The paper investigates the convergence of a certain Hodge-Dirac operator \(D_h = d_h + d_h^\ast\) on the lattice \(h \mathbb{Z}^n\) to the continuous Hodge-Dirac operator on \(\mathbb{R}^n\) in the limit \(h\to 0\).
The authors first introduce an abstract framework called “combinatorial differential complexes” \(X\) and associated exterior derivatives. These notions generalize simplicial complexes and their associated exterior derivatives. This concept allows them to consider \(h\mathbb{Z}^n\) as a combinatorial differential complex \(X(h \mathbb{Z}^n)\) of dimension \(n\) (as a simplicial complex, \(h\mathbb{Z}^n\) is a graph and has dimension \(1\)). Vaguely speaking, hypercubes of dimension \(j\) in the lattice \(h \mathbb{Z}^n\) correspond to faces of dimension \(j\) in the combinatorial differential complex \(X(h \mathbb{Z}^n)\). The associated exterior derivative \(d_h \colon \ell^2(X(h \mathbb{Z}^n)) \to \ell^2(X(h \mathbb{Z}^n))\) then gives rise to the Dirac operator \(D_h = d_h+ d_h^\ast\).
The exterior derivative \(d_h\) is shown to be unitarily equivalent to a certain discrete Hodge-Dirac operator \(\tilde d_h = \sum_{j=0}^n \tilde d_{j,h}\) on \(\bigoplus_{j=0}^n \ell^2(h\mathbb{Z}^n; \bigwedge^j (h \mathbb{Z}^n))\), where each \(\tilde d_{j,h} \colon \ell^2(h\mathbb{Z}^n; \bigwedge^j (h \mathbb{Z}^n)) \to \ell^2(h\mathbb{Z}^n; \bigwedge^{j+1} (h \mathbb{Z}^n))\) is a discrete exterior derivative defined on a suitable space of discrete differential forms \(\ell^2(h\mathbb{Z}^n; \bigwedge^j (h \mathbb{Z}^n))\) of order \(j\).
The authors then prove that, as \(h \to 0\), the discrete Hodge-Dirac operator \(D_h\) converges in a generalized norm resolvent sense to the Hodge-Dirac operator \(D = d +d^\ast\) on \(\bigoplus_{j=0}^n L^2(\mathbb{R}^n; \mathbb{C}^{\binom{n}{k}})\). Other versions of discrete Dirac operators are known to converge to their continuous counter-parts only in the (generalized) strong resolvent sense, see, e.g., [H. Cornean et al., J. Spectr. Theory 12, No. 4, 1589–1622 (2022; Zbl 1530.47016)].