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 (∑ala=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 Pn. We discuss how one recovers the McKay quiver using the gauged linear sigma model (GLSM) near the orbifold or Gepner point in Kähler moduli space.