On the Heterotic World-sheet Instanton Superpotential and its individual Contributions

TL;DR

This paper investigates the conditions under which the superpotential contributions from rational curves in heterotic string compactifications vanish simultaneously, revealing cases where these conditions are conceptually compatible and related to moduli space transitions.

Contribution

It demonstrates that for certain cases, the zero conditions of individual Pfaffians can be simultaneously satisfied, especially in spectral cover bundles, linking to moduli space transitions.

Findings

Conditions for simultaneous vanishing of contributions are identified.

Spectral cover bundles exhibit special moduli space solutions.

These solutions relate to transitions changing the generation number.

Abstract

For supersymmetric heterotic string compactifications on a Calabi-Yau threefold $X$ endowed with a vector bundle $V$ the world-sheet superpotential $W$ is a sum of contributions from isolated rational curves $\C$ in $X$; the individual contribution is given by an exponential in the K\"ahler class of the curve times a prefactor given essentially by the Pfaffian which depends on the moduli of $V$ and the complex structure moduli of $X$. Solutions of $DW=0$ (or even of $DW=W=0$) can arise either by nontrivial cancellations between the individual terms in the summation over all contributing curves or because each of these terms is zero already individually. Concerning the latter case conditions on the moduli making a single Pfaffian vanish (for special moduli values) have been investigated. However, even if corresponding moduli - fulfilling these constraints - for the individual contribution of one curve are known it is not at all clear whether {\em one} choice of moduli exists which fulfills the corresponding constraints {\em for all contributing curves simultaneously}. Clearly this will in general happen only if the conditions on the 'individual zeroes' had already a conceptual origin which allows them to fit together consistently. We show that this happens for a class of cases. In the special case of spectral cover bundles we show that a relevant solution set has an interesting location in moduli space and is related to transitions which change the generation number.