Identities between dimer partition functions on different surfaces. (26th October 2016) - Cimasoni, David; Pham, Anh Minh
Abstract: Given a weighted graph G embedded in a non-orientable surface Σ, one can consider the corresponding weighted graph G ̃ embedded in the so-called orientation cover Σ ̃ of Σ . We prove identities relating twisted partition functions of the dimer model on these two graphs. When Σ is the Möbius strip or the Klein bottle, then Σ ̃ is the cylinder or the torus, respectively, and under some natural assumptions, these identities imply relations between the genuine dimer partition functions Z ( G ) and Z ( G ̃ ) . For example, we show that if G is a locally but not globally bipartite graph embedded in the Möbius strip, then Z ( G ̃ ) is equal to the square of Z ( G ). This extends results for the square lattice previously obtained by various authors.