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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.

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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).

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  • $\begingroup$ Okay, that was my suspicion. However, one question remains. The asymptotic symmetry group of a spacetime depends on the boundary condition we impose on the solutions. How is this choice dictated by the holographic dictionary ? If I am not mistaken, we nee to impose fefferman graham boundary conditions to recover the Virasoro asymptotic symmetry, but how is this motivated a priori from ads/cft ? $\endgroup$
    – Frotaur
    Commented Jul 19, 2022 at 12:55
  • $\begingroup$ It absolutely does! Boundary conditions define the theory! Different boundary conditions lead to different symmetry groups and different boundary theories as well. The precise choice of boundary conditions depends on type of questions you are trying to answer. The F-G gauge is sufficient to answer most questions of interest in AdS/CFT and in this case the ASG is Virasoro. $\endgroup$
    – Prahar
    Commented Jul 19, 2022 at 13:06
  • $\begingroup$ But will we still have AdS/CFT if we choose other boundary conditions ? Probably not, since if the ASG is not Virasoro we cannot have dual which is a CFT. So in some sense the F-G boundary conditions are required for holography, no ? Anyway I will accept the answer as it responds to the question asked in the main post. $\endgroup$
    – Frotaur
    Commented Jul 19, 2022 at 13:40
  • $\begingroup$ There will likely still be some version holography but it may not be AdS/CFT. you could put boundary conditions which break conformal invariance and then the dual theory won’t be a CFT (perhaps AdS/CMT). You could even preserve all symmetries but still change the dual theory by changing boundary cond. for instance, Vasiliev higher spin theory is dual to O(N) vector model with Neumann bc, but it’s dual to tricritical Ising mode with Dirichlet bc. $\endgroup$
    – Prahar
    Commented Jul 19, 2022 at 14:48
  • $\begingroup$ @Prahar "Consequently, the isometries of AdSd+1 are meaningless to talk about." Could it still be that they are somehow relevant for holography? Would you agree that one would expect that the isometries of the vacuum of the bulk are isomorphic to the symmetries of the vacuum of the boundary (at least in standard contexts)? $\endgroup$
    – ungerade
    Commented Aug 28, 2022 at 22:48

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