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29 March 2024
 
  » arxiv » 1509.0880

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Calabi-Yau completions and orbifold equivalences
Nils Carqueville ; Alexander Quintero Velez ;
Date 2 Sep 2015
AbstractCalabi-Yau algebras are particularly symmetric differential graded algebras. There is a construction due to Keller 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 Calabi-Yau completion and equivariant completion are compatible. More precisely, we prove that any orbifold equivalence between two homologically smooth and proper dg algebras lifts to an orbifold equivalence between their Calabi-Yau completions. As a corollary we obtain orbifold equivalences between Ginzburg algebras of Dynkin quivers.
Source arXiv, 1509.0880
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