  We study geometric engineering of Argyres-Douglas superconformal theories
realized by type IIB strings propagating in singular Calabi-Yau threefolds. We
use this construction to count the degeneracy of light BPS states under small
perturbations away from the conformal point, by computing the degeneracy of
D3-branes wrapped around supersymmetric 3-cycles in the Calabi-Yau. We find
finitely many BPS states, the number of which depends on how this deformation
is done, similarly to the degeneracy of kink solutions for the deformation of
N=2 Landau-Ginzburg superconformal theories in two dimensions. Also, some
aspects of worldsheet theories near general Calabi-Yau singularities are
discussed.
