Philip Candelas, Xenia de la Ossa, Pyry Kuusela, Joseph McGovern
SciPost Phys. 15, 146 (2023) ·
published 9 October 2023

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We find continuous families of supersymmetric flux vacua in IIB CalabiYau 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 weighttwo modular forms, these manifolds being said to be weighttwo 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 Ftheory on elliptically fibred CalabiYau fourfolds. If conjectural expressions for Deligne's periods are true, then these imply that the Ftheory fibre is complexisomorphic 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.
Philip Candelas, Xenia de la Ossa, Pyry Kuusela, Joseph McGovern
SciPost Phys. 15, 144 (2023) ·
published 6 October 2023

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We study the mirrors of fiveparameter CalabiYau threefolds first studied by Hulek and Verrill in the context of observed modular behaviour of the zeta functions for CalabiYau manifolds. Toric geometry allows for a simple explicit construction of these mirrors, which turn out to be familiar manifolds. These are elliptically fibred in multiple ways. By studying the singular fibres, we are able to identify the rational curves of low degree on the mirror manifolds. This verifies the mirror symmetry prediction obtained by studying the mirror map near large complex structure points. We undertake also an extensive study of the periods of the HulekVerrill manifolds and their monodromies. On the mirror, we compute the genuszero and one instanton numbers, which are labelled by 5 indices, as $h^{1,1}\!=5$. There is an obvious permutation symmetry on these indices, but in addition there is a surprising repetition of values. We trace this back to an $S_{6}$ symmetry made manifest by certain constructions of the complex structure moduli space of the HulekVerrill manifold. Among other consequences, we see in this way that the moduli space has six large complex structure limits. It is the freedom to expand the prepotential about any one of these points that leads to this symmetry in the instanton numbers. An intriguing fact is that the group that acts on the instanton numbers is larger than $S_6$ and is in fact an infinite hyperbolic Coxeter group, that we study. The group orbits have a "web" structure, and with certain qualifications the instanton numbers are only nonzero if they belong to what we term "positive webs". This structure has consequences for instanton numbers at all genera.