Special Lagrangian Fibrations I: Topology

TL;DR

This paper explores the topological aspects of special Lagrangian fibrations in Calabi-Yau manifolds, assuming their existence, and investigates the relationships between their cohomologies, spectral sequences, and mirror symmetry conjectures.

Contribution

It provides topological insights into mirror symmetry by analyzing cohomology, spectral sequences, and couplings under the assumption of existing special Lagrangian fibrations.

Findings

Large complex radius limit couplings match between mirror pairs.

Leray and weight filtrations coincide in three-dimensional cases.

Results depend on a conjecture about monodromy diffeomorphisms at large complex structure limits.

Abstract

In 1996, Strominger, Yau and Zaslow made a conjecture about the geometric relationship between two mirror Calabi-Yau manifolds. Roughly put, if X and Y are a mirror pair of such manifolds, then X should possess a special Lagrangian torus fibration $f:X\to B$ such that Y is obtained by dualizing the fibration f. This leaves a huge amount to be done to verify such conjectures. This paper takes a speculative point of view, in that it assumes that a special Lagrangian torus fibration exists on X. We address a number of questions of a topological nature: what is the relationship between the cohomology of X and the cohomology of the dual fibration? what kind of information does the Leray spectral sequence for f contain? what is the relationship between the topological (1,1) couplings of the dual of f and the (1,n-1)-couplings of X in the large complex structure limit? These questions are shown to have nice answers if a key conjecture about the monodromy diffeomorphisms about a large complex structure limit point holds. Roughly put, this conjecture says that monodromy about a large complex structure limit point can be described as a very natural generalization of a Dehn twist for an elliptic curve. Given this conjecture, we show, among other results, that the large complex radius limit of the (1,n-1) couplings on X coincide with the topological (1,1) couplings on Y, and if dim X=3, the Leray filtration and weight filtrations of the mixed Hodge structure coincide, as conjectured by myself and P.M.H. Wilson, and independently by D. Morrison.