The Master Equation for the Prepotential

TL;DR

This paper derives the one-loop correction to the N=2 prepotential in heterotic string compactifications using a direct method, providing new differential equations and modular property insights for various models.

Contribution

It introduces a direct calculation method for the one-loop prepotential in heterotic strings, extending previous amplitude-based results and including differential equations and modular analysis.

Findings

Derived the differential equation for the third derivative of the prepotential.

Calculated the one-loop prepotential using modular properties.

Extended results to models with specific modular groups and non-decomposable lattices.

Abstract

The perturbative prepotential and the K\"ahler metric of the vector multiplets of the N=2 effective low-energy heterotic strings is calculated directly in N=1 six-dimensional toroidal compactifications of the heterotic string vacua. This method provides the solution for the one loop correction to the N=2 vector multiplet prepotential for compactifications of the heterotic string for any rank three and four models, as well for compactifications on $K_3 \times T^2$. In addition, we complete previous calculations, derived from string amplitudes, by deriving the differential equation for the third derivative of the prepotential with respect of the usual complex structure U moduli of the $T^2$ torus. Moreover, we calculate the one loop prepotential, using its modular properties, for N=2 compactifications of the heterotic string exhibiting modular groups similar with those appearing in N=2 sectors of N=1 orbifolds based on non-decomposable torus lattices and on N=2 supersymmetric Yang-Mills.