Vacuum varieties, holomorphic bundles and complex structure stabilization in heterotic theories

Abstract

We discuss the use of gauge fields to stabilize complex structure moduli in Calabi-Yau three-fold compactifications of heterotic string and M-theory. The requirement that the gauge fields in such models preserve supersymmetry leads to a complicated landscape of vacua in complex structure moduli space. We develop methods to systematically map out this multi-branched vacuum space, in a computable and explicit manner. In analysing the resulting vacua, it is found that the associated Calabi-Yau three-folds are sometimes stabilized at a value of complex structure resulting in a singular compactification manifold. We describe how it is possible to resolve these singularities, in some cases, while maintaining computational control over the moduli stabilization mechanism. The discussion is illustrated throughout the paper with explicit worked examples.

Gauge backgrounds and zero-mode counting in F-theory

“…They are the manifestation of the lifting of vectorlike pairs as we vary the vacuum expectation value of some of the chiral fields of the model. Analogous effects have been studied intensively for heterotic compactifications such as [115][116][117] and references therein. It would be exciting to determine the minimal number of vectorlike pairs for a given topological type of F-theory model as we vary the complex structure moduli and to interpret this result from an effective field theory point of view.…”

Algebroids, heterotic moduli spaces and the Strominger system

“…In order to explain the structure we will present, it is helpful to make a comparison to a case which is already well known in the literature -that of the Atiyah class stabilization of complex structure moduli in Calabi-Yau threefold compactifications of heterotic theories [2][3][4]49]. Gauge fields in such a compactification must obey the Hermitian Yang-Mills equation at zero slope:…”

“…The constraint of bundle holomorphy, equation (2.19), has already be analyzed, in the fashion being discussed here, in the literature [2][3][4]. In addition, it will be useful in what follows to add the equation describing how the holomorphic tangent bundle remains holomorphic under deformations.…”

“…Naively, the moduli of such a heterotic solution are the metric and bundle moduli, which, via theorems by Yau, Donaldson, and Uhlenbeck and Yau, are counted by the dimensions of the cohomology groups H 1 (T X), H 1 (T X ∨ ) and H 1 (End 0 (V )). Even in this case, there are some subtleties in counting the number of massless fields due to the gauge field structure in the problem, as has recently been discussed in [2][3][4].…”

The moduli space of heterotic line bundle models: a case study for the tetra-quadric

“…The interested reader can find in ref. [26] an outline of the basic techniques for computing line bundle cohomology on complete intersection CalabiYau manifolds in products of projective spaces. We find that the map H 0 (X, B ⊗ C) −→ H 0 (X, S 2 C) has rank 94, while the map between H 1 (X, B ⊗ C) −→ H 1 (X, S 2 C) has maximal rank 48.…”