Knowlpedia

Definition

KO-dimension sign table for a real spectral triple

The mod-eight rule assigning the commutation signs of the real structure with the Dirac operator and grading.

Definition

For a JJ, the KO-dimension sign table assigns signs to

J2=ε,JD=ε′DJ,JΓ=ε′′ΓJJ^2=\varepsilon,\qquad JD=\varepsilon'DJ,\qquad J\Gamma=\varepsilon''\Gamma J

as a function of n∈Z/8Zn\in\mathbb Z/8\mathbb Z. In the Connes–Marcolli convention the eight sign pairs (ε,ε′)(\varepsilon,\varepsilon'), for n=0,…,7n=0,\ldots,7, are

(+,+),(+,−),(−,+),(−,+),(−,+),(−,−),(+,+),(+,+).(+,+),(+,-),(-,+),(-,+),(-,+),(-,-),(+,+),(+,+).

The grading sign ε′′\varepsilon'' exists only in even dimension and equals +,−,+,−+,-,+,- for n=0,2,4,6n=0,2,4,6, respectively. Thus KO-dimension is discrete real-parity data; it is independent of the metric or of the triple.

Expanded lookup

The convention above gives the following complete lookup:

  • n=0n=0: (ε,ε′,ε′′)=(+,+,+)(\varepsilon,\varepsilon',\varepsilon'')=(+,+,+).
  • n=1n=1: (ε,ε′)=(+,−)(\varepsilon,\varepsilon')=(+,-).
  • n=2n=2: (ε,ε′,ε′′)=(−,+,−)(\varepsilon,\varepsilon',\varepsilon'')=(-,+,-).
  • n=3n=3: (ε,ε′)=(−,+)(\varepsilon,\varepsilon')=(-,+).
  • n=4n=4: (ε,ε′,ε′′)=(−,+,+)(\varepsilon,\varepsilon',\varepsilon'')=(-,+,+).
  • n=5n=5: (ε,ε′)=(−,−)(\varepsilon,\varepsilon')=(-,-).
  • n=6n=6: (ε,ε′,ε′′)=(+,+,−)(\varepsilon,\varepsilon',\varepsilon'')=(+,+,-).
  • n=7n=7: (ε,ε′)=(+,+)(\varepsilon,\varepsilon')=(+,+).

This is Connes and Marcolli, Definition 1.124.

Conventions and use

The table packages the real Clifford-algebra periodicity behind KO-homology. In particular, KO-dimension 22 gives J2=−1J^2=-1, JD=DJJD=DJ, and JΓ=−ΓJJ\Gamma=-\Gamma J, while KO-dimension 66 changes only J2J^2 among these three relations.

References
  1. A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, American Mathematical Society, 2008. DOI record. Relevant: Definition 1.124, especially equation (1.470) and its mod-eight table.
  2. A. Connes, “Noncommutative Geometry and Reality,” Journal of Mathematical Physics 36 (1995), 6194–6231. DOI record. Relevant: §2 on real K-cycles and dimension-dependent sign relations.