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

by Philip Candelas, Xenia de la Ossa, Pyry Kuusela, Joseph McGovern

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Submission summary

Authors (as registered SciPost users): Pyry Kuusela · Joseph McGovern · Xenia de la Ossa
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-07-27
Date submitted: 2023-07-20 03:46
Submitted by: McGovern, Joseph
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


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.

Published as SciPost Phys. 15, 146 (2023)

Author comments upon resubmission

We thank the referees for their careful reading of the manuscript, helpful suggestions, and corrections.

We have incorporated all requested and suggested changes, apart from not adding Sturm bounds to our tables.

This is because we do not identify modular forms from a known conductor, and so feel that this inclusion would give a misleading impression of our search method. Since it is not strictly necessary in our approach, we do not see the utility in including Sturm bounds.

Best wishes,
Philip, Xenia, Pyry, and Joseph.

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Anonymous Report 1 on 2023-7-21 (Invited Report)


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Anonymous on 2023-07-23  [id 3830]

I recommend the paper for publication as it is.