Abstract
We present models of heterotic compactification on Calabi-Yau threefolds and compute the non-perturbative superpotential for vector bundle moduli. The key feature of these models is that the threefolds, which are elliptically fibered over del Pezzo surfaces, have homology classes with a unique holomorphic, isolated genus-zero curve. Using the spectral cover construction, we present vector bundles for which we can explicitly calculate the Pfaffians associated with string instantons on these curves. These are shown to be non-zero, thus leading to a non-vanishing superpotential in the 4D effective action. We discuss, in detail, why such compactifications avoid the Beasley-Witten residue theorem.
Highlights
As we will outline in detail, it has been much more difficult to calculate the potentials for the moduli associated with the holomorphic vector bundles on the CalabiYau threefold
We present models of heterotic compactification on Calabi-Yau threefolds and compute the non-perturbative superpotential for vector bundle moduli
Using a “residue theorem” [36], these authors have shown that, for vacua satisfying certain explicit properties which we list below, the worldsheet instanton contributions to the superpotential arising from different isolated rational curves with the same complexified area AC, that is, curves in the same homology class, must sum to zero
Summary
Throughout these notes, we will consider Calabi-Yau manifolds X →π B that are elliptically fibered over complex surface B. If the base is a del Pezzo surface dPr, the spectral cover is irreducible if and only if the following conditions are satisfied [49]: 1) η · E ≥ 0 for any generator E of the Mori cone 2) η − n KB is effective in B, with KB the anti-canonical divisor of B. To gain some intuition for the Fourier-Mukai transformation, we note that the restriction of V to the elliptic fiber is a sum of n degree zero line bundles Li. On an elliptic curve, such a line bundle is always dual to a divisor of the form Qi − p, where p is the origin, i.e., the point marked by the zero section.
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