Derived category, Mirror symmetry and McKay correspondence seminar (DMM seminar)

Speaker: Akira Ishii (Hiroshima University)
Title: Dimer models and exceptional collections
Date (JST): Mon, Apr 26, 2010, 14:00 - 17:00
Place: Seminar Room A
Abstract: Part1: I will talk about a joint work with Kazushi Ueda, a construction of exceptional collections via dimer models, which is based on a proposal by physicists. A dimer model is a bicolored graph on a real two-torus which encodes the information of a quiver with
relations, and which determines a lattice polygon. If we specify an interior lattice point together with several boundary lattice points, we can consider the toric weak Fano suface (stack) associated with it. Assuming that the dimer model is "consistent", we can construct a full strong exceptional collection consisting of line bundles on the
surface. Moreover, we can describe the total endomorphism algebra in terms of quiver with relations, which should be relevant to
homological mirror symmetry.

Part 2: I will talk about relations between dimer models and 3-dimensional affine toric Gorenstein varieties. Under some conditions, the moduli space of quiver representations will be a crepant resolution of the toric variety. We discuss consistency conditions on dimer models which ensures derived equivalence between the crepant resolution and the path algebra. We further describe an operation on dimer models corresponding to removing a vertex from the lattice polygon.
Remarks: we have a break at 15:00-15:30.