BPS Black Holes Research Articles

Do all BPS black hole microstates carry zero angular momentum?

From the analysis of the near horizon geometry and supersymmetry algebra it has been argued that all the microstates of single centered BPS black holes with four unbroken supersymmetries carry zero angular momentum in the region of the moduli space where the black hole description is valid. A stronger form of the conjecture would be that the result holds for any sufficiently generic point in the moduli space. In this paper we set out to test this conjecture for a class of black hole microstates in type II string theory on $T^6$, represented by four stacks of D-branes wrapped on various cycles of $T^6$. For this system the above conjecture translates to the statement that the moduli space of classical vacua must be a collection of points. Explicit analysis of systems carrying a low number of D-branes supports this conjecture.

F-Theory, spinning black holes and multi-string branches

We study 5d supersymmetric black holes which descend from strings of generic $\mathcal{N}=(1,0)$ supergravity in 6d. These strings have an F-theory realization in 6d as D3 branes wrapping smooth genus $g$ curves in the base of elliptic 3-folds. They enjoy $(0,4)$ worldsheet supersymmetry with an extra $SU(2)_L$ current algebra at level $g$ realized on the left-movers. When the smooth curves degenerate they lead to multi-string branches and we find that the microscopic worldsheet theory flows in the IR to disconnected 2d CFTs having different central charges. The single string sector is the one with maximal central charge, which when wrapped on a circle, leads to a 5d spinning BPS black hole whose horizon volume agrees with the leading entropy prediction from the Cardy formula. However, we find new phenomena where this branch meets other branches of the CFT. These include multi-string configurations which have no bound states in 6 dimensions but are bound through KK momenta when wrapping a circle, as well as loci where the curves degenerate to spheres. These loci lead to black hole configurations which can have total angular momentum relative to a Taub-Nut center satisfying $J^2 > M^3$ and whose number of states, though exponentially large, grows much slower than those of the large spinning black hole.

Functional determinants, index theorems, and exact quantum black hole entropy

The exact quantum entropy of BPS black holes can be evaluated using localization in supergravity. An important ingredient in this program, that has been lacking so far, is the one-loop effect arising from the quadratic fluctuations of the exact deformation (the $Q\mathcal{V}$ operator). We compute the fluctuation determinant for vector multiplets and hyper multiplets around $Q$-invariant off-shell configurations in four-dimensional $\mathcal{N}=2$ supergravity with $AdS_{2} \times S^{2}$ boundary conditions, using the Atiyah-Bott fixed-point index theorem and a subsequent zeta function regularization. Our results extend the large-charge on-shell entropy computations in the literature to a regime of finite charges. Based on our results, we present an exact formula for the quantum entropy of BPS black holes in $\mathcal{N}=2$ supergravity. We explain cancellations concerning $\frac18$-BPS black holes in $\mathcal{N}=8$ supergravity that were observed in arXiv:1111.1161. We also make comments about the interpretation of a logarithmic term in the topological string partition function in the low energy supergravity theory.

Hair-brane ideas on the horizon

We continue an examination of the microstate geometries program begun in arXiv:1409.6017, focussing on the role of branes that wrap the cycles which degenerate when a throat in the geometry deepens and a horizon forms. An associated quiver quantum mechanical model of minimally wrapped branes exhibits a non-negligible fraction of the gravitational entropy, which scales correctly as a function of the charges. The results suggest a picture of AdS_3/CFT_2 duality wherein the long string that accounts for BTZ black hole entropy in the CFT description, can also be seen to inhabit the horizon of BPS black holes on the gravity side.

Supersymmetric localization for BPS black hole entropy: 1-loop partition function from vector multiplets

We use the techniques of supersymmetric localization to compute the BPS black hole entropy in N=2 supergravity. We focus on the n_v+1 vector multiplets on the black hole near horizon background which is AdS_2 x S^2 space. We find the localizing saddle point of the vector multiplets by solving the localization equations, and compute the exact one loop partition function on the saddle point. Furthermore, we propose the appropriate functional integration measure. Through this measure, the one loop determinant is written in terms of the radius of the physical metric, which depends on the localizing saddle point value of the vector multiplets. The result for the one loop determinant is consistent with the logarithmic corrections to the BPS black hole entropy from vector multiplets.

