Abstract
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, Ṗ) 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 given by homogeneous polynomials 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 Ṗ = 0, such as L(q, P), 1 ⩽ q ⩽ 3 and P ⩾ 1, or L(q, 1), 4 ⩽ q ⩽ 9, and one model with Ṗ = 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) P ⩾ 2 non-symmetric models of mathcal{N} = 2, D = 4 supergravity.
Highlights
In the last decades, the theoretical and phenomenological implications of the physics of black holes [73,74,75] had a profound and fertile impact on many branches of science, from astrophysics, cosmology, particle physics, to mathematical physics [100], quantum information theory [48,49,50], and, recently, number theory [15]
The singularity theorems proved by Penrose and Hawking [84] imply that the black holes are an unavoidable consequence of Einstein’s theory of General Relativity, as well as of its modern generalizations such as supergravity [20,21,22, 44,45,46, 120, 121], superstrings and M-theory [23,24,25,26,27,28]
The present paper is devoted to the determination of the explicit expression of two purely charge-dependent quantities, characterizing the physics of BPS extremal black holes: the Bekenstein-Hawking entropy S (p, q) and the attractor values, collectively denoted by φH (p, q), acquired by the scalar fields when approaching the event horizon
Summary
The theoretical and phenomenological implications of the physics of black holes [73,74,75] had a profound and fertile impact on many branches of science, from astrophysics, cosmology, particle physics, to mathematical physics [100], quantum information theory [48,49,50], and, recently, number theory [15]. The present paper is devoted to the determination of the explicit expression of two purely charge-dependent quantities, characterizing the physics of BPS extremal black holes: the Bekenstein-Hawking entropy S (p, q) and the attractor values, collectively denoted by φH (p, q), acquired by the scalar fields when approaching the (unique) event horizon (regardless of the boundary conditions of their evolution dynamics). Within the validity of such a condition (which we will prove to hold for an infinite, countable number of models), we will explicitly determine the expression of the Bekenstein-Hawking (semi)classical black hole entropy S (p, q) as well as of the purely charge-dependent attractor values φH (p, q) acquired by the scalar fields at the event horizon of asymptotically flat, spherically symmetric, static, dyonic, extremal BPS black holes. Some outlook and hints for further developments are provided in the concluding section 15
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