Modular Vector Fields Attached to Dwork Family: s l 2 ( C ) Lie algebra

Abstract

This paper aims to show that a certain moduli space T, which arises from the so-called Dwork family of Calabi-Yau n-folds, carries a special complex Lie algebra containing a copy of sl 2 (C). In order to achieve this goal, we introduce an algebraic group G acting from the right on T and describe its Lie algebra Lie(G). We observe that Lie(G) is isomorphic to a Lie subalgebra of the space of the vector fields on T. In this way, it turns out that Lie(G) and the modular vector field R generate another Lie algebra G, called AMSY-Lie algebra, satisfying dim(G) = dim(T). We find a copy of sl 2 (C) containing R as a Lie subalgebra of G. The proofs are based on an algebraic method calling "Gauss-Manin connection in disguise". Some explicit examples for n = 1, 2, 3, 4 are stated as well.

A survey on modular vector fields and CY modular forms attached to Dwork family

“…To understand better the GMCD one can start reading the paper [5] which applies the method to the families of elliptic curves, and then continue with the paper [6] or the book [7]. The author in a joint work with Movasati [8] applied GMCD to a family of CY n-folds, n ∈ N, arising from the Dwork family, and then pushed the studies forward in subsequent papers [9,10,11].…”

“…The present article gives a survey of [8,9,10,11] and states explicitly the essential ingredients and objects in dimensions n = 1, 2, 3, 4.…”

“…The present article is prepared as follows. In Section 2 we first construct a moduli space arising from the Dwork family, and then establish and discuss the main results of [8,9,10] for any dimension n. We state explicitly the modular vector field, the associated sl 2 (C)-module structure, solution components and some other related facts in dimensions n = 1, 2, 3, 4, respectively, in Section 3, Section 4, Section 5, Section 6.…”

“…We conjecture that component solutions of D can be expressed as q-expansions with integer coefficients (this is verified for n = 1, 2, 3, 4, which are stated in the next sections). For all n, we can find vector fields W := W n and δ := δ n in T which along with D form a copy of sl 2 (C) (see [9,Theorem 1.4]), i.e. :…”

“…where [•, •] refers to the Lie bracket of vector fields. Note that the vector fields D, W, δ in [8,9,10,11] were denoted by R, H, F, respectively. Indeed, we observe that W and δ are in the following forms:…”