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Special Polynomial Rings, Quasi Modular Forms and Duality of Topological Strings
High Energy Physics - Theory · Submitted 31 May 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that these rings are isomorphic to the rings of quasi modular forms for threefolds with duality groups for which these are known.
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The abstract, sentence by sentence
We study the differential polynomial rings which are defined using the special geometry of the moduli spaces of Calabi-Yau threefolds. The higher genus topological string amplitudes are expressed as polynomials in the generators of these rings, giving them a global description in the moduli space. At particular loci, the amplitudes yield the generating functions of Gromov-Witten invariants. We show that these rings are isomorphic to the rings of quasi modular forms for threefolds with duality groups for which these are known. For the other cases, they provide generalizations thereof. We furthermore study an involution which acts on the quasi modular forms. We interpret it as a duality which exchanges two distinguished expansion loci of the topological string amplitudes in the moduli space. We construct these special polynomial rings and match them with known quasi modular forms for non-compact Calabi-Yau geometries and their mirrors including local P^2 and local del Pezzo geometries with E_5,E_6,E_7 and E_8 type singularities. We provide the analogous special polynomial ring for the quintic.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the differential polynomial rings which are defined using the special geometry of the moduli spaces of Calabi-Yau threefolds. | contribution 0.97 | 0.87 | 0.04 | asserted 0.96 |
| s1 | The higher genus topological string amplitudes are expressed as polynomials in the generators of these rings, giving them a global description in the moduli space. | result 0.73 | 0.85 | 0.18 | asserted 1.00 |
| s2 | At particular loci, the amplitudes yield the generating functions of Gromov-Witten invariants. | result 0.96 | 0.50 | 0.07 | asserted 1.00 |
| s3 | We show that these rings are isomorphic to the rings of quasi modular forms for threefolds with duality groups for which these are known. | result 0.98 | 0.92 | 0.33 | asserted 1.00 |
| s4 | For the other cases, they provide generalizations thereof. | result 0.72 | 0.65 | 0.09 | asserted 0.98 |
| s5 | We furthermore study an involution which acts on the quasi modular forms. | contribution 1.00 | 0.93 | 0.06 | asserted 0.97 |
| s6 | We interpret it as a duality which exchanges two distinguished expansion loci of the topological string amplitudes in the moduli space. | interpretation 1.00 | 0.88 | 0.18 | asserted 0.99 |
| s7 | We construct these special polynomial rings and match them with known quasi modular forms for non-compact Calabi-Yau geometries and their mirrors including local P^2 and local del Pezzo geometries with E_5,E_6,E_7 and E_8 type singularities. | contribution 0.75 | 0.95 | 0.22 | asserted 1.00 |
| s8 | We provide the analogous special polynomial ring for the quintic. | contribution 0.84 | 0.95 | 0.10 | asserted 1.00 |
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The paper
Original abstract
We study the differential polynomial rings which are defined using the special geometry of the moduli spaces of Calabi-Yau threefolds. The higher genus topological string amplitudes are expressed as polynomials in the generators of these rings, giving them a global description in the moduli space. At particular loci, the amplitudes yield the generating functions of Gromov-Witten invariants. We show that these rings are isomorphic to the rings of quasi modular forms for threefolds with duality groups for which these are known. For the other cases, they provide generalizations thereof. We furthermore study an involution which acts on the quasi modular forms. We interpret it as a duality which exchanges two distinguished expansion loci of the topological string amplitudes in the moduli space. We construct these special polynomial rings and match them with known quasi modular forms for non-compact Calabi-Yau geometries and their mirrors including local P^2 and local del Pezzo geometries with E_5,E_6,E_7 and E_8 type singularities. We provide the analogous special polynomial ring for the quintic.
Human review
References
How the literature uses it
16 arXiv papers cite this in 49 sentences
Papers citing it · 38 sentences without a confident label
Uses its method or tool 1 paper
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Using the polynomial recursion technique [ref] , one can then prove [ref] by induction that the topological partition functions, as solutions to the holomorphic anomaly equations, are almost-holomorphic modular forms of weight $0$.
Differential Rings from Special K\"ahler Geometry · Section · 0.87
Compares with it 1 paper
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The matching with the functions of Ref. [ref] , provided for CY threefolds in Ref. [ref] , was obtained by enhancing the differential ring to the special differential ring, given for one dimensional moduli spaces in Ref. [ref] and generalized in Ref. [ref] .
Algebraic structure of $tt^*$ equations for Calabi-Yau sigma models · Differential rings · 0.86
Cites it as background 8 papers
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We refer the interested readers to [ref] and references therein for the detailed discussion on other families.
Differential Rings from Special K\"ahler Geometry · Picard-Fuchs equations for families of Calabi-Yau manifolds · 0.98
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We give the expressions of the modular objects that appear in this work, more details can be found in Ref. [ref] and references therein.
Algebraic structure of $tt^*$ equations for Calabi-Yau sigma models · Quasi modular forms and differential rings · 0.89
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We refer to [ref] for earlier studies in the literature.[We are grateful to Thorsten Schimannek for his help about the material in this section.]
