A Morse-theoretic approach to family Floer homology

Abstract

This dissertation introduces a new model of the family Floer approach to Kontsevich's homological mirror symmetry conjecture constructed via Morse theoretic technology. Homological mirror symmetry (HMS) asserts a derived equivalence between the Fukaya category of a symplectic manifold X and the category of coherent sheaves on its mirror Xˇ. On the other hand, the family Floer program gives a modern reinterpretation of the construction of a Strominger--Yau--Zaslow (SYZ) mirror, and this mirror space typically comes equipped with a functor from the Fukaya category of X into coherent sheaves on Xˇ which can be used to prove HMS as asserted. In order to give an analogous presentation of this story, we define the Morse--Fukaya algebra A associated to a suitable class of SYZ fibrations π : X → B; this is a curved A∞-algebra determined by a Morse function on the total space X, taking coefficients in analytic functions on its rigid analytic mirror space. For an appropriate choice of Morse function, A can be understood as a (suitably deformed) algebra of Čech cochains valued in polyvector fields on Xˇ. We then construct an A∞-functor from (a suitable subcategory of) the Fukaya category of X into the category mod-A of modules over A implementing the expected correspondence. Along the way we record comparison maps which together witness invariance of our constructions under a change of auxiliary technical choices.