Instantons and Hilbert Functions

TL;DR

This paper investigates conditions under which superpotentials from worldsheet instantons in heterotic Calabi-Yau compactifications are non-zero, revealing rarity in some cases and commonality in others, based on algebraic criteria.

Contribution

It derives necessary conditions for non-vanishing instanton superpotentials and introduces a sufficient criterion using affine Hilbert functions for certain vector bundles.

Findings

Non-vanishing superpotentials are rare and require specific bundle patterns.

A sufficient criterion for non-vanishing superpotentials is established using affine Hilbert functions.

The criterion is computationally accessible via commutative algebra methods.

Abstract

We study superpotentials from worldsheet instantons in heterotic Calabi-Yau compactifications for vector bundles constructed from line bundle sums, monads and extensions. Within a certain class of manifolds and for certain second homology classes, we derive simple necessary conditions for a non-vanishing instanton superpotential. These show that non-vanishing instanton superpotentials are rare and require a specific pattern for the bundle construction. For the class of monad and extension bundles with this pattern, we derive a sufficient criterion for non-vanishing instanton superpotentials based on an affine Hilbert function. This criterion shows that a non-zero instanton superpotential is common within this class. The criterion can be checked using commutative algebra methods only and depends on the topological data defining the Calabi-Yau X and the vector bundle V.