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There are two complementary correspondences to black holes. One is a metric where far from the black hole horizon there is $AdS_3$ spacetime. The other occurs quite oppositely where the $AdS_2\times S^2$ spacetime occurs in the near horizon condition of an extremal black hole. For the asymptotic condition, usually seen as the AdS black hole one simply has ...


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Well while it has similarities with the OPE, it is more than that. In fact, it satisfies the OPE limit when $z\to w_j$ for any $j$, since the OPE you are talking about tells you only the singular terms, while there are also infinitely many non-singular terms, i.e., schematically $$ T(z)\phi(w,\bar w)=\frac{h_\phi}{(z-w)^2}\phi(w,\bar w)+\frac{1}{z-w}\...



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