Formal geometry and Tamarkin--Tsygan calculi of dg manifolds

TL;DR

This paper explores the formal geometry of dg manifolds, constructing Fedosov dg foliations and proving isomorphisms between their associated calculi, advancing the understanding of Tamarkin--Tsygan structures in derived geometry.

Contribution

It introduces a Fedosov dg foliation for dg manifolds and establishes homotopy contractions that demonstrate isomorphisms of Tamarkin--Tsygan calculi.

Findings

Constructed Fedosov dg foliation for dg manifolds

Established homotopy contractions between key algebraic structures

Proved isomorphism of Tamarkin--Tsygan calculi

Abstract

The main goal of this paper is to study the formal geometry of dg manifolds \`a la Fedosov. For any dg manifold $(\mathcal{M}, Q)$, we construct a Fedosov dg foliation (or dg Lie algebroid) $\mathcal{F}_Q \to \mathcal{N}_Q$. We establish homotopy contractions between their respective spaces of polyvector fields, differential forms, polydifferential operators, and polyjets. As a consequence, we prove that their respective Cartan calculi and noncommutative calculi, in the sense of Tamarkin--Tsygan, are isomorphic.