Odd spectral triple - Imported Knowlpedia Content

Definition

An odd spectral triple is a spectral triple $(\mathcal A,H,D)$ regarded without a compatible $\mathbb Z/2$ -grading: the data are the represented involutive algebra $\mathcal A$ , Hilbert space $H$ , and self-adjoint operator $D$ , subject to the compactness and bounded-commutator axioms of a spectral triple. “Odd” names its parity in analytic K-homology; it does not mean that $D$ has odd degree on an unstated grading. If a grading $\Gamma$ commuting with $\mathcal A$ and anticommuting with $D$ is included, the resulting object is instead an even spectral triple.

Example: the circle

For the circle, take

\mathcal A=C^\infty(S^1),\qquad H=L^2(S^1),\qquad D=-i\frac{d}{d\theta}

with periodic domain $H^1(S^1)$ . Multiplication by $f\in C^\infty(S^1)$ preserves the domain, and

[D,f]=-i\,f'

is bounded multiplication. The Fourier basis diagonalizes $D$ , and its eigenvalues tend to both positive and negative infinity, so $D$ has compact resolvent. This is the basic odd spectral triple associated with a one-dimensional closed spin manifold.

Bounded transform

The bounded transform

F=D(1+D^2)^{-1/2}

is self-adjoint and bounded. Under the spectral-triple hypotheses, it satisfies the compactness relations used in an odd bounded Fredholm module. This construction connects the unbounded geometric cycle with an odd K-homology class. It does not say that $D$ itself is bounded or Fredholm in the bounded-operator sense.

Parity convention

Many texts define an odd cycle simply by omitting grading data. This does not assert that no compatible grading could possibly exist; it asserts that none is part of the odd cycle. Conversely, forgetting the grading from an even triple loses essential parity data and is not a neutral identification of its K-homology class.

References