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The monopole-dimer model is a signed variant of the monomer-dimer model whichhas determinantal structure. We extend the monopole-dimer model for planargraphs introduced by the second author (Math. Phys. Anal. Geom., 2015) toCartesian products thereof and show that the partition function of this modelcan be expressed as a determinant of a generalised signed adjacency matrix. Wethen show that the partition function is independent of the orientations of theplanar graphs so long as the orientations are Pfaffian. When these planargraphs are bipartite, we show that the computatation of the partition functionbecomes especially simple. We then give an explicit product formula for thepartition function of three-dimensional grid graphs a la Kasteleyn andTemperley--Fischer, which turns out to be fourth power of a polynomial when allgrid lengths are even. Finally, we generalise this product formula to $d$dimensions, again obtaining an explicit product formula. We conclude with adiscussion on asymptotic formulas for the free energy and monopole densities.

The monopole-dimer model on Cartesian products of plane graphs