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.
Related
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BibTeX
@misc{wang2026cliffordweighted,
title = {{Clifford-weighted residues, partial spectral curvature, and torsion}},
author = {Yong Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34256},
eprint = {2610.34256},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34256},
}
RIS
TY - UNPB
TI - Clifford-weighted residues, partial spectral curvature, and torsion
AU - Yong Wang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34256
UR - https://arxiv.org/abs/2610.34256
AB - 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.
ER -