Brane Tilings and Homological Mirror Symmetry
Quiver gauge theories are realized in string theory as configuration of D5/NS5-branes, whose data is summarized concisely by a dimer on torus, which is known as brane tiling. One of the virtues of brane tilings is that it makes the connection with Calabi-Yau geometry manifest. This means brane tilings are also useful for studying Calabi-Yau geometry itself. We have applied the technology of brane tiling to prove homological mirror symmetry, which is a version of mirror symmetry proposed by Kontsevich. The surprising fact is that complicated mathematical problem can be understood quite intuitively from brane perspective. Along the way I will also describe recent developments in the theory of brane tilings.