BPS black holes in a non-homogeneous deformation of the stu model of N = 2, D = 4 gauged supergravity

We consider a deformation of the well-known stu model of N = 2, D = 4 supergravity, characterized by a non-homogeneous special Kahler manifold, and by the smallest electric-magnetic duality Lie algebra consistent with its upliftability to five dimensions. We explicitly solve the BPS attractor equations and construct static supersymmetric black holes with radial symmetry, in the context of U(1) dyonic Fayet-Iliopoulos gauging, focussing on axion-free solutions. Due to non-homogeneity of the scalar manifold, the model evades the analysis recently given in the literature. The relevant physical properties of the resulting black hole solution are discussed.

Conformal quivers and melting molecules

Quiver quantum mechanics describes the low energy dynamics of a system of wrapped D-branes. It captures several aspects of single and multicentered BPS black hole geometries in four-dimensional $$ \mathcal{N} $$ = 2 supergravity such as the presence of bound states and an exponential growth of microstates. The Coulomb branch of an Abelian three node quiver is obtained by integrating out the massive strings connecting the D-particles. It allows for a scaling regime corresponding to a deep AdS2 throat on the gravity side. In this scaling regime, the Coulomb branch is shown to be an SL(2, ℝ) invariant multi-particle superconformal quantum mechanics. Finally, we integrate out the strings at finite temperature — rather than in their ground state — and show how the Coulomb branch ‘melts’ into the Higgs branch at high enough temperatures. For scaling solutions the melting occurs for arbitrarily small temperatures, whereas bound states can be metastable and thus long lived. Throughout the paper, we discuss how far the analogy between the quiver model and the gravity picture, particularly within the AdS2 throat, can be taken.

Static BPS black holes in AdS4 with general dyonic charges

We complete the study of static BPS, asymptotically AdS$_4$ black holes within N=2 FI-gauged supergravity and where the scalar manifold is a homogeneous very special Kahler manifold. We find the analytic form for the general solution to the BPS equations, the horizon appears as a double root of a particular quartic polynomial whereas in previous work this quartic polynomial further factored into a pair of double roots. A new and distinguishing feature of our solutions is that the phase of the supersymmetry parameter varies throughout the black hole. The general solution has $2n_v$ independent parameters; there are two algebraic constraints on $2n_v+2$ charges, matching our previous analysis on BPS solutions of the form $AdS_2\times \Sigma_g$. As a consequence we have proved that every BPS geometry of this form can arise as the horizon geometry of a BPS AdS$_4$ black hole. When specialized to the STU-model our solutions uplift to M-theory and describe a stack of M2-branes wrapped on a Riemman surface in a Calabi-Yau fivefold with internal angular momentum.

The holographic supersymmetric Rényi entropy in five dimensions

We compute the supersymmetric Renyi entropy across an entangling three-sphere for five-dimensional superconformal field theories using localization. For a class of USp(2N) gauge theories we construct a holographic dual 1/2 BPS black hole solution of Euclidean Romans F(4) supergravity. The large N limit of the gauge theory results agree perfectly with the supergravity computations.

Indefinite theta functions for counting attractor backgrounds

In this note, we employ indefinite theta functions to regularize canonical par- tition functions for single-center dyonic BPS black holes. These partition functions count dyonic degeneracies in the Hilbert space of four-dimensional toroidally compactified het- erotic string theory, graded by electric and magnetic charges. The regularization is achieved by viewing the weighted sums of degeneracies as sums over charge excitations in the near- horizon attractor geometry of an arbitrarily chosen black hole background, and eliminating the unstable modes. This enables us to rewrite these sums in terms of indefinite theta func- tions. Background independence is then implemented by using the transformation property of indefinite theta functions under elliptic transformations, while modular transformations are used to make contact with semi-classical results in supergravity.

