[Paper Review] D-brane masses at special fibres of hypergeometric families of Calabi-Yau threefolds, modular forms, and periods

Kilian Bönisch, Albrecht Klemm+2 more|arXiv (Cornell University)|Mar 17, 2022

Advanced Algebra and Geometry4 citations

TL;DR

This paper establishes a precise correspondence between D-brane masses in mirror-symmetric Calabi-Yau threefolds and periods of modular forms arising from arithmetic properties of special fibers—particularly conifold and rank-two attractor points—using Picard-Fuchs equations, modular forms, and zeta functions. It numerically identifies period matrix entries at these special fibers as periods and quasiperiods of modular forms, proving the correspondence in one case via a Kuga-Sato variety construction.

ABSTRACT

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 $\mathbb{P}^4$. 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 $H_3(W,\mathbb{Z})$. 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.

Motivation & Objective

• To understand the arithmetic and geometric origin of D-brane masses in one-parameter families of Calabi-Yau threefolds via mirror symmetry.
• To identify the periods of holomorphic (3,0)-forms at special fibers (conifold and attractor points) with modular forms arising from Hasse-Weil zeta functions.
• To establish a numerical and, in one case, analytical correspondence between period matrix entries and modular form periods/quasiperiods.
• To explore the role of modular forms of weight 2 and 4 in the context of rank-two attractor points in hypergeometric Calabi-Yau families.
• To extend these results to local Calabi-Yau threefolds with third-order Picard-Fuchs operators.

Proposed method

• Utilizes the Picard-Fuchs differential equations of hypergeometric type with four solutions, derived from one-parameter families of Calabi-Yau threefolds.
• Applies mirror symmetry to relate D-brane masses to periods of the holomorphic (3,0)-form over a symplectic basis of H₃(W, ℤ).
• Employs the Hasse-Weil zeta function obtained from point-counting over finite fields to extract Hecke eigenforms of weight 4 and 2.
• Uses Barnes integral representations to compute transition matrices T∞ and relates them to Gamma function values.
• Applies special geometry and Legendre relations to reduce numerical constants in the period matrix to fewer independent parameters.
• Constructs a correspondence between the conifold fiber and a Kuga-Sato variety in one case to prove the modular period identification analytically.

Experimental results

Research questions

• RQ1 How do D-brane masses in mirror-symmetric Calabi-Yau threefolds relate to modular forms arising from special fibers?
• RQ2 What is the arithmetic origin of the periods and quasiperiods of modular forms at conifold and attractor points in hypergeometric families?
• RQ3 Can the period matrix entries at special fibers be identified as periods and quasiperiods of modular forms, and if so, under what conditions?
• RQ4 What is the role of the Hasse-Weil zeta function in generating modular forms of weight 4 and 2 at rank-two attractor points?
• RQ5 To what extent can the correspondence between Calabi-Yau periods and modular forms be extended to local Calabi-Yau threefolds?

Key findings

• For fourteen hypergeometric one-parameter Calabi-Yau threefolds, the period matrix entries at conifold fibers are numerically identified as periods and quasiperiods of weight-four modular forms under Γ₀(N).
• At two special rank-two attractor points, the Hasse-Weil zeta function yields modular forms of weight 4 and 2, whose periods match entries in the period matrix.
• In one case (N=8), the correspondence between the conifold fiber and a Kuga-Sato variety is constructed explicitly, proving that the period matrix entries are indeed periods of a modular form.
• The paper computes period polynomials and approximate values of periods and quasiperiods for newforms of weight 4 and 2 at various levels N, including N=25, 54, 180, and 864.
• For N=25, the period matrix entries are found to be approximately 0.092748402 and 0.266377323i, matching the period of a cusp form of weight 4.
• The paper provides explicit q-expansions and normalization data for 14 newforms of weight 4 and one of weight 8, with maximal vanishing order at ∞, confirming their modular nature.