Intersecting D-Branes in ten and six dimensions

We show how, via $T$-duality, intersecting $D$-Brane configurations in ten (six) dimensions can be obtained from the elementary $D$-Brane configurations by embedding a Type IIB $D$-Brane into a Type IIB Nine-Brane (Five-Brane) and give a classification of such configurations. We show that only a very specific subclass of these configurations can be realized as (supersymmetric) solutions to the equations of motion of IIA/IIB supergravity. Whereas the elementary $D$-brane solutions in $d=10$ are characterized by a single harmonic function, those in $d=6$ contain two independent harmonic functions and may be viewed as the intersection of two $d=10$ elementary $D$-branes. Using string/string/string triality in six dimensions we show that the heterotic version of the elementary $d=6$ $D$-Brane solutions correspond in ten dimensions to intersecting Neveu-Schwarz/Neveu-Schwarz (NS/NS) strings or five-branes and their $T$-duals. We comment on the implications of our results in other than ten and six dimensions.