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Published in:

2015 | OriginalPaper | Chapter

# Lectures on BCOV Holomorphic Anomaly Equations

Authors: Atsushi Kanazawa, Jie Zhou

Publisher: Springer New York

## 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.
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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