Charge of $D$-branes on singular varieties

Abstract

Considering the $D$-branes on a variety $Z$ as the objects of the derived category $D^b(Z)$, we propose a definition for the charge of $D$-branes on not necessarily smooth varieties. We define the charge $Q({\mathcal G})$ of ${\mathcal G}\in D^b(Z)$ as an element of the homology of $Z$, so that the mapping $Q$ is compatible with the pushforward by proper maps between varieties. Given a generic anticanonical hypersurface $Y$ of a toric variety $X$ defined by a reflexive polytope, we express the charge of a line bundle on $Y$ defined by a divisor $D$ of $X$ in terms of intersections of $D$ with cycles determined by the polytope faces.