Koszul duality and Calabi-Yau structures

Abstract: We show that Koszul duality between differential graded categories and pointed curved coalgebras interchanges smooth and proper Calabi-Yau structures. This result is a generalization and conceptual explanation of the following two applications. For a finite-dimensional Lie algebra a smooth Calabi-Yau structure on the universal enveloping algebra is equivalent to a proper Calabi-Yau structure on the Chevalley-Eilenberg chain coalgebra, which exists if and only if Poincare duality is satisfied. For a topological space X having the homotopy type of a finite complex we show an oriented Poincare duality structure (with local coefficients) on X is equivalent to a proper Calabi-Yau structure on the dg coalgebra of chains on X and to a smooth Calabi-Yau structure on the dg algebra of chains on the based loop space of X.