Koszul duality and Calabi-Yau structures

TL;DR

The paper develops a comprehensive Koszul duality framework between dg categories and pointed curved coalgebras, showing that proper Calabi–Yau structures on coalgebras correspond to smooth Calabi–Yau structures on their cobar dg categories, and conversely. It provides explicit bar–cobar/cosymbol constructions and a coderived–derived perspective to prove this duality, including a version applicable over principal ideal domains. Two main applications illustrate the power of the theory: (i) unimodular finite-dimensional Lie algebras, where a proper CY structure on the Chevalley–Eilenberg coalgebra $C_*(rak g)$ is equivalent to a smooth CY structure on $ ext{U}(rak g)$, and (ii) spaces of finite-type, where Poincaré duality (with local coefficients) on a space $X$ yields a proper CY structure on $C_*(X)$ and a smooth CY structure on $C_*( ext{Ω}X)$ (or on chains of the loop space). The results unify and extend known dualities in algebra and topology, provide chain-level realizations, and offer a natural framework for string topology and Poincaré duality phenomena within Koszul duality. The approach yields explicit correspondences via the functors $ ext{Ω}$ and $B$, the negative cyclic/coHochschild homologies, and the derived/coderived category equivalences, with broad potential for applications in geometry, representation theory, and higher 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.

Theorems & Definitions (89)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Theorem \ref{['thm:liekoszul']}
• Remark 1.4
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Definition 2.5
• Definition 2.6
• Proposition 3.1