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

TL;DR

This paper introduces a mutation-based method for constructing exceptional sheaves on Calabi-Yau manifolds, connecting geometric and algebraic structures through quivers, and verifies it with examples including singular ambient spaces.

Contribution

It develops a mutation-based approach to construct exceptional sheaves linked to Gepner model orbits, extending the Beilinson quiver concept to Calabi-Yau contexts.

Findings

Constructed exceptional sheaves via mutations of helices.

Verified the method with examples involving singular ambient spaces.

Connected the approach to McKay quivers using GLSM.

Abstract

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 $(\sum_al_a=0)$ 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 $\BP^n$. 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.