Tag - Mirror symmetry

It has been a continuing interest, often with profound importance, in understanding the geometric and topological relationship between a Hamiltonian G-manifold Y and a symplectic quotient X. In this talk, we shall provide precise relations between their (equivariant) Lagrangian Floer theory. In particular, we will address a conjecture of Teleman, motivated by 3d mirror symmetry, on the 2d mirror construction of X from that of Y, which generalises Givental-Hori-Vafa mirror construction for toric varieties. The key technical ingredient is the Kim-Lau-Zheng’s equivariant extension of Fukaya’s Lagrangian correspondence tri-modules over equivariant Floer complexes.

The Hochschild cohomology of the Floer algebra of a Lagrangian L, and the associated closed-open string map, play an important role in the generation criterion for the Fukaya category and in deformation theory approaches to mirror symmetry. I will explain how, in the monotone setting, one can build a map from the Floer cohomology of L with certain local coefficients to (a version of) Hochschild cohomology. This map makes things much more geometric, by transferring the algebraic complexity to the world of matrix factorisations, and is an isomorphism when L is a torus.