Calabi-Yau algebras and the shifted noncommutative symplectic structure

Abstract In this paper we show that for a Koszul Calabi-Yau algebra, there is a shifted bi-symplectic structure in the sense of Crawley-Boevey-Etingof-Ginzburg [15] , on the cobar construction of its co-unitalized Koszul dual coalgebra, and hence its DG representation schemes, in the sense of Berest-Khachatryan-Ramadoss [3] , have a shifted symplectic structure in the sense of Pantev-Toen-Vaquie-Vezzosi [28] .