Matrix Factorisation of Morse-Bott functions

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 $Y$.

10 pages. Added an appendix on curved formal deformations. Proofs clarified, references added, other minor corrections

References in corpus (1)

• Generators in formal deformations of categories

Cited by in corpus (2)

• Topological Strings on Non-Commutative Resolutions
• Mirror symmetry for Berglund-Hübsch Milnor fibers