Floer cohomology of $\mathfrak{g}$-equivariant Lagrangian branes. (17th December 2015) - Lekili, Yankı
Abstract : Building on Seidel and Solomon's fundamental work [ Symplectic cohomology and $q$ - intersection numbers, Geom. Funct. Anal.22 (2012), 443–477], we define the notion of a $\mathfrak{g}$ -equivariant Lagrangian brane in an exact symplectic manifold $M$, where $\mathfrak{g}\subset SH^{1}(M)$ 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 $SH^{1}(M)$ . 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.