  We consider the holomorphic twist of the worldvolume theory of flat
D$(2k-1)$-branes transversely probing a Calabi--Yau manifold. A chain complex,
constructed using the BV formalism, computes the local observables in the
holomorphically twisted theory. Generalizing earlier work in the case $k=2$, we
find that this complex can be identified with the Ginzburg dg algebra
associated to the Calabi--Yau. However, the identification is subtle; the
complex is the space of fields contributing to the holomorphic twist of the
free theory, and its differential arises from interactions. For $k=1$, this
holomorphically twisted theory is related to the elliptic genus. We give a
general description for D1-branes probing a Calabi--Yau fourfold singularity,
and for $\mathcal{N}=(0,2)$ quiver gauge theories. In addition, we propose a
relation between the equivariant Hirzebruch $\chi_y$ genus of large-$N$
symmetric products and cyclic homology.
