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The K\"ahler-Dirac fermion, recognized as an elegant geometric approach,offers an alternative to traditional representations of relativistic fermions.Recent studies have demonstrated that symmetric mass generation (SMG) canprecisely occur with two copies of K\"ahler-Dirac fermions across any spacetimedimensions. This conclusion stems from the study of anomaly cancellation withinthe fermion system. Our research provides an alternative understanding of thisphenomenon from a condensed matter perspective, by associating the interactingK\"ahler-Dirac fermion with the boundary of bosonic symmetry-protectedtopological (SPT) phases. We show that the low-energy bosonic fluctuations in asingle copy of the K\"ahler-Dirac fermion can be mapped to the boundary modesof a $\mathbb{Z}_2$-classified bosonic SPT state, protected by an inversionsymmetry universally across all dimensions. This implies that two copies ofK\"ahler-Dirac fermions can always undergo SMG through interactions mediated bythese bosonic modes. This picture aids in systematically designing SMGinteractions for K\"ahler-Dirac fermions in any dimension. We present the exactlattice Hamiltonian of these interactions and validate their efficacy indriving SMG.

Symmetric Mass Generation of K\"ahler-Dirac Fermions from the Perspective of Symmetry-Protected Topological Phases