More on Penrose limits and non-local theories

We obtain the Penrose limit of six dimensional Non-Commutative Open String(NCOS$_6$) theory and show that in the neighborhood of a particular nullgeodesic it leads to an exactly solvable string theory (unlike theircounterparts in four or in other dimensions). We describe the phase structureof this theory and discuss the Penrose limit in different phases including OpenD-string (OD1) theory. We compute the string spectrum and discuss theirrelations with the states of various theories at different phases. We alsoconsider the case of general null geodesic for which the Penrose limit leads tostring theory in the time dependent pp-wave background and comment on therenormalization group flow in the dual theory.Comment: latex, 22 pages, minor corrections, references added, published versio