Non-supersymmetric attractors in string theory

Lecture Notes in Physics, 2008

We present the main features of the physics of extremal black holes embedded in supersymmetric theories of gravitation, with a detailed analysis of the attractor mechanism for BPS and non-BPS black-hole solutions in four dimensions.

Journal of High Energy Physics, 2007

We study the attractor mechanism for N = 1 supergravity coupled to vector and chiral multiplets and compute the attractor equations of these theories. These equations may have solutions depending on the choice of the holomorphic symmetric matrix f ΛΣ which appears in the kinetic lagrangian of the vector sector. Models with non trivial electric-magnetic duality group which have or have not attractor behavior are exhibited. For a particular class of models, based on an N = 1 reduction of homogeneous special geometries, the attractor equations are related to the theory of pure spinors.

Classical and Quantum Gravity, 2007

We express the d = 4, N = 2 black hole effective potential for cubic holomorphic F functions and generic dyonic charges in terms of d = 5 real special geometry data. The 4d critical points are computed from the 5d ones, and their relation is elucidated. For symmetric spaces, we identify the BPS and non-BPS classes of attractors and the respective entropies. These always derive from simple interpolating formulae between four and five dimensions, depending on the volume modulus and on the 4d magnetic (or electric) charges.

Physical Review D, 2006

We derive and solve the black hole attractor conditions of N = 8 supergravity by finding the critical points of the corresponding black hole potential. This is achieved by a simple generalization of the symplectic structure of the special geometry to all extended supergravities with N > 2. There are two classes of solutions for regular black holes, one for 1/8 BPS ones and one for the non-BPS. We discuss the solutions of the moduli at the horizon for BPS attractors using N = 2 language. An interpretation of some of these results in N = 2, STU black hole context helps to clarify the general features of the black hole attractors.

Physical Review D, 2009

We apply the entropy formalism to the study of the near-horizon geometry of extremal black p-brane intersections in D > 5 dimensional supergravities.

Physical Review D, 2008

We consider extremal black hole attractors (both BPS and non-BPS) for N = 3 and N = 5 supergravity in d = 4 space-time dimensions.

Physical Review D, 2009

We consider the Bekenstein-Hawking entropy-area formula in four dimensional extended ungauged supergravity and its electric-magnetic duality property.

Entropy, 2008

The general solutions of the radial attractor flow equations for extremal black holes, both for non-BPS with non-vanishing central charge Z and for Z = 0, are obtained for the so-called stu model, the minimal rank-3 N = 2 symmetric supergravity in d = 4 space-time dimensions.

Journal of Mathematical Physics, 2011

We classify 2-center extremal black hole charge configurations through duality-invariant homogeneous polynomials, which are the generalization of the unique invariant quartic polynomial for single-center black holes based on homogeneous symmetric cubic special Kähler geometries.

Physical Review D, 2008

We generalize the description of the d = 4 Attractor Mechanism based on an effective black hole (BH) potential to the presence of a gauging which does not modify the derivatives of the scalars and does not involve hypermultiplets. The obtained results do not rely necessarily on supersymmetry, and they can be extended to d > 4, as well.

General Relativity and Gravitation, 2008

We investigate four-dimensional spherically symmetric black hole solutions in gravity theories with massless, neutral scalars non-minimally coupled to gauge fields. In the non-extremal case, we explicitly show that, under the variation of the moduli, the scalar charges appear in the first law of black hole thermodynamics. In the extremal limit, the near horizon geometry is AdS 2 × S 2 and the entropy does not depend on the values of moduli at infinity. We discuss the attractor behaviour by using Sen's entropy function formalism as well as the effective potential approach and their relation with the results previously obtained through special geometry method. We also argue that the attractor mechanism is at the basis of the matching between the microscopic and macroscopic entropies for the extremal non-BPS Kaluza-Klein black hole.

Journal of Mathematical Physics, 2011

We reconsider the sub-leading quantum perturbative corrections to N = 2 cubic special Kähler geometries. Imposing the invariance under axion-shifts, all such corrections (but the imaginary constant one) can be introduced or removed through suitable, lower unitriangular symplectic transformations, dubbed Peccei-Quinn (PQ) transformations.

Journal of High Energy Physics, 2008

We study the attractor equations for a quantum corrected prepotential F = t 3 + iλ, with λ ∈ R,which is the only correction which preserves the axion shift symmetry and modifies the geometry.

