Mirror symmetry via 3-tori for a class of Calabi-Yau threefolds

Abstract

The authors of [16] have proposed a conjectural construction of mirror symmetry for Calabi-Yau threefolds. They argue from the physics that in a neighbourhood of the large complex structure limit (see [11] for the definition of large complex structure limits), any Calabi-Yau threefold X with a mirror Y should admit a family of supersymmetric toroidal 3-cycles. In mathematical terminology, this says that there should be a fibration on X whose general fibre is a special Lagrangian 3-torus T 3 .We recall from [8] the notion of a special Lagrangian submanifold.Definition. Let X be a Kähler manifold of dimension n, with complex structure I and Kähler metric g, whose holonomy is contained in SU (n). Recall that this latter condition is equivalent to the existence of a covariant constant holomorphic n-form Ω on X. We say that Ω is normalized if (−1) n(n−1)/2 (i/2) n Ω ∧Ω = ω n /n! the volume form on X, where ω denotes the Kähler form -thus the normalized holomorphic n-form on X is unique up to a phase factor e iθ . We say that a submanifold M of X is special Lagrangian (we shall take M to be embedded, although more generally one only needs it to be immersed) if a normalized holomorphic n-form Ω can be chosen on X whose real part Re Ω restricts to the volume form on M -the normalization has been chosen here so that Re Ω is a calibration in the sense of [8]. This definition is equivalent to M being an oriented submanifold of real dimension n such that the restriction of ω to M is zero, i.e. M is Lagrangian, together with the existence of a holomorphic n-form Ω whose imaginary part Im Ω also restricts to zero on M [8]. Given a special Lagrangian submanifold M of X, it is shown in [10] that the deformations of M as a special Lagrangian submanifold are unobstructed and that the tangent * Supported in part by NSF grant DMS-9400873 and Trinity College, Cambridge space to the local deformation space may be identified with the harmonic 1-forms on M . The dimension of the local deformation space is therefore dim R H 1 (M, R). For X aCalabi-Yau n-fold and T a special Lagrangian n-torus submanifold, T therefore moves in an n-dimensional family.Definition. A Calabi-Yau n-fold X is said to have a special Lagrangian n-torus fibration if there exists a map of topological manifolds f : X → B whose general fibre is a special Lagrangian n-torus.We have taken B as a topological manifold, but from the results of [10], it will be locally a differentiable manifold with a natural Riemannian metric [10], (3.10), except perhaps at points corresponding to singular fibres. In §3, we shall generalise the notion of special Lagrangian n-torus fibration to the case when the metric on X is allowed to degenerate in a suitably nice way.Let us now take X to be a Calabi-Yau threefold. If X contains a special Lagrangian T 3 , then it moves locally in a three dimensional family. There are hard questions which need to be addressed concerning whether the family obtained foliates the manifold and whether it can be suitably compactified so as to obtain a special...

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