Almost-commutative spectral triple - Imported Knowlpedia Content

Definition

Let $M$ be a closed even-dimensional Riemannian spin manifold with canonical spin spectral triple $(C^\infty(M),L^2(M,S),\not D_M,\Gamma_M)$ , and let $(\mathcal A_F,H_F,D_F,\Gamma_F)$ be a finite-dimensional even spectral triple. Their product

\left( C^\infty(M)\odot\mathcal A_F,\, L^2(M,S)\widehat\otimes H_F,\, \not D_M\otimes1+\Gamma_M\otimes D_F,\, \Gamma_M\otimes\Gamma_F \right)

is an almost-commutative spectral triple. The manifold factor supplies the continuous geometry, while the finite factor supplies finitely many internal degrees of freedom.

Algebra-bundle interpretation

For a trivial finite factor, the algebra is the smooth section algebra of the trivial bundle $M\times\mathcal A_F$ . More general globally almost-commutative geometries replace it by smooth sections of a locally trivial bundle of finite-dimensional star-algebras and replace the product Dirac operator by a compatible twisted Dirac-type operator. This extension allows nontrivial internal algebra bundles while retaining a classical manifold as the base.

Examples and gauge-theoretic role

Taking $\mathcal A_F=\mathbb C$ , $H_F=\mathbb C$ , and $D_F=0$ recovers the canonical spin spectral triple. Taking a noncommutative finite algebra, such as a sum of complex matrix algebras, gives matrix-valued functions over $M$ . Inner fluctuations of the product Dirac operator then split into ordinary gauge fields along $M$ and finite-direction scalar fields; this is the mechanism used in spectral-triple models of gauge theories van Suijlekom, chapter “Almost-Commutative Manifolds and Gauge Theories,” pp. 137–158.

Additional structures and scope

Applications commonly equip both factors with real structures and impose the order-zero and first-order conditions. Their product involves KO-dimension-dependent signs, and orientability or regularity must be checked separately. Those data are not part of the bare complex definition above.

“Almost commutative” does not mean that $C^\infty(M)\odot\mathcal A_F$ is nearly commutative in a metric sense. It means specifically that all noncommutativity is confined to a finite-dimensional internal factor, or to its finite-algebra-bundle generalization.

References

Walter D. van Suijlekom, Noncommutative Geometry and Particle Physics, Springer, 2015. Publisher record. Relevant: “Almost-Commutative Manifolds and Gauge Theories,” pp. 137–158.
Alain Connes and Matilde Marcolli, Noncommutative Geometry, Quantum Fields and Motives, American Mathematical Society and Hindustan Book Agency, 2008. AMS record. Relevant: the almost-commutative spectral geometry underlying the noncommutative-geometric Standard Model.