Modeling Rozansky-Witten Theory with Sheaves of Categories

Abstract

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