In the context of the Holographic correspondence, $AdS_{n}/CFT_{n-1}$, it is often cited as a "confirmation" that the two symmetry groups of the theories correspond. Indeed, in dimensions $n>3$, the isometry group of $AdS_{n}$ is $SO(2,n-1)$ and there is a known isomorphism between the conformal group in $n-1$ dimensions and $SO(2,n-1)$.
However, I am struggling to see the same "matching" if we set $n=3$. Indeed, in this special case, the $CFT_2$ symmetry is enhanced to $Virasoro \times Virasoro$, whereas the isometries of $AdS_3$ remain finite dimensional, as is $SO(2,2)$. If we rewrite $SO(2,2)\cong SL(2,\mathbb{R})\times SL(2,\mathbb{R})$, we can identify the "global" part of $CFT_2$ in the isometries of $AdS_3$. However, there is a really big part missing.
I have a partial answer to this question when looking at the "asymptotic symmetries" of $AdS_3$, such as in this famous paper. In this case, we indeed find that the "asymptotic symmetry group" of $AdS_3$ is Virasoro, but I don't feel very satisfied with this answer, as it seems we are indeed computing the symmetry group of the dual CFT.
So my question is this : given a Conformal transformation on the dual $CFT_2$, is there a dictionary that tells us to what transformations it corresponds in the bulk, $AdS_3$. I suspect that it will correspond to any diffeomorphism that generates the correct "asymptotic symmetry action", but I am just guessing, as I haven't found any source on this subject.
Best Answer
The bulk is a gravitational theory, meaning that the bulk geometry is always fluctuating and is never purely $\text{AdS}_{d+1}$. Consequently, the isometries of $\text{AdS}_{d+1}$ are meaningless to talk about. In a holographic context, we should ALWAYS look for asymptotic symmetries.
It just so happens that for $d>2$, the asymptotic symmetry group of $\text{AdS}_{d+1}$ is $SO(1,d+1)$ which is of course also its isometry group. On the other hand, the asymptotic symmetry group for $\text{AdS}_3$ is Virasoro$\,\times\,$Virasoro.
A similar thing happens in asymptotically flat spacetimes as well. In this case, the ASG is the so-called BMS group which is an infinite-dimensional extension of the Poincare group (in all dimensions).