The monopole-dimer model on high-dimensional cylindrical, toroidal, M\"obius and Klein grids

TL;DR

This paper extends the understanding of the monopole-dimer model by deriving explicit partition function formulas for high-dimensional cylindrical, toroidal, M"obius, and Klein grid graphs, revealing dimension-dependent behaviors.

Contribution

It provides new product formulas for the monopole-dimer model on higher-dimensional grids and explores their limitations and relations across different topologies.

Findings

Product formulas for the monopole-dimer partition function on high-dimensional cylindrical and toroidal grids.

Explicit evaluation of the model on three-dimensional M"obius and Klein grids.

The formulas do not extend straightforwardly to higher dimensions.

Abstract

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 M\"obius 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 M\"obius 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 M\"obius grid.