The Duflo formula for Q-manifolds

Let $X$ be a smooth $Q$-manifold, with $Q$ an odd vector field of degree $+1$ satisfying $[Q,Q]=0$. Let $T^\bullet_{\mathrm{poly}}(X,Q)$ be its differential graded algebra of polyvector fields, let $\varphi_{\mathrm{HKR}}$ be the Hochschild–Kostant–Rosenberg map, and let

$$\varphi_{\mathrm{strange}}=\exp\left(\sum_{k\geq 1}\alpha_{2k}c_{2k}\right)$$

be the characteristic correction defined from the Atiyah class, with the coefficients $\alpha_{2k}$ as in the source. Let $\mathcal U_Q$ be the tangent map at $Q$ of a formality quasi-isomorphism.

Duflo formula for Q-manifolds.

1. The map

$$[\varphi_{\mathrm{HKR}}\circ\varphi_{\mathrm{strange}}]:H^\bullet(T^\bullet_{\mathrm{poly}}(X,Q))\to HH^\bullet(C^\infty(X),Q)$$

is an isomorphism of algebras. 2. The map $[\mathcal U_Q]$ induced by the tangent map $\mathcal U_Q$ coincides with $[\varphi_{\mathrm{HKR}}\circ\varphi_{\mathrm{strange}}]$. This conjecture extends the classical Duflo formula and the corresponding formula for complex manifolds to arbitrary smooth $Q$-manifolds. The source uses it to give an explicit description of the canonical tangent-map isomorphism.