Supersymmetric D3 brane and N=4 SYM actions in plane wave backgrounds

The explicit (all-order in fermions) form of the kappa-symmetric $D3$ brane probe action was previously found in the two maximally supersymmetric type IIB vacua: flat space and $AdS_5 \times S^5$. Here we present the form of the action in the third maximally supersymmetric type IIB background: gravitational plane wave supported by constant null 5-form strength. We study $D3$ brane action in both covariant and light cone kappa symmetry gauges. Like the fundamental string action, the $D3$ brane action takes a simple form once written in the light cone kappa-symmetry gauge. We also consider the $\mathcal{N}=4$ SYM theory in 4d plane wave background. Since some (super)symmetries of plane wave SYM action are friendly to (super)symmetries of the type IIB superstring in plane wave Ramond-Ramond background we suggest this SYM model may be useful in the context of AdS/CFT duality. We develop the Hamiltonian light cone gauge formulation for this theory.