Monodromy calculations of fourth order equations of Calabi-Yau type
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Bibliographic record
Abstract
This paper contains a preliminary study of the monodromy of certain fourth order differential equations, that were called of Calabi-Yau type in [3]. 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 oneparameter models. Furthermore, we present a conjectural identification of the Picard-Fuchs equation for 5 new examples from Borceas list 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. 1
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