Modeling Rozansky-Witten Theory with Sheaves of Categories

We model Rozansky-Witten theory of $T^{\ast }X$ using modules over $\mathrm{Perf}(X\times {\mathbb{A}}^1)$ viewed as sheaves of categories over $X\times {\mathbb{A}}^1$. This is in parallel to Tamarkin’s approach to nonconic Lagrangians in $T^{\ast }X$ via the singular support of sheaves on $X\times \mathbb{R}$. More specifically, we construct the objects of Rozansky-Witten theory of $T^{\ast }X$ given by hybrid Lagrangians of which graphs and conormals are special cases. As a consistency check, we show that $\mathrm{Hom}$s between such objects are given by suitable matrix factorizations.