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Title:
Floer-Theoretic Corrections to the Geometry of Moduli Spaces of Lagrangian Tori
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Abstract:
Given a Lagrangian torus fibration on the complement of an anticanonical divisor in a Kahler manifold, one usually constructs a mirror space by gluing local charts (moduli spaces of objects of the Fukaya category supported on generic torus fibers) via wall-crossing transformations determined by counts of Maslov index 0 holomorphic discs; this mirror also comes equipped with a regular function (the superpotential) which enumerates Maslov index 2 holomorphic discs. Holomorphic discs of negative Maslov index deform this picture by introducing inconsistencies in the wall-crossing transformations; the geometric features of the corrected moduli space of objects of the Fukaya category can be understood in the language of extended deformations of Landau-Ginzburg models. We will illustrate this phenomenon on an explicit example (a 4-fold obtained by blowing up a toric variety), and, if time permits, discuss a family Floer approach to the geometry of the corrected mirror in this setting.
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