Abstract
We study to what extent Wilson lines in heterotic Calabi-Yau compactifications lead to non-trivial H-flux via Chern-Simons terms. Wilson lines are basic ingredients for Standard Model constructions but their induced H-flux may affect the consistency of the leading order background geometry and of the two-dimensional worldsheet theory. Moreover H-flux in heterotic compactifications would play an important role for moduli stabilization and could strongly constrain the supersymmetry breaking scale. We show how to compute H-flux and the corresponding superpotential, given an explicit complete intersection Calabi-Yau compactification and choice of Wilson lines. We do so by classifying special Lagrangian submanifolds in the Calabi-Yau, understanding how the Wilson lines project onto these submanifolds, and computing their Chern-Simons invariants. We illustrate our procedure with the quintic hypersurface as well as the split-bicubic, which can provide a potentially realistic three generation model.
Highlights
On vector bundles could lift many of the flat directions already at tree-level [8,9,10,11,12]
Discrete Wilson lines are a key ingredient in heterotic Standard Model constructions based on Calabi-Yau compactifications
They are introduced to break grand unified gauge groups down to the standard model whilst maintaining supersymmetry and the control that this provides. They can sometimes induce a non-trivial fractional H-flux via their Chern-Simons contributions, 18Note that this does not imply that all the A-type and C-type special Lagrangian submanifolds (sLags) are homologically equivalent, but only that the number of linearly independent homology elements covered by the cycles is at least zero
Summary
We will discuss two seemingly contradictory results that are important to bear in mind when considering H-flux in heterotic string compactifications. The gaugino condensate in 4D is expected to descend from a non-vanishing expectation value of Σ This would imply that gaugino condensation is not compatible with the supersymmetry conditions on CY internal spaces. Showed that an AdS supersymmetric solution is possible in the resulting 4D effective field theory provided that threshold corrections are taken into account so that the gauge coupling function takes the form f = S + βT,. If we have non-trivial H-flux together with a 10D fermion bilinear, both non-supersymmetric Minkowski × CY compactifications [30] and supersymmetric AdS × non-CY compactifications [19, 20, 26, 27, 36] are possible Matching these solutions to a corresponding solution obtained directly in 4D (with gaugino condensates) is non-trivial and not fully understood. A 10D construction of these solutions has so far not been obtained, as — at leading order — H-flux and fermion bilinears in the equations of motion are not compatible with vanishing supersymmetry transformations
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