We study two problems of real aspects in symplectic topology. The first problem concerns the topology of real Lagrangian submanifolds in a toric symplectic manifold.
Real Lagrangians we consider come from involutive symmetries on the moment polytope of a toric symplectic manifold.
We establish a real analogue of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data of the polytope.
As an application, we realize all possible diffeomorphism types of connected real Lagrangians in toric symplectic del Pezzo surfaces.
In the second problem, we deal with a real analogue of the symplectic mapping class group of a monotone $Q:=S^2\times S^2$, the set $\pi_{0}\mathcal{I}(Q,\ow ,S^2)$ of the isotopy classes of antisymplectic involutions of $Q$ having a Lagrangian sphere as the fixed point set. It is shown that $\pi_{0}\mathcal{I}(Q,\ow ,S^2)$ has a single element. This follows from a stronger result, namely that any two anti-symplectic involutions in that space are Hamiltonian isotopic.
