We discuss brane webs preserving eight supercharges and derive geometries produced by them. Consistency conditions of supergravity are shown to impose certain requirements on the locations of the sources, and these restrictions are found to be in a perfect agreement with results of the probe analysis. In particular, solutions of IIB SUGRA describing (p,q) stings are inconsistent, unless the web consists of straight line segments whose orientation is correlated with charges of the string. The geometries produced by membranes and D3 branes are only consistent if brane profiles are holomorphic. Using perturbation theory, we show that a unique gravity solution exists for any allowed distribution of sources. We also revisit 1/4-BPS geometries with AdS_p x S^q asymptotics and derive the boundary conditions leading to regular geometries. All degenerate limits of regular solutions are shown to agree with expectations from the brane probe analysis.
In the Coulomb phase of the 6-dim (2,0) superconformal theories, the 1/4, 1/8, 1/16 BPS selfdual string webs are argued to exist such that the spatial SO(5) and internal SO(5) rotations are correlated. The basic constituents are 1/2 BPS strings and 1/4 BPS string junctions. One support comes from the existence of the similar BPS dyonic monostring webs in 5-dim maximally supersymmetric gauge theories. Another comes from the study of the supersymmetry of the intersecting M2 brane stripes terminating on M5 branes. We also discuss the related BPS webs in little string theories and other theories.
We argue for the existence of many new 1/4 BPS states in N=4 SU(N_c) Super-Yang-Mills theory with N_c>=3, by constructing them from supersymmetric string webs whose external strings terminate on parallel D3-branes. The masses of the string webs are shown to agree with the BPS bound for the corresponding states in SYM. We identify the curves of marginal stability, at which these states decay into other BPS states. We find the bosonic and fermionic zero modes of the string webs, and thereby the degeneracy and spin content of some of the BPS states. States of arbitrarily high spin are predicted i…
We analyze more explicitly four sectors of 1/4 BPS geometries and 1/8 BPS geometries, corresponding to BPS states in N=4 SYM, constructed previously. These include the states with several SO(6) angular momenta as well as those with SO(4) AdS spins. We also discuss their relations to the dual gauge theory.
We study the BPS spectrum of four dimensional N=2 SU(2) theory with massive fundamental matters using the D3-brane probe. Since the BPS states are realized by string webs subject to the BPS conditions, we determine explicitly the configurations of such webs. It is observed that there appear BPS string webs with multiple of junctions corresponding to the fact that the curves of marginal stability in massive theory are infinitely nested. In terms of the string configurations, various properties of the curves of marginal stability are explained intuitively.
We study the BPS spectrum of supersymmetric 5 dimensional field theories and their representations as string webs. It is found that a state of given charges exists when it has a representation as an irreducible string web. Its spin is determined by the string web. The number of fermionic zero modes is 8g+4b, where g is the number of internal faces and b is the number of boundaries. In the lift to M theory of 4d field theories such states are described by membranes ending on the 5-brane, breaking SUSY from 8 to 4, and g becomes the genus of the membrane. Mathematically, we obtain a diagrammatic…
We derive a non-BPS linear ansatz using the charged Weyl formalism in string and M-theory backgrounds. Generic solutions are static and axially-symmetric with an arbitrary number of non-BPS sources corresponding to various brane, momentum and KKm charges. Regular sources are either four-charge non-extremal black holes or smooth non-BPS bubbles. We construct several families such as chains of non-extremal black holes or smooth non-BPS bubbling geometries and study their physics. The smooth horizonless geometries can have the same mass and charges as non-extremal black holes. Furthermore, we fin…
In the search for a classification of BPS backgrounds with flux, we look at geometries that arise when M-branes wrap supersymmetric cycles in Calabi-Yau manifolds. We find constraints on the differential forms in the back-reacted manifolds and discover that the calibration corresponding to the (background generating) M-brane is a co-closed form.
Recently a new class of 1/8-BPS regular geometries in type IIB string theory was constructed in arXiv:1503.01463. In this paper we provide a precise description of the semiclassical states dual, in the AdS/CFT sense, to these geometries. In explicit examples we show that the holographic 1-point functions and the Ryu-Takayanagi's Entanglement Entropy for a single small interval match the corresponding CFT calculations performed by using the proposed dual states. We also discuss several new examples of such precision holography analysis in the 1/4-BPS sector and provide an explicit proof that th…
We construct the general 1/2 BPS M2 giant graviton solutions asymptotic to the eleven-dimensional maximally supersymmetric plane wave background, based on the recent work of Lin, Lunin and Maldacena. The solutions have null singularity and we argue that it is unavoidable to have null singularity in the proposed framework, although the solutions are still physically relevant. They involve an arbitrary function F(x) which is shown to have a correspondence to the 1/2 BPS states of the BMN matrix model. A detailed map between the 1/2 BPS states of both sides is worked out.
