# 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