A one parameter family of Calabi-Yau manifolds with attractor points of rank two

Abstract

In the process of studying the ζ-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 ζ-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 ζ-function and are led to conjecture that the coefficients in this polynomial are distributed according to the statistics of random USp(4) matrices.