Geometric structures on moduli spaces of special lagrangian submanifolds

Abstract

Let M, N be a pair of Calabi-Yau 3-folds: we say that N is a mirror partner of M if the Amodel string theory of M is isomorphic to the B-model string theory of N and so, in particular, conformal field theories associated to mirror partners are equivalent; this is a general pattern, up to now lacking precise mathematical formulation, one aspect of which is, roughly speaking, that ~counting the numbers of rational curves of various degree in M~ is the same as ~calculating the period integrals of holomorphic forms in N~ (and, consequently, we have the famous relations between the Hodge numbers of the mirror parmers) (cf. e. g. [7, 10, 11]). Few years ago, Andrew Strominger, Shing-Tung Yau and Eric Zaslow [12] argued that every Calabi-Yau 3-fold M with mirror partner N admits a family of <~supersymmetric toroidal 3-cycles)) (i.e. special Lagrangian tori equipped with fiat U( 1 )-bundles) and N arises precisely as the compactification of the moduli space of such cycles, the latter being nothing but the complexification of the moduli space of special Lagrangian tori. Immediately after, Nigel Hitchin ([6]) provided an excellent description of SYZ construction for arbitrary dimension. Pushing a bit forward Hitchin's result, in the present paper we show how the non compactified moduli space of special Lagrangian submanifols and its complexification can be endowed with a very tamed geometric structure; in particular, they allow atlases of local coordinates whose transition functions are restrictions of elements of a finite presentation subgroup of the special linear group (see (2.13) and (4.3)). This fact seems to be of general interest: in fact, on one hand, compactification at the present time appears out of reach, on the other hand, the geometric structure allows us to