Calabi-Yau algebras and the shifted noncommutative symplectic structure

TL;DR

This paper demonstrates that Koszul Calabi-Yau algebras possess a shifted bi-symplectic structure on their cobar constructions, leading to shifted symplectic structures on their DG representation schemes, connecting algebraic and geometric frameworks.

Contribution

It establishes a new link between Koszul Calabi-Yau algebras and shifted symplectic structures via their cobar constructions and representation schemes.

Findings

Existence of shifted bi-symplectic structure on cobar construction

Shifted symplectic structure on DG representation schemes

Bridges algebraic structures with geometric symplectic geometry

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, on the cobar construction of its co-unitalized Koszul dual coalgebra, and hence its DG representation schemes, in the sense of Berest-Khachatryan-Ramadoss, have a shifted symplectic structure in the sense of Pantev-To\"en-Vaqui\'e-Vezzosi.