Counting Curves with Modular Forms

TL;DR

This paper explores the enumeration of algebraic curves in a specific Calabi-Yau space using modular forms, linking string theory compactification data to mathematical structures.

Contribution

It introduces a modular form framework to compute the prepotential in a string compactification on a degree 12 hypersurface in weighted projective space.

Findings

Prepotential expressed via modular covariant functions.

Functions determined by monodromy and singularity analysis.

Establishes symmetry constraints on the functions.

Abstract

We consider the type IIA string compactified on the Calabi-Yau space given by a degree 12 hypersurface in the weighted projective space ${\bf P}^4_{(1, 1, 2,2, 6)}$. We express the prepotential of the low-energy effective supergravity theory in terms of a set of functions that transform covariantly under $PSL(2, \mathbb{Z})$ modular transformations. These functions are then determined by monodromy properties, by singularities at the massless monopole point of the moduli space, and by $S \leftrightarrow T$ exchange symmetry.