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Lagrangian classes in K-theory

Dongwook Choa, Jeongseok Oh

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 Knörrer 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.

Lagrangian classes in K-theory

Abstract

For a -shifted Lagrangian in a critical locus, we construct a homomorphism from the -group of matrix factorisations of the critical locus to the -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 -shifted symplectic space is a -shifted Lagrangian of a point, whose -group is . The image of 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 Knörrer 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.
Paper Structure (1 section, 4 theorems, 14 equations)

This paper contains 1 section, 4 theorems, 14 equations.

Table of Contents

  1. Introduction

Key Result

Theorem 1

Let $L$ be an oriented $(-1)$-Lagrangian of $(-1)$-symplectic $M$ which is the critical locus of a critical model $(U,f)$. Suppose that Then we construct a homomorphism, called the $K$-theoretic Lagrangian class, If $(U,f)=(\operatorname{Spec}\mathbb C,0)$, then it takes $\mathcal{O}_{\operatorname{Spec}\mathbb C}$ to $\widehat{\mathcal{O}}^{\operatorname{vir}}_L$In Ku, Kuhn constructed a lift o

Theorems & Definitions (4)

  • Theorem : Theorem \ref{['main2']}
  • Theorem : Theorem \ref{['thm: spfunctor']}
  • Theorem : Theorem \ref{['thm:critind']}
  • Theorem : Theorem \ref{['main4']}