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2015 | OriginalPaper | Chapter

# Polynomial Structure of Topological String Partition Functions

Author: Jie Zhou

Publisher: Springer New York

## Abstract

We review the polynomial structure of the topological string partition functions as solutions to the holomorphic anomaly equations. We also explain the connection between the ring of propagators defined from special Kähler geometry and the ring of almost-holomorphic modular forms defined on modular curves.
Footnotes
1
Throughout the note, we shall simply call it Kähler structure by abuse of language.

2
See also [38, 29, 30, 33, 45, 31, 27, 1, 26, 4, 19, 6, 22, 21, 42, 5, 32, 7] for related works.

3
The quantity $$\mathcal{F}^{(g)}$$ is really a section rather than a function, but in the literature it is termed topological string partition function which we shall follow in this note.

4
In this note, we shall use $$\bar{\partial }_{\bar{\imath }}$$ and $$\partial _{\bar{\imath }}$$ interchangeably to denote $$\frac{\partial } {\partial \bar{z}^{\bar{\imath }}}$$ for some local complex coordinates $$z =\{ z^{i}\}_{i=1}^{\mathrm{dim}\mathcal{M}}$$ chosen on the moduli space $$\mathcal{M}$$.

5
This assumption is reasonable since these quantities have different singular behaviors when written in the canonical coordinates at the large complex structure.

6
See also [38, 29, 30, 33, 45, 31, 27, 1, 26, 4, 19, 6, 22, 21, 42, 5, 32, 7] for related works.

7
This is due to properties of special Kähler geometry and the particular form for the Picard-Fuchs equation, see  for details.

8
This is related to the ψ coordinate in  by $$z = (5\psi )^{-5}$$.

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