D-brane Masses at Special Fibres of Hypergeometric Families of Calabi–Yau Threefolds, Modular Forms, and Periods

• Published in: Communications in mathematical physics 2024-06, Vol.405 (6), Article 134
• Main Authors: Bönisch, Kilian, Klemm, Albrecht, Scheidegger, Emanuel, Zagier, Don
• Format: Article
• Language: English
• Subjects: Analytic functions; Branes; Classical and Quantum Gravitation; Complex Systems; Eigenvalues; Fields (mathematics); Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-024-05006-6

We consider the fourteen families W of Calabi–Yau threefolds with one complex structure parameter and Picard–Fuchs equation of hypergeometric type, like the mirror of the quintic in P 4 . Mirror symmetry identifies the masses of even-dimensional D-branes of the mirror Calabi–Yau M with four periods of the holomorphic (3, 0)-form over a symplectic basis of H 3 ( W , Z ) . It was discovered by Chad Schoen that the singular fiber at the conifold of the quintic gives rise to a Hecke eigenform of weight four under Γ 0 ( 25 ) , whose Hecke eigenvalues are determined by the Hasse–Weil zeta function which can be obtained by counting points of that fiber over finite fields. Similar features are known for the thirteen other cases. In two cases we further find special regular points, so called rank two attractor points, where the Hasse–Weil zeta function gives rise to modular forms of weight four and two. We numerically identify entries of the period matrix at these special fibers as periods and quasiperiods of the associated modular forms. In one case we prove this by constructing a correspondence between the conifold fiber and a Kuga–Sato variety. We also comment on simpler applications to local Calabi–Yau threefolds.