Essential tori in spaces of symplectic embeddings

Given two $2n$–dimensional symplectic ellipsoids whose symplectic sizes satisfy certain inequalities, we show that a certain map from the $n$–torus to the space of symplectic embeddings from one ellipsoid to the other induces an injective map on singular homology with mod $2$ coefficients. The proof uses parametrized moduli spaces of $J$–holomorphic cylinders in completed symplectic cobordisms.