Local Zeta Functions of Multiparameter Calabi-Yau Threefolds from the Picard-Fuchs Equations

TL;DR

This paper extends the deformation approach for computing zeta functions from one-parameter to multiparameter Calabi-Yau threefolds, introducing new formalism, addressing singularities, and providing practical computational tools with examples.

Contribution

It develops an improved formalism for multiparameter Calabi-Yau threefolds, enabling practical zeta function computations and addressing previous open issues.

Findings

Successfully computed zeta functions for complex multiparameter Calabi-Yau examples.

Reproduced and extended previous theoretical results.

Provided a Mathematica package for practical computations.

Abstract

The deformation approach of arXiv:2104.07816 for computing zeta functions of one-parameter Calabi-Yau threefolds is generalised to cover also multiparameter manifolds. Consideration of the multiparameter case requires the development of an improved formalism. This allows us, among other things, to make progress on some issues left open in previous work, such as the treatment of apparent and conifold singularities and changes of coordinates. We also discuss the efficient numerical computation of the zeta functions. As examples, we compute the zeta functions of the two-parameter mirror octic, a non-symmetric split of the quintic threefold also with two parameters, and the $S_5$ symmetric five-parameter Hulek-Verrill manifolds. These examples allow us to exhibit the several new types of geometries for which our methods make practical computations possible. They also act as consistency checks, as our results reproduce and extend those of arXiv:hep-th/0409202 and arXiv:math/0304169. To make the methods developed here more approachable, a Mathematica package "CY3Zeta" for computing the zeta functions of Calabi-Yau threefolds, which is attached to this paper, is presented.