Spectral Covers, Integrality Conditions, and Heterotic/F-theory Duality

TL;DR

This paper explores the duality between heterotic string theory and F-theory, focusing on Calabi-Yau fourfolds, spectral cover constructions, and integrality conditions, revealing new geometric challenges and potential resolutions.

Contribution

It introduces a systematic algorithmic approach to construct dual geometries and investigates cases where the standard duality map fails, analyzing integrality conditions and Picard groups.

Findings

Identified cases where dual heterotic and F-theory geometries appear incompatible.

Analyzed the integrality condition of spectral covers and its geometric implications.

Computed Hodge numbers and bounded the Picard number of spectral surfaces.

Abstract

In this work we review a systematic, algorithmic construction of dual heterotic/F-theory geometries corresponding to 4-dimensional, N = 1 supersymmetric compactifications. We look in detail at a class of well-defined Calabi-Yau fourfolds for which the standard formulation of the duality map appears to fail, leading to dual heterotic geometry which appears naively incompatible with the spectral cover construction of vector bundles. In the simplest class of examples the F-theory background consists of a generically singular elliptically fibered Calabi-Yau fourfold with E7 symmetry. The vector bundles arising in the corresponding heterotic theory appear to violate an integrality condition of an SU(2) spectral cover. A possible resolution of this puzzle is explored by studying the most general form of the integrality condition. This leads to the geometric challenge of determining the Picard group of surfaces of general type. We take an important first step in this direction by computing the Hodge numbers of an explicit spectral surface and bounding the Picard number.