Involutions, obstructions and mirror symmetry

TL;DR

This paper proves that certain symmetric Lagrangian submanifolds have unobstructed, formal Floer cohomology, aligning with de Rham cohomology, which advances understanding in mirror symmetry and Lagrangian fibrations.

Contribution

It establishes the formality and unobstructedness of Floer cohomology for a class of symmetric Lagrangians, extending to cases with specific cohomological conditions.

Findings

Floer and de Rham cohomologies coincide for these Lagrangians.

The result applies to smooth fibers of singular Lagrangian fibrations.

Supports approaches to mirror symmetry involving SYZ and family Floer cohomology.

Abstract

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 $A_\infty$ 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 $4.$