By using the approach introduced in arXiv:2107.09677 we construct non-BPS solutions of 6D $(1,0)$ supergravity coupled to two tensor multiplets as a perturbation of AdS$_3\times S^3$. These solutions are both regular and asymptotically AdS$_3\times S^3$, so according to the standard holographic framework they must have a dual CFT interpretation as non-supersymmetric heavy operators of the D1-D5 CFT. We provide quantitative evidence that such heavy CFT operators are bound states of a large number of light BPS operators that are mutually non-BPS.
We investigate the 1/8 BPS geometries with SU(2) x U(1) x SO(4) x R symmetry in IIB supergravity which were classified by Gava et al, (hep-th/0611065). It is desirable to have a complete set of differential equations imposed on the controlling functions such that they are not only necessary but also sufficient to produce supergravity solutions with those symmetries. We work on this issue and find a new differential equation for the controlling functions. For a special case, we exhaust all the remaining constraints and show that they reduce to one Liouville equation. The solutions of this equat…
Recently Lin, Lunin and Maldacena (LLM) (hep-th/0409174) explicitly mapped 1/2 BPS excitations of type IIB supergravity on AdS_5 x S^5 into free fermion configurations. We discuss thermal coarse-gaining of LLM geometries by explicitly mapping the corresponding equilibrium finite temperature fermion configuration into supergravity. Following Mathur conjecture, a prescription of this sort should generate a horizon in the geometry. We did not find a horizon in finite temperature equilibrium LLM geometry. This most likely is due to the fact that coarse-graining is performed only in a half-BPS sect…
We use a multi-species supertube solution to construct an example of a scaling microstate geometry for non-BPS black rings in five dimensions. We obtain the asymptotic charges of the microstate geometry and show how the solution is related to the corresponding non-BPS black ring. The supersymmetry is broken in a very controlled manner using holonomy and this enables a close comparison with a scaling, BPS microstate geometry. Requiring that there are no closed time-like curves near the supertubes places additional restrictions on the moduli space of physical, non-BPS solutions when compared to …
We discuss how to take a Penrose limit in bubbling 1/2 BPS geometries at the stage of a single function z(x_1,x_2,y). By starting from the z of the AdS_5 x S^5 we can directly derive that of the pp-wave via the Penrose limit. In that time the function z for the pp-wave with 1/R^2-corrections is obtained. We see that it surely reproduces the pp-wave with 1/R^2 terms. In addition we consider the Penrose limit in the configuration of the concentric rings.
We consider all 1/2 BPS excitations of $AdS \times S$ configurations in both type IIB string theory and M-theory. In the dual field theories these excitations are described by free fermions. Configurations which are dual to arbitrary droplets of free fermions in phase space correspond to smooth geometries with no horizons. In fact, the ten dimensional geometry contains a special two dimensional plane which can be identified with the phase space of the free fermion system. The topology of the resulting geometries depends only on the topology of the collection of droplets on this plane. These so…
In hep-th/0409174, Lin, Lunin and Maldacena constructed a set of regular 1/2 BPS geometries in IIB theory. These remarkable `bubbling AdS' geometries have a natural interpretation as duals of chiral primary operators with weight Delta=J in N=4 super-Yang Mills theory. Although these geometries have been assumed to be complete, from a purely supergravity point of view, additional 1/2 BPS configurations may potentially exist with a preferred null isometry. We explore this possibility and prove that the only additional class of 1/2 BPS solutions with SO(4) x SO(4) isometry are the familiar IIB pp…
We carry out a general analysis of the representations of the superconformal algebras SU(2,2/N), OSp(8/4,R) and OSp(8^*/4) and give their realization in superspace. We present a construction of their UIR's by multiplication of the different types of massless superfields ("supersingletons"). Particular attention is paid to the so-called "short multiplets". Representations undergoing shortening have "protected dimension" and correspond to BPS states in the dual supergravity theory in anti-de Sitter space. These results are relevant for the classification of multitrace operators in boundary confo…
