Consider a Maslov zero Lagrangian submanifold diffeomorphic to a Lie group on which an anti-symplectic involution acts by the inverse map of the group. We show that the Fukaya A∞ endomorphism algebra of such a Lagrangian is quasi-isomorphic to its de Rham cohomology tensored with the Novikov field. In particular, it is unobstructed, formal, and its Floer and de Rham cohomologies coincide. Our result implies that the smooth fibers of a large class of singular Lagrangian fibrations are unobstructed and their Floer and de Rham cohomologies coincide. This is a step in the SYZ and family Floer cohomology approaches to mirror symmetry. More generally, our result continues to hold if the Lagrangian has cohomology the free graded algebra on a graded vector space V concentrated in odd degree, and the anti-symplectic involution acts on the cohomology of the Lagrangian by the induced map of negative the identity on V. It suffices for the Maslov class to vanish modulo 4.
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The author would like to thank G. Tian for his constant encouragement and P. Seidel for a conversation that deeply influenced the present work. The author would also like to thank R. Bezrukavnikov, D. Kaledin, D. Kazhdan, M. Temkin, and Y. Varshavsky, for helpful conversations, and an anonymous referee for helpful suggestions. The author was partially supported by ERC Starting Grant 337560 and ISF Grant 569/18.
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- A algebra
- Mirror symmetry