Spectral zeta function of a spectral triple - Imported Knowlpedia Content

Definition

Let $(\mathcal A,H,D)$ be a spectral triple. Write $|D|^{-s}$ for the complex power that is zero on $\ker D$ and equals $\lambda^{-s}$ on the positive spectral value $\lambda$ . The spectral zeta function is

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

at those $s\in\mathbb C$ for which $|D|^{-s}$ is trace class, using the canonical operator trace. More generally, for a bounded operator $b$ , the weighted spectral zeta function is $\zeta_b(s)=\operatorname{Tr}(b|D|^{-s})$ wherever the product is trace class. Meromorphic continuations, when they exist, are continuations of these initially convergent functions.

Convergence and spectral growth

If the nonzero eigenvalues of $|D|$ , repeated with multiplicity, are $\lambda_1\leq\lambda_2\leq\cdots$ , then

\zeta_D(s)=\sum_{n}\lambda_n^{-s}

in its half-plane of absolute convergence. Thus the convergence abscissa measures eigenvalue growth. Finite-dimensional kernels do not affect that growth, but the convention on $\ker D$ must be fixed before negative powers are written.

Heat-kernel relation

For $\operatorname{Re}s$ sufficiently large, Mellin transformation gives

\Gamma(s/2)\zeta_D(s) =\int_0^\infty t^{s/2-1} \operatorname{Tr}\!\left(e^{-tD^2}-P_{\ker D}\right)\,dt.

This relation connects small-time heat asymptotics with poles and residues of the zeta function. Under regularity and suitable asymptotic hypotheses, weighted zeta functions are the analytic input for the dimension spectrum and local index formula Connes–Moscovici, §II.

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 on zeta functions, residues, and dimension spectrum.
J. M. Gracia-Bondía, J. C. Várilly, and H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser, 2001. Publisher record. Relevant: §10.5 on spectral dimension and zeta functions.