We consider the supergravity backgrounds that correspond to supersymmetric Wilson line operators in the context of AdS/CFT correspondence. We study the gravitino and dilatino conditions of the IIB supergravity under the appropriate ansatz, and obtain some necessary conditions for a supergravity background that preserves the same symmetry as the supersymmetric Wilson lines. The supergravity solutions are characterized by continuous version of maya diagrams. This diagram is related to the eigenvalue distribution of the Gaussian matrix model. We also consider the similar backgrounds of the 11-dim…
We study semiclassical string solutions on the 1/2 BPS geometry of type IIB string theory characterized by concentric rings on the boundary plane. We consider both folded rotating strings carrying nonzero R-charge and circular pulsating strings. We find that unlike rotating strings, as far as circular pulsating strings are concerned, the dynamics remains qualitatively unchanged when the concentric rings replace AdS_5\times S^5. Using the gravity dual we have also studied the Wilson loop of the corresponding gauge theory. The result is qualitatively the same as that in AdS_5\times S^5 in the gl…
We carry out a general analysis of the representations of the superconformal algebras OSp(8/4,R) and OSp(8*/2N) in terms of harmonic superspace. We present a construction of their highest-weight UIR's by multiplication of the different types of massless conformal superfields ("supersingletons"). Particular attention is paid to the so-called "short multiplets". Representations undergoing shortening have "protected dimension" and may correspond to BPS states in the dual supergravity theory in anti-de Sitter space. These results are relevant for the classification of multitrace operators in bound…
Webs of domain walls are constructed as 1/4 BPS states in d=4, N=2 supersymmetric U(Nc) gauge theories with Nf hypermultiplets in the fundamental representation. Web of walls can contain any numbers of external legs and loops like (p,q) string/5-brane webs. We find the moduli space M of a 1/4 BPS equation for wall webs to be the complex Grassmann manifold. When moduli spaces of 1/2 BPS states (parallel walls) and the vacua are removed from M, the non-compact moduli space of genuine 1/4 BPS wall webs is obtained. All the solutions are obtained explicitly and exactly in the strong gauge coupling…
Recently, 1/2-BPS AdS bubble solutions have been obtained by Lin, Lunin and Maldacena, which correspond to Fermi droplets in phase space in the dual CFT picture. They can be thought of as generalisations of 1/2-BPS AdS black hole solutions in five or seven dimensional gauged supergravity. In this paper, we extend these solutions by invoking additional gauge fields and scalar fields in the supergravity Lagrangians, thereby obtaining AdS bubble generalisations of the previously-known multi-charge AdS black solutions of gauged supergravity. We also obtain analogous AdS bubble solutions in four-di…
This paper focuses on supergravity duals of BPS states in N=4 super Yang-Mills. In order to describe these duals, we begin with a sequence of breathing mode reductions of IIB supergravity: first on S^3, then S^3 x S^1, and finally on S^3 x S^1 x CP^1. We then follow with a complete supersymmetry analysis, yielding 1/8, 1/4 and 1/2 BPS configurations, respectively (where in the last step we take the Hopf fibration of S^3). The 1/8 BPS geometries, which have an S^3 isometry and are time-fibered over a six-dimensional base, are determined by solving a non-linear equation for the Kahler metric on …
We continue to study 5d N=1 supersymmetric field theories and their compactifications on a circle through brane configurations. We develop a model, which we call (p,q) Webs, which enables simple geometrical computations to reproduce the known results, and facilitates further study. The physical concepts of field theory are transparent in this picture, offering an interpretation for global symmetries, local symmetries, the effective (running) coupling, the Coulomb and Higgs branches, the monopole tensions, and the mass of BPS particles. A rule for the dimension of the Coulomb branch is found by…
