Real spectral triple - Imported Knowlpedia Content

Definition

A real spectral triple of KO-dimension $n$ is a spectral triple $(\mathcal A,H,D)$ , even or odd, together with a real structure $J$ having the signs prescribed by $n\bmod 8$ . Writing

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

the data must satisfy the order-zero condition

[a,b^\circ]=0

and the first-order condition

\bigl[\, [D,a],b^\circ\,\bigr]=0

for all $a,b\in\mathcal A$ . In even parity the grading $\Gamma$ commutes with $\mathcal A$ , anticommutes with $D$ , and obeys the KO-dimensional relation with $J$ . All commutators involving $D$ use their bounded extensions.

Bimodule and parity structure

The order-zero condition makes $H$ an $\mathcal A$ -bimodule, with $a\xi b=ab^\circ\xi$ . The first-order condition says that each bounded commutator $[D,a]$ is right $\mathcal A$ -linear. Together they abstract the fact that a Dirac operator is first-order.

The parity of the spectral triple and its KO-dimension are distinct: parity records whether a grading is present, whereas KO-dimension records the signs of $J$ with $D$ and, when present, $\Gamma$ Connes and Marcolli, Definition 1.124.

Canonical example and scope

For a closed Riemannian spin manifold $M$ , the data

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

form a real spectral triple, with chirality added in even dimension. Functions act by multiplication, $J$ is charge conjugation, and $[\not D,f]$ is Clifford multiplication by $df$ ; hence it commutes with the right action by functions.

References

A. Connes, “Noncommutative Geometry and Reality,” Journal of Mathematical Physics 36 (1995), 6194–6231. DOI record. Relevant: §2 on real K-cycles and the opposite action.
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).