Shifted bisymplectic and double Poisson structures on non-commutative derived prestacks

Abstract

We introduce the notions of shifted bisymplectic and shifted double Poisson structures on differential graded associative algebras, and more generally on non-commutative derived moduli functors with well-behaved cotangent complexes. For smooth algebras concentrated in degree $0$, these structures recover the classical notions of bisymplectic and double Poisson structures, but in general they involve an infinite hierarchy of higher homotopical data, ensuring that they are invariant under quasi-isomorphism. The structures induce shifted symplectic and shifted Poisson structures on the underlying commutative derived moduli functors, and also on underlying representation functors. We show that there are canonical equivalences between the spaces of shifted bisymplectic structures and of non-degenerate $n$-shifted double Poisson structures. We also give canonical shifted bisymplectic and bi-Lagrangian structures on various derived non-commutative moduli functors of modules over Calabi--Yau dg categories. Unlike their commutative counterparts, these structures enjoy a formal integration property, which we exploit to show that Calabi--Yau and pre-Calabi--Yau structures on a dg algebra correspond respectively to bisymplectic and double Poisson structures on its quotient prestacks by the adjoint $\mathbb{G}_m$-action.