Eighth-BPS local operators in N=4 SYM are dual to quantum states arising from the quantization of a moduli space of giant gravitons in AdS5xS5. Earlier results on the quantization of this moduli space give a Hilbert space of multiple harmonic oscillators in 3 dimensions. We use these results, along with techniques from fuzzy geometry, to develop a map between quantum states and brane geometries. In particular there is a map between the oscillator states and points in a discretization of the base space in the toric fibration of the moduli space. We obtain a geometrical decomposition of the spac…
We consider the supergravity back-reaction of M2 branes wrapping around the space-time cycles in 1/8-BPS microstate geometries. We show that such brane wrappings will generically break all the supersymmetries. In particular, all the supersymmetries will be broken if there are such wrapped branes but the net charge of the wrapped branes is zero. We show that if M2 branes wrap a single cycle, or if they wrap a several of co-linear cycles with the same orientation, then the solution will be 1/16-BPS, having two supersymmetries. We comment on how these results relate to using W-branes to understan…
We describe and study families of BPS microstate geometries, namely, smooth, horizonless asymptotically-flat solutions to supergravity. We examine these solutions from the perspective of earlier attempts to find solitonic solutions in gravity and show how the microstate geometries circumvent the earlier "No-Go" theorems. In particular, we re-analyse the Smarr formula and show how it must be modified in the presence of non-trivial second homology. This, combined with the supergravity Chern-Simons terms, allows the existence of rich classes of BPS, globally hyperbolic, asymptotically flat, micro…
We construct a three-parameter family of non-extremal microstate geometries, or "microstrata," that are dual to states and deformations of the D1-D5 CFT. These families are non-extremal analogues of superstrata. We find these microstrata by using a Q-ball-inspired Ansatz that reduces the equations of motion to solving for eleven functions of one variable. We then solve this system both perturbatively and numerically and the results match extremely well. We find that the solutions have normal mode frequencies that depend upon the amplitudes of the excitations. We also show that, at higher order…
Solutions of type IIB supergravity which preserve half of the supersymmetries have a dual description in terms of free fermions, as elucidated by the "bubbling AdS" construction of Lin, Lunin and Maldacena. In this paper we study the half-BPS geometry associated with a gas of free fermions in thermodynamic equilibrium obeying the Fermi-Dirac distribution. We consider both regimes of low and high temperature. In the former case, we present a detailed computation of the ADM mass of the supergravity solution and find agreement with the thermal energy of the fermions. The solution has a naked null…
We study curves of marginal stability (CMS) in five-dimensional N=1 E_N theories compactified on a circle using the D3-brane probe realization. In this realization, BPS states correspond to string webs in the affine E_N 7-brane background and junction positions of the webs deteremine CMS. We find that there exist string webs involving infinitely many junctions. Consequently the E_N theories have infinitely many CMS. We also find that there exists a transition from open strings to string webs involving loops. The transition describes a new phenomenon occurring on CMS.
Topological transitions in bubbling half-BPS Type IIB geometries with SO(4) x SO(4) symmetry can be decomposed into a sequence of n elementary transitions. The half-BPS solution that describes the elementary transition is seeded by a phase space distribution of fermions filling two diagonal quadrants. We study the geometry of this solution in some detail. We show that this solution can be interpreted as a time dependent geometry, interpolating between two asymptotic pp-waves in the far past and the far future. The singular solution at the transition can be resolved in two different ways, relat…
The fivebrane worldvolume theory in eleven dimensions is known to contain BPS threebrane solitons which can also be interpreted as a fivebrane whose worldvolume is wrapped around a Riemann surface. By considering configurations of intersecting fivebranes and hence intersecting threebrane solitons, we determine the Bogomol'nyi equations for more general BPS configurations. We obtain differential equations, generalising Cauchy-Riemann equations, which imply that the worldvolume of the fivebrane is wrapped around a calibrated geometry.
