Equivariant Lagrangian correspondence and a conjecture of Teleman

Abstract

In this paper, we study the Floer theory of equivariant Lagrangian correspondences and apply it to derive precise relations between the disc potential of an invariant Lagrangian submanifold and that of its quotient, thereby addressing a conjecture of Teleman. Furthermore, we proved that their (equivariant) Lagrangian Floer cohomologies are isomorphic. In particular, the functor by equivariant Lagrangian correspondence induces a quasi-isomorphism between the equivariant derived Fukaya category at a regular moment-map level and the derived Fukaya category of the corresponding symplectic quotient. A key step is to extend Fukaya's construction of an $A_\infty$ tri-module for Lagrangian correspondences to Borel spaces. We demonstrate that the equivariant obstruction of a Lagrangian correspondence plays an essential role, which leads to quantum corrections in the disc potentials of the quotients. We computed the disc potential of the Lagrangian correspondence in the toric setup and relate it with mirror maps for compact semi-Fano toric manifolds.