Canonical spectral triple of a compact spin manifold

Definition

Let $M$ be a closed Riemannian manifold with a spin structure and complex spinor bundle $S$ . Its canonical spin spectral triple is

\bigl(C^\infty(M),\,L^2(M,S),\,\not D\bigr),

where $C^\infty(M)$ acts faithfully by pointwise multiplication and $\not D$ is the self-adjoint closure of the spin Dirac operator. The operator $\not D$ has compact resolvent, and for every $f\in C^\infty(M)$ ,

[\not D,f]=c(df)

extends to a bounded operator. These facts make the displayed data a spectral triple.

Why the spectral-triple axioms hold

Ellipticity on the compact manifold implies that the resolvent of $\not D$ is compact. The first-order Leibniz rule identifies its commutator with multiplication by the Clifford field $c(df)$ , which is bounded because $M$ is compact. Smooth functions are used rather than all of $C(M)$ : a merely continuous function need not preserve the domain of $\not D$ or have a bounded commutator.

Parity and additional structures

If $\dim M$ is even, chirality grades $L^2(M,S)$ , commutes with functions, and anticommutes with $\not D$ , producing an even spectral triple. In odd dimension the canonical triple is ungraded. Charge conjugation can supply a real structure, but that operator and its dimension-dependent signs are additional data, not part of the three-component spectral triple defined in the core.

Geometry recovered from the triple

The growth of the eigenvalues of $|\not D|$ records the dimension through Weyl asymptotics. Connes's distance formula recovers the Riemannian geodesic distance from

d(p,q)=\sup\{|f(p)-f(q)|:\lVert[\not D,f]\rVert\leq1\}.

The reconstruction theorem requires further regularity, finiteness, orientability, and reality hypotheses; compact resolvent and bounded commutators alone do not characterize canonical manifold triples.

References

Alain Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted book. Relevant: Part VI.1 on the spectral geometry of compact spin manifolds.
José M. Gracia-Bondía, Joseph C. Várilly, and Héctor Figueroa, Elements of Noncommutative Geometry, Birkhäuser, 2001. DOI record. Relevant: §10.2 on the canonical commutative spectral triple.