Matrix factorizations, Reality and Kn\"orrer periodicity

TL;DR

This paper extends Kn"orrer periodicity to categories of Real matrix factorizations, connecting algebraic geometry, topology, and physics through periodicity theorems and Real categorical representation theory.

Contribution

It introduces and studies Real matrix factorizations, generalizing Kn"orrer periodicity with equivariance and torsion twists, bridging multiple mathematical and physical theories.

Findings

Generalization of Kn"orrer periodicity to Real matrix factorizations

Incorporation of equivariance and discrete torsion into the theory

Structural similarity to known periodicities in $KR$-theory and Grothendieck-Witt theory

Abstract

Motivated by periodicity theorems for Real $K$-theory and Grothendieck--Witt theory and, separately, work of Hori-Walcher on the physics of Landau-Ginzburg orientifolds, we introduce and study categories of Real matrix factorizations. Our main results are generalizations of Kn\"{o}rrer periodicity to categories of Real matrix factorizations. These generalizations are structurally similar to $(1,1)$-periodicity for $KR$-theory and $4$-periodicity for Grothendieck-Witt theory. We use techniques from Real categorical representation theory which allow us to incorporate into our main results equivariance for a finite group and discrete torsion twists.