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Graph theory and Pfaffian representations of Ising partition function

A well known theorem due to Kasteleyn states that the partition function of an Ising model on an arbitrary planar graph can be represented as the Pfaffian of a skew-symmetric matrix associated to the graph. This results both embodies the free fermionic nature of any planar Ising model and eventually gives an effective way of computing its partition functions in closed form. An extension of this result to non planar models expresses the partition function as a sum of Pfaffians which number is related to the genus of the oriented surface on which the graph can be embedded. In graph theory, McLane's theorem (1937) gives a characterization of planarity as a property of the cycle space of a graph, and recently, Diestel et al. (2009) extended this approach to embeddings in arbitrary surfaces. Here we show that McLane's approach naturally leads to Kasteleyn's results: McLane characterization of planar graphs is just what is needed to turn an Ising partition function into a Pfaffian. Using this approach, we prove that the Ising partition function on an arbitrary non planar graph can be written as \emph{the real part}of the Pfaffian of a single matrix with coefficients taken in a multicomplex algebra $\C_{\tilde g}$, where $\tilde g$ is the non-orientable genus, or crosscap number, of the embedding surface. Known representations as sums of Pfaffians follow from this result. In particular, Kasteleyn's result which involves $4^g$ matrices with real coefficients, $g$ orientable genus, is also recovered through some algebraic reduction.