Holomorphic field theories and Calabi–Yau algebras

Abstract

We consider the holomorphic twist of the worldvolume theory of flat D[Formula: see text]-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 [Formula: see text], 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 [Formula: see text], 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 [Formula: see text] quiver gauge theories. In addition, we propose a relation between the equivariant Hirzebruch [Formula: see text] genus of large-[Formula: see text] symmetric products and cyclic homology.