M Theory Lesson 207

A Kasteleyn matrix for a bipartite graph with $n$ vertices, drawn on the integer lattice in the plane, is an $n \times n$ adjacency matrix with non zero entries corresponding to edges $E_{ij}$, given by $K_{ij} = 1$ for horizontal edges and $K_{ij} = i$ for vertical edges. The square root of the determinant of $K$ counts the domino tilings of the checkerboard underlying the graph. An interesting paper by Stienstra includes examples of $6 \times 6$ generalised Kasteleyn matrices associated to $\mathbb{C}^{3} \backslash \mathbb{Z}_{6}$, where the row index corresponds to black vertices and the column index to white vertices, such as

1 -1 0 -1 0 0
0 1 -1 0 -1 0
-1 0 1 0 0 -1
-1 0 0 1 -1 0
0 -1 0 0 1 -1
0 0 -1 -1 0 1

which we observe is of the form

(1) (2)
(2) (1)

using $3 \times 3$ 1-circulants. It obeys the relation $K^{2} = 2K + [(312), 2(231)]$, where the final term is the obvious simple $6 \times 6$ matrix in terms of the permutation basis. Labelling edges with general complex units allows complex units as matrix entries. This example is derived from the hexagonal graph where the opposite sides of the hexagon are glued. That is, there are really only 18 edges, which is the number of non zero entries in $K$. If we did not glue edges there would be 24 non zero entries, based on 12 non zero entries for a pair of $3 \times 3$ circulants, just like the neutrino mixing matrix.