Polynomial Structure of Topological String Partition Functions

Zhou, Jie

doi:10.1007/978-1-4939-2830-9_14

Cited by 4 publications

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“…The polynomiality for the topological string amplitudes F g provides a significant enhancement for practical computations and also equips the ring generated by the propagators and Kähler derivatives with interesting mathematical structures. A more detailed overview of this subject, as well as the connection of the ring to modular forms [1,28,5,41,3], can be found in a separate expository article [42].…”

Section: Propagators and Polynomialitymentioning

confidence: 99%

Lectures on BCOV Holomorphic Anomaly Equations

Kanazawa

Zhou

2015

Calabi-Yau Varieties: Arithmetic, Geometry and Physics

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The present article surveys some mathematical aspects of the BCOV holomorphic anomaly equations introduced by Bershadsky, Cecotti, Ooguri and Vafa [8,9]. It grew from a series of lectures the authors gave at the Fields Institute in the Thematic Program of Calabi-Yau Varieties in the fall of 2013.A candidate of the higher genus B-model was provided by Bershadsky, Cecotti, Ooguri and Vafa in the seminal papers [8,9] (BCOV theory). Among other things, they derived a set of equations, now called the BCOV holomorphic anomaly equa-In this section, we give a brief summary of the basics of special Kähler geometry that we need throughout this article. Special Kähler geometry is a basic computational tool used in the calculations in mirror symmetry. This section also serves to set conventions and notations. Standard references are [37,8,19]. Special Coordinates and PrepotentialLet M be the moduli space of complex structures of a smooth Calabi-Yau threefold X of dimension n := dim M = h 2,1 (X). The vector bundle H := R 3 π * C ⊗ O M of rank 2n + 2 comes equipped with the Gauss-Manin connection ∇ and the natural Hodge filtration F • of weight 3. The Hodge filtration F • yields the smooth decompositionis called the vacuum bundle. We also fix a reference point [X] ∈ M and smoothly identify 1 the fibers of H with H 3 (X, C). We endow H 3 (X, C) with the symplectic pairing (α, β ) := √ −1´X α ∪ β . Then the period domain D is defined by3 (X, C). More concretely, by fixing a symplectic basis {α I , β J } n I,J=0 of H 3 (X, Z) and its dual basis {A I , B J } n I,J=0 of H 3 (X, Z), the period map P is written in terms of a section Ω = {Ω z } z∈M of the vacuum bundle L as 2 P(z) := φ I (z)α I + F J (z)β J , where φ I (z) :=´A I Ω z and F J (z) :=´BJ Ω z . Proposition 1. With the notation above, the following hold:

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