BPS Dendroscopy on Local $P^2$ · Periods as Eichler integrals · 0.99
What it cites · 120 sentences without a confident label
Builds on 3 papers
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Our new addition to these previous discussions consists of the explicit identification of the rings of quasi modular forms of $Γ_0(3)$ which are adapted to this specific moduli space as well as their translation to the special geometry ring of polynomial generators of Ref. [ref] .
Polynomial Structure of the (Open) Topological String Partition Function · Applications · 0.91
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The space-time derivation of [ref] is not restricted to the conifold case and applies also to the case $m>1$ singularities which give rise to a different spectrum of extra massless vector and hypermultiplets in space-time.
M-Theory and Topological Strings--II · Conifold-like loci · 0.90
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The space-time derivation of [ref] is not restricted to the conifold case and applies also to the case $m>1$ singularities which give rise to a different spectrum of extra massless vector and hypermultiplets in space-time.
M-Theory and Topological Strings--I · Conifold-like loci · 0.90
Uses the method or tool of 2 papers
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This has been done using methods of analytic continuation in Refs. [ref] , the use of the duality operation simplifies this considerably.
Integrability of the holomorphic anomaly equations · Higher genus amplitudes and duality · 0.80
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This has been done using methods of analytic continuation in Refs. [ref] , the use of the duality operation simplifies this considerably.
Global Properties of Topological String Amplitudes and Orbifold Invariants · Higher genus amplitudes and duality · 0.80
Uses data or values from 2 papers
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The topological invariants for the corresponding A-model non-compact CY threefolds can be found in [ref] [The Chern number $c_2$ is the integral of the Chern class previously denoted by $∫c_2J$ in ([ref]).] [equation] Near the large complex structure limit, the genus one amplitude, denoted by $F_lcs^(1)$, is given by [equation] The constant $a$ is universal and is given by $a=-112$, while $b=-c_224$.
Non-Critical Strings, Del Pezzo Singularities And Seiberg-Witten Curves · Genus 1 · 0.87
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The topological invariants for the corresponding A-model non-compact CY threefolds can be found in [ref] [The Chern number $c_2$ is the integral of the Chern class previously denoted by $∫c_2J$ in ([ref]).] [equation] Near the large complex structure limit, the genus one amplitude, denoted by $F_lcs^(1)$, is given by [equation] The constant $a$ is universal and is given by $a=-112$, while $b=-c_224$.
Local Mirror Symmetry: Calculations and Interpretations · Genus 1 · 0.87
Cites as support 6 papers
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It was furthermore shown in explicit examples [ref] that there are choices of $ℰ^zz,ℰ^z,ℰ$ such that $h^zz_z,h^z_z,h_zz,h_z,h_zzzz$ are rational functions in $z$ of the form $p(z)/Δ$ where $Δ$ is the discriminant, and $p(z)$ is a polynomial in the algebraic complex structure coordinates.
Polynomial Structure of the (Open) Topological String Partition Function · Polynomial ring of sections in arbitrary coordinates · 0.97
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It was furthermore shown in explicit examples [ref] that there are choices of $ℰ^zz,ℰ^z,ℰ$ such that $h^zz_z,h^z_z,h_zz,h_z,h_zzzz$ are rational functions in $z$ of the form $p(z)/Δ$ where $Δ$ is the discriminant, and $p(z)$ is a polynomial in the algebraic complex structure coordinates.
Global Properties of Topological String Amplitudes and Orbifold Invariants · Polynomial ring of sections in arbitrary coordinates · 0.97
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It was furthermore shown in explicit examples [ref] that there are choices of $ℰ^zz,ℰ^z,ℰ$ such that $h^zz_z,h^z_z,h_zz,h_z,h_zzzz$ are rational functions in $z$ of the form $p(z)/Δ$ where $Δ$ is the discriminant, and $p(z)$ is a polynomial in the algebraic complex structure coordinates.
BCOV ring and holomorphic anomaly equation · Polynomial ring of sections in arbitrary coordinates · 0.97
Cites as background 16 papers
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The Yukawa coupling is given by [For the mirror quintic in terms of an algebraic coordinate on the moduli space, and adopting the convention of multiplying lower tensorial indices by $z$.] [equation] We fix the holomorphic functions appearing in ([ref]) as in Refs. [ref] : [equation] The generator of rational functions ([ref]) becomes: [equation] giving the special ring relations: [equation]
Lectures on Mirror Symmetry and Topological String Theory · Special polynomial ring · 0.89
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The Yukawa coupling is given by [For the mirror quintic in terms of an algebraic coordinate on the moduli space, and adopting the convention of multiplying lower tensorial indices by $z$.] [equation] We fix the holomorphic functions appearing in ([ref]) as in Refs. [ref] : [equation] The generator of rational functions ([ref]) becomes: [equation] giving the special ring relations: [equation]
BCOV ring and holomorphic anomaly equation · Special polynomial ring · 0.89
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Fixing a complex structure on $ℳ$, at any given point in the base manifold the fibre of the bundle $ℋ$ can be decomposed into the following form [ref] (For the B-model this gives the familiar Hodge decomposition): [equation] where $ℒ$ is the Hodge line bundle, $Tℳ$ is the holomorphic tangent bundle and $ℒ$ and $Tℳ$ are their complex conjugates.
Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes · Special geometry ring · 0.89
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