Abstract
We discuss the period geometry and the topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homological mirror symmetry we then obtain Bridgelands involution and its monodromy action on the integral basis for non-singular elliptically fibered fourfolds. The full monodromy group contains a subgroup that acts as PSL(2,Z) on the Kähler modulus of the fiber and we analyze the consequences of this modularity for the genus zero and genus one amplitudes as well as the associated geometric invariants. We find holomorphic anomaly equations for the amplitudes, reflecting precisely the failure of exact PSL(2,Z) invariance that relates them to quasi-modular forms. Finally we use the integral basis of periods to study the horizontal flux superpotential and the leading order Kähler potential for the moduli fields in F-theory compactifications globally on the complex structure moduli space. For a particular example we verify attractor behaviour at the generic conifold given an aligned choice of flux which we expect to be universal. Furthermore we analyze the superpotential at the orbifold points but find no stable vacua.
Highlights
At present F-theory compactifications on elliptic Calabi-Yau fourfolds provide the richest class of explicit N = 1 effective theories starting from string theory
Ignoring the details of how this happens for the concrete geometry under consideration it has been shown that by degenerating the fourfold in a controlled way viable phenomenological low energy particle spectra will emerge in four dimensions as was worked out in the F-theory revival starting with the papers of [3,4,5,6]
We described a very efficient method to obtain the integral flux superpotential using the central charge formula defined in terms of the Γclass
Summary
We study the global structure of the properly quantized horizontal flux superpotential for a particular example To this end we analytically continue the integral periods to the generic conifold locus, the generic orbifold and the Gepner point. Note added: after this article appeared on the arxiv, Georg Oberdieck pointed out that our results in section 4.5 match with his and Aaron Pixton’s conjectured holomorphic anomaly equation on Calabi-Yau n-folds appearing in [23, 24]. He explained to us the explicit form of the generalized holomorphic anomaly equation for the Gromov-Witten potentials on Calabi-Yau fourfolds, which we include in appendix B. We performed further non-trivial checks of his conjecture with our data beyond the material that appeared already in appendix A.5
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