Journal of High Energy Physics, 2007

We examine in details the entropy function formalism for non-extremal D3, M 2, and M 5-branes that their throat approximation is given by Schwarzschild black hole in AdS p+2 × S D−(p+2) . We show that even though there is no attractor mechanism in the non-extremal black holes/branes, the entropy function formalism does work and the entropy is given by the entropy function at its saddle point.

Physical Review D, 2008

We investigate extremal dyon black holes in the Einstein-Maxwell-dilaton (EMD) theory with higher curvature corrections in the form of the Gauss-Bonnet density coupled to the dilaton. In the same theory without the Gauss-Bonnet term the extremal dyon solutions exist only for discrete values of the dilaton coupling constant a. We show that the Gauss-Bonnet term acts as a dyon hair tonic enlarging the allowed values of a to continuous domains in the plane (a, qm) the second parameter being the magnetic charge. In the limit of the vanishing curvature coupling (a large magnetic charge) the dyon solutions obtained tend to the Reissner-Nordström solution but not to the extremal dyons of the EMD theory. Both solutions have the same values of the horizon radius as a function of charges. The entropy of new dyonic black holes interpolates between the Bekenstein-Hawking value in the limit of the large magnetic charge (equivalent to the vanishing Gauss-Bonnet coupling) and twice this value for the vanishing magnetic charge. Although an expression for the entropy can be obtained analytically using purely local near-horizon solutions, its interpretation as the black hole entropy is legitimate only once the global black hole solution is known to exist, and we obtain numerically the corresponding conditions on the parameters. Thus, a purely local analysis is insufficient to fully understand the entropy of the curvature corrected black holes. We also find dyon solutions which are not asymptotically flat, but approach the linear dilaton background at infinity. They describe magnetic black holes on the electric linear dilaton background. the present paper for extremal dyon black holes with the AdS 2 × S 2 horizon in the Gauss-Bonnet gravity.

Journal of Physics A: Mathematical and Theoretical, 2010

Reconsidering the analysis of the moduli space of N = 2 eight dimensional supergravity coupled to seven scalars, we propose a new scalar manifold factorization given by SO(2,2) SO(2)× SO(2) × SO(2,1) SO(2) × SO(1,1). This factorization is supported by the appearance of three solutions of Type IIA extremal black p-branes (p = 0, 1, 2) with AdS p+2 × S 6−p nearhorizon geometries in eight dimensions. We analyze the corresponding attractor mechanism.

Journal of High Energy Physics, 2016

We show the equality between macroscopic and microscopic black hole entropy for a class of four dimensional non-supersymmetric black holes in N = 2 supergravity theory, up to the first subleading order in their charges. This solves a long standing entropy puzzle for this class of black holes. The macroscopic entropy has been computed in the presence of a newly derived higher-derivative supersymmetric invariant of [1], connected to the five dimensional supersymmetric Weyl squared Lagrangian. Microscopically, the crucial role in obtaining the equivalence is played by the anomalous gauge gravitational Chern-Simons term.

Physical Review D, 2006

We present explicit U-duality invariants for the R, C, Q, O (real, complex, quaternionic and octonionic) magic supergravities in four and five dimensions using complex forms with a reality condition. From these invariants we derive an explicit entropy function and corresponding stabilization equations which we use to exhibit stationary multi-center 1/2 BPS solutions of these N = 2 d = 4 theories, starting with the octonionic one with E 7(−25) duality symmetry. We generalize to stationary 1/8 BPS multicenter solutions of N = 8, d = 4 supergravity, using the consistent truncation to the quaternionic magic N = 2 supergravity. We present a general solution of non-BPS attractor equations of the STU truncation of magic models. We finish with a discussion of the BPS-non-BPS relations and attractors in N = 2 versus N = 5, 6, 8.

Physical Review D, 2007

Following the recent trend we develop further the black hole analogy between quantum information theory and the theory of extremal stringy black hole solutions. We show that the three-qubit interpretation of supersymmetric black hole solutions in the STU model can be extended also to include non-supersymmetric ones. First we show that the black hole potential can be expressed as one half the norm of a suitably chosen three-qubit entangled state containing the quantized charges and the moduli. The extremization of the black hole potential in terms of this entangled state amounts to either supressing bit flip errors (BPS-case) or allowing very special types of flips transforming the states between different classes of non-BPS solutions. We are illustrating our results for the example of the D2 − D6 system. In this case the bit flip errors are corresponding to sign flip errors of the charges originating from the number of D2 branes. After moduli stabilization the states depending entirely on the charges are maximally entangled graph-states (of the triangle graph) well-known from quantum information theory. An N = 8 interpretation of the STU-model in terms of a mixed state with fermionic purifications is also given.

