Monodromy calculatons of fourth order equations of Calabi-Yau type

TL;DR

This paper investigates the monodromy of specific fourth order differential equations of Calabi-Yau type, using numerical methods and homological mirror symmetry insights to explore their geometric interpretations and identify new Calabi-Yau examples.

Contribution

It provides a preliminary numerical study of monodromies for Calabi-Yau type equations and proposes conjectural identifications of new Picard-Fuchs equations and Calabi-Yau threefolds.

Findings

Determined monodromies for several models using numerical methods

Conjectured new Picard-Fuchs equations for five examples

Suggested existence of new Calabi-Yau threefolds

Abstract

This paper contains a preliminary study of the monodromy of certain fourth order differential equations, that were called of Calabi-Yau type in math.NT/0402386. Some of these equations can be interpreted as the Picard-Fuchs equations of a Calabi-Yau manifold with one complex modulus, which links up the observed integrality to the conjectured integrality of the Gopakumar-Vafa invariants. A natural question is if in the other cases such a geometrical interpretation is also possible. Our investigations of the monodromies are intended as a first step in answering this question. We use a numerical approach combined with some ideas from homological mirror symmetry to determine the monodromy for some further one-parameter models. Furthermore, we present a conjectural identification of the Picard-Fuchs equation for 5 new examples from Borcea's list and one constructed by Tonoli and conjecture the existence of some new Calabi-Yau three folds. The paper does not contain any theorems or proofs but is, we think, nevertheless of interest.