D-geometric Structure of Orbifolds

We study D-branes on abelian orbifolds C^d/Z_N for d=2, 3. The toric data describing the D-brane vacuum moduli space, which represents the geometry probed by D-branes, has certain redundancy compared with the classical geometric description of the orbifolds. We show that the redundancy has a simple combinatorial structure and find analytic expressions for degrees of the redundancy. For d=2 the structure of the redundancy has a connection with representations of SU(N) Lie algebra, which provides a new correspondence between geometry and representation theory. We also prove that non-geometric phases do not appear in the Kahler moduli space for d=2.