Calabi-Yau algebras

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].

few comments added, misprints corrected

References in corpus (3)

• Donaldson-Thomas invariants via microlocal geometry
• Algebraic structure of Yang-Mills theory
• Finite representations of a quiver arising from string theory

Cited by in corpus (20)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions
• Deformed Calabi-Yau Completions
• On the noncommutative Donaldson-Thomas invariants arising from brane tilings
• Theorie homotopique des DG-categories
• Crepant resolutions and brane tilings I: Toric realization
• On Serre duality for compact homologically smooth DG algebras
• D-Branes on Toric Calabi-Yau Varieties
• Quivers with potentials and their representations I: Mutations
• The string topology BV algebra, Hochschild cohomology and the Goldman bracket on surfaces
• A theory of generalized Donaldson-Thomas invariants. II. Multiplicative identities for Behrend functions
• Counting invariants for the ADE McKay quivers
• Superpotentials and Higher Order Derivations
• The A-infinity Deformation Theory of a Point and the Derived Categories of Local Calabi-Yaus
• The calculus structure of the Hochschild homology/cohomology of preprojective algebras of Dynkin quivers
• Noncommutative Tangent Cones and Calabi Yau Algebras
• A Note on Support in Triangulated Categories
• Crepant Resolutions, Quivers and GW/NCDT Duality
• Cocommutative Calabi-Yau Hopf algebras and deformations
• Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms