In 2012, Tseng and Yau introduced several Laplacians on symplectic manifolds that are related to a system of supersymmetric equations from physics.  In this talk, we will discuss these "symplectic Laplacians" and their relations with cohomologies on compact symplectic manifolds with boundary.  For this purpose, we will introduce new boundary conditions for differential forms on symplectic manifolds.  Their properties and importance will be discussed.