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.
Highlights
Heterotic/F-theory duality has proven to be a robust and useful tool in the determination of Ftheory effective physics as well as a remarkable window into the structure of the string landscape
In this work we aim to broaden the consideration of background geometry of manifolds/bundles arising in heterotic compactifications with an aim towards extending the validity and understanding of heterotic/F-theory duality
We will focus on elliptically fibered Calabi-Yau geometries arising in heterotic theories in the context of the so-called Fourier Mukai transforms of vector bundles on elliptically fibered manifolds
Summary
Heterotic/F-theory duality has proven to be a robust and useful tool in the determination of Ftheory effective physics as well as a remarkable window into the structure of the string landscape. To be understood in the context of the fiber-wise duality (induced from 8-dimensional correspondence), the data of the vector bundle must be presented “fiber by fiber” in Xn over the base Bn−1 To this end, the work of Friedman, Morgan and Witten [3] introduced the tools of FourierMukai Transforms into heterotic theories. We generalize these results to the case of elliptically fibered manifolds with fibral divisors in Section 3 and geometries with additional sections to the elliptic fibration in Sections 4 and 5. For more information about the applications of Fourier-Mukai functors in studying the moduli space of stable sheaves over elliptically fibered manifolds, the interested reader is referred to [34]
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