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

TL;DR

This paper investigates flux vacua in type IIB Calabi-Yau compactifications with a $bZ_2$ symmetry, providing computational evidence for their modularity linked to elliptic curves and modular forms, and exploring implications for F-theory.

Contribution

It demonstrates the modularity of flux vacua in $bZ_2$ symmetric Calabi-Yau manifolds and extends methods to analyze their zeta functions and modular forms.

Findings

Numerators of local zeta functions have quadratic factors linked to weight-two modular forms.

Flux vacua are related to a continuous family of elliptic curves and can be lifted to F-theory.

Agreement between zeta function coefficients and modular form Fourier coefficients across multiple examples.

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.