Matrix factorization of Morse–Bott functions

Abstract

For a function $W\in \mathbb{C}\left[X\right]$ on a smooth algebraic variety $X$ with Morse–Bott critical locus $Y\subset X$, Kapustin, Rozansky, and Saulina suggest that the associated matrix factorization category $MF(X;W)$ should be equivalent to the differential graded category of $2$-periodic coherent complexes on $Y$ (with a topological twist from the normal bundle). We confirm their conjecture in the special case when the first neighborhood of $Y$ in $X$ is split and establish the corrected general statement. The answer involves the full Gerstenhaber structure on Hochschild cochains. This note was inspired by the failure of the conjecture, observed by Pomerleano and Preygel, when $X$ is a general $1$-parameter deformation of a $K3$ surface $Y$.