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Triangulated categories of singularities and D-branes in Landau-Ginzburg models

Dmitri Orlov

TL;DR

The paper introduces triangulated categories of singularities $D_{Sg}(X)$ as quotients of derived categories by perfect complexes and proves their locality and finiteness properties, connecting them to D-branes in Landau-Ginzburg models. It then establishes Knörrer periodicity: adjoining a quadratic term to the superpotential preserves the singularity category of fibers, enabling dimension lifting between LG models of different ambient dimensions. A central bridge is built between Kontsevich's category of B-branes in LG models and $D_{Sg}$ of LG fibers via matrix factorizations and cokernel constructions, culminating in an explicit calculation for an $A_{n-1}$-type singularity. The results provide a computational framework for B-branes through singularity theory and have implications for Homological Mirror Symmetry in non-Calabi–Yau settings.

Abstract

In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a connection of these categories with D-branes in Landau-Ginzburg models.

Triangulated categories of singularities and D-branes in Landau-Ginzburg models

TL;DR

The paper introduces triangulated categories of singularities as quotients of derived categories by perfect complexes and proves their locality and finiteness properties, connecting them to D-branes in Landau-Ginzburg models. It then establishes Knörrer periodicity: adjoining a quadratic term to the superpotential preserves the singularity category of fibers, enabling dimension lifting between LG models of different ambient dimensions. A central bridge is built between Kontsevich's category of B-branes in LG models and of LG fibers via matrix factorizations and cokernel constructions, culminating in an explicit calculation for an -type singularity. The results provide a computational framework for B-branes through singularity theory and have implications for Homological Mirror Symmetry in non-Calabi–Yau settings.

Abstract

In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a connection of these categories with D-branes in Landau-Ginzburg models.

Paper Structure

This paper contains 9 sections, 30 theorems, 73 equations.

Table of Contents

  1. Singularities and triangulated categories
  2. Triangulated categories and localizations
  3. Triangulated categories of singularities
  4. Triangulated categories of singularities for Gorenstein schemes
  5. Knörrer periodicity
  6. Triangulated category of D-branes of type B in Landau-Ginzburg model
  7. Kontsevich's proposal and categories of pairs
  8. Categories of pairs and categories of singularities
  9. Some simple calculations

Key Result

Lemma 1.1

The category ${\mathcal{D}}/{\mathcal{N}}$ becomes a triangulated category by taking for exact triangles such that are isomorphic to the images of exact triangles in ${\mathcal{D}}$. The quotient functor $Q:{\mathcal{D}}\longrightarrow {\mathcal{D}}/{\mathcal{N}}$ annihilates ${\mathcal{N}}$. Moreov

Theorems & Definitions (42)

  • Lemma 1.1
  • Lemma 1.2
  • Proposition 1.3
  • Lemma 1.4
  • Definition 1.5
  • Lemma 1.6
  • Remark 1.7
  • Definition 1.8
  • Remark 1.9
  • Lemma 1.10
  • ...and 32 more