Membrane and Noncommutativity

We analyse the dynamics of an open membrane, both for the free case and when it is coupled to a background three-form, whose boundary is attached to $p$-branes. The role of boundary conditions and constraints in the Nambu-Goto and Polyakov formulations is studied. The low-energy approximation that effectively reduces the membrane to an open string is examined in detail. Noncommutative features of the boundary string coordinates, where the cylindrical membrane is attached to the D$p$-branes, are revealed by algebraic consistency arguments and not by treating boundary conditions as primary constraints, as is usually done. The exact form of the noncommutative algebra is obtained in the low-energy limit.