D-branes on Stringy Calabi-Yau Manifolds

We argue that D-branes corresponding to rational B boundary states in a Gepner model can be understood as fractional branes in the Landau-Ginzburg orbifold phase of the linear sigma model description. Combining this idea with the generalized McKay correspondence allows us to identify these states with coherent sheaves, and to calculate their K-theory classes in the large volume limit, without needing to invoke mirror symmetry. We check this identification against the mirror symmetry results for the example of the Calabi-Yau hypersurface in ${\text{WP}}^{1,1,2,2,2}$.