We construct large classes of non-BPS smooth horizonless geometries that are asymptotic to AdS$_3\times$S$^3\times$T$^4$ in type IIB supergravity. These geometries are supported by electromagnetic flux corresponding to D1-D5 charges. We show that Einstein equations for systems with eight commuting Killing vectors decompose into a set of Ernst equations, thereby admitting an integrable structure. This feature, which can a priori be applied to other AdS$_D\times\mathcal{C}$ settings in supergravity, allows us to use solution-generating techniques associated with the Ernst formalism. We explicitl…
We review recent results of emergent geometries in the BMN matrix model, a one-dimensional gauge theory considered as a non-perturbative formulation of M-theory on the plane-wave geometry. A key to understand the emergent geometries is the eigenvalue distribution of a BPS operator. Gauge-theory calculation shows that the BPS operator reproduces the corresponding supergravity solutions in the gauge/gravity duality and also brane geometries in the M-brane picture. At finite temperatures, these geometries should be realised in a non-trivial way. Monte Carlo simulations of this gauge theory reveal…
These are notes for some introductory lectures about microstate geometries and their construction. The first lecture considers BPS black holes in four dimensions as a way to introduce what one should expect from the BPS equations. The second lecture discusses the "no solitons without topology" theorem. The subsequent lectures move on to the construction and properties of bubbled microstate geometries in five dimensions. Since these are graduate lectures, they involve a strongly computational narrative intended to build some proficiency and understanding of supersymmetric solitons in five-dimen…
We discuss collective coordinate quantization of the half-BPS geometries of Lin, Lunin and Maldacena (hep-th/0409174). The LLM geometries are parameterized by a single function $u$ on a plane. We treat this function as a collective coordinate. We arrive at the collective coordinate action as well as path integral measure by considering D3 branes in an arbitrary LLM geometry. The resulting functional integral is shown, using known methods (hep-th/9309028), to be the classical limit of a functional integral for free fermions in a harmonic oscillator. The function $u$ gets identified with the cla…
Ambi-polar metrics, defined so as to allow the signature to change from +4 to -4 across hypersurfaces, are a mainstay in the construction of BPS microstate geometries. This paper elucidates the cohomology of these spaces so as to simplify greatly the construction of infinite families of fluctuating harmonic magnetic fluxes. It is argued that such fluxes should come from scalar, harmonic pre-potentials whose source loci are holomorphic divisors. This insight is obtained by exploring the Kahler structure of ambi-polar Gibbons-Hawking spaces and it is shown that differentiating the pre-potentials…
We study in detail the moduli space of solutions discovered in LLM relaxing the constraint that guarantees the absence of singularities. The solutions fall into three classes, non-singular, null-singular and time machines with a time-like naked singularity. We study the general features of these metrics and prove that there are actually just two generic classes of space-times - those with null singularities are in the same class as the non-singular metrics. AdS/CFT seems to provide a dual description only for the first of these two types of space-time in terms of a unitary CFT indicating the p…
Supersymmetric U(Nc) gauge theory with Nf massive hypermultiplets in the fundamental representation admits various BPS solitons like domain walls and their webs. In the first part we show as a review of the previous paper hep-th/0412024 that domain walls are realized as kinky fractional D3-branes interpolating between separated D7-branes. In the second part we discuss brane configurations for domain wall webs. This is a contribution to the conference based on the talk given by MN.
We construct the first known example of a near horizon supergravity solution for a pair of NS5-branes, intersecting on R^{1,3} and localised in all directions except a single transverse circle. We do this by establishing an explicit map between the conifold metric and the near horizon geometry of two intersecting NS5-branes, clarifying and correcting a number of open issues in the literature en route. Our technique is general in nature and may be applied to a whole class of 1/4-BPS five-brane webs and their geometric duals. These 1/4-BPS solutions may have an interesting holographic interpreta…
We study the Coulomb branches of the rank-one 4d $\mathcal{N} = 2$ quantum field theories, including the KK theories obtained from the circle compactification of the 5d $\mathcal{N}= 1$ $E_n$ Seiberg theories. The focus is set on the relation between their Seiberg-Witten geometries and rational elliptic surfaces, with more attention being given to the modular surfaces, which we completely classify using the classification of subgroups of the modular group ${\rm PSL}(2,\mathbb{Z})$. We derive closed-form expressions for the modular functions for all congruence and some of the non-congruence sub…
We construct the first smooth horizonless supergravity solutions that have two topologically-nontrivial three-cycles supported by flux, and that have the same mass and charges as a non-extremal D1-D5-P black hole. Our configurations are solutions to six-dimensional ungauged supergravity coupled to a tensor multiplet, and uplift to solutions of Type IIB supergravity. The solutions represent multi-center generalizations of the non-BPS solutions of Jejjala, Madden, Ross, and Titchener, which have over-rotating angular momenta. By adding an additional Gibbons-Hawking center, we succeed in lowering…
We study supersymmetric branes in AdS_7 x S^4 and AdS_4 x S^7. We show that in the former case the membranes should be viewed as M5 branes with fluxes and we identify two types of such fivebranes (they are analogous to giant gravitons and to dual giants). In AdS_4 x S^7 we find both M5 branes with fluxes and freestanding stacks of membranes. We also go beyond probe approximation and construct regular supergravity solutions describing geometries produced by the branes. The metrics are completely specified by one function which satisfies either Laplace or Toda equation and we give a complete cla…
Two charge BPS horizon free supergravity geometries are important in proposals for understanding black hole microstates. In this paper we construct a new class of geometries in the NS1-P system, corresponding to solitonic strings carrying fermionic as well as bosonic condensates. Such geometries are required to account for the full microscopic entropy of the NS1-P system. We then briefly discuss the properties of the corresponding geometries in the dual D1-D5 system.
