Ishii, Akira; Ueda, Kazushi A note on consistency conditions on dimer models. (English) Zbl 1270.16012 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. Cited in 9 Documents 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 PDF BibTeX XML Cite \textit{A. Ishii} and \textit{K. Ueda}, RIMS Kôkyûroku Bessatsu B24, 143--164 (2011; Zbl 1270.16012) Full Text: arXiv