On scalar propagators of three-dimensional higher-spin black holes

We explore some aspects of three-dimensional higher-spin holography by studying scalar fluctuations in the background of higher-spin black holes. We furnish an independent derivation of the bulk-boundary propagator by purely invoking a well-known infinite dimensional matrix representation of $hs[λ]$ algebra related to its construction as a quotient of the universal enveloping algebra of $sl(2)$, thus evading the need in previous literature to perform an analytic continuation from some integer to $λ$. The propagator and the boundary two-point functions are derived for black hole solutions in $hs[λ]\times hs[λ]$ Chern-Simons theory with spin-3 and spin-4 charges up to second-order in the potentials. We match them with three- and four-point torus correlation functions of the putative dual conformal field theory which has $\mathcal{W}_\infty [λ]$ symmetry and is deformed by higher-spin currents.

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