Abstract
We discuss the non-perturbative superpotential in \({{E}_{8}} \times {{E}_{8}}\) heterotic string theory on a non-simply connected Calabi–Yau manifold X, as well as on its simply connected covering space \(\tilde {X}.\) The superpotential is induced by the string wrapping holomorphic, isolated, genus zero curves. We show, in a specific example, that the superpotential is non-zero both on \(\tilde {X}\) and on X avoiding the no-go residue theorem of Beasley and Witten. On the non-simply connected manifold X, we explicitly compute the leading contribution to the superpotential from all holomorphic, isolated, genus zero curves with minimal area. The reason for the non-vanishing of the superpotental on X is that the second homology class contains a finite part called discrete torsion. As a result, the curves with the same area are distributed among different torsion classes and their contributions do not cancel each other.
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Notes
As usual, we split the six-dimensional vector index along \(X\) into its holomorphic and anti-holomorphic parts.
The results of Beasley and Witten are also expected to be valid for complete intersections in toric spaces.
For simplicity, we will work for a fixed complex structure which makes the Pfaffians to be the only non-trivial one-loop determinants.
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ACKNOWLEDGMENTS
This work was supported by the ARC Future Fellowship FT120100466. The author would like to thank B. A. Ovrut for collaborations.
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Buchbinder, E.I. Non-Perturbative Superpotentials and Discrete Torsion. Phys. Part. Nuclei 49, 835–840 (2018). https://doi.org/10.1134/S1063779618050088
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DOI: https://doi.org/10.1134/S1063779618050088