Knörrer periodicity and Bott periodicity

Abstract

The goal of this article is to explain a precise sense in which Knörrer periodicity in commutative algebra and Bott periodicity in topological $K$-theory are compatible phenomena. Along the way, we prove an 8-periodic version of Knörrer periodicity for real isolated hypersurface singularities, and we construct a homomorphism from the Grothendieck group of the homotopy category of matrix factorizations of a complex (real) polynomial $f$ into the topological $K$-theory of its Milnor fiber (positive or negative Milnor fiber).