Extremal higher spin black holes

Abstract

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) ChernSimons 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.

Generalized Chern–Simons higher-spin gravity theories in three dimensions

“…Even though the absence of local degrees of freedom, CS theories are known to possess a rich structure that makes them worth to be studied. In fact, similarly to what happens in the pure gravity case, the SL (n, R) × SL (n, R) CS theory has interesting solutions, such as HS black holes [27][28][29][30][31][32][33][34] and conical singularities [35,36]. Moreover, the asymptotic symmetry of the theory realizes two copies of the W n algebra [26,37,38], which led to conjecture the duality between threedimensional HS theories and a W n minimal model CFT in a large-N limit [39,40].…”

Correlators in higher-spin AdS3 holography from Wilson lines with loop corrections

“…In particular, an N = 2 supersymmetric version of duality was proposed in [42], and the bulk description of its semiclassical limit was argued to be given by sl(N + 1|N ) Chern-Simons gauge theory [43]. See [44][45][46][47][48] for conical defect or black hole solutions in higher spin supergravity. We think that supersymmetric extension is important for the following two reasons.…”

Superconformal blocks from Wilson lines with loop corrections

“…Here any restriction is not yet assigned on s, and the condition for s will be obtained by requiring the geometry preserving supersymmetry below. In order to obtain the supersymmetric condition, we apply the method developed for the supersymmetric geometry in the sl(N + 1|N ) Chern-Simons theory as in [16,[49][50][51][52][53]. We look for spinors, which satisfy the Killing spinor equation,…”