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Register a new userBPS Black Hole Entropy and Attractors in Very Special Geometry. Cubic Forms, Gradient Maps and their Inversion
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We consider Bekenstein-Hawking entropy and attractors in extremal BPS black holes of $mathcal{N}=2$, $D=4$ ungauged supergravity obtained as reduction of minimal, matter-coupled $D=5$ supergravity. They are generally expressed in terms of solutions to an inhomogeneous system of coupled quadratic equations, named BPS system, depending on the cubic prepotential as well as on the electric-magnetic fluxes in the extremal black hole background. Focussing on homogeneous non-symmetric scalar manifolds (whose classification is known in terms of $L(q,P,dot{P})$ models), under certain assumptions on the Clifford matrices pertaining to the related cubic prepotential, we formulate and prove an invertibility condition for the gradient map of the corresponding cubic form (to have a birational inverse map which is an homogeneous polynomial of degree four), and therefore for the solutions to the BPS system to be explicitly determined, in turn providing novel, explicit expressions for the BPS black hole entropy and the related attractors as solution of the BPS attractor equations. After a general treatment, we present a number of explicit examples with $dot{P}=0$, such as $L(q,P)$, $1leqslant qleqslant 3$ and $Pgeqslant 1$,or $L(q,1)$, $4leqslant qleqslant 9$, and one model with $dot{P}=1$, namely $L(4,1,1)$. We also briefly comment on Kleinian signatures and split algebras. In particular, we provide, for the first time, the explicit form of the BPS black hole entropy and of the related BPS attractors for the infinite class of $L(1,P)$ $Pgeqslant 2$ non-symmetric models of $mathcal{N}=2$, $D=4$ supergravity.
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BPS black hole degeneracies can be expressed in terms of an inverse Laplace transform of a partition function based on a mixed electric/magnetic ensemble, which involves a non-trivial integration measure. This measure has been evaluated for black holes with various degrees of supersymmetry and for N=4 supersymmetric black holes all results agree. It generally receives contributions from non-holomorphic corrections. An explicit evaluation of these corrections in the context of the effective action of the FHSV model reveals that these are related to, but quantitatively different from, the non-holomorphic corrections to the topological string, indicating that the relation between the twisted partition functions of the latter and the effective action is more subtle than has so far been envisaged. The effective action result leads to a duality invariant BPS free energy and arguments are presented for the existence of consistent non-holomorphic deformations of special geometry that can account for these effects. A prediction is given for the measure based on semiclassical arguments for a class of N=2 black holes. Furthermore an attempt is made to confront some of the results of this paper to a recent proposal for the microstate degeneracies of the STU model.
We study the attractor equations for a quantum corrected prepotential F=t^3+ilambda, with lambda in R,which is the only correction which preserves the axion shift symmetry and modifies the geometry. By performing computations in the ``magnetic charge configuration, we find evidence for interesting phenomena (absent in the classical limit of vanishing lambda). For a certain range of the quantum parameter lambda we find a ``separation of attractors, i.e. the existence of multiple solutions to the Attractor Equations for fixed supporting charge configuration. Furthermore, we find that, away from the classical limit, a ``transmutation of the supersymmetry-preserving features of the attractors takes place when lambda reaches a particular critical value.
These lectures provide a pedagogical, introductory review of the so-called Attractor Mechanism (AM) at work in two different 4-dimensional frameworks: extremal black holes in N=2 supergravity and N=1 flux compactifications. In the first case, AM determines the stabilization of scalars at the black hole event horizon purely in terms of the electric and magnetic charges, whereas in the second context the AM is responsible for the stabilization of the universal axion-dilaton and of the (complex structure) moduli purely in terms of the RR and NSNS fluxes. Two equivalent approaches to AM, namely the so-called ``criticality conditions and ``New Attractor ones, are analyzed in detail in both frameworks, whose analogies and differences are discussed. Also a stringy analysis of both frameworks (relying on Hodge-decomposition techniques) is performed, respectively considering Type IIB compactified on $CY_{3}$ and its orientifolded version, associated with $frac{CY_{3}times T^{2}}{mathbb{Z}_{2}}$. Finally, recent results on the U-duality orbits and moduli spaces of non-BPS extremal black hole attractors in $3leqslant Nleqslant 8$, d=4 supergravities are reported.
By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory compactified on the same space. Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor $cal D$, at the large volume attractor point. For $cal D$ irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on $cal D$ and is therefore known to be modular. Instead, when $cal D$ is the sum of $n$ irreducible divisors ${cal D}_i$, we show that the generating function acquires a modular anomaly. We characterize this anomaly for arbitrary $n$ by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions. As a result, the generating function turns out to be a (mixed) mock modular form of depth $n-1$.
We give a brief overview of black hole entropy, covering a few main developments since Bekensteins original proposal
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