Extending the Geometry of Heterotic Spectral Cover Constructions

TL;DR

This paper generalizes the spectral cover construction for vector bundles on elliptically fibered Calabi-Yau manifolds, including cases with fibral divisors and multiple sections, using Fourier-Mukai techniques to expand heterotic/F-theory duality understanding.

Contribution

It introduces a geometric formalism extending spectral cover methods to more complex fibrations with multiple sections and fibral divisors, employing Fourier-Mukai transforms.

Findings

Developed new Fourier-Mukai tools for stable bundles

Constructed examples of chirality-changing small instanton transitions

Extended spectral cover formalism to non-Weierstrass fibrations

Abstract

In this work we extend the well-known spectral cover construction first developed by Friedman, Morgan, and Witten to describe more general vector bundles on elliptically fibered Calabi-Yau geometries. In particular, we consider the case in which the Calabi-Yau fibration is not in Weierstrass form, but can rather contain fibral divisors or multiple sections (i.e. a higher rank Mordell-Weil group). In these cases, general vector bundles defined over such Calabi-Yau manifolds cannot be described by ordinary spectral data. To accomplish this we employ well established tools from the mathematics literature of Fourier-Mukai functors. We also generalize existing tools for explicitly computing Fourier-Mukai transforms of stable bundles on elliptic Calabi-Yau manifolds. As an example of these new tools we produce novel examples of chirality changing small instanton transitions. The goal of this work is to provide a geometric formalism that can substantially increase the understood regimes of heterotic/F-theory duality.