Rozansky–Witten conjecture on monoidal deformations of coherent-sheaf categories

Let $Y$ be the complex manifold under consideration. Write $T^*_fY$ for its formal holomorphic cotangent neighborhood, and let $\beta$ be an inhomogeneous $(0,q)$-form with odd $q$ and values in the graded bundle

$$\bigoplus_{p=2}^{\infty} \operatorname{Sym}^p(TY).$$

Assume that $\beta$ satisfies the Maurer–Cartan equation

$$\overline{\partial}\beta+\frac{1}{2}[\beta,\beta]=0,$$

where the bracket combines wedge product with the Poisson-induced Lie bracket on fiberwise-holomorphic functions on $T^*_fY$. Let ${\mathsf M}_Y$ denote the space of such solutions, and let gauge transformations have infinitesimal action

$$a:\ \beta\mapsto\beta+\overline{\partial}a+[\beta,a],$$

with $a$ a section of $\bigoplus_{p=1}^{\infty}\operatorname{Sym}^p(TY)$. Rozansky–Witten conjecture. There is a surjective map from ${\mathsf M}_Y$ to the space of monoidal deformations of the category $D_{{\mathbb Z}_2}(\operatorname{Coh}(Y))$. Two monoidal deformations are equivalent if and only if the corresponding Maurer–Cartan solutions are related by a gauge transformation with the infinitesimal action above. The Rozansky–Witten model is expected to identify formal holomorphic symplectic deformations of the neighborhood of $Y$ with monoidal deformations of its coherent-sheaf category. The conjecture gives the asserted parametrization and equivalence relation; the supplied text does not provide a resolution status.