Abstract
We study aspects of heterotic/F-theory duality for compactifications with Abelian gauge symmetries. We consider F-theory on general Calabi-Yau manifolds with a rank one Mordell-Weil group of rational sections. By rigorously performing the stable degeneration limit in a class of toric models, we derive both the Calabi-Yau geometry as well as the spectral cover describing the vector bundle in the heterotic dual theory. We carefully investigate the spectral cover employing the group law on the elliptic curve in the heterotic theory. We find in explicit examples that there are three different classes of heterotic duals that have U(1) factors in their low energy effective theories: split spectral covers describing bundles with S(U(m) x U(1)) structure group, spectral covers containing torsional sections that seem to give rise to bundles with SU(m) x Z_k structure group and bundles with purely non-Abelian structure groups having a centralizer in E_8 containing a U(1) factor. In the former two cases, it is required that the elliptic fibration on the heterotic side has a non-trivial Mordell-Weil group. While the number of geometrically massless U(1)'s is determined entirely by geometry on the F-theory side, on the heterotic side the correct number of U(1)'s is found by taking into account a Stuckelberg mechanism in the lower-dimensional effective theory. In geometry, this corresponds to the condition that sections in the two half K3 surfaces that arise in the stable degeneration limit of F-theory can be glued together globally.
Highlights
Introduction and summary of resultsThe study of effective theories of string theory in lower dimensions with minimal supersymmetry are both of conceptual and phenomenological relevance
We find three different classes of examples of how a Up1q gauge group is obtained in the heterotic string: one class of examples has a split spectral cover, which is a well-known ingredient for obtaining Up1q gauge groups in the heterotic literature starting with [33] and the F-theory literature, see e.g. [34,35,36]; another class of models have a spectral cover containing a torsional section of the heterotic Calabi-Yau manifold Zn, where duality suggests that this should describe zero-size instantons of discrete holonomy, as considered in [37]; in a last set of examples, the Up1q arises as the commutant inside E8 of vector bundles with purely non-Abelian structure groups
This paper is organized in the following way: in section 2, we provide a brief review of the key points of heterotic/F-theory duality as well as a discussion of the new insights gained in this work into spectral covers and half K3-fibrations for vector bundles with non- connected structure groups
Summary
The study of effective theories of string theory in lower dimensions with minimal supersymmetry are both of conceptual and phenomenological relevance. We will apply the simple and unifying description on the F-theory side in terms of elliptically fibered Calabi-Yau manifolds Xn1 to study explicitly, using stable degeneration, the structure of spectral covers yielding heterotic vector bundles that give rise to Up1q gauge symmetry in the lower-dimensional effective theory, continuing the analysis explained in the 2010 talk [21].1. [34,35,36]; another class of models have a spectral cover containing a torsional section of the heterotic Calabi-Yau manifold Zn, where duality suggests that this should describe zero-size instantons of discrete holonomy, as considered in [37]; in a last set of examples, the Up1q arises as the commutant inside E8 of vector bundles with purely non-Abelian structure groups. Lower-dimensional dualities are obtained, applying the adiabatic argument [40], by fibering the eight-dimensional duality over a base manifold Bn1 of complex dimension n 1 that is common to both theories of the duality
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