Physical Review D, 2011

We link the recently discovered black hole-qubit correspondence to the structure of extra dimensions. In particular we show that for toroidal compactifications of type IIB string theory simple qubit systems arise naturally from the geometrical data of the tori parametrized by the moduli. We also generalize the recently suggested idea of the attractor mechanism as a distillation procedure of GHZ-like entangled states on the event horizon, to moduli stabilization for flux attractors in F-theory compactifications on elliptically fibered Calabi-Yau four-folds. Finally using a simple example we show that the natural arena for qubits to show up is an embedded one within the realm of fermionic entanglement of quantum systems with indistinguishable constituents.

Physical Review D, 2012

According to the Black Hole/Qubit Correspondence (BHQC) certain black hole entropy formulas in supergravity can be related to multipartite entanglement measures of quantum information. Here we show that the origin of this correspondence is a connection between Hitchin functionals used as action functionals for form theories of gravity related to topological strings, and entanglement measures for systems with a small number of constituents. The basic idea acting as a unifying agent in these seemingly unrelated fields is stability connected to the mathematical notion of special prehomogeneous vector spaces associated to Freudenthal systems coming from simple Jordan algebras. It is shown that the nonlinear function featuring these functionals and defining Calabi-Yau and generalized Calabi-Yau structures is the Freudenthal dual a concept introduced recently in connection with the BHQC. We propose to use the Hitchin invariant for three-forms in 7 dimensions as an entanglement measure playing a basic role in classifying three-fermion systems with seven modes. The representative of the class of maximal tripartite entanglement is the threeform used as a calibration for compactification on manifolds with G 2 holonomy. The idea that entanglement measures are related to action functionals from which the usual correspondence of the BHQC follows at the tree level suggests that one can use the BHQC in a more general context.

Journal of High Energy Physics, 2013

We apply the H-FGK formalism to the study of some properties of the general class of black holes in N = 2 supergravity in four dimensions that correspond to the harmonic and hyperbolic ansätze and obtain explicit extremal and non-extremal solutions for the t 3 model with and without a quantum correction. Not all solutions of the corrected model (quantum black holes), including in particular a solution with a single q 1 charge, have a regular classical limit.

Physical Review D, 2010

Small black holes in heterotic string theory have vanishing horizon area at the supergravity level, but the horizon is stretched to the finite radius AdS2 × S D−2 geometry once higher curvature corrections are turned on. This has been demonstrated to give good agreement with microscopic entropy counting. Previous considerations, however, were based on the classical local solutions valid only in the vicinity of the event horizon. Here we address the question of global existence of extremal black holes in the D-dimensional Einstein-Maxwell-Dilaton theory with the Gauss-Bonnet term introducing a variable dilaton coupling a as a parameter. We show that asymptotically flat black holes exist only in a bounded region of the dilaton couplings 0 < a < acr where acr depends on D. For D ≥ 5 (but not for D = 4) the allowed range of a includes the heterotic string values. For a > acr numerical solutions meet weak naked singularities at finite radii r = rcusp (spherical cusps), where the scalar curvature diverges as |r − rcusp| −1/2. For D ≥ 7 cusps are met in pairs, so that solutions can be formally extended to asymptotically flat infinity choosing a suitable integration variable. We show, however, that radial geodesics cannot be continued through the cusp singularities, so such a continuation is unphysical.

Physical Review D, 2007

We show that four-dimensional Einstein-Maxwell-Dilaton-Gauss-Bonnet gravity admits asymptotically flat black hole solutions with a degenerate event horizon of the Reissner-Nordström type AdS2 × S 2. Such black holes exist for the dilaton coupling constant within the interval 0 ≤ a 2 < a 2 cr. Black holes must be endowed with an electric charge and (possibly) with magnetic charge (dyons) but they can not be purely magnetic. Purely electric solutions are constructed numerically and the critical dilaton coupling is determined acr ≃ 0.488219703. For each value of the dilaton coupling a within this interval and for a fixed value of the Gauss-Bonnet coupling α we have a family of black holes parameterized by their electric charge. Relation between the mass, the electric charge and the dilaton charge at both ends of the allowed interval of a is reminiscent of the BPS condition for dilaton black holes in the Einstein-Maxwell-Dilaton theory. The entropy of the DGB extremal black holes is twice the Bekenstein-Hawking entropy.

