The Hodge–Dirac operator and Dabrowski–Sitarz–Zalecki-type theorems for manifolds with boundary

Dabrowski et al. Spectral metric and Einstein functionals for Hodge–Dirac operator, preprint (2023), arXiv:2307.14877 gave spectral Einstein bilinear functionals of differential forms for the Hodge–Dirac operator Formula: see text on an oriented even-dimensional Riemannian manifold. In this paper, we generalize the results of Dabrowski et al. to the cases of 4-dimensional oriented Riemannian manifolds with boundary. Furthermore, we give the proof of Dabrowski–Sitarz–Zalecki-type theorems associated with the Hodge–Dirac operator for manifolds with boundary.