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Twistors and Black Holes
High Energy Physics - Theory · Submitted 23 Jan 2007
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The results bear a simple relation to the Hesse potential \Sigma of the special Kahler manifold M_s, and hence to the Bekenstein-Hawking entropy for BPS black holes.
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The abstract, sentence by sentence
Motivated by black hole physics in N=2, D=4 supergravity, we study the geometry of quaternionic-Kahler manifolds M obtained by the c-map construction from projective special Kahler manifolds M_s. Improving on earlier treatments, we compute the Kahler potentials on the twistor space Z and Swann space S in the complex coordinates adapted to the Heisenberg symmetries. The results bear a simple relation to the Hesse potential \Sigma of the special Kahler manifold M_s, and hence to the Bekenstein-Hawking entropy for BPS black holes. We explicitly construct the ``covariant c-map'' and the ``twistor map'', which relate real coordinates on M x CP^1 (resp. M x R^4/Z_2) to complex coordinates on Z (resp. S). As applications, we solve for the general BPS geodesic motion on M, and provide explicit integral formulae for the quaternionic Penrose transform relating elements of H^1(Z,O(-k)) to massless fields on M annihilated by first or second order differential operators. Finally, we compute the exact radial wave function (in the supergravity approximation) for BPS black holes with fixed electric and magnetic charges.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Motivated by black hole physics in N=2, D=4 supergravity, we study the geometry of quaternionic-Kahler manifolds M obtained by the c-map construction from projective special Kahler manifolds M_s. | contribution 0.67 | 0.69 | 0.06 | asserted 0.91 |
| s1 | Improving on earlier treatments, we compute the Kahler potentials on the twistor space Z and Swann space S in the complex coordinates adapted to the Heisenberg symmetries. | contribution 0.67 | 0.95 | 0.08 | asserted 1.00 |
| s2 | The results bear a simple relation to the Hesse potential \Sigma of the special Kahler manifold M_s, and hence to the Bekenstein-Hawking entropy for BPS black holes. | interpretation 0.75 | 0.81 | 0.28 | asserted 1.00 |
| s3 | We explicitly construct the ``covariant c-map'' and the ``twistor map'', which relate real coordinates on M x CP^1 (resp. M x R^4/Z_2) to complex coordinates on Z (resp. S). | contribution 0.77 | 0.95 | 0.09 | asserted 1.00 |
| s4 | As applications, we solve for the general BPS geodesic motion on M, and provide explicit integral formulae for the quaternionic Penrose transform relating elements of H^1(Z,O(-k)) to massless fields on M annihilated by first or second order differential operators. | contribution 0.55 | 0.96 | 0.15 | asserted 1.00 |
| s5 | Finally, we compute the exact radial wave function (in the supergravity approximation) for BPS black holes with fixed electric and magnetic charges. | result 0.86 | 0.96 | 0.20 | asserted 1.00 |
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The paper
Original abstract
Motivated by black hole physics in N=2, D=4 supergravity, we study the geometry of quaternionic-Kahler manifolds M obtained by the c-map construction from projective special Kahler manifolds M_s. Improving on earlier treatments, we compute the Kahler potentials on the twistor space Z and Swann space S in the complex coordinates adapted to the Heisenberg symmetries. The results bear a simple relation to the Hesse potential \Sigma of the special Kahler manifold M_s, and hence to the Bekenstein-Hawking entropy for BPS black holes. We explicitly construct the ``covariant c-map'' and the ``twistor map'', which relate real coordinates on M x CP^1 (resp. M x R^4/Z_2) to complex coordinates on Z (resp. S). As applications, we solve for the general BPS geodesic motion on M, and provide explicit integral formulae for the quaternionic Penrose transform relating elements of H^1(Z,O(-k)) to massless fields on M annihilated by first or second order differential operators. Finally, we compute the exact radial wave function (in the supergravity approximation) for BPS black holes with fixed electric and magnetic charges.
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References
How the literature uses it
37 arXiv papers cite this in 121 sentences
Papers citing it · 95 sentences without a confident label
Builds on it 3 papers
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The purpose of this note is to extend the techniques of [ref] to the case of $=4$ supergravity with $n_v$ vector multiplets in 3+1 dimensions.
N=4 BPS black holes and octonionic twistors · Introduction · 0.98
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These results obtained first for quaternionic symmetric spaces [ref] extend to general quaternionic manifolds that are in the C-map [ref] in complete parallel to the situation in going from five to four dimensions.
Spectrum Generating Conformal and Quasiconformal U-Duality Groups, Supergravity and Spherical Vectors · Quasiconformal Group Actions and Quartic Light-cones · 0.93
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Then for any $γ∈$ we write the locally defined function[This formula was first obtained in joint work with Boris Pioline, and is essentially the rigid limit of a formula in [ref] for the quaternionic-Kähler case.
Four-dimensional wall-crossing via three-dimensional field theory · Twistorial construction of the semiflat geometry · 0.83
Uses its method or tool 8 papers
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A simple calculation using [ref] then shows that the Darboux coordinates in the patch $_0$ are given by [ref]
Modularity, Quaternion-Kahler spaces and Mirror Symmetry · The metric and twistorial construction · 0.83
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One introduces Darboux coordinates [ref]
Heterotic-type II duality in twistor space · Type II Darboux coordinates · 0.88
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Using [ref], we find that the Darboux coordinates in the patch $_0$ are given by [ref]
S-duality in Twistor Space · Twistorial description of $_H$ · 0.93
Compares with it 5 papers
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In parallel with the treatment of BPS black holes in $D=4, =2$ supergravity [ref] , we have shown that such configurations correspond to supersymmetric geodesics on the three-dimensional symmetric moduli space $_3$.
N=4 BPS black holes and octonionic twistors · Discussion · 0.83
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Then using the result of [ref] we are able to establish agreement with the expression for $χ$ given in [ref] for hyperkähler cones which generically arise via the c-map.
E7(7) formulation of N=2 backgrounds · Introduction · 0.97
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Such coordinates in fact exist for all $c$-map spaces [ref] , and [ref] agrees with the “twistor map” derived in that context in [ref] .
Quasi-conformal actions, quaternionic discrete series and twistors: SU(2,1) and G_2(2) · Twistor space · 0.89
Cites it as support 1 paper
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This reduced phase of BPS black holes can be identified with the twistor space of $ℳ_3$ of complex dimension $(2n_V+3)$ [ref] .
Lectures on Spectrum Generating Symmetries and U-duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace · $3D$ U-duality Groups as Spectrum Generating Quasiconformal Groups of $4D$ Supergravity Theories and Quantum Attractor F · 0.98
Cites it as background 5 papers
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The twistorial description of the HM moduli space was worked out at tree-level in [ref] , the one-loop correction was included in [ref] and D-instantons were incorporated in [ref] .
Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces · Twistorial description of $$ · 0.94
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This has been understood in [ref] , based on the previous results of [ref] .
S-duality in Twistor Space · Twistorial description of $_H$ · 0.91
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These two constructions are summarized in Sections 2.1 and 2.2 following [ref] .
Linear perturbations of quaternionic metrics · Quaternionic-Kähler geometry and twistors · 0.81
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