Flux Vacua and Modularity for ${\mathbb{Z}}_2$ Symmetric Calabi-Yau Manifolds

Abstract

We find continuous families of supersymmetric flux vacua in IIB Calabi-Yau compactifications for multiparameter manifolds with an appropriate ${\mathbb{Z}}_2$ symmetry. We argue, supported by extensive computational evidence, that the numerators of the local zeta functions of these compactification manifolds have quadratic factors. These factors are associated with weight-two modular forms, and these manifolds are said to be weight-two modular. Our evidence supports the flux modularity conjecture of Kachru, Nally, and Yang. The modular forms are related to a continuous family of elliptic curves. The flux vacua can be lifted to F-theory on elliptically fibred Calabi-Yau fourfolds. If conjectural expressions for Deligne's periods are true, then these imply that the F-theory fibre is complex-isomorphic to the modular curve. In three examples, we compute the local zeta function of the internal geometry using an extension of known methods, which we discuss here and in more detail in a companion paper. With these techniques, we are able to compare the zeta function coefficients to modular form Fourier coefficients for hundreds of manifolds in three distinct families, finding agreement in all cases. Our techniques enable us to study not only parameters valued in $\mathbb{Q}$ but also in algebraic extensions of $\mathbb{Q}$, so exhibiting relations to Hilbert and Bianchi modular forms. We present in appendices the zeta function numerators of these manifolds, together with the corresponding modular forms.

includes 61 pages of tables; v2: typos and notation corrected, a section on conventions added, references added

References in corpus (9)

• Flux Compactification
• Elliptic integral evaluations of Bessel moments
• New F-theory lifts
• The Langlands Program and String Modular K3 Surfaces
• D(-1)-Instanton Superpotential In String Theory
• Time reversal and $\mathbf{CP}$ invariance in Calabi-Yau compactifications
• Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations
• Mirror Symmetry for Five-Parameter Hulek-Verrill Manifolds
• D-brane masses at special fibres of hypergeometric families of Calabi-Yau threefolds, modular forms, and periods

Cited by in corpus (3)

• A Calabi-Yau-to-Curve Correspondence for Feynman Integrals
• Flux vacua in type IIB compactifications on orbifolds: their finiteness and minimal string coupling
• Classical Weight-Four L-value Ratios as Sums of Calabi--Yau Invariants