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Calabi-Yau completions and orbifold equivalences

Nils Carqueville, Alexander Quintero Velez

Abstract

Calabi-Yau algebras are particularly symmetric differential graded algebras. There is a construction called `Calabi-Yau completion' which produces a canonical Calabi-Yau algebra from any homologically smooth dg algebra. Homologically smooth dg algebras also form a 2-category to which the construction of `equivariant completion' can be applied. In this theory two objects are called `orbifold equivalent' if there is a 1-morphism $X$ with invertible quantum dimensions between them. Any such relation entails a whole family of equivalences between categories. We show that an orbifold equivalence between two homologically smooth and proper dg algebras lifts to an orbifold equivalence between their Calabi-Yau completions under certain conditions on $X$.

Calabi-Yau completions and orbifold equivalences

Abstract

Calabi-Yau algebras are particularly symmetric differential graded algebras. There is a construction called `Calabi-Yau completion' which produces a canonical Calabi-Yau algebra from any homologically smooth dg algebra. Homologically smooth dg algebras also form a 2-category to which the construction of `equivariant completion' can be applied. In this theory two objects are called `orbifold equivalent' if there is a 1-morphism with invertible quantum dimensions between them. Any such relation entails a whole family of equivalences between categories. We show that an orbifold equivalence between two homologically smooth and proper dg algebras lifts to an orbifold equivalence between their Calabi-Yau completions under certain conditions on .

Paper Structure

This paper contains 21 sections, 28 theorems, 122 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Background on differential graded algebras
  3. Perfect categories and duality
  4. The derived category of a dg algebra
  5. The perfect category of a dg algebra
  6. Duality for perfect dg modules
  7. Calabi-Yau completions and Ginzburg algebras
  8. Calabi-Yau completions
  9. Ginzburg algebras
  10. Equivariant completion and orbifold equivalence
  11. General theory
  12. Bicategorical algebra
  13. Equivariant completion
  14. Applications to Landau-Ginzburg models
  15. Bicategory of affine Landau-Ginzburg models
  16. ...and 6 more sections

Key Result

Proposition 2.1

Let $A$ and $B$ be dg algebras and let $F \colon \mathbf{D}(A) \rightarrow \mathbf{D}(B)$ be a triangulated functor such that $F(A)$ is in $\operatorname{Perf}(B)$. Then, for any $M \in \operatorname{Perf}(A)$, its image $F(M)$ is in $\operatorname{Perf}(B)$.

Figures (1)

  • Figure 3.1: ADE Dynkin diagrams with vertex label convention of kst0511155

Theorems & Definitions (46)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Theorem 2.8
  • Definition 3.1
  • Definition 3.2
  • ...and 36 more