Product of spectral triples - Imported Knowlpedia Content

Definition

Let $(\mathcal A_i,H_i,D_i,\Gamma_i)$ , $i=1,2$ , be even spectral triples over complex Hilbert spaces. Their product spectral triple has

\mathcal A=\mathcal A_1\odot\mathcal A_2,\qquad H=H_1\widehat\otimes H_2,\qquad D=\overline{D_1\otimes1+\Gamma_1\otimes D_2}, \qquad \Gamma=\Gamma_1\otimes\Gamma_2.

The algebraic tensor product algebra acts on the Hilbert tensor product in the evident way, and the bar denotes the self-adjoint closure from the natural tensor-product core. The grading makes the two unbounded summands anticommute, so $D^2=D_1^2\otimes1+1\otimes D_2^2$ on that core.

Why the formula works

For $a_i\in\mathcal A_i$ ,

[D,a_1\otimes a_2] =[D_1,a_1]\otimes a_2 +\Gamma_1a_1\otimes[D_2,a_2],

which is bounded. The anticommutation in the square combines compact resolvents to give compact resolvent for $D$ under the standard compact spectral-triple hypotheses. These signs are precisely the graded-operator signs; omitting $\Gamma_1$ generally destroys the simple square formula Connes, Part VI.3.

Geometric example

For closed even-dimensional spin manifolds $M_1$ and $M_2$ , the product of their canonical spin spectral triples is unitarily equivalent to the canonical triple of $M_1\times M_2$ , after the standard identification of product spinors. The displayed Dirac formula becomes the familiar product formula for spin Dirac operators.

Parity and conventions

If only the first factor is even, the same operator formula defines the even–odd product, which is odd and has no product grading. Odd–odd products require an auxiliary two-dimensional Clifford module or an equivalent matrix convention. Different choices are unitarily equivalent after the convention is fixed, but writing $D_1\otimes1+1\otimes D_2$ without a grading sign is not the graded product.

For $C^*$ -completions of $\mathcal A_1\odot\mathcal A_2$ , the intended tensor norm must be specified separately; the spectral-triple axioms are imposed on the chosen dense star-subalgebra.

References

Alain Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted book. Relevant: Part VI.3 on products in spectral geometry.
Walter D. van Suijlekom, Noncommutative Geometry and Particle Physics, Springer, 2015. Publisher record. Relevant: the product construction used in the chapter “Almost-Commutative Manifolds and Gauge Theories,” pp. 137–158.