Calabi–Yau contraction isomorphism conjecture for smooth dg algebras

Let $A$ be a non-negatively graded connected smooth compact Calabi–Yau dg $k$-algebra, and let $v_1$ be a non-zero element of $H_w(\Omega_\bullet(A),d)$. Under the formality assumption, contraction by $v_1$ is defined on Hochschild cochains and chains by $X\mapsto\iota_Xv_1$.

Calabi–Yau contraction isomorphism conjecture. Under the formality assumption, the induced morphism of dg $k$-modules

$$(\mathcal{T}^\bullet_{\mathrm{poly}}(A),d)\longrightarrow(\Omega_{w-\bullet}(A),d),\qquad X\longmapsto i_Xv_1,$$

is an isomorphism.

This is the dg-categorical analogue of contraction with a holomorphic volume form, identifying polyvector fields with differential forms in the Calabi–Yau setting. The source states the result under its formality assumption and gives no separate resolution status.