Even spectral triple

An even spectral triple is a spectral triple $(\mathcal A,H,D)$ together with a grading operator $\Gamma$ on $H$ such that

\Gamma=\Gamma^*,\qquad \Gamma^2=1,\qquad \Gamma a=a\Gamma,\qquad \Gamma D=-D\Gamma

for every $a\in\mathcal A$ , with $\Gamma\operatorname{Dom}(D)=\operatorname{Dom}(D)$ . Thus the algebra representation is even and $D$ is an odd operator in the unbounded, domain-sensitive sense. Equivalently, $H=H^+\oplus H^-$ , every $a$ preserves the two summands, and $D$ interchanges them.

Block form

Relative to $H^+\oplus H^-$ , the data have the form

a= \begin{pmatrix} a^+&0\\ 0&a^- \end{pmatrix}, \qquad D= \begin{pmatrix} 0&D^-\\ D^+&0 \end{pmatrix},

where self-adjointness gives $D^-=(D^+)^*$ , with the corresponding domains understood. The off-diagonal form is precisely the anticommutation relation $\Gamma D=-D\Gamma$ .

Geometric example

On a closed even-dimensional Riemannian spin manifold, the complex spinor bundle splits as

S=S^+\oplus S^-.

The chirality operator supplies $\Gamma$ , functions preserve the splitting, and the spin Dirac operator exchanges positive and negative spinors. The canonical spectral triple is therefore even.

Index pairing

For a projection $p$ over the algebra, the represented projection preserves the graded subspaces. Under the usual compactness hypotheses, the compression of the positive part,

pD^+p:pH^+\longrightarrow pH^-,

has a Fredholm realization, and its integer index gives the even pairing with K-theory. Precise domain and matrix-amplification conventions are part of the pairing construction, not extra axioms in the definition above.

Graded-algebra variant

The standard definition assumes that $\mathcal A$ is trivially graded, which is why every represented $a$ commutes with $\Gamma$ . If $\mathcal A$ itself is graded, a graded representation and graded commutators replace these ordinary parity relations. That convention is broader and should be stated explicitly.

References