A complete embedding is a symplectic embedding $\iota :Y\to M$ of a geometrically bounded symplectic manifold $Y$ into another geometrically bounded symplectic manifold $M$ of the same dimension. When $Y$ satisfies an additional finiteness hypothesis, we prove that the truncated relative symplectic cohomology of a compact subset $K$ inside $Y$ is naturally isomorphic to that of its image $\iota (K)$ inside $M$. Under the assumption that the torsion exponents of $K$ are bounded, we deduce the same result for relative symplectic cohomology. We introduce a technique for constructing complete embeddings using what we refer to as integrable anti-surgery. We apply these to study symplectic topology and mirror symmetry of symplectic cluster manifolds and other examples of symplectic manifolds with singular Lagrangian torus fibrations satisfying certain completeness conditions.

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.

The width of a Lagrangian is the largest capacity of a ball that can be symplectically embedded into the ambient manifold such that the ball intersects the Lagrangian exactly along the real part of the ball. Due to Dimitroglou Rizell, finite width is an obstruction to a Lagrangian admitting an exact Lagrangian cap in the sense of Eliashberg–Murphy. In this paper we introduce a new method for bounding the width of a Lagrangian $Q$ by considering the Lagrangian Floer cohomology of an auxiliary Lagrangian $L$ with respect to a Hamiltonian whose chords correspond to geodesic paths in $Q$. This is formalized as a wrapped version of the Floer–Hofer–Wysocki capacity and we establish an associated energy–capacity inequality with the help of a closed–open map. For any orientable Lagrangian $Q$ admitting a metric of non-positive sectional curvature in a Liouville manifold, we show the width of $Q$ is bounded above by four times its displacement energy.