proceedings Article New black hole solutions in N = 2 and N = 8 gauged supergravity Antonio Gallerati 1,2* 1 Politecnico di Torino, Dipartimento di Scienza Applicata e Tecnologia, corso Duca degli Abruzzi 24, 10129 Torino, Italy 2 Istituto Nazionale di Fisica Nucleare, Sezione di Torino, via Pietro Giuria 1, 10125 Torino, Italy * Correspondence: [email protected] (A.G.) Abstract: We present a N = 2 supergravity model that interpolates between all the single dilaton truncations of the gauged SO(8), N = 8 supergravity. We provide new explicit non-extremal charged black hole solutions and their supersymmetric limits, exploiting the non-trivial transformation of the Fayet-Iliopoulos parameters under electromagnetic duality to connect the electric and the magnetic configurations. We also provide the asymptotic charges, thermodynamics and boundary conditions of these black hole configurations. We then construct a new supersymmetric truncation of the maximal supergravity, this new sector featuring non-extremal and supersymmetric black holes. Keywords: supergravity; black hole solutions; black holes thermodynamics; BPS black holes. 1. Introduction Anti-de Sitter (AdS) black hole configurations are an amazing and stimulating field of research due to the role they play in high energy theory as well as in the phenomenology of the AdS/CFT conjecture [1]. In particular, classical AdS black hole solutions can reveal specific features about the dual, strongly coupled gauge theory, providing then a possible description of many condensed matter phenomena. Citation: . Proceedings 2021, 1, 0. The thermodynamic properties of AdS black holes where first analyzed in [2] and sub- sequently extended to various other AdS black hole configurations [3–6]. These researches Academic Editor: described how black holes feature specific phase structures, giving rise to critical phenom- Received: ena analogous to other common thermodynamic systems. Of particular interest are black Accepted: hole solutions preserving a certain amount of supersymmetry, since they allow to map a Published: weak (string) coupling description of the system thermodynamics to the strong-coupling regime, where a formulation in terms of a black hole configuration is valid [7]. Moreover, Publisher’s Note: MDPI stays neu- these solutions can be exploited to study the BPS attractor flows in AdS spacetime [8–21]. tral with regard to jurisdictional claims In the following we will discuss the new exact charged hairy black hole solutions in in published maps and institutional gauged N = 2, D = 4 supergravity of [22]1, interpolating between four single dilaton affiliations. truncations of the maximal SO(8), N = 8 supergravity. In particular, we will provide the explicit expressions for two new different families of non-extremal black hole solutions, analysing the duality relation between them. We will then investigate the thermody- namic properties of our new solutions, together with the analysis of the related boundary Copyright: © 2020 by the authors. Li- conditions. We will also study which conditions for the parameters give rise to BPS config- censee MDPI, Basel, Switzerland. This article is an open access article distributed uration ones. Finally, we will characterize certain models within the general class under under the terms and conditions of the consideration as consistent truncations of the maximal N = 8, D = 4 gauged supergravity. Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). 1 these new solutions generalize the uncharged configurations of [23,24] The 1st Electronic Conference on Universe The 1st Electronic Conference on Universe 2 of 14 2. Results Let us consider an extended N = 2 supergravity theory in D = 4, coupled to nv vector multiplets and no hypermultiplets, in the presence of Fayet–Iliopoulos (FI) terms. L The model describes nv + 1 vector fields Am ,(L = 0, ... , nv) and ns = nv complex scalar i fields z (i = 1, . , ns). The bosonic gauged Lagrangian has the explicit form 1 R 1 1 p L = − + g ¶ zi ¶mz¯¯ + I (z, z¯) FL FS mn + p R (z, z¯) #mnrs FL FS − V(z, z¯) , −g BOS 2 i¯ m 4 LS mn 8 −g LS mn rs (1) where g = det(gmn) and with the nv + 1 vector field strengths: L L L Fmn = ¶m An − ¶n Am . (2) i The ns complex scalars z couple to the vector fields in a non-minimal way through the real symmetric matrices ILS(z, z¯), RLS(z, z¯) and span a special Kähler manifold MSK, while the scalar potential V(z, z¯) originates from electric-magnetic FI terms. 