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On the Scalar Spectrum of the Y^{p,q} Manifolds
High Energy Physics - Theory · Submitted 25 Jan 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim However, despite this analytical obstacle, we manage to get both lower and upper bounds on the eigenvalues of the scalar spectrum by comparing the eigenvalue problem with a simpler, solvable system.
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The abstract, sentence by sentence
The spectra of supergravity modes in anti de Sitter (AdS) space on a five-sphere endowed with the round metric (which is the simplest 5d Sasaki-Einstein space) has been studied in detail in the past. However for the more general class of cohomogeneity one Sasaki-Einstein metrics on S^2 x S^3, given by the Y^{p, q} class, a complete study of the spectra has not been attempted. Earlier studies on scalar spectrum were restricted to only the first few eigenstates. In this paper we take a step in this direction by analysing the full scalar spectrum on these spaces. However it turns out that finding the exact solution of the corresponding eigenvalue problem in closed form is not feasible since the computation of the eigenvalues of the Laplacian boils down to the analysis of a one-dimensional operator of Heun type, whose spectrum cannot be computed in closed form. However, despite this analytical obstacle, we manage to get both lower and upper bounds on the eigenvalues of the scalar spectrum by comparing the eigenvalue problem with a simpler, solvable system. We also briefly touch upon various other new avenues such as non-commutative and dipole deformations as well as possible non-conformal extensions of these models.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The spectra of supergravity modes in anti de Sitter (AdS) space on a five-sphere endowed with the round metric (which is the simplest 5d Sasaki-Einstein space) has been studied in detail in the past. | background 1.00 | 0.02 | 0.02 | asserted 1.00 |
| s1 | However for the more general class of cohomogeneity one Sasaki-Einstein metrics on S^2 x S^3, given by the Y^{p, q} class, a complete study of the spectra has not been attempted. | gap 0.99 | 0.04 | 0.03 | asserted 1.00 |
| s2 | Earlier studies on scalar spectrum were restricted to only the first few eigenstates. | gap 0.65 | 0.04 | 0.02 | asserted 1.00 |
| s3 | In this paper we take a step in this direction by analysing the full scalar spectrum on these spaces. | contribution 1.00 | 0.80 | 0.10 | asserted 0.96 |
| s4 | However it turns out that finding the exact solution of the corresponding eigenvalue problem in closed form is not feasible since the computation of the eigenvalues of the Laplacian boils down to the analysis of a one-dimensional operator of Heun type, whose spectrum cannot be computed in closed form. | limitation 0.99 | 0.84 | 0.09 | asserted 1.00 |
| s5 | However, despite this analytical obstacle, we manage to get both lower and upper bounds on the eigenvalues of the scalar spectrum by comparing the eigenvalue problem with a simpler, solvable system. | result 0.56 | 0.96 | 0.65 | asserted 1.00 |
| s6 | We also briefly touch upon various other new avenues such as non-commutative and dipole deformations as well as possible non-conformal extensions of these models. | future work 0.80 | 0.67 | 0.04 | hedged possibility 0.43 |
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The paper
Original abstract
The spectra of supergravity modes in anti de Sitter (AdS) space on a five-sphere endowed with the round metric (which is the simplest 5d Sasaki-Einstein space) has been studied in detail in the past. However for the more general class of cohomogeneity one Sasaki-Einstein metrics on S^2 x S^3, given by the Y^{p, q} class, a complete study of the spectra has not been attempted. Earlier studies on scalar spectrum were restricted to only the first few eigenstates. In this paper we take a step in this direction by analysing the full scalar spectrum on these spaces. However it turns out that finding the exact solution of the corresponding eigenvalue problem in closed form is not feasible since the computation of the eigenvalues of the Laplacian boils down to the analysis of a one-dimensional operator of Heun type, whose spectrum cannot be computed in closed form. However, despite this analytical obstacle, we manage to get both lower and upper bounds on the eigenvalues of the scalar spectrum by comparing the eigenvalue problem with a simpler, solvable system. We also briefly touch upon various other new avenues such as non-commutative and dipole deformations as well as possible non-conformal extensions of these models.
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How the literature uses it
2 arXiv papers cite this in 2 sentences
Papers citing it · 1 sentence without a confident label
Cites it as background 1 paper
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The scalar Laplacian on $Y^p,q$ and aspects of the spectrum have been investigated in [ref] .
The shortened KK spectrum of IIB supergravity on Y^{p,q} · The shortened spectrum of IIB supergravity on $Y^p,q$ · 0.94
What it cites · 44 sentences without a confident label
Builds on 1 paper
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In this paper we study spectrum of Sasaki-Einstein manifold $Y^p,q$, using spectral-theoretic methods, continuing the work of [ref] .
Global causal propagator for the Klein-Gordon equation on a class of supersymmetric AdS backgrounds · Introduction · 0.94
Uses the method or tool of 3 papers
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In order to circumvent this problem the steps taken in [ref] is to start by constructing a Fourier-type decomposition of the space of square-integrable functions on $Y^p,q$ adapted to the global structure of the manifold and to the action of the Laplacian.
Global causal propagator for the Klein-Gordon equation on a class of supersymmetric AdS backgrounds · Scalar modes in $Y^p, q$ · 0.96
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We will also take the sechsbein $e^a$ to be the ones given in eq (2.8) of [ref] with appropriate redefinations of the variables therein.
Toward NS5 Branes on the Resolved Cone over Y^{p,q} · Resolution and deformation of the cones over $Y^p,q$ · 0.92
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For example computing the super-conformal index along the lines of [ref] , or even going beyond the supergravity modes a-la [ref] .
On the Superconformal Index of N=1 IR Fixed Points: A Holographic Check · Conclusion and open question · 0.80
Compares with 1 paper
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The form of the background [ref] is similar to the one that we had in [ref] except now the internal space is different.
Supersymmetric Configurations, Geometric Transitions and New Non-Kahler Manifolds · D5 branes on the resolved $Y^p, q$ manifold · 0.96
Cites as support 1 paper
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However despite this obstacle, one thing is clear from the analysis of [ref] : the resultant metric will not be a Kähler manifold, in fact, it may not even be a complex manifold.
Supersymmetric Configurations, Geometric Transitions and New Non-Kahler Manifolds · Resolution and deformation of the cones over $Y^p,q$ · 0.97
Cites as background 2 papers
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Recently the NS5 brane picture has been studied in [ref] .
Toward NS5 Branes on the Resolved Cone over Y^{p,q} · D5 branes on the resolved $Y^p, q$ manifold · 0.89
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Here we will introduce and motivate the study of Sasaki-Einstein manifolds, mostly summarizing the extensive reviews in [ref] in a language slightly more appropriate for the physicists.
Sasaki-Einstein Manifolds · Motivation to study Sasaki-Einstein manifolds · 0.99
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