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Dimer Models and Calabi-Yau Algebras

Nathan Broomhead Leibniz University Hannover, Hannover, Germany
Available Formats:
Electronic ISBN: 978-0-8218-8514-7
Product Code: MEMO/215/1011.E
List Price: $70.00 MAA Member Price:$63.00
AMS Member Price: $42.00 Click above image for expanded view Dimer Models and Calabi-Yau Algebras Nathan Broomhead Leibniz University Hannover, Hannover, Germany Available Formats:  Electronic ISBN: 978-0-8218-8514-7 Product Code: MEMO/215/1011.E  List Price:$70.00 MAA Member Price: $63.00 AMS Member Price:$42.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 2152011; 86 pp
MSC: Primary 14; Secondary 82;

In this article the author uses 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. He further shows 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. The author considers two types of ‘consistency’ conditions on dimer models, and shows that a ‘geometrically consistent’ dimer model is ‘algebraically consistent’. He proves that the algebras obtained from algebraically consistent dimer models are 3-dimensional Calabi-Yau algebras. This is the key step which allows him to prove that these algebras are NCCRs of the Gorenstein affine toric threefolds associated to the dimer models.

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
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Accessibility – to request an alternate format of an AMS title
Volume: 2152011; 86 pp
MSC: Primary 14; Secondary 82;

In this article the author uses 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. He further shows 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. The author considers two types of ‘consistency’ conditions on dimer models, and shows that a ‘geometrically consistent’ dimer model is ‘algebraically consistent’. He proves that the algebras obtained from algebraically consistent dimer models are 3-dimensional Calabi-Yau algebras. This is the key step which allows him to prove that these algebras are NCCRs of the Gorenstein affine toric threefolds associated to the dimer models.

• 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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