Floer cohomology of $\mathfrak{g}$-equivariant Lagrangian branes

Abstract

Building on Seidel-Solomon's fundamental work, we define the notion of a $\mathfrak{g}$-equivariant Lagrangian brane in an exact symplectic manifold $M$ where $\mathfrak{g}\subset S{H}^1(M)$ is a sub-Lie algebra of the symplectic cohomology of $M$. When $M$ is a (symplectic) mirror to an (algebraic) homogeneous space $G/P$, homological mirror symmetry predicts that there is an embedding of $\mathfrak{g}$ in $S{H}^1(M)$. This allows us to study a mirror theory to classical constructions of Borel-Weil and Bott. We give explicit computations recovering all finite dimensional irreducible representations of ${\mathfrak{sl}}_2$ as representations on the Floer cohomology of an ${\mathfrak{sl}}_2$-equivariant Lagrangian brane and discuss generalizations to arbitrary finite-dimensional semisimple Lie algebras.

46 pages, 2 figures. To appear in Compositio Mathematica

References in corpus (1)

• Disjoinable Lagrangian spheres and dilations

Cited by in corpus (2)

• Bounds on spectral norms and barcodes
• Picard-Lefschetz theory and dilating C^*-actions