Extremal higher spin black holes

The gauge sector of three-dimensional higher spin gravities can be formulated as a Chern-Simons theory. In this context, a higher spin black hole corresponds to a flat connection with suitable holonomy (smoothness) conditions which are consistent with the properties of a generalized thermal ensemble. Building on these ideas, we discuss a definition of black hole extremality which is appropriate to the topological character of 3d higher spin theories. Our definition can be phrased in terms of the Jordan class of the holonomy around a non-contractible (angular) cycle, and we show that it is compatible with the zero-temperature limit of smooth black hole solutions. While this notion of extremality does not require supersymmetry, we exemplify its consequences in the context of sl(3|2) ⊕ sl(3|2) Chern-Simons theory and show that, as usual, not all extremal solutions preserve supersymmetries. Remarkably, we find in addition that the higher spin setup allows for non-extremal supersymmetric black hole solutions. Furthermore, we discuss our results from the perspective of the holographic duality between sl(3|2) ⊕ sl(3|2) Chern-Simons theory and two-dimensional CFTs with W(3|2) symmetry, the simplest higher spin extension of the N = 2 super-Virasoro algebra. In particular, we compute W(3|2) BPS bounds at the full quantum level, and relate their semiclassical limit to extremal black hole or conical defect solutions in the 3d bulk. Along the way, we discuss the role of the spectral flow automorphism and provide a conjecture for the form of the semiclassical BPS bounds in general N = 2 two-dimensional CFTs with extended symmetry algebras.