Curved D-brane geometry and zero modes

We present a path integral formalism to review the effective dynamics of an arbitrary and curved Dirichlet p-brane (Dp-brane) in open bosonic string a nd subsequently in type I superstring theory. We obtain the perturbative co rrections to the Dirac-Born-Infeld dynamics up to O(alpha') in the presence of an antisymmetric two-form B field. A part of the corrections can be see n to be associated with an ultraviolet divergence. Renormalization of the D p-brane collective coordinates is performed to show that the residual corre ctions lead to a noncommutative description on the world volume. The Dp-bra ne modes are analyzed and we show that its zero modes play a vital role in noncommutative geometry. Our analysis facilitates the noncommutative world volume description for an arbitrary B field.