Extended generalized geometry and a DBI-type effective action for branes ending on branes

Starting from the Nambu-Goto bosonic membrane action, we develop a geometric description suitable for p -brane backgrounds. With tools of generalized geometry we derive the pertinent generalization of the string open-closed relations to the p -brane case. Nambu-Poisson structures are used in this context to generalize the concept of semi-classical noncommutativity of D -branes governed by a Poisson tensor. We find a natural description of the correspondence of recently proposed commutative and noncommutative versions of an effective action for p -branes ending on a p ′ -brane. We calculate the power series expansion of the action in background independent gauge. Leading terms in the double scaling limit are given by a generalization of a (semi-classical) matrix model