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Investigations in Calabi-Yau modularity and mirror symmetry
High Energy Physics - Theory · Submitted 1 Dec 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The main results include a study, and finding an application to the higher genus problem, of infinite Coxeter symmetries in the sets of Gopakumar-Vafa invariants; provision of a new class of solutions to the supersymmetric flux vacuum equations which have elsewhere been conjectured to give weight-two modular manifolds; provision of two new conjectural examples of weight-four modular varieties (rank-two attractors); and discussion of a set of numerical relations between infinite sums of Gromov-Witten invariants and critical L-values.
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This is the author's PhD thesis. Two main sections address various aspects of mirror symmetry for compact Calabi-Yau threefolds and the roles that classically modular varieties play in string theory compactifications. The main results include a study, and finding an application to the higher genus problem, of infinite Coxeter symmetries in the sets of Gopakumar-Vafa invariants; provision of a new class of solutions to the supersymmetric flux vacuum equations which have elsewhere been conjectured to give weight-two modular manifolds; provision of two new conjectural examples of weight-four modular varieties (rank-two attractors); and discussion of a set of numerical relations between infinite sums of Gromov-Witten invariants and critical L-values.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This is the author's PhD thesis. | background 0.98 | 0.13 | 0.01 | asserted 0.99 |
| s1 | Two main sections address various aspects of mirror symmetry for compact Calabi-Yau threefolds and the roles that classically modular varieties play in string theory compactifications. | contribution 0.47 | 0.42 | 0.05 | asserted 0.99 |
| s2 | The main results include a study, and finding an application to the higher genus problem, of infinite Coxeter symmetries in the sets of Gopakumar-Vafa invariants; provision of a new class of solutions to the supersymmetric flux vacuum equations which have elsewhere been conjectured to give weight-two modular manifolds; provision of two new conjectural examples of weight-four modular varieties (rank-two attractors); and discussion of a set of numerical relations between infinite sums of Gromov-Witten invariants and critical L-values. | result 0.54 | 0.92 | 0.78 | asserted 0.88 |
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The paper
Original abstract
This is the author's PhD thesis. Two main sections address various aspects of mirror symmetry for compact Calabi-Yau threefolds and the roles that classically modular varieties play in string theory compactifications. The main results include a study, and finding an application to the higher genus problem, of infinite Coxeter symmetries in the sets of Gopakumar-Vafa invariants; provision of a new class of solutions to the supersymmetric flux vacuum equations which have elsewhere been conjectured to give weight-two modular manifolds; provision of two new conjectural examples of weight-four modular varieties (rank-two attractors); and discussion of a set of numerical relations between infinite sums of Gromov-Witten invariants and critical L-values.
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References
How the literature uses it
What this paper does with the arXiv papers it cites
What it cites · 96 sentences without a confident label
Builds on 3 papers
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Building on the work of [ref] , where such divergences were not further addressed, we identify Padé resummation as the appropriate scheme for summing these series for larger values of $y_0$.
Attractors with Large Complex Structure for One-Parameter Families of Calabi-Yau Manifolds · Summation identities: L-value ratios from GW invariants · 0.80
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In another attempt to generalise the results of [ref] , we can see manifolds with Coxeter symmetries that are not CICYs.
Symmetries of Calabi-Yau Prepotentials with Isomorphic Flops · Coxeter groups · 1.00
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Chapter Three describes ongoing work, building on ideas that appeared in [ref] where the appearance of the Coxeter groups was noted.
Mirror Symmetry for Five-Parameter Hulek-Verrill Manifolds · Introduction · 0.92
Uses the method or tool of 16 papers
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Following [ref] , we can express the Frobenius periods as linear combinations of the two critical L-values associated to a weight-four modular form.
A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two · Summation identities: L-value ratios from GW invariants · 0.86
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Following [ref] , we can express the Frobenius periods as linear combinations of the two critical L-values associated to a weight-four modular form.
D-brane masses at special fibres of hypergeometric families of Calabi-Yau threefolds, modular forms, and periods · Summation identities: L-value ratios from GW invariants · 0.86
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In IIB Calabi-Yau compactifications those scalars give the complex structure moduli of the Calabi-Yau $X$, and the fixed points of attractor flows are the solutions of Strominger's equations [equation] In this section we will describe a method, first worked through in [ref] and later fully developed in [ref] , for finding rank-two attractors.
Local Zeta Functions From Calabi-Yau Differential Equations · New weight four modular manifolds · 0.94
Uses data or values from 3 papers
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The real parts of the critical L-values associated to 34.4.b.a have decimal expansions beginning [equation] The complete L-values are [equation] Introduce symbols for the following numbers: [equation] In terms of the real parts $λ(1), λ(2)$ of the L-values, the periods as computed in [ref] are (again, conjecturally with 1000 digits of precision found numerically by those authors) [equation] In the $t$-plane, the attractor is [equation] The attractor equations are solved for the charge vectors [equation] The period vector is orthogonal to the charge vectors [equation] $Q_4$ shall be used to obtain an identity.
A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two · $z3̄3+8√(17)$ · 0.99
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The other key source of information was the thesis [ref] of Kadir, in which tables of zeta functions were given, with various choices of moduli, for the mirror of the octic hypersurface in $_1,1,2,2,2$.
The Arithmetic of Calabi--Yau Manifolds and Mirror Symmetry · Modularity of supersymmetric vacua in flux compactifications · 0.94
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When the tables of genus 0 and 1 instanton numbers in [ref] were produced, a number of repetitions of values were noted.
Mirror Symmetry for Five-Parameter Hulek-Verrill Manifolds · Coxeter groups · 0.91
Compares with 3 papers
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We incorporate permutation symmetries in our discussion, so our presented Coxeter groups differ to those of [ref] .
Symmetries of Calabi-Yau Prepotentials with Isomorphic Flops · Coxeter groups · 0.97
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These numbers were computed on a 6-CPU desktop machine with 16GB of RAM, and so we do not reach so high a degree as was done for the five parameter model [ref] in [ref] .
Mirror Symmetry for Five-Parameter Hulek-Verrill Manifolds · Performing the computation efficiently · 0.99
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In earlier work by Bershadsky, Cecotti, Ooguri and Vafa [ref] , the prescription differed slightly (counting what they called “primitive elliptic curves") and so some care is needed when comparing curve counts across the literature.
Holomorphic Anomalies in Topological Field Theories · Genus 1 mirror symmetry: Counting elliptic curves · 0.98
Cites as support 4 papers
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We will study supersymmetric vacuum configurations, and provide new examples in support of the flux modularity conjecture [ref] .
Supersymmetric Flux Compactifications and Calabi-Yau Modularity · Resuming the computation · 0.96
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We will study supersymmetric vacuum configurations, and provide new examples in support of the flux modularity conjecture [ref] .
Flux Modularity, F-Theory, and Rational Models · Resuming the computation · 0.96
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For example, the works [ref] were able to express D-brane masses and semiclassical black hole entropies using Mellin transforms of the modular forms associated to modular geometries.
A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two · Motivation · 0.98
Cites as background 28 papers
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The first of these is the identification of attractor points of rank-two, a program instigated by Moore in [ref] .
Attractors and Arithmetic · Modularity of Calabi-Yau threefolds · 0.84
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The first of these is the identification of attractor points of rank-two, a program instigated by Moore in [ref] .
Arithmetic and Attractors · Modularity of Calabi-Yau threefolds · 0.84
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We shall briefly review his construction, informed also by [ref] .
Flux Modularity, F-Theory, and Rational Models · F-theory lifts · 0.81
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