Electronic ISBN:  9780821885147 
Product Code:  MEMO/215/1011.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 215; 2011; 86 ppMSC: 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 3dimensional CalabiYau algebras. The CalabiYau property appears throughout geometry and string theory and is increasingly being studied in algebra. He further shows that the algebras constructed are examples of noncommutative 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 noncommutative algebras, as the path algebra of a quiver with relations obtained from a ‘superpotential’. Some examples are CalabiYau 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 3dimensional CalabiYau 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. Zigzag flows and perfect matchings

5. Toric algebras and algebraic consistency

6. Geometric consistency implies algebraic consistency

7. CalabiYau algebras from algebraically consistent dimers

8. Noncommutative crepant resolutions


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In this article the author uses techniques from algebraic geometry and homological algebra, together with ideas from string theory to construct a class of 3dimensional CalabiYau algebras. The CalabiYau property appears throughout geometry and string theory and is increasingly being studied in algebra. He further shows that the algebras constructed are examples of noncommutative 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 noncommutative algebras, as the path algebra of a quiver with relations obtained from a ‘superpotential’. Some examples are CalabiYau 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 3dimensional CalabiYau 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. Zigzag flows and perfect matchings

5. Toric algebras and algebraic consistency

6. Geometric consistency implies algebraic consistency

7. CalabiYau algebras from algebraically consistent dimers

8. Noncommutative crepant resolutions