Categorical Saito theory, I: A comparison result

TL;DR

This paper constructs an explicit minimal $A_infty$ model for matrix factorizations of hypersurface singularities, using deformation quantization and formality maps to compare categorical and geometric structures.

Contribution

It introduces a new explicit model for matrix factorizations and establishes a comparison theorem linking categorical and geometric structures via deformation techniques.

Findings

The categorical variation of semi-infinite Hodge structure is isomorphic to Saito's primitive form.

The comparison theorem confirms an analogue of Caldararu's conjecture for matrix factorizations.

An explicit cyclic minimal $A_infty$ model is constructed for $ ext{MF}(W)$.

Abstract

In this paper, we present an explicit cyclic minimal $A_\infty$ model for the category of matrix factorizations $\MF(W)$ of an isolated hypersurface singularity. The key observation is to use Kontsevich's deformation quantization technique. Pushing this idea further, we use the Tsygan formality map to obtain a comparison theorem that the categorical Variation of Semi-infinite Hodge Structure of $\MF(W)$ is isomorphic to Saito's original geometric construction in primitive form theory. An immediate corollary of this comparison result is that the analogue of Caldararu's conjecture holds for the category $\MF(W)$.