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The isometries of AdS in $D+1$ dimensions and the conformal symmetries in $D$ are isomorphic as Lie algebras. However, the generators on each side have a physical interpretation. In the bulk we have spatial and temporal translations $P_a^{AdS}$ and $H^{AdS}$, boosts $B_a^{AdS}$ and rotations $J_{ab}^{AdS}$. On the boundary, we have again spatial ($P_a^{CFT}$) and temporal translations ($H^{CFT}$), rotations ($J_{ab}^{CFT}$), boosts ($B_a^{CFT}$) and addtionally special conformal transformations $K_a^{CFT}$ and dilatations $D$.

How do I precisely match the generators of the bulk and the boundary in the AdS/CFT correspondence?

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  • $\begingroup$ Of course, a resource that provides this precise matching is also appreciated. $\endgroup$ – ungerade Jul 13 at 20:42
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    $\begingroup$ Are you familiar with the construction of AdS as a hyperboloid in a higher-dimensional flat spacetime $M$ (with two time dimensions)? In that construction, isometries of AdS are given by origin-preserving Lorentz transformations in $M$. Points on the "boundary" $B$ of AdS correspond to light-rays from the origin in $M$, and those same Lorentz transformations in $M$ give conformal transformations of $B$, making the relationship between isometries of AdS and conformal transformations of $B$ is relatively intuitive. $\endgroup$ – Chiral Anomaly Jul 14 at 19:50
  • $\begingroup$ I'm not very familiar with it, but that sound very interesting. Do the precise relationships follow from this construction? Do you maybe have an instructive resource that explains this in detail? Of course, if you have already the intuition maybe you could provide an answer with the detailed match on the Lie algebra ;). In any case thanks for the hint $\endgroup$ – ungerade Jul 14 at 20:23

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