A note on consistency conditions on dimer models. (English) Zbl 1270.16012

Ishii, Akira; Ueda, Kazushi

RIMS Kôkyûroku Bessatsu B24, 143-164 (2011).

Summary: We show that for a non-degenerate dimer model, both the first consistency condition of S. Mozgovoy and M. Reineke [Adv. Math. 223, No. 5, 1521-1544 (2010; Zbl 1191.14008)] and the properly-orderedness condition of D. R. Gulotta [[J. High Energy Phys. 2008, No. 10, Article ID 014 (2008; Zbl 1245.81091)] are equivalent to a condition on zigzag paths, which goes back to A. Hanany and D. Vegh [J. High Energy Phys. 2007, No. 10, Article ID 029 (2007)]. The last condition is used by the authors [in Dimer models and the special McKay correspondence, arXiv:0905.0059] to study the behavior of a dimer model under the operation of removing a vertex from the lattice polygon and taking the convex hull of the rest.

MSC:

16G20	Representations of quivers and partially ordered sets
14D21	Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
81T30	String and superstring theories; other extended objects (e.g., branes) in quantum field theory
83E30	String and superstring theories in gravitational theory
14J32	Calabi-Yau manifolds (algebro-geometric aspects)
82B20	Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics

Keywords:

non-degenerate dimer models; consistency conditions; properly-orderedness condition; zigzag paths; lattice polygons