Involutions, obstructions and mirror symmetry

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 ∞ 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 antisymplectic 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.

Homological Lagrangian monodromy for some monotone tori

“…It was pointed out to us by Jake Solomon that this argument could be extended to other settings where HF * (L, L) is known to be graded commutative, e.g. [8,15,19], and where the PSS map allows the action on H * (L) to be deduced from the action on HF * (L, L).…”

Family Floer superpotential's critical values are eigenvalues of quantum product by c_1

“…Remark 1.3 We justify the assumption as follows. First, it holds if there is an anti-symplectic involution that preserves L [Sol20]. This case already includes many special Lagrangian submanifolds and those in [CBM09].…”

The branched deformations of the special Lagrangian submanifolds

“…We do, however, have an additional R-symmetry given by the antiholomorphic involution, which allows us to check the assumption (iii) for specific situations. When dim(L) = 2, according to [42], the condition (i) and (iii) holds for L = S 2 and T 2 . For a single special Lagrangian submanifold, we could introduce topological constraint to avoid obstruction, which give us a Betti number constraint for the branched covering.…”