Matrix factorizations and motivic measures

Abstract

This article is the continuation of [17]. We use categories of matrix factorizations to define a morphism of rings (= a Landau–Ginzburg motivic measure) from the (motivic) Grothendieck ring of varieties over ${\mathbb{A}}^1$ to the Grothendieck ring of saturated dg categories (with relations coming from semi-orthogonal decompositions into admissible subcategories). Our Landau–Ginzburg motivic measure is the analog for matrix factorizations of the motivic measure in [5] whose definition involved bounded derived categories of coherent sheaves. On the way we prove smoothness and a Thom–Sebastiani theorem for enhancements of categories of matrix factorizations.