Vector Bundle Moduli Superpotentials in Heterotic Superstrings and M-Theory

TL;DR

This paper develops a method to explicitly compute the non-perturbative superpotential in heterotic superstrings by analyzing the kernel of a linear map related to vector bundle moduli, enabling precise determination of superpotentials.

Contribution

It introduces a novel approach to compute the Pfaffian superpotential as a holomorphic function of vector bundle moduli using linear algebra techniques.

Findings

Explicit computation of the Pfaffian determines the superpotential.

The method applies to non-trivial examples, confirming its effectiveness.

Provides a complete characterization of the superpotential dependence on moduli.

Abstract

The non-perturbative superpotential generated by a heterotic superstring wrapped once around a genus-zero holomorphic curve is proportional to the Pfaffian involving the determinant of a Dirac operator on this curve. We show that the space of zero modes of this Dirac operator is the kernel of a linear mapping that is dependent on the associated vector bundle moduli. By explicitly computing the determinant of this map, one can deduce whether or not the dimension of the space of zero modes vanishes. It is shown that this information is sufficient to completely determine the Pfaffian and, hence, the non-perturbative superpotential as explicit holomorphic functions of the vector bundle moduli. This method is illustrated by a number of non-trivial examples.