We prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact manifolds.

Recently Dabrowski et al. [Adv. Math. 427, 109128 (2023)] obtained the metric and Einstein functionals by two vector fields and Laplace-type operators over vector bundles, giving an interesting example of the spinor connection and square of the Dirac operator. Pfäffle and Stephan [Commun. Math. Phys. 321, 283–310 (2013)] considered orthogonal connections with arbitrary torsion on compact Riemannian manifolds and computed the spectral action. Motivated by the spectral functionals and Dirac operators with torsion, we give some new spectral functionals which is the extension of spectral functionals to the noncommutative realm with torsion, and we relate them to the noncommutative residue for manifolds with boundary. Our method of producing these spectral functionals is the noncommutative residue and Dirac operators with torsion.