Pre-Calabi-Yau structures and moduli of representations

Abstract

We establish a system of formal noncommutative calculus for differential forms and polyvector fields, which forms the foundations for the study of pre-Calabi-Yau categories. Using an explicit trace map, we show that any $n$-Calabi-Yau structure on a non-positively graded dg algebra $A$ induces a $(2-n)$-shifted symplectic structure on its derived moduli stack of representations; while any $n$-pre-Calabi-Yau structure on $A$ induces a $(2-n)$-shifted Poisson structure on this derived moduli stack.

32 pages, 3 figures. Comments welcome

References in corpus (5)

• Calabi-Yau algebras
• Duality between Lagrangian and Legendrian invariants
• Relative Calabi-Yau completions
• Structures symplectiques et de Poisson sur les champs en catégories
• Computing a categorical Gromov-Witten invariant

Cited by in corpus (6)

• Multiplicative preprojective algebras are 2-Calabi-Yau
• Erratum to "Deformed Calabi-Yau completions"
• McKay correspondence, cohomological Hall algebras and categorification
• Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology
• Shifted symplectic and Poisson structures on global quotients
• Twisted bi-symplectic structure on Koszul twisted Calabi-Yau algebras