Acknowledgements .............................................. vii
Chapter 1 Introduction ......................................... 1
1.1 Overview ................................................... 1
1.2 Structure of the article and main results .................. 2
1.3 Related results ............................................ 5
Chapter 2 Introduction to the dimer model ...................... 7
2.1 Quivers and algebras from dimer models ..................... 7
2.2 Symmetries ................................................ 13
2.3 Perfect matchings ......................................... 14
Chapter 3 Consistency ......................................... 19
3.1 A further condition on the R-symmetry ..................... 19
3.2 Rhombus tilings ........................................... 20
3.3 Zig-zag flows ............................................. 24
3.4 Constructing dimer models ................................. 28
3.5 Some consequences of geometric consistency ................ 31
Chapter 4 Zig-zag flows and perfect matchings ................. 35
4.1 Boundary flows ............................................ 35
4.2 Some properties of zig-zag flows .......................... 36
4.3 Right and left hand sides ................................. 38
4.4 Zig-zag fans .............................................. 39
4.5 Constructing some perfect matchings ....................... 43
4.6 The extremal perfect matchings ............................ 45
4.7 The external perfect matchings ............................ 48
Chapter 5 Toric algebras and algebraic consistency ............ 53
5.1 Toric algebras ............................................ 53
5.2 Some examples ............................................. 54
5.3 Some properties of toric algebras ......................... 56
5.4 Algebraic consistency for dimer models .................... 58
5.5 Example ................................................... 58
Chapter 6 Geometric consistency implies algebraic
consistency ......................................... 61
6.1 Flows which pass between two vertices ..................... 61
6.2 Proof of Proposition 6.2 .................................. 64
6.3 Proof of Theorem 6.1 ...................................... 71
Chapter 7 Calabi-Yau algebras from algebraically consistent
dimers .............................................. 73
7.1 Calabi-Yau algebras ....................................... 73
7.2 The one sided complex ..................................... 75
7.3 Key lemma ................................................. 77
7.4 The main result ........................................... 78
Chapter 8 Non-commutative crepant resolutions ................. 81
8.1 Reflexivity ............................................... 81
8.2 Non-commutative crepant resolutions ....................... 83
Bibliography ................................................... 85
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