We study the half-BPS sector of the AdS/CFT correspondence. In the full sector of the correspondence, on-shell actions in type IIB supergravity have been played crucial roles, for example in the GKP-Witten relation. We derive, therefore, an on-shell action that reproduces a class of half-BPS solutions found by Lin, Lunin and Maldacena. These solutions describe the bubbling geometries in the supergravity. We show that this on-shell action for the LLM solutions is dual to the classical limit of an observable of the N free fermions in a harmonic oscillator potential in 1+1 dimensions. The observa…
We discuss the BPS configurations of IIB strings in the 7-brane background from M-theory viewpoint. We first obtain the hyperkahler geometry background of M-theory expected from the 7-brane solutions of the type IIB supergravity. We choose the appropriate complex structures of the background geometries and embed a membrane of M-theory holomorphically to obtain a BPS string configuration. The recently discussed BPS string configurations such as 3-string junctions and string networks in the flat background are generalized to the cases with the 7-brane backgrounds. The property of the BPS string …
In this paper we classify the ten dimensional half BPS solutions of the type IIB supergravity which have SO(4) X SO(4) X U(1) isometry found by Lin-Lunin-Maldacena (LLM). Our classification is based on their asymptotic behavior and causal structure according which they fall into two classes: 1) those with R X S^3 boundary and 2) those with one dimensional light-like boundary. Each class can be divided into some subclasses depending on the asymptotic characteristics of the solutions, which in part specify the global charges defining the geometry. We analyze each of these classes in some detail …
The string theory description of BPS states in D-brane world-volume field theories may undergo transitions from open strings to string webs, as well as between different string webs, as one moves in the field theory moduli space. These transitions are driven by the string creation phenomenon. We demonstrate such transitions in the D3-brane realization of N=2 SU(2) Super-Yang-Mills theory.
Motivated by a recent observation that the LLM geometries admit 1/4-BPS M5-brane probes with worldvolume AdS3 X Σ_2 X S1 preserving the R-symmetry, we initiate a classification of the most general AdS3 X S2 geometries in M-theory. We retain all field strengths consistent with symmetry and derive the torsion conditions for M_6 in terms of two linearly independent spinors. Surprisingly, we identify three Killing directions for M_6, but only two of these generate isometries of the overall ansatz. We show that the existence of this third direction depends on the norm of the spinors. Then, speciali…
We review the issue of chronology protection and show how string theory can solve it in the half BPS sector of AdS/CFT. According to the LLM prescription, half BPS excitations of AdS_5 x S^5 geometries in type IIB string theory can be mapped into free fermion configurations. We show that unitarity of the theory describing these fermions is intimately related to the protection of the chronology in the dual geometries.
We perform a systematic study, in eleven dimensional supergravity, of the geometry of wrapped brane configurations admitting $AdS_2$ limits. Membranes wrapping holomorphic curves in Calabi-Yau manifolds are found to exhibit some novel features; in particular, for fourfolds or threefolds, the gravitational effect of the branes on the overall transverse space is only weakly restricted by the kinematics of the Killing spinor equation. We also study the $AdS_2$ limits of the wrapped brane supergravity descriptions. For membranes wrapped in a two-fold, we derive a set of $AdS_2$ supersymmetry condi…
For a given attractor black hole with generic D6-D4-D2-D0-charges in four dimensions we identify the set of supersymmetric branes, static or stationary in global coordinates, of the corresponding eleven-dimensional near horizon geometry. The set of these BPS states, which includes branes which partially or fully wrap the horizon, should play a role in understanding the partition function of black holes with D6-charges.
We review recent results in the study of attractor horizon geometries (with non-vanishing Bekenstein-Hawking entropy) of dyonic extremal d=4 black holes in supergravity. We focus on N=2, d=4 ungauged supergravity coupled to a number n_{V} of Abelian vector multiplets, outlining the fundamentals of the special Kaehler geometry of the vector multiplets' scalar manifold (of complex dimension n_{V}), and studying the 1/2-BPS attractors, as well as the non-BPS (non-supersymmetric) ones with non-vanishing central charge. For symmetric special Kaehler geometries, we present the complete classificatio…
