Flux Modularity, F-Theory, and Rational Models

Abstract

In recent work, we conjectured that Calabi-Yau threefolds defined over $\mathbb{Q}$ and admitting a supersymmetric flux compactification are modular, and associated to (the Tate twists of) weight-two cuspidal Hecke eigenforms. In this work, we will address two natural follow-up questions, of both a physical and mathematical nature, that are surprisingly closely related. First, in passing from a complex manifold to a rational variety, as we must do to study modularity, we are implicitly choosing a "rational model" for the threefold; how do different choices of rational model affect our results? Second, the same modular forms are associated to elliptic curves over $\mathbb{Q}$; are these elliptic curves found anywhere in the physical setup? By studying the F-theory uplift of the supersymmetric flux vacua found in the compactification of IIB string theory on (the mirror of) the Calabi-Yau hypersurface $X$ in $\mathbb{P}(1,1,2,2,2)$, we find a one-parameter family of elliptic curves whose associated eigenforms exactly match those associated to $X$. Actually, we find two such families, corresponding to two different choices of rational models for the same family of Calabi-Yaus.

arXiv comment25 pages, 1 figure, 6 tables. v2: Minor changes, typos corrected

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• Flux Compactifications on Calabi-Yau Threefolds
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• String-theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication
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• Flux Vacua and Modularity for ${\mathbb{Z}}_2$ Symmetric Calabi-Yau Manifolds
• W=0 Complex Structure Moduli Stabilization on CM-type K3 x K3 Orbifolds:---Arithmetic, Geometry and Particle Physics---
• Classical Weight-Four L-value Ratios as Sums of Calabi--Yau Invariants