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Dimer Models and Calabi-Yau Algebras
 
Nathan Broomhead Leibniz University Hannover, Hannover, Germany
Dimer Models and Calabi-Yau Algebras
eBook 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
Dimer Models and Calabi-Yau Algebras
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Dimer Models and Calabi-Yau Algebras
Nathan Broomhead Leibniz University Hannover, Hannover, Germany
eBook 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.

  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    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 publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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