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Black hole partition functions and duality
High Energy Physics - Theory · Submitted 16 Jan 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this way we confirm the presence of a measure in the duality invariant inverse Laplace integrals.
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The abstract, sentence by sentence
The macroscopic entropy and the attractor equations for BPS black holes in four-dimensional N=2 supergravity theories follow from a variational principle for a certain `entropy function'. We present this function in the presence of R^2-interactions and non-holomorphic corrections. The variational principle identifies the entropy as a Legendre transform and this motivates the definition of various partition functions corresponding to different ensembles and a hierarchy of corresponding duality invariant inverse Laplace integral representations for the microscopic degeneracies. Whenever the microscopic degeneracies are known the partition functions can be evaluated directly. This is the case for N=4 heterotic CHL black holes, where we demonstrate that the partition functions are consistent with the results obtained on the macroscopic side for black holes that have a non-vanishing classical area. In this way we confirm the presence of a measure in the duality invariant inverse Laplace integrals. Most, but not all, of these results are obtained in the context of semiclassical approximations. For black holes whose area vanishes classically, there remain discrepancies at the semiclassical level and beyond, the nature of which is not fully understood at present.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The macroscopic entropy and the attractor equations for BPS black holes in four-dimensional N=2 supergravity theories follow from a variational principle for a certain `entropy function'. | background 0.99 | 0.14 | 0.15 | asserted 1.00 |
| s1 | We present this function in the presence of R^2-interactions and non-holomorphic corrections. | contribution 0.99 | 0.88 | 0.06 | asserted 0.99 |
| s2 | The variational principle identifies the entropy as a Legendre transform and this motivates the definition of various partition functions corresponding to different ensembles and a hierarchy of corresponding duality invariant inverse Laplace integral representations for the microscopic degeneracies. | contribution 0.57 | 0.71 | 0.29 | asserted 0.98 |
| s3 | Whenever the microscopic degeneracies are known the partition functions can be evaluated directly. | method 0.70 | 0.21 | 0.05 | asserted 0.97 |
| s4 | This is the case for N=4 heterotic CHL black holes, where we demonstrate that the partition functions are consistent with the results obtained on the macroscopic side for black holes that have a non-vanishing classical area. | result 0.85 | 0.90 | 0.21 | asserted 0.87 |
| s5 | In this way we confirm the presence of a measure in the duality invariant inverse Laplace integrals. | result 0.70 | 0.83 | 0.21 | asserted 1.00 |
| s6 | Most, but not all, of these results are obtained in the context of semiclassical approximations. | limitation 1.00 | 0.74 | 0.06 | asserted 0.98 |
| s7 | For black holes whose area vanishes classically, there remain discrepancies at the semiclassical level and beyond, the nature of which is not fully understood at present. | limitation 0.98 | 0.40 | 0.07 | asserted 0.97 |
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Original abstract
The macroscopic entropy and the attractor equations for BPS black holes in four-dimensional N=2 supergravity theories follow from a variational principle for a certain `entropy function'. We present this function in the presence of R^2-interactions and non-holomorphic corrections. The variational principle identifies the entropy as a Legendre transform and this motivates the definition of various partition functions corresponding to different ensembles and a hierarchy of corresponding duality invariant inverse Laplace integral representations for the microscopic degeneracies. Whenever the microscopic degeneracies are known the partition functions can be evaluated directly. This is the case for N=4 heterotic CHL black holes, where we demonstrate that the partition functions are consistent with the results obtained on the macroscopic side for black holes that have a non-vanishing classical area. In this way we confirm the presence of a measure in the duality invariant inverse Laplace integrals. Most, but not all, of these results are obtained in the context of semiclassical approximations. For black holes whose area vanishes classically, there remain discrepancies at the semiclassical level and beyond, the nature of which is not fully understood at present.
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References
How the literature uses it
29 arXiv papers cite this in 52 sentences
Papers citing it · 43 sentences without a confident label
Uses its method or tool 3 papers
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Our analysis will follow the one given in [ref] ; this in turn is based on the earlier analysis of [ref] .
Black Hole Entropy Function, Attractors and Precision Counting of Microstates · The full partition function · 0.96
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To be more thorough, we should take into account all the fields in the theory – this can be done by using duality invariance as a criterion for the measure as in [ref] .
Functional determinants, index theorems, and exact quantum black hole entropy · Exact formulas for $=2$ quantum black hole entropy and the relation to topological strings · 0.93
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The procedure to obtain the asymptotic behaviour has been developed in [ref] , which we summarize briefly [We follow the discussion in [ref] .].
Dyon degeneracies from Mathieu moonshine · $1/4$ BPS dyon partition functions · 0.99
Cites it as background 6 papers
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The presence of $Ω$ leads to various modifications of [ref], as discussed in [ref] and reviewed in [ref] .
The gravitational path integral for $ N=4$ BPS black holes from black hole microstate counting · The quantum entropy function · 0.88
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With this notation and in the $λ→∞$ limit, the path-integral reduces to Z^sugra (M_4) = ∫[ d ϕ] exp(-S^sugra (ϕ)) Z^sugra_1-loop (ϕ) , where the one-loop contribution comes from integrating over the quadratic fluctuations of the fields around the localization locus, and one should carefully understand the measure of the remaining integral, as dictated by consistency of the quantum theory (see [ref] for discussions on this point).
Factorization of log-corrections in AdS$_4$/CFT$_3$ from supergravity localization · The one-loop set-up · 0.88
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The macroscopic entropy of a static, supersymmetric black hole computed [ref] from the effective lagrangian is given by [equation] where [equation] There exist a proposal [ref] for the BPS entropy function whose stationary points are attractor equations and the value of the entropy function at the attractor points equals the macroscopic entropy ([ref]).
Duality covariant variables for STU-model in presence of non-holomorphic corrections · Introduction · 0.93
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