D-branes, Exceptional Sheaves and Quivers on Calabi-Yau manifolds: From Mukai to McKay

We present a method based on mutations of helices which leads to the construction (in the large volume limit) of exceptional coherent sheaves associated with the $({\sum }_a{l}_a=0)$ orbits in Gepner models. This is explicitly verified for a few examples including some cases where the ambient weighted projective space has singularities not inherited by the Calabi-Yau hypersurface. The method is based on two conjectures which lead to the analog,in the general case, of the Beilinson quiver for ${\text{BP}}^n$. We discuss how one recovers the McKay quiver using the gauged linear sigma model (GLSM) near the orbifold or Gepner point in K"ahler moduli space.