**Memoirs of the American Mathematical Society**

2011;
86 pp;
Softcover

MSC: Primary 14;
Secondary 82

Print ISBN: 978-0-8218-5308-5

Product Code: MEMO/215/1011

List Price: $70.00

Individual Member Price: $42.00

**Electronic ISBN: 978-0-8218-8514-7
Product Code: MEMO/215/1011.E**

List Price: $70.00

Individual Member Price: $42.00

# Dimer Models and Calabi-Yau Algebras

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*Nathan Broomhead*

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

# Table of Contents

## Dimer Models and Calabi-Yau Algebras

- Acknowledgements vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. Introduction to the dimer model 716
- Chapter 3. Consistency 1928
- Chapter 4. Zig-zag flows and perfect matchings 3544
- Chapter 5. Toric algebras and algebraic consistency 5362
- Chapter 6. Geometric consistency implies algebraic consistency 6170
- Chapter 7. Calabi-Yau algebras from algebraically consistent dimers 7382
- Chapter 8. Non-commutative crepant resolutions 8190
- Bibliography 8594