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2015 | OriginalPaper | Chapter

Lectures on BCOV Holomorphic Anomaly Equations

Authors: Atsushi Kanazawa, Jie Zhou

Published in: Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Publisher: Springer New York

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Abstract

The present article surveys some mathematical aspects of the BCOV holomorphic anomaly equations introduced by Bershadsky et al. (Nucl Phys B 405:279–304, 1993; Comm Math Phys 165:311–428, 1994). 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.
previous chapter Introduction to Modular Forms
next chapter Polynomial Structure of Topological String Partition Functions
Footnotes
1
We take a universal covering of \(\mathcal{M}\) if necessary but most of what follows works in a local setting.
 
2
We use the Einstein summation convention.
 
3
We have \(N_{g}(0) =\int _{\overline{M}_{ g}\times X^{\vee }}c_{top}(Ob) = (-1)^{g}\frac{\chi (X^{\vee })} {2} \int _{\overline{M}_{ g}}c_{g-1}^{3}(\mathcal{H}_{g})\), where \(Ob \rightarrow \overline{M}_{g,0}(X, 0)\mathop{\cong}\overline{M}_{g} \times X^{\vee }\) is the obstruction bundle and \(\mathcal{H}_{g} \rightarrow \overline{M}_{g}\) is the Hodge bundle [17].
 
4
See [14] which proposes a rigorous definition for the \(\mathcal{F}_{g}\)’s.
 
5
This case is somewhat misleading because an elliptic curve is a self-mirror manifold. However, we believe this is still a good example the reader should keep in mind.
 
6
We have to take care of the first term of the second line, see [16].
 
7
Computationally, for genus one amplitude, we need to take its derivative to get rid of the anti-holomorphic terms. Also the generating function of genus one Gromo-Witten invariants with one insertion, which is given by the first derivative of \(\mathcal{F}_{1}\), is more natural due to stability reasons.
 
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Metadata
Title
Lectures on BCOV Holomorphic Anomaly Equations
Authors
Atsushi Kanazawa
Jie Zhou
Copyright Year
2015
Publisher
Springer New York
Book
Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Print ISBN: 978-1-4939-2829-3

Electronic ISBN: 978-1-4939-2830-9

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-1-4939-2830-9_13

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