Dimer Models and Calabi-Yau Algebras.

Intro -- Contents -- Acknowledgements -- Chapter 1. Introduction -- 1.1. Overview -- 1.2. Structure of the article and main results -- 1.3. Related results -- Chapter 2. Introduction to the dimer model -- 2.1. Quivers and algebras from dimer models -- 2.2. Symmetries -- 2.3. Perfect matchings -- Chapter 3. Consistency -- 3.1. A further condition on the R-symmetry -- 3.2. Rhombus tilings -- 3.3. Zig-zag flows -- 3.4. Constructing dimer models -- 3.5. Some consequences of geometric consistency -- Chapter 4. Zig-zag flows and perfect matchings -- 4.1. Boundary flows -- 4.2. Some properties of zig-zag flows -- 4.3. Right and left hand sides -- 4.4. Zig-zag fans -- 4.5. Constructing some perfect matchings -- 4.6. The extremal perfect matchings -- 4.7. The external perfect matchings -- Chapter 5. Toric algebras and algebraic consistency -- 5.1. Toric algebras -- 5.2. Some examples -- 5.3. Some properties of toric algebras -- 5.4. Algebraic consistency for dimer models -- 5.5. Example -- Chapter 6. Geometric consistency implies algebraic consistency -- 6.1. Flows which pass between two vertices -- 6.2. Proof of Proposition 6.2 -- 6.3. Proof of Theorem 6.1 -- Chapter 7. Calabi-Yau algebras from algebraically consistent dimers -- 7.1. Calabi-Yau algebras -- 7.2. The one sided complex -- 7.3. Key lemma -- 7.4. The main result -- Chapter 8. Non-commutative crepant resolutions -- 8.1. Reflexivity -- 8.2. Non-commutative crepant resolutions -- Bibliography.

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