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Floer cohomology of
$\mathfrak{g}$-equivariant Lagrangian branes
Published online by Cambridge University Press: 17 December 2015
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.
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- © The Authors 2015
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