The monopole-dimer model on high-dimensional cylindrical, toroidal, Möbius and Klein grids

The dimer (monomer-dimer) model deals with weighted enumeration of perfect matchings (matchings). The monopole-dimer model is a signed variant of the monomer-dimer model whose partition function is a determinant. In 1999, Lu and Wu evaluated the partition function of the dimer model on two-dimensional grids embedded on a Möbius strip and a Klein bottle. While the partition function of the dimer model has been known for the two-dimensional grids with different boundary conditions, we present a similar product formula for the partition function of the monopole-dimer model on higher dimensional cylindrical and toroidal grid graphs. We also evaluate the same for the three-dimensional Möbius and Klein grid graphs and show that the formula does not generalise for the higher dimensions. Further, we present a relation between the product formula for the three-dimensional cylindrical and Möbius grid.