Calabi-Yau algebras
Victor Ginzburg
TL;DR
The paper develops Calabi–Yau algebras as noncommutative CY analogues of geometric CY spaces, providing a universal construction via a noncommutative symplectic DG resolution and a potential $\Phi$ that defines ${\mathfrak A}(F,\Phi)$. In dimension 3, a DG resolution ${\mathfrak D}(F,\Phi)$ encodes a noncommutative BV structure; representation varieties are linked to critical loci and vanishing cycles through trace maps, BRST/BV formalisms, and the extended cotangent complex. The work connects CY algebras to quiver/McKay data, crepant resolutions, Sklyanin-like algebras, and group/Chern–Simons constructions, providing tools for computing invariants via matrix integrals and outlining universal CY constructions from symplectic data. It also ties CY properties to Serre duality and derived-category dualities, bridging noncommutative algebra with Calabi–Yau geometry and mirror symmetry. The framework yields broad applications, including 3D McKay correspondences, fundamental group algebras, and noncommutative Hessians that underpin BV-type calculus in this setting.
Abstract
We introduce some new algebraic structures arising naturally in the geometry of Calabi-Yau manifolds and mirror symmetry. We give a universal construction of Calabi-Yau algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative potential. Representation varieties of the Calabi-Yau algebra are intimately related to the set of critical points, and to the sheaf of vanishing cycles of the potential. Numerical invariants, like ranks of cyclic homology groups, are expected to be given by `matrix integrals' over representation varieties. We discuss examples of Calabi-Yau algebras involving quivers, 3-dimensional McKay correspondence, crepant resolutions, Sklyanin algebras, hyperbolic 3-manifolds and Chern-Simons. Examples related to quantum Del Pezzo surfaces will be discussed in [EtGi].