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Quantum Black Holes, Elliptic Genera and Spectral Partition Functions
High Energy Physics - Theory · Submitted 9 Aug 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.
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The abstract, sentence by sentence
We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lie in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi-Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. | contribution 0.99 | 0.90 | 0.15 | asserted 0.99 |
| s1 | The common feature in our examples lie in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. | interpretation 0.62 | 0.65 | 0.09 | asserted 1.00 |
| s2 | We present new calculations of supergravity elliptic genera on local Calabi-Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. | contribution 0.82 | 0.94 | 0.09 | asserted 1.00 |
| s3 | We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology. | interpretation 0.95 | 0.64 | 0.68 | speculative 1.00 |
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The paper
Original abstract
We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lie in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi-Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.
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How the literature uses it
6 arXiv papers cite this in 8 sentences
Papers citing it · 3 sentences without a confident label
Cites it as background 4 papers
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For the next step let us introduce the Ruelle spectral function $ℛ(s)$ associated with hyperbolic three-geometry [ref] .
Partition functions for supersymmetric gauge theories on spheres · Restricted specializations · 0.83
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From this point of view, it is important that Macdonald polynomials possess a generalization not only in the MacMahon (DIM) direction, but also in a seemingly different one, to the Kerov functions [ref] (see [ref] for early applications and [ref] for a recent review).
On Hamiltonians for Kerov functions · Introduction · 0.84
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In this paper, we continue our study of theory of Kerov functions[The Kerov function should not be confused with much better known "Kerov character polynomials", also associated with the name of S.Kerov [ref] : to avoid the confusion, we use the term functions throughout the present text.] [ref] and their applications.
Kerov functions for composite representations and Macdonald ideal · Introduction · 0.98
What it cites · 50 sentences without a confident label
Uses the method or tool of 7 papers
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To apply the blow-up formulas in this case, we first compute the partition function ([ref]) on $M=^2$ using the localization formula to get [ref] [equation] Equivalently, by applying the localization formula to the spectral function representation ([ref]) we have [equation] Following [ref] , we can sum this series explicitly by identifying this partition function with the generating function for orbifold equivariant elliptic genera of symmetric products given by _^2(t_1,t_2;τ,ρ,σ) = ∑_n=0^∞ χ(^n^2 ; y,p)(t_1,t_2) q^n .
math.AG/0412089 · (2,2) field theory elliptic genus · 0.92
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To apply the blow-up formulas in this case, we first compute the partition function ([ref]) on $M=^2$ using the localization formula to get [ref] [equation] Equivalently, by applying the localization formula to the spectral function representation ([ref]) we have [equation] Following [ref] , we can sum this series explicitly by identifying this partition function with the generating function for orbifold equivariant elliptic genera of symmetric products given by _^2(t_1,t_2;τ,ρ,σ) = ∑_n=0^∞ χ(^n^2 ; y,p)(t_1,t_2) q^n .
Matrix Models, Geometric Engineering and Elliptic Genera · (2,2) field theory elliptic genus · 0.92
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To apply the blow-up formulas in this case, we first compute the partition function ([ref]) on $M=^2$ using the localization formula to get [ref] [equation] Equivalently, by applying the localization formula to the spectral function representation ([ref]) we have [equation] Following [ref] , we can sum this series explicitly by identifying this partition function with the generating function for orbifold equivariant elliptic genera of symmetric products given by _^2(t_1,t_2;τ,ρ,σ) = ∑_n=0^∞ χ(^n^2 ; y,p)(t_1,t_2) q^n .
Elliptic Genera of Symmetric Products and Second Quantized Strings · (2,2) field theory elliptic genus · 0.92
Compares with 2 papers
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We will now show that ([ref]) is in perfect agreement with the microscopic M5-brane computation of the black hole entropy in the two-dimensional $(4,0)$ superconformal field theory derived by [ref] ; it also agrees with the macroscopic supergravity result [ref] .
Instanton on toric singularities and black hole countings · Microscopic black hole entropy · 0.99
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We will now show that ([ref]) is in perfect agreement with the microscopic M5-brane computation of the black hole entropy in the two-dimensional $(4,0)$ superconformal field theory derived by [ref] ; it also agrees with the macroscopic supergravity result [ref] .
Black Hole Entropy in M-Theory · Microscopic black hole entropy · 0.99
Cites as background 5 papers
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Let us introduce next the Ruelle zeta-functions $(s)$ [ref] .
Spectral Functions for BTZ Black Hole Geometry · Spectral functions of hyperbolic three-manifolds · 0.90
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In physics, a two-variable elliptic genus can be associated with any two-dimensional $= (2,2)$ superconformal field theory [ref] ; it is given by [equation] where $q$ and $y$ are complex parameters, $L_0$ (resp. $L_0$) is the Virasoro generator of left (resp. right) movers, and $J_0$ is the $U(1)$ charge operator of left moving modes.
Elliptic Genera and N=2 Superconformal Field Theory · Elliptic genera · 0.84
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Consider the single particle spectrum, and suppose that it starts at a conformal weight $h_min = h_min+s$ for some $s$.[ The appropriate complete spectrum of single particle primary states can be found in e.g. [ref] .
Partition functions and elliptic genera from supergravity · Supergravity boson-fermion correspondence · 0.81
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