Attractors and the Holomorphic Anomaly

Erik Verlinde

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Attractors and the Holomorphic Anomaly

Erik Verlinde

High Energy Physics - Theory · Submitted 14 Dec 2004

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Our analysis results in a generalized formula for the relation between the microscopic density of black hole states and topological strings valid for all backgrounds.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.21 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Motivated by the recently proposed connection between N=2 BPS black holes and topological strings, I study the attractor equations and their interplay with the holomorphic anomaly equation. The topological string partition function is interpreted as a wave-function obtained by quantizing the real cohomology of the Calabi-Yau. In this interpretation the apparent background dependence due to the holomorphic anomaly is caused by the choice of complex polarization. The black hole attractor equations express the moduli in terms of the electric and magnetic charges, and lead to a real polarization in which the background dependence disappears. Our analysis results in a generalized formula for the relation between the microscopic density of black hole states and topological strings valid for all backgrounds.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Motivated by the recently proposed connection between N=2 BPS black holes and topological strings, I study the attractor equations and their interplay with the holomorphic anomaly equation.	contribution 0.74	0.85	0.04	asserted 0.72
s1	The topological string partition function is interpreted as a wave-function obtained by quantizing the real cohomology of the Calabi-Yau.	interpretation 0.48	0.55	0.06	asserted 1.00
s2	In this interpretation the apparent background dependence due to the holomorphic anomaly is caused by the choice of complex polarization.	interpretation 0.94	0.55	0.21	asserted 1.00
s3	The black hole attractor equations express the moduli in terms of the electric and magnetic charges, and lead to a real polarization in which the background dependence disappears.	result 0.49	0.60	0.25	asserted 1.00
s4	Our analysis results in a generalized formula for the relation between the microscopic density of black hole states and topological strings valid for all backgrounds.	result 0.98	0.95	0.78	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

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P abstract

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S sections
T body sentences
C citations
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The paper

Original abstract

Motivated by the recently proposed connection between N=2 BPS black holes and topological strings, I study the attractor equations and their interplay with the holomorphic anomaly equation. The topological string partition function is interpreted as a wave-function obtained by quantizing the real cohomology of the Calabi-Yau. In this interpretation the apparent background dependence due to the holomorphic anomaly is caused by the choice of complex polarization. The black hole attractor equations express the moduli in terms of the electric and magnetic charges, and lead to a real polarization in which the background dependence disappears. Our analysis results in a generalized formula for the relation between the microscopic density of black hole states and topological strings valid for all backgrounds.

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References

Forward citations

Cited by 9 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Split States, Entropy Enigmas, Holes and Halos
hep-th 2007-02 conditional novelty 8.0 of 10

A wall-crossing and factorization machinery for BPS indices yields a refined OSV formula with an extra measure factor and a cutoff, plus a two-centered configuration whose entropy grows as the cube of the charge and m...
Black Hole Entropy Function and the Attractor Mechanism in Higher Derivative Gravity
hep-th 2005-06 accept novelty 8.0 of 10

Extremizing a single entropy function built from the Lagrangian over the near-horizon geometry yields both the horizon data and the entropy of an extremal black hole, with no appeal to supersymmetry.
Revisiting the Quantum Geometry of Torus-fibered Calabi-Yau Threefolds
hep-th 2025-10 conditional novelty 7.0 of 10

The conjectured Jacobi modularity of topological string amplitudes on torus-fibered Calabi-Yau threefolds is derived conditionally from the wave-function property under the relative conifold monodromy, which also maps...
Direct Integration of the Topological String
hep-th 2007-02 conditional novelty 7.0 of 10

A direct-integration method solves the holomorphic anomaly equations for topological string amplitudes more efficiently than Feynman graph expansion, yielding closed results up to genus six on the Enriques Calabi-Yau ...
Topological String Theory on Compact Calabi-Yau: Modularity and Boundary Conditions
hep-th 2006-12 conditional novelty 7.0 of 10

Boundary conditions from the conifold gap, orbifold regularity, and Castelnuovo bounds fix the holomorphic ambiguity of the topological string on one-parameter compact Calabi-Yau spaces, yielding high-genus amplitudes...

Show 4 more citing papers

New Connections Between 4D and 5D Black Holes
hep-th 2005-03 conditional novelty 7.0 of 10

The BPS partition function of a 4D Calabi-Yau black hole equals that of a 5D spinning black hole in Taub-NUT space, which in turn equals the squared N=2 topological string partition function.
Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces
hep-th 2010-10 conditional novelty 6.0 of 10

The hypermultiplet moduli space at fixed string coupling is a circle bundle over the intermediate Jacobian with curvature set by the Euler number, and NS5-brane instanton corrections form a non-Gaussian theta series b...
Topological Strings and (Almost) Modular Forms
hep-th 2006-07 conditional novelty 6.0 of 10

The genus-g B-model topological string amplitude is a quasi-modular (or almost holomorphic modular) form of the monodromy group, and the BCOV propagators are its Eisenstein-series generators.
Partition function of non-supersymmetric black holes in the supergravity limit
hep-th 2006-02 conditional novelty 6.0 of 10

The paper proposes Z_ns = exp(pi^2 D(p)/(2*phi0)) as the mixed partition function of non-supersymmetric attractors, engineered so its Legendre transform equals the Bekenstein-Hawking entropy.

How the literature uses it

12 arXiv papers cite this in 20 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 15 sentences without a confident label

Cites it as background 5 papers

First, let us recall the wave function interpretation of the topological string partition function, following [ref] .

Revisiting the Quantum Geometry of Torus-fibered Calabi-Yau Threefolds · Wave-function property of the topological string partition function · 0.91
This review section is based on [ref] .

Geometric Transition as a Change of Polarization · Wavefuncion interpretation of closed topological strings · 0.98
These advances have also led to much progress in our understanding of the topological string amplitude, in particular with regard to its wave function character [ref] .

Topological wave functions and the 4D-5D lift · Introduction · 0.86

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