2.1. The model Let us now focus on a N = 2 theory with no hypermultiplets and a single vector multiplet (nv = 1), with a complex scalar field z. The geometry of the special Kähler manifold is characterized by a prepotential of the form: i n 2−n F(X L) = − X 0 X 1 , (3) 4 X L(z) being components of a holomorphic section of the symplectic bundle over the manifold and the coordinate z being identified with the ratio X 1/X 0, in a local patch in which X 0 6= 0. For special values of n, the model is a consistent truncation of the STU model2. If we set X 0 = 1, the holomorphic section WM of the model reads: 0 1 1 B z C B C M B i C W = B − n z2−n C , (4) B 4 C @ i A − (2 − n) z1−n 4 and the Kähler potential K has the expression 1 e−K = z1−n n z − (n − 2) z¯ + c.c. (5) 4 The theory is deformed with the introduction of abelian electric-magnetic FI terms defined by a constant symplectic vector qM = (q1, q2, q3, q4), encoding the gauge parameters of the model. The scalar potential V(z, z¯) can be then obtained from: 1 V = gi¯ U M U N − 3 V M V N q q = − q MMN q − 4 V M V Nq q , (6) i ¯ M N 2 M N M N M K/2 M M M where V = e W , Ui = Di V and M(f) is the symplectic, symmetric, negative definite matrix encoding the non-minimal couplings of the scalars to the vector fields of the theory. 2 the STU model [25–27] is a N p= 2 supergravity coupled to nv = 3 vector multiplets and characterized, in a suitable symplectic frame, by the L i 0 1 2 3 3 prepotential FSTU (X ) = − 4 X X X X , together with symmetric scalar manifold of the form MSTU = (SL(2, R)/SO(2)) spanned by the three complex scalars zi = X i/X 0 (i = 1, 2, 3); this model is in turn a consistent truncation of the maximal N = 8 theory in four-dimensions with SO(8) gauge group [28–30] The 1st Electronic Conference on Universe 3 of 14 Writing z = el f + i c , the truncation to the dilaton field f (i.e. c = 0) is consistent provided: (2 − n) q1 q3 − n q2 q4 = 0 , (7) and the metric restricted to the dilaton reads: 2 1 2 2 ds = 2 gzz¯ dz dz¯ = l n (2 − n) df , (8) c=0 2 dc=0 q < < = 2 and is positive provided 0 n 2. Choosing l n (2−n) the kinetic term for f is then canonically normalized. As a function of the dilaton only, the scalar potential has the following explicit form: 2 n − 1 2 n − 3 V(f) = − 2 el f (n−2) q2 + 4 q q el f + q2 e2 l f − n 1 1 2 n − 2 2 1 − e−l f (n−2) (2 n − 1) n q2 − 4 q q n (n − 2) e−l f + (n − 2)(2 n − 3) q2 e−2 l f . 8 3 3 4 4 (9) 2.1.1. Redefinitions Let us now make the shift 2 n f ! f − log(q x) , (10) l (n + 1) 2 and redefine the FI terms as: n + 1 − n−1 2 n n−1 2 a n+1 − n+1 n+1 q1 = q2 x , q3 = 2 a(x q2) s , q4 = , (11) n − 1 q2 x s having defined the quantity n = (n − 1)−1 and having also introduced the parameters a, s and 2 L n 1 x = p , (12) n − 1 1 − a2 L2 expressed in terms of the AdS radius L. Let us recall that the truncation to the dilaton is consistent provided equation (7) is satisfied and, in light of the new parametrization (11), this condition requires p (s2 − 1)(n2 − 1) a 1 − L2 a2 = 0 , (13) which is solved, excluding values n = 0 and n = 2, either for pure electric FI terms (a = 0) or for s = ±1. Since we are interested in dyonic FI terms, we shall restrict ourselves to the latter case. After the shift (10), the scalar field z is expressed as 2 n − + l f z = (q2 x) n 1 e , (14) and the same redefinition for the potential (with s = ±1) yields a2 (n − 1)(n − 2) (n + 1)(n + 2) V(f) = − e−f ` (n+1) + 2(n2 − 1) e−f ` + ef ` (n−1) + n2 2 2 a2 − L−2 (n − 1)(n − 2) (n + 1)(n + 2) + ef ` (n+1) + 2(n2 − 1) ef ` + e−f ` (n−1) , n2 2 2 (15) l where ` = and having disposed of q by the above redefinitions. n 2 The 1st Electronic Conference on Universe 4 of 14 After the restriction to the dilaton truncation, the matrix RLS vanishes and the action has the form Z m 1 p R ¶mf ¶ f 1 S = d4x −g − + + I (f) FL FS mn − V(f) , (16) 8pG 2 2 4 LS mn If we define the canonically normalized gauge fields r r 1 1 + n 1−n 1 −1 + n F¯1 = (q x) 1+n F1 , F¯2 = (q x) F2 , (17) 2 n 2 2 n 2 the action can be expressed as Z m 1 p R ¶mf ¶ f 1 (− + ) ` 2 1 −( + ) ` 2 S = d4x −g − + − e 1 n f F¯1 − e 1 n f F¯2 − V(f) .