Holomorphic discs of negative Maslov index and extended deformations in mirror symmetry

The SYZ approach to mirror symmetry for log Calabi-Yau manifolds starts from a Lagrangian torus fibration on the complement of an anticanonical divisor. A mirror space is constructed by gluing local charts (moduli spaces of local systems on generic torus fibers) via wall-crossing transformations which account for corrections to the analytic structure of moduli spaces of objects of the Fukaya category induced by bubbling of Maslov index 0 holomorphic discs, and made into a Landau-Ginzburg model by equipping it with a regular function (the superpotential) which enumerates Maslov index 2 holomorphic discs. When they occur, holomorphic discs of negative Maslov index deform this picture by introducing inconsistencies in the wall-crossing transformations, so that the mirror is no longer an analytic space; the geometric features of the corrected mirror can be understood in the language of extended deformations of Landau-Ginzburg models. We illustrate this phenomenon (and show that it actually occurs) by working through the construction for an explicit example (a log Calabi-Yau 4-fold obtained by blowing up a toric variety), and discuss a family Floer approach to the geometry of the corrected mirror in this setting. Along the way, we introduce a Morse-theoretic model for family Floer theory which may be of independent interest.Comment: 47 page