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2015 | OriginalPaper | Chapter

Lectures on BCOV Holomorphic Anomaly Equations

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Authors: Atsushi Kanazawa, Jie Zhou

Publisher: Springer New York

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

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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.

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We take a universal covering of \(\mathcal{M}\) if necessary but most of what follows works in a local setting.

We use the Einstein summation convention.

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].

See [ 14] which proposes a rigorous definition for the \(\mathcal{F}_{g}\)’s.

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.

We have to take care of the first term of the second line, see [ 16].

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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Title

Lectures on BCOV Holomorphic Anomaly Equations

Book

Calabi-Yau Varieties: Arithmetic, Geometry and Physics

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

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

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

DOI

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

Authors:

Atsushi Kanazawa
Jie Zhou

Publisher

Springer New York

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