Matrix Factorisation of Morse-Bott functions

TL;DR

This paper proves a conjecture relating matrix factorisation categories to coherent complexes on Morse-Bott critical loci, confirming it in specific cases and addressing its general form with Gerstenhaber structures.

Contribution

It confirms and corrects a conjecture connecting matrix factorisations with coherent complexes on Morse-Bott loci, including the full Gerstenhaber structure.

Findings

Confirmed the conjecture when the first neighbourhood of Y in X is split.

Established the corrected general statement of the conjecture.

Addressed the failure of the original conjecture in deformations of K3 surfaces.

Abstract

For a function $W\in \mathbb{C}[X]$ on a smooth algebraic variety $X$ with Morse-Bott critical locus $Y\subset X$, Kapustin, Rozansky and Saulina suggest that the associated matrix factorisation category $\mathrm{MF}(X;W)$ should be equivalent to the differential graded category of $2$-periodic coherent complexes on $Y$ (with a topological twist from the normal bundle of $Y$). I confirm their conjecture in the special case when the first neighbourhood 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 one-parameter deformation of a $K3$ surface $Y$.