On the partition functions of higher spin black holes

We reconsider black hole solutions of D = 3 higher-spin gravity in the $hs[λ]\oplus hs[λ]$ Chern-Simons formulation. A suitable generalisation of the BTZ black hole has a spin-3 chemical potential $α$, and non-zero values of all the conserved charges associated with the asymptotic $\mathcal W_\infty[λ]$ symmetry. We extend the available perturbative expansion of the partition function to order $\mathcal O(α^{18})$ for generic values of the $λ$ parameter. The result matches the CFT prediction at $λ=0$ and at $λ=1$ where we provide the exact all-order expansion of the partition function. The perturbative series is then analysed in the interesting non-trivial limit $λ\to\infty$ and we derive the exact analytical expressions of the partition function and the spin-4 charge in closed form as functions of $α$. Also, the first subleading correction at large $λ$ is shown to be simply related to the leading contribution.

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