On counting special Lagrangian homology 3-spheres

We attempt to define a new invariant I of (almost) Calabi-Yau 3-folds M, by counting special Lagrangian rational homology 3-spheres N in M in each 3-homology class, with a certain weight w(N) depending on the topology of N. This is motivated by the Gromov-Witten invariants of a symplectic manifold, which count the J-holomorphic curves in each 2-homology class. In order for this invariant to be interesting, it should either be unchanged by deformations of the underlying (almost) Calabi-Yau structure, or else transform according to some rigid set of rules as the periods of the almost Calabi-Yau structure pass through some topologically determined hypersurfaces in the cohomology of M. As we deform the underlying almost Calabi-Yau 3-fold, the collection of special Lagrangian homology 3-spheres only change when they become singular. Thus, to determine the stability of the invariant under deformations we need know about the singular behaviour of special Lagrangian 3-folds, which is not well understood. We describe two kinds of singular behaviour of special Lagrangian 3-folds, and derive identities on the weight function w(N) for I to be unchanged or transform well under them. The weight function w(N)=|H_1(N,Z)| satisfies these identities. We conjecture that an invariant I defined with this weight is independent of the Kahler class, and changes in certain ways as the holomorphic 3-form passes through some real hypersurfaces in H^3(M,C). Finally we consider connections with String Theory. We argue that our invariant I counts isolated 3-branes, and that it should play a part in the Mirror Symmetry story for Calabi-Yau 3-folds.