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Consider an elliptic parameter $k$; we introduce a family of $Z^u$-Diracoperators $(\mathsf{K}(u))_{u\in\Re(\mathbb{T}(k))}$, relate them to the$Z$-massive Laplacian of [BdTR17b], and extend to the full $Z$-invariant casethe results of Kenyon [Ken02] on discrete holomorphic and harmonic functions,which correspond to the case $k=0$. We prove, in a direct statistical mechanicsway, how and why the $Z^u$-Dirac and $Z$-massive Laplacian operators appear inthe $Z$-invariant Ising model, considering the case of infinite and finiteisoradial graphs. More precisely, consider the dimer model on the Fisher graph${\mathsf{G}}^{\scriptscriptstyle{\mathrm{F}}}$ arising from a $Z$-invariantIsing model. We express coefficients of the inverse Fisher Kasteleyn operatoras a function of the inverse $Z^u$-Dirac operator and also as a function of the$Z$-massive Green function; in particular this proves a (massive) random walkrepresentation of important observables of the Ising model. We prove that thesquared partition function of the Ising model is equal, up to a constant, tothe determinant of the $Z$-massive Laplacian operator with specific boundaryconditions, the latter being the partition function of rooted spanning forests.To show these results, we relate the inverse Fisher Kasteleyn operator and thatof the dimer model on the bipartite graph${\mathsf{G}}^{\scriptscriptstyle{\mathrm{Q}}}$ arising from the XOR-Isingmodel, and we prove matrix identities between the Kasteleyn matrix of${\mathsf{G}}^{\scriptscriptstyle{\mathrm{Q}}}$ and the $Z^u$-Dirac operator,that allow to reach inverse matrices as well as determinants.

The $Z$-Dirac and massive Laplacian operators in the $Z$-invariant Ising model