Atiyah classes and Kontsevich–Duflo type theorem for dg manifolds

In this expository paper, we survey some recent works on the Atiyah class and other characteristic classes of dg manifolds. In particular, we describe a Kontsevich–Duflo type theorem for dg manifolds: For every finite-dimensional dg manifold $({\cal M},Q)$, the composition ${\rm hkr}\circ({\rm td}_{({\cal M},Q)})^{{1}/{2}}$ is an isomorphism of Gerstenhaber algebras from $H^{\bullet}({\rm tot}_\oplus({\cal T}_{\rm poly}({\cal M}),{\cal Q}))$ to $H^{\bullet}({\rm tot}({\cal D}_{\rm poly}({\cal M})),d_{\mathscr H}+{\cal Q})$ — the square root of the Todd class of the dg manifold ${\rm td}_{({\cal M},Q)}^{{1}/{2}}\in\prod_{k\geqslant 0} H^{k}((\Omega^k({\cal M}))^\bullet,{\cal Q})$ acts on $H^{\bullet}({\rm tot}_\oplus ({\cal T}_{\rm poly}({\cal M})),{\cal Q})$, by contraction. The Duflo theorem of Lie theory and the Kontsevich theorem regarding the Hochschild cohomology of complex manifolds can both be derived as special cases of this Kontsevich–Duflo type theorem for dg manifolds. The paper ends with a discussion of extensions of this theorem.