Metric dimension of a spectral triple - Imported Knowlpedia Content

Definition

For a spectral triple $(\mathcal A,H,D)$ , its metric dimension is the critical summability exponent

\dim_{\mathrm{met}}(D)= \inf\left\{p>0: \operatorname{Tr}\!\left((1+D^2)^{-p/2}\right)<\infty \right\},

with value $+\infty$ if the set is empty. Equivalently, it is the infimum of exponents for which the regularized inverse $(1+D^2)^{-1/2}$ belongs to the corresponding Schatten ideal. The infimum need not be attained and need not be an integer. It depends on the spectral growth of $D$ , including eigenvalue multiplicities, rather than on the algebra alone.

Counting-function interpretation

Let $N_D(\Lambda)$ count eigenvalues of $|D|$ not exceeding $\Lambda$ , with multiplicity. Polynomial growth $N_D(\Lambda)=O(\Lambda^d)$ implies summability for every $p>d$ . Under a matching two-sided Weyl asymptotic $N_D(\Lambda)\sim C\Lambda^d$ , the critical exponent equals $d$ , and $(1+D^2)^{-1/2}$ has singular values of order $n^{-1/d}$ . It then lies at the endpoint in the weak Schatten ideal $\mathcal L^{d,\infty}$ , although generally not in $\mathcal L^d$ .

Without regular variation or comparable estimates, a critical exponent alone does not imply weak-ideal membership at the endpoint.

Canonical manifold case

For the canonical spin spectral triple of a closed $d$ -dimensional Riemannian spin manifold, Weyl's law gives metric dimension $d$ . The triple is strictly $p$ -summable for every $p>d$ , while $|\not D|^{-d}$ , with the kernel removed, is of weak trace class. This analytic dimension agrees with manifold dimension and supports the noncommutative integral Connes, Chapter VI, §1.

Rescaling $D$ by a nonzero constant changes metric lengths but not the critical exponent. Taking direct sums can change the dimension to the larger of the component dimensions when both have polynomial spectral growth.

Distinction from dimension spectrum

Some authors reserve “spectral dimension” for a heat-kernel scaling exponent or for the abscissa of convergence of $\operatorname{Tr}(|D|^{-s})$ . These agree with the core under standard compact-resolvent conventions, after removing the kernel, but borderline and infinite-dimensional cases require care.

References

A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted text. Relevant: Chapters IV and VI on infinitesimal order, summability, and the dimension of a canonical manifold triple.
J. M. Gracia-Bondía, J. C. Várilly, and H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser, 2001. DOI record. Relevant: §§10.1 and 10.5 on summability, spectral dimension, and zeta functions.