# Formality theorem for differential graded manifolds

## Meeting Details

Abstract: The Atiyah class of a dg manifold $(\mathcal{M},Q)$ is the obstruction to the existence of an affine connection on the graded manifold $\mathcal{M}$ that is compatible with the homological vector field $Q$. The Todd class of dg manifolds extends both the classical Todd class of complex manifolds and the Duflo element of Lie theory. Using Kontsevich's famous formality theorem, Liao, Xu and I established a formality theorem for smooth dg manifolds: given any finite-dimensional dg manifold $(\mathcal{M},Q)$, there exists an $L_\infty$ quasi-isomorphism of dglas from an appropriate space of polyvector fields $\mathcal{T}_{\oplus,\operatorname{poly}}^{\bullet}(\mathcal{M})$ endowed with the Schouten bracket $[-,-]$ and the differential $[Q,-]$ to an appropriate space of polydifferential operators $\mathcal{D}_{\oplus,\operatorname{poly}}^{\bullet}(\mathcal{M})$ endowed with the Gerstenhaber bracket $\llbracket -,- \rrbracket$ and the differential $\llbracket m+Q,- \rrbracket$, whose first Taylor coefficient (1) is equal to the composition of the action of the square root of the Todd class of the dg manifold $(\mathcal{M},Q)$ on $\mathcal{T}_{\oplus,\operatorname{poly}}^{\bullet}(\mathcal{M})$ with the Hochschild--Kostant--Rosenberg map and (2) preserves the associative algebra structures on the level of cohomology. As an application, we proved the Kontsevich--Shoikhet conjecture: a Kontsevich--Duflo type theorem holds for all finite-dimensional smooth dg manifolds. This last result shows that, when understood in the unifying framework of dg manifolds, the classical Duflo theorem of Lie theory and the Kontsevich--Duflo theorem for complex manifolds are really just one and the same phenomenon.