A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two

TL;DR

The authors study a one-parameter Calabi–Yau family (AESZ34) and uncover persistent factorisations of the zeta-function numerator into two quadrics at special algebraic parameter values, signaling rank-two attractor points. They show these points correspond to a splitting of the H^3 Hodge structure and are tied to modular forms of weights 2 and 4 via the associated Frobenius polynomials, with the periods at attractor values expressible through critical L-values. The work combines Picard–Fuchs period calculations, Dwork-type deformation methods, and arithmetic in quadratic fields to exhibit explicit period relations and instanton identities, and it discusses possible geometric origins of the splitting in terms of a self-correspondence and elliptic modular surfaces. The results illuminate deep connections between Calabi–Yau geometry, number theory (Serre modularity, L-values), and black-hole physics, and they suggest probabilistic structure (random USp(4) statistics) in Frobenius data for these attractor points.

Abstract

In the process of studying the zeta-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, first studied by Verrill, for which the quartic numerator of the zeta-function factorises into two quadrics remarkably often. Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in Q, so independent of any particular prime. Such factorisations are expected to be modular with each quadratic factor associated to a modular form. If the parameter is defined over Q this modularity is assured by the proof of the Serre Conjecture. We identify three values of the parameter that give rise to persistent factorisations, one of which is defined over Q, and identify, for all three cases, the associated modular groups. We note that these factorisations are due a splitting of Hodge structure and that these special values of the parameter are rank two attractor points in the sense of IIB supergravity. To our knowledge, these points provide the first explicit examples of non-singular, non-rigid rank two attractor points for Calabi-Yau manifolds of full SU(3) holonomy. The values of the periods and their covariant derivatives, at the attractor points, are identified in terms of critical values of the L-functions of the modular groups. Thus the critical L-values enter into the calculation of physical quantities such as the area of the black hole in the 4D spacetime. In our search for additional rank two attractor points, we perform a statistical analysis of the numerator of the zeta-function and are led to conjecture that the coefficients in this polynomial are distributed according to the statistics of random USp(4) matrices.

Figures (25)

• Figure 1: The four-loop banana graph that is related to the Hulek-Verrill manifold.
• Figure 2: Attractor flow associated to the charge vector $Q=(4,-15,-5,0)$ in the $\varphi$-plane. The red dot represents the attractor point $\varphi=-1/7$, the hollow black dot is the large complex structure point $\varphi=0$ and the solid black dots represent the two nearest conifolds at $\varphi=1/25$ and $\varphi=1/9$. The flow lines are discontinuous across branch cuts which illustrates the fact that the flow takes place on a Riemann surface that is a multi-sheeted cover of the $\varphi$-plane.
• Figure 3: A sketch of the (four dimensional) space $H^3(X,\mathbbl{R})$ for generic $\varphi$, showing the two planes generated by $\text{Re}\,\Omega$ and $\text{Im}\,\Omega$ and by charge vectors $\Gamma_1$ and $\Gamma_2$. As $\varphi$ varies, the plane generated by $\text{Re}\,\Omega$ and $\text{Im}\,\Omega$ moves and, when $\varphi{\;=\;}\varphi_*$ is an attractor point of rank two, the two planes coincide.
• Figure 4: The flows for $\varphi=\varphi(\rho)$ for the charges $Q=(0,0,2,1)$ (above) and $Q=(-4,15,5,0)$ (below) leading to attractor point at $\varphi{\;=\;}-1/7$. The point of maximal unipotent monodromy at $\varphi=0$ is indicated by a hollow black dot while the solid black dots represent conifold singularities.
• Figure 5: The upper plot shows the number of factorisations into two quadrics as $\varphi$ varies over each $\mathbbl{F}_p$, $7\leq p\leq3583$, for the manifold AESZ34. For comparison, the lower plot provides the same information for the mirror of the quintic which explains why it is difficult to find rank two attractor points on this family.