Dimension spectrum of a spectral triple - Imported Knowlpedia Content

Definition

Let $(\mathcal A,H,D)$ be a regular spectral triple, and let $\mathcal B$ be the algebra generated by all $\delta^k(a)$ and $\delta^k([D,a])$ , where $a\in\mathcal A$ , $k\geq0$ , and $\delta(T)=[|D|,T]$ . A discrete set $\Sigma\subset\mathbb C$ is a dimension spectrum if, for every $b\in\mathcal B$ , the initially defined

\zeta_b(s)=\operatorname{Tr}(b|D|^{-s})

extends meromorphically to $\mathbb C$ with all poles contained in $\Sigma$ , and $\Sigma$ is minimal with this property. The triple has simple dimension spectrum when all those poles are simple. Thus the definition concerns a family of weighted spectral zeta functions, not only $\zeta_D$ .

Geometric meaning

Pole locations encode the exponents in high-energy or small-time asymptotic expansions. For the canonical Dirac spectral triple on a closed $d$ -dimensional manifold, the dimension spectrum is contained in an arithmetic sequence of integers bounded above by $d$ ; parity and vanishing of heat coefficients can remove some candidate poles. The largest actual pole often recovers the metric dimension, but the full set contains finer information.

Role in the local index formula

The local index formula uses residues of the functions $\zeta_b$ to build cyclic cocycles. Regularity alone supplies the algebra $\mathcal B$ , but it does not imply the required meromorphic continuation. Discreteness and bounds on pole order are separate analytic hypotheses Connes–Moscovici, §§II and III.

Conventions and scope

References

A. Connes and H. Moscovici, “The Local Index Formula in Noncommutative Geometry,” Geometric and Functional Analysis 5 (1995), 174–243. DOI record. Relevant: §§II–III, dimension spectrum and residue cocycles.
N. Higson, “The Local Index Formula in Noncommutative Geometry,” in Contemporary Developments in Algebraic K-Theory, ICTP Lecture Notes 15, 2004. Author-hosted manuscript. Relevant: zeta-function asymptotics and dimension-spectrum hypotheses.