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Floer cohomology of
$\mathfrak{g}$-equivariant Lagrangian branes
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 5 / May 2016
- Published online by Cambridge University Press:
- 17 December 2015, pp. 1071-1110
- Print publication:
- May 2016
-
- Article
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- You have access
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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.
