant. (I didn't have time to watch the entire lecture, you could perhaps update your link to point to a specific time in the video where this is discussed.) Feb 7 '17 at 8:26
• @PeterKravchuk It's basically the beginning of the lecture when he introduces the problem. Mar 16 '17 at 19:20