Physical Review D, 2018

We study dilaton-gravity theories in 2-dimensions obtained by dimensional reduction of higher dimensional nonrelativistic theories. Focussing on certain families of extremal charged hyperscaling violating Lifshitz black branes in Einstein-Maxwell-scalar theories with an extra gauge field in 4-dimensions, we obtain AdS 2 backgrounds in the near horizon throats. We argue that these backgrounds can be obtained in equivalent theories of 2-dim dilaton-gravity with an extra scalar, descending from the higher dimensional scalar, and an interaction potential with the dilaton. A simple subcase here is the relativistic black brane in Einstein-Maxwell theory. We then study linearized fluctuations of the metric, dilaton and the extra scalar about these AdS 2 backgrounds. The coefficient of the leading Schwarzian derivative term is proportional to the entropy of the (compactified) extremal black branes.

Physical Review D, 2014

A new general class of solutions of ungauged four-dimensional Supegravity, in one-to-one correspondence with spherically symmetric, static black-hole solutions and Lifshitz solutions with Hyperscaling violation (hvLif) is studied. The causal structure of the space-time is then elucidated.

Journal of High Energy Physics, 2012

In this paper we apply in a systematic way a previously developed integration algorithm of the relevant Lax equation to the construction of spherical symmetric, asymptotically flat black hole solutions of N = 2 supergravities with symmetric Special Geometry. Our main goal is the classification of these black-holes according to the H orbits in which the space of possible Lax operators decomposes. By H one denotes the isotropy group of the coset U D=3 /H which appears in the time-like dimensional reduction of supergravity from D = 4 to D = 3 dimensions. The main result of our investigation is the construction of three universal tensors, extracted from quadratic and quartic powers of the Lax operator, that are capable of classifying both regular and nilpotent H orbits of Lax operators. Our tensor based classification is compared, in the case of the simple one-field model S 3 , to the algebraic classification of nilpotent orbits and it is shown to provide a simple discriminating method. In particular we present a detailed analysis of the S 3 model, constructing explicitly its solutions and discussing the Liouville integrability of the corresponding dynamical system. By means of the Kostant-representation of a generic Lie algebra element, we were able to develop an algorithm which produces the necessary number of hamiltonians in involution required by Liouville integrability of generic orbits. The degenerate orbits correspond to extremal blackholes and are nilpotent. We present an in depth discussion of their identification and of the construction of the corresponding supergravity solutions. We dwell on the relation between H orbits and critical points of the geodesic potential showing that there is correspondence yet not one-to-one. Finally we present the conjecture that our newly identified tensor classifiers are universal and able to label all regular and nilpotent orbits in all homogeneous symmetric Special Geometries. 1 Presently Prof. Fré holds the office of Scientific Counselor of the Italian Embassy in the Russian Federation.

Journal of High Energy Physics, 2013

Four dimensional supergravity theories whose scalar manifold is a symmetric coset manifold U D=4 /H c are arranged into a finite list of Tits Satake universality classes. Stationary solutions of these theories, spherically symmetric or not, are identified with those of an euclidian three-dimensional σ-model, whose target manifold is a Lorentzian coset U D=3 /H ⋆ and the extremal ones are associated with H ⋆ nilpotent orbits in the K ⋆ representation emerging from the orthogonal decomposition of the algebra U D=3 with respect to H ⋆ . It is shown that the classification of such orbits can always be reduced to the Tits-Satake projection and it is a class property of the Tits Satake universality classes. The construction procedure of Bossard et al of extremal multicenter solutions by means of a triangular hierarchy of integrable equations is completed and converted into a closed algorithm by means of a general formula that provides the transition from the symmetric to the solvable gauge. The question of the relation between H ⋆ orbits and charge orbits W of the corresponding black holes is addressed and also reduced to the corresponding question within the Tits Satake projection. It is conjectured that on the vanishing locus of the Taub-NUT current the relation between H ⋆ -orbit and W-orbit is rigid and one-to-one. All black holes emerging from multicenter solutions associated with a given H ⋆ orbit have the same W-type. For the S 3 model we provide a complete survey of its multicenter solutions associated with all of the previously classified nilpotent orbits of sl(2) × sl(2) within g 2,2 . We find a new intrinsic classification of the W -orbits of this model that might provide a paradigm for the analogous classification in all the other Tits Satake universality classes. 1 Prof. Fré is presently fulfilling the duties of Scientific Counselor of the Italian Embassy in the Russian Federation,

