Holomorphic field theories and Calabi--Yau algebras

TL;DR

This paper explores the holomorphic twist of D-brane worldvolume theories in Calabi--Yau backgrounds, linking the resulting algebraic structures to Ginzburg dg algebras and elliptic genera, with implications for string theory and algebraic geometry.

Contribution

It generalizes the connection between holomorphic twists of D-brane theories and Calabi--Yau algebras, extending previous work to higher dimensions and relating to elliptic genera and cyclic homology.

Findings

Identifies the holomorphic twist complex with Ginzburg dg algebra for Calabi--Yau cases.

Relates the $k=1$ case to elliptic genus computations.

Proposes a link between equivariant Hirzebruch genus and cyclic homology.

Abstract

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.