Lagrangian classes in K-theory

Abstract

For a $(-1)$-shifted Lagrangian in a critical locus, we construct a homomorphism from the $K$-group of matrix factorisations of the critical locus to the $K$-group of the Lagrangian, partially answering the Joyce-Safronov conjecture. The key step is the construction of a specialisation functor for categories of matrix factorisations along the deformation to the normal cone. Any $(-2)$-shifted symplectic space is a $(-1)$-shifted Lagrangian of a point, whose $K$-group is $\mathbb{Z}$. The image of $1\in \mathbb{Z}$ under the above homomorphism is the virtual structure sheaf. We prove that two equivalent critical models of a given critical locus induce homomorphisms that commute via Kn\"orrer periodicity. When a torus acts on the Lagrangian, we further prove a localisation formula, namely the commutativity of the homomorphisms associated with the Lagrangian and its fixed locus.