First-order flows and stabilisation equations for non-BPS extremal black holes

Pietro Galli 3 Kevin Goldstein 1 Stefanos Katmadas 2 Jan Perz 0 0 Instituto de Fsica Teorica UAM/CSIC , C/ Nicolas Cabrera, 13-15, C.U. Cantoblanco, 28049 Madrid, Spain 1 National Institute for Theoretical Physics (NITHeP), School of Physics and Centre for Theoretical Physics, University of the Witwatersrand , WITS 2050, Johannesburg, South Africa 2 Institute for Theoretical Physics and Spinoza Institute, Universiteit Utrecht , Postbus 80.195, 3508 TD Utrecht, The Netherlands 3 Departament de Fsica Teo`rica and IFIC (CSIC-UVEG), Universitat de Val`encia , C/ Dr. Moliner, 50, 46100 Burjassot (Val`encia), Spain We derive a generalised form of flow equations for extremal static and rotating non-BPS black holes in four-dimensional ungauged N = 2 supergravity coupled to vector multiplets. For particular charge vectors, we give stabilisation equations for the scalars, analogous to the BPS case, describing full known solutions. Based on this, we propose a generic ansatz for the stabilisation equations, which surprisingly includes ratios of harmonic functions. 1 Introduction 2 3 4 Bosonic action and special geometry Flow equations from the action 3.1 The action as a sum of squares 3.2 Solving the equations 4.1 Special solutions 4.2 The ansatz 5.1 The static flow equations 5.2 The fake superpotential Conclusions and outlook A On inverse harmonic functions Introduction The study of black holes in theories with eight or more supercharges resulting from string theory compactifications has proved to be a very useful tool in uncovering some of the structure of the underlying statistical systems. For supersymmetric black holes this task is facilitated by the fact that they exhibit the attractor mechanism and full supersymmetry enhancement near the event horizon [13]. Using the constraints imposed by supersymmetry, general stationary asymptotically flat solutions have been found in ungauged N = 2 Einstein-Maxwell supergravity, including higher-derivative corrections, both in four and five dimensions [49]. The spatial profile of scalars in these solutions follows a first-order gradient flow, which is integrable to (non-differential) stabilisation equations, expressing the scalars in terms of harmonic functions. On the event horizon (the endpoint of the flow), the values of scalars are dictated by the charges through the attractor equations, independently of the asymptotic boundary conditions (the beginning of the flow).1 In contrast, when the requirement that the solutions must preserve some supersymmetry is abandoned, much less is known about the general structure of the supergravity solutions and the microscopic theory behind them. The simplest generalisation of BPS 1Some authors interchange the meaning of the terms stabilisation equations and attractor equations. black holes to consider are the extremal black holes which do not preserve any supersymmetry (see [10]). These are known to share some desirable features with the BPS branch, most importantly the attractor phenomenon [1113]. For theories with 8 supercharges coupled to vector multiplets in four and five dimensions,2 the general structure of these non-BPS extremal solutions is unclear, since only partial results are known. In the static case, a restricted set of examples can be found by simply changing the sign of a subset of the charges, which breaks supersymmetry [14, 15]. It was found that the non-BPS solutions exhibit flat directions in the scalar sector, in the sense that the scalars are not completely fixed at the horizon once the charges are chosen [14]. However, these examples are not generic enough they contain one less than the minimum number of parameters required for the most general solution to be derived from them by dualities. A solution that does contain enough parameters is called a seed solution. For cubic prepotentials, an appropriate seed was found in [16, 17] and the full duality orbit for the stu model was subsequently derived in [18]. This full example clarifies how the non-supersymmetric solutions differ from their BPS counterparts in more than simply changing the signs of charges. In particular, they have flat directions that are subject to symmetries that act along the full flow, including the horizon [1921]. If one allows for angular momentum, there are two types of single-centre extremal solutions which display attractor behaviour [22]. The over-rotating (or ergo) branch are very different from the BPS solutions, as they feature an ergoregion and are continuously connected to the Kerr solution [2325]. In contrast, the under-rotating (or ergo-free) black holes have a continuous limit to static charged black holes and seem to be tractable using BPS-inspired techniques. Recently, the single-centre under-rotating seed solution and various multi-centred generalisations were found in [2630]. In these cases, the nontrivial parameter appearing in the static seed solutions can be viewed as the constant part of a harmonic function describing rotation. Despite the existence of these known solutions, finding an organising principle for their general structure has proven challenging. The best developed approaches are based on fourdimensional supergravity, where electric-magnetic duality limits the possible structures. One such framework is provided by the timelike dimensional reduction of Breitenlohner, Maison and Gibbons [31], which relates black holes, regardless of supersymmetry (or even extremality), to geodesics on the (augmented) scalar manifold. Given sufficient symmetry on the scalar manifold, solutions, including multi-centre black holes, may be generated with powerful group-theoretical methods, cf. [3237] and references therein. Unfortunately, this comes at the expense of the results being expressed less explicitly. A more direct perspective has been offered by the fake superpotential approach of Ceresole and DallAgata [38]. They noticed that the rewriting of the effective black hole potential for the scalars [11] as a sum of squares is not unique, leading to more than one type of first-order flow for the scalar fields. The flow, which in the supersymmetric case is 2Since the two are related by dimensional reduction, we do not make a distinction between them in this introduction. governed by the absolute value of the central charge, may be more generally controlled by a different function, called the fake superpotential. The derivation of first-order equations based on a superpotential has been subsequently extended to static non-extremal black holes and for a number of models superpotentials have been identified explicitly [3943] (see [44] for a synopsis of these developments and [45] for earlier related work). The superpotential method has been first applied to multi-centre black holes in [46], which directly generalised [5]. However, simplifying assumptions restricted the non-supersymmetric solutions, as in [33], to threshold-bound configurations with mutually local charges (...truncated)