Holomorphic Discs of Negative Maslov Index and Extended Deformations in Mirror Symmetry

Explore a 58-minute lecture by Denis Auroux from Harvard University on holomorphic discs of negative Maslov index and extended deformations in mirror symmetry. Delve into the construction of mirror spaces using Lagrangian torus fibrations on complements of anticanonical divisors in Kähler manifolds. Examine how local charts are glued via wall-crossing transformations determined by Maslov index 0 holomorphic disc counts, and learn about the superpotential function enumerating Maslov index 2 discs. Investigate the impact of negative Maslov index holomorphic discs on this construction, introducing inconsistencies in wall-crossing transformations. Understand the resulting mirror's geometric features through the lens of extended deformations of Landau-Ginzburg models. Study an explicit example involving a 4-fold obtained by blowing up a Calabi-Yau toric variety, and explore a family Floer approach to the geometry of the corrected mirror in this context.