Static BPS black holes in U(1) gauged supergravity

We consider the flow equations for 1/4-BPS asymptotically AdS4 static black holes in Fayet-Iliopoulos gauged supergravity, using very special geometry identities to obtain a simplified form in the most general case. Under mild assumptions on the form of the solution, we analyse the flow equations and find an explicit solution for arbitrary gauging and charge vectors. Comparing with the corresponding attractor equations, we find that the solution is given in terms of exactly the same vector of parameters, implying that all regular attractors can be extended to full black hole solutions. We present explicit examples of black hole solutions with all complex scalars and allowed charges turned on, within the STU model and its truncations.

Microscopic and macroscopic entropy of extremal black holes in string theory

This is a short review summarizing the current status of the comparison between microscopic and macroscopic entropy of extremal BPS black holes in string theory.

Heat kernels on the AdS2 cone and logarithmic corrections to extremal black hole entropy

We develop new techniques to efficiently evaluate heat kernel coefficients for the Laplacian in the short-time expansion on spheres and hyperboloids with conical singularities. We then apply these techniques to explicitly compute the logarithmic contribution to black hole entropy from an N=4 vector multiplet about a Z(N) orbifold of the near-horizon geometry of quarter--BPS black holes in N=4 supergravity. We find that this vanishes, matching perfectly with the prediction from the microstate counting. We also discuss possible generalisations of our heat kernel results to higher-spin fields over Z(N) orbifolds of higher-dimensional spheres and hyperboloids.

BPS black hole horizons in N=2 gauged supergravity

We study static BPS black hole horizons in four dimensional N=2 gauged supergravity coupled to $n_v$-vector multiplets and with an arbitrary cubic prepotential. We work in a symplectically covariant formalism which allows for both electric and magnetic gauging parameters as well as dyonic background charges and obtain the general solution to the BPS equations for horizons of the form $AdS_2\times \Sigma_g$. In particular this means we solve for the scalar fields as well as the metric of these black holes as a function of the gauging parameters and background charges. When the special Kahler manifold is a symmetric space, our solution is completely explicit and the entropy is related to the familiar quartic invariant. For more general models our solution is implicit up to a set of holomorphic quadratic equations. For particular models which have known embeddings in M-theory, we derive new horizon geometries with dyonic charges and numerically construct black hole solutions. These correspond to M2-branes wrapped on a Riemann surface in a local Calabi-Yau five-fold with internal spin.

New higher-derivative invariants in N = 2 supergravity and the Gauss-Bonnet term

A new class of N = 2 locally supersymmetric higher-derivative invariants is constructed based on logarithms of conformal primary chiral superfields. They characteristically involve a coupling to $ {{\mathcal{R}}_{uv}}^2-\frac{1}{3}{{\mathcal{R}}^2} $ , which equals the non-conformal part of the Gauss-Bonnet term. Upon combining one such invariant with the known supersymmetric version of the square of the Weyl tensor, one obtains the supersymmetric extension of the Gauss-Bonnet term. The construction is carried out in the context of both conformal superspace and the superconformal multiplet calculus. The new class of supersymmetric invariants resolves two open questions. The first concerns the proper identification of the 4D supersymmetric invariants that arise from dimensional reduction of the 5D mixed gauge-gravitational Chern-Simons term. The second is why the pure Gauss-Bonnet term without supersymmetric completion has reproduced the correct result in calculations of the BPS black hole entropy in certain models.