Physical Review D, 2022

Freudenthal duality (F-duality), an anti-involution of charge vectors, keeps the entropy and attractor solutions invariant for an extremal supersymmetric black hole. We analyze the effect of F-duality on the entropy of a near-extremal STU black hole in N ¼ 2 ungauged, four-dimensional supergravity. We consider double-extremal black holes, whose attractor solutions are fixed in terms of the black hole charges throughout the moduli space. It is well known that Jackiw-Teitelboim (JT) gravity governs the dynamics of the near-horizon regions of higher-dimensional, near-extremal black holes. Owing to this fact, we reduce the four-dimensional supergravity theory to two dimensions to construct a JT gravity-like model and compute the near-extremal entropy. We then analyze the effect of F-duality on this entropy. We show that the F-duality breaks down for the case of near-extremal solutions if one considers the duality operation generated through near-extremal entropy rather than the extremal one.

Physical Review D, 2011

We relate the U-duality invariants characterizing two-center extremal black hole solutions in the stu, st 2 and t 3 models of N = 2, d = 4 supergravity to the basic invariants used to characterize entanglement classes of four-qubit systems. For the elementary example of a D0D4-D2D6 composite in the t 3 model we illustrate how these entanglement invariants are related to some of the physical properties of the two-center solution. Next we show that it is possible to associate elliptic curves to charge configurations of two-center composites. The hyperdeterminant of the hypercube, a four-qubit polynomial invariant of order 24 with 2894276 terms, is featuring the j invariant of the elliptic curve. We present some evidence that this quantity and its straightforward generalization should play an important role in the physics of two-center solutions.

Journal of High Energy Physics, 2014

In this paper we study spherically symmetric single-centered attractors in N = 2 supergravity in four dimensions. The attractor points are obtained by extremising the effective black hole potential in the moduli space. Both supersymmetric as well as non-supersymmetric attractors exist in mutually exclusive domains of the charge lattice. We construct axion free supersymmetric as well as non-supersymmetric multiple attractors in a simple two parameter model. We further obtain explicit examples of two distinct nonsupersymmetric attractors in type IIA string theory compactified on K3 × T 2 carrying D0 − D4 − D6 charges. We compute the entropy of these attractors and analyse their stability in detail.

Modern Physics Letters A, 2016

In this paper, we consider the four-dimensional N = 2 supergravity theory arising from the compactification of type IIA string theory on a Calabi–Yau manifold. We analyze the supersymmetric flow equations for static, spherically symmetric, single-centered black holes. These flow equations are solved by a set of algebraic equations involving the holomorphic sections and harmonic functions. We examine black hole configurations with D0–D4–D6 charge for which the most general solution of these algebraic equations are considered. Though the black hole solution is unique for a given value of the charges, we find new phases of the black hole solutions upon varying them.

Journal of High Energy Physics, 2014

With reference to the effective three-dimensional description of stationary, single center solutions to (ungauged) symmetric supergravities, we complete a previous analysis on the definition of a general geometrical mechanism for connecting global symmetry orbits (duality orbits) of non-extremal solutions to those of extremal black holes. We focus our attention on a generic representative of these orbits, providing its explicit description in terms of D = 4 fields. As a byproduct, using a new characterization of the angular momentum in terms of quantities intrinsic to the geometry of the D = 3 effective model, we are able to prove on general grounds its invariance, as a function of the boundary data, under the D = 4 global symmetry. In the extremal under-rotating limit it becomes moduli-independent. We also discuss the issue of the fifth parameter characterizing the four-dimensional seed solution, showing that it can be generated by a transformation in the global symmetry group which is manifest in the D = 3 effective description.

Journal of High Energy Physics, 2021

We analyze the charge-to-mass structure of BPS states in general infinite-distance limits of $$ \mathcal{N} $$ N = 2 compactifications of Type IIB string theory on Calabi-Yau three-folds, and use the results to sharpen the formulation of the Swampland Conjectures in the presence of multiple gauge and scalar fields. We show that the BPS bound coincides with the black hole extremality bound in these infinite distance limits, and that the charge-to-mass vectors of the BPS states lie on degenerate ellipsoids with only two non-degenerate directions, regardless of the number of moduli or gauge fields. We provide the numerical value of the principal radii of the ellipsoid in terms of the classification of the singularity that is being approached. We use these findings to inform the Swampland Distance Conjecture, which states that a tower of states becomes exponentially light along geodesic trajectories towards infinite field distance. We place general bounds on the mass decay rate λ of thi...