00 6 Calabi - Yau Algebras

We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY 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 CY 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 CY algebras involving quivers, 3-dimensional McKay correspondence , crepant resolutions, Sklyanin algebras, hyperbolic 3-manifolds and Chern-Simons. Examples related to quantum Del Pezzo surfaces are discussed in [EtGi].