Clifford-weighted residues, partial spectral curvature, and torsion

Abstract

We study Clifford-weighted noncommutative residues associated with Dirac operators and connections. On closed spin manifolds a double commutator of the square of the Dirac operator, paired with a Clifford commutator, recovers the localized Riemann curvature tensor. We prove presentation independence for its extension to arbitrary classical one-forms, recompute four-Clifford functionals, and give explicit Weyl, Schouten and Einstein curvature combinations. For spectral triples with an admissible residue calculus we define a partial spectral curvature functional on two arbitrary represented one-forms and two exact represented one-forms. Its well-definedness is distinguished from module tensoriality: explicit flat noncommutative-torus computations exhibit both a failure of general presentation descent and nonzero partial pairings. A connection-based curvature pairing provides a separate construction, including noncentral pure Clifford-degree-two examples. We also derive ordered first- and second-derivative torsion residues on free torus modules and prove an obstruction to ordinary alternation for unrestricted noncentral inputs. All constants, coefficient orders and hypotheses are kept explicit.