Spectral triple - Imported Knowlpedia Content

Definition

A compact spectral triple $(\mathcal A,H,D)$ consists of a unital involutive algebra $\mathcal A$ , a Hilbert space $H$ carrying a bounded star-representation of $\mathcal A$ , and a densely defined self-adjoint operator $D$ . The operator has compact resolvent, and every $a\in\mathcal A$ preserves $\operatorname{Dom}(D)$ , with the commutator there extending to a bounded operator on $H$ . The representation is usually suppressed, so $a$ denotes its represented operator. It is often required to be faithful, or $\mathcal A$ is replaced by its represented quotient.

What the axioms encode

The algebra $\mathcal A$ plays the role of a smooth algebra of functions, while $H$ is the space on which geometry is represented. The operator $D$ supplies metric and differential information. Compact resolvent gives discrete spectral behavior analogous to an elliptic operator on a compact manifold. Boundedness of

[D,a]=Da-aD

is the abstract first-order regularity condition: multiplication by a smooth function changes a first-order differential operator only by an operator of order zero.

The adjective “spectral” does not mean that the spectrum alone determines every aspect of the triple. The representation of $\mathcal A$ and its commutators with $D$ are part of the data.

Canonical commutative example

Let $M$ be a closed Riemannian spin manifold, let $\mathcal A=C^\infty(M)$ , let $H=L^2(M,S)$ be the square-integrable sections of its spinor bundle, and let $D$ be the spin Dirac operator. Functions act by multiplication. Ellipticity and compactness of $M$ give compact resolvent, while

[D,f]=c(df)

is bounded Clifford multiplication by the differential of $f$ . Thus $(C^\infty(M),L^2(M,S),D)$ is a spectral triple.

Parity

An even spectral triple includes a compatible grading on $H$ for which the algebra acts evenly and $D$ acts oddly. An odd spectral triple is ungraded. Parity is additional structure and is separate from summability, regularity, reality, and first-order axioms that may be imposed in more elaborate versions.

Nonunital and locally compact variants

For a nonunital algebra, compact resolvent is generally too strong. A common locally compact convention replaces it by

a(1+D^2)^{-1/2}\in K(H) \qquad\text{for every }a\in\mathcal A.

For unital $\mathcal A$ , taking $a=1$ recovers compactness of $(1+D^2)^{-1/2}$ , hence compact resolvent. Semifinite, real, twisted, and graded-algebra spectral triples modify other parts of the definition; hypotheses from one variant should not be presumed in another.

References