Real structure on a spectral triple - Imported Knowlpedia Content

Definition

Let $(\mathcal A,H,D)$ be a spectral triple. A real structure of KO-dimension $n$ is an antiunitary $J:H\to H$ for which

J^2=\varepsilon,\qquad JD=\varepsilon' DJ,

and, in the even case with grading $\Gamma$ ,

J\Gamma=\varepsilon''\Gamma J.

The signs $\varepsilon,\varepsilon',\varepsilon''\in\{1,-1\}$ are prescribed by $n\bmod 8$ . The operator also gives a complex-linear representation of the opposite algebra by

b^{\circ}=Jb^*J^{-1}.

To obtain the standard real spectral-triple axioms, this right action must satisfy the order-zero and first-order conditions. Those conditions are additional compatibility requirements, not consequences of antiunitarity or of the sign relations.

Meaning of the data

The antiunitary $J$ abstracts charge conjugation on spinors. The formula for $b^\circ$ turns the Hilbert space into a candidate $\mathcal A$ -bimodule: the represented algebra acts from the left and its opposite algebra acts from the right. The KO-sign relations record the real Clifford-module behavior of $J$ , $D$ , and the grading Connes and Marcolli, Definition 1.124.

Multiplying $J$ by a complex scalar of modulus one does not change $J^2$ , because $J$ is antilinear.

Canonical example and scope

For the canonical spin spectral triple of a closed Riemannian spin manifold, charge conjugation on the complex spinor bundle supplies $J$ . Its commutation signs depend only on the dimension modulo eight. The opposite action of a function is again multiplication by that function, so it commutes with the left action.

References

A. Connes, “Noncommutative Geometry and Reality,” Journal of Mathematical Physics 36 (1995), 6194–6231. DOI record. Relevant: §2 on the real structure, opposite action, and sign relations.
A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, American Mathematical Society, 2008. DOI record. Relevant: Definition 1.124 and equations (1.470)–(1.473).