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Dimer models and Calabi-Yau algebras
About this Title
Nathan Broomhead, Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 215, Number 1011
ISBNs: 978-0-8218-5308-5 (print); 978-0-8218-8514-7 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00617-9
Published electronically: April 6, 2011
Keywords: Dimer models,
Calabi-Yau algebras,
Noncommutative crepant resolutions
MSC: Primary 14M25, 14A22; Secondary 82B20
Table of Contents
Chapters
- Acknowledgements
- 1. Introduction
- 2. Introduction to the dimer model
- 3. Consistency
- 4. Zig-zag flows and perfect matchings
- 5. Toric algebras and algebraic consistency
- 6. Geometric consistency implies algebraic consistency
- 7. Calabi-Yau algebras from algebraically consistent dimers
- 8. Non-commutative crepant resolutions
Abstract
In this article we use techniques from algebraic geometry and homological algebra, together with ideas from string theory to construct a class of 3-dimensional Calabi-Yau algebras. The Calabi-Yau property appears throughout geometry and string theory and is increasingly being studied in algebra. We further show that the algebras constructed are examples of non-commutative crepant resolutions (NCCRs), in the sense of Van den Bergh, of Gorenstein affine toric threefolds.
Dimer models, first studied in theoretical physics, give a way of writing down a class of non-commutative algebras, as the path algebra of a quiver with relations obtained from a ‘superpotential’. Some examples are Calabi-Yau and some are not. We consider two types of ‘consistency’ conditions on dimer models, and show that a ‘geometrically consistent’ dimer model is ‘algebraically consistent’. We prove that the algebras obtained from algebraically consistent dimer models are 3-dimensional Calabi-Yau algebras. This is the key step which allows us to prove that these algebras are NCCRs of the Gorenstein affine toric threefolds associated to the dimer models.
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