Cusped hyperbolic Lagrangians as mirrors to lines in three‐space

Abstract We construct a Lagrangiansubmanifold in the cotangent bundle of a 3‐torus whose projection to the fiber is a neighborhood of a tropical curve with a single 4‐valent vertex. This Lagrangian submanifold has an isolated conical singular point, and its smooth locus is diffeomorphic to the minimally twisted five‐component chain link complement, a cusped hyperbolic 3‐manifold. From this singular Lagrangian, we construct a Lagrangian immersion, and identify a moduli space of objects in the wrapped Fukaya category that it supports. We show that for a generic line in projective 3‐space, there is a local system on this Lagrangian immersion such that the resulting object of the wrapped Fukaya category is homologically mirror to an object of the derived category supported on the line. In the course of the proof, we construct a version of the wrapped Fukaya category with objects supported on Lagrangian immersions, which may be of independent interest.