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 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.

中文翻译：

---

对合、障碍物和镜像对称

摘要 考虑一个马斯洛夫零拉格朗日子流形微分同胚于一个李群，在该群上的反辛对合由该群的逆映射作用。我们证明了这种拉格朗日量的 Fukaya A ∞ 内同态代数与它与诺维科夫场张量的 de Rham 上同调是准同构的。特别是，它是通畅的、形式化的，并且它的 Floer 和 de Rham 上同调重合。我们的结果意味着一大类奇异拉格朗日纤维的光滑纤维是畅通无阻的，并且它们的 Floer 和 de Rham 上同调重合。这是镜像对称的 SYZ 和 Floer 族上同调方法中的一个步骤。更一般地，如果拉格朗日函数具有集中在奇数次的分级向量空间 V 上的自由分级代数上同调，我们的结果继续成立，