Quantum black hole entropy and the holomorphic prepotential of $ \mathcal{N}=2 $ supergravity

Abstract Supersymmetric terms in the effective action of $ \mathcal{N}=2 $ supergravity in four dimensions are generically classified into chiral-superspace integrals and full-superspace integrals. For a theory of $ \mathcal{N}=2 $ vector multiplets coupled to supergravity, a special class of couplings is given by chiral-superspace integrals that are governed by a holomorphic prepotential function. The quantum entropy of BPS black holes in such theories depends on the prepotential according to a known integral formula. We show, using techniques of localization, that a large class of full-superspace integrals in the effective action of $ \mathcal{N}=2 $ supergravity do not contribute to the quantum entropy of BPS black holes at any level in the derivative expansion. Our work extends similar results for semi-classical supersymmetric black hole entropy, and goes towards providing an explanation of why the prepotential terms capture the exact microscopic quantum black hole entropy.

BPS black holes in AdS 4 from M-theory

Abstract We study supersymmetric black holes in AdS 4 in the framework of four dimensional gauged $ \mathcal{N}=2 $ supergravity coupled to hypermultiplets. We derive the flow equations for a general electrically gauged theory where the gauge group is Abelian and, restricting them to the fixed points, we derive the gauged supergravity analogue of the attractor equations for theories coupled to hypermultiplets. The particular models we analyze are consistent truncations of M-theory on certain Sasaki-Einstein seven-manifolds. We study the space of horizon solutions of the form AdS 2 × Σ g with both electric and magnetic charges and find a four-dimensional solution space when the theory arises from a reduction on Q 111. For other SE 7 reductions, the solutions space is a subspace of this. We construct explicit examples of spherically symmetric black holes numerically.

Extremal limits of rotating black holes

We consider non-extremal, stationary, axion-dilaton solutions to ungauged symmetric supergravity models, obtained by Harrison transformations of the non-extremal Kerr solution. We define a general algebraic procedure, which can be viewed as an Inonu-Wigner contraction of the Noether charge matrix associated with the effective D=3 sigma-model description of the solution, yielding, through different singular limits, the known BPS and non-BPS extremal black holes (which include the under-rotating non-BPS one). The non-extremal black hole can thus be thought of as "interpolating" among these limit-solutions. The algebraic procedure that we define generalizes the known Rasheed-Larsen limit which yielded, in the Kaluza-Klein theory, the first instance of under-rotating extremal solution. As an example of our general result, we discuss in detail the non-extremal solution in the T^3-model, with either (q_0, p^1) or (p^0, q_1) charges switched on, and its singular limits. Such solutions, computed in D=3 through the solution-generating technique, is completely described in terms of D=4 fields, which include the fully integrated vector fields.

Supersymmetry on three-dimensional Lorentzian curved spaces and black hole holography

We study N <= 2 superconformal and supersymmetric theories on Lorentzian threemanifolds with a view toward holographic applications, in particular to BPS black hole solutions. As in the Euclidean case, preserved supersymmetry for asymptotically locally AdS solutions implies the existence of a (charged) "conformal Killing spinor" on the boundary. We find that such spinors exist whenever there is a conformal Killing vector which is null or timelike. We match these results with expectations from supersymmetric four-dimensional asymptotically AdS black holes. In particular, BPS bulk solutions in global AdS are known to fall in two classes, depending on their graviphoton magnetic charge, and we reproduce this dichotomy from the boundary perspective. We finish by sketching a proposal to find the dual superconformal quantum mechanics on the horizon of the magnetic black holes.

Ungauging black holes and hidden supercharges

Abstract We embed the general solution for non-BPS extremal asymptotically flat static and under-rotating black holes in abelian gauged D = 4 $ \mathcal{N}=2 $ supergravity, in the limit where the scalar potential vanishes but the gauging does not. Using this result, we show explicitly that some supersymmetries are preserved in the near horizon region of all the asymptotically flat solutions above, in the gauged theory. This reveals a deep relation between microscopic entropy counting of extremal black holes in Minkowski and BPS black holes in AdS. Finally, we discuss the relevance of this construction to the structure of asymptotically AdS4 black holes, as well as the possibility of including hypermultiplets.