nLab zeta function of an elliptic differential operator

Idea

For a suitable linear operator $H$ (say on section of a line bundle over a Riemann surface), its zeta function is the analytic continuation of the trace

$$\zeta_H(s) \coloneqq Tr(H^{-s})$$

of the $-s$ power of $H$, which, if $H$ is suitably self-adjoint, is the sum of the $-s$-powers of all its eigenvalues, as a function of $s$. This is analogous to the Riemann zeta function and the Dedekind zeta function (or would be if there were something like a Laplace operator on Spec(Z) or more generally on an arithmetic curve, see at function field analogy).

The exponential of the derivative of the zeta function at $n = 0$ also encodes the functional determinant of $H$, a regularized version (“zeta function regularization”) of the naive and generally ill-defined product of all eigenvalues. As such, zeta functions play a central role in quantum field theory.

Generally, the values of $\zeta_H(s)$ of interest in physics (when regarding $H$ as a Hamilton operator) are those for (low) integral $s$. These are just the special values of L-functions.

Definition

The zeta function

Given an elliptic differential operator with positive lower bound $c$, write $H$ for its self-adjoint extension and write

$$0 \lt \lambda_1 \leq \lambda_2 \leq \cdots$$

for its eigenvalues.

Definition

The zeta function of $H$ is the holomorphic function defined by the series

$$\begin{aligned} \zeta_H(s) & \coloneqq Tr( H^{-s} ) \\ & \coloneqq \underoverset{n = 1}{\infty}{\sum} \frac{1}{(\lambda_n)^s} \end{aligned} \,.$$

where this converges and then extended by analytic continuation.

(e.g. (Duistermaat-Guillemin 75 (2.13), Berline-Getzler-Vergne 04, section 9.6 ) ).

Functional determinant and zeta-function regularization

Notice that the first derivative $\zeta^\prime_H$ of this zeta function is, where the original series converges, given by

$$\zeta_H^\prime(s) = \sum_{n = 1}^\infty \frac{- \ln \lambda_n}{ (\lambda_n)^s} \,.$$

Therefore one says (Ray-Singer 71) that the functional determinant of $H$ is the exponential of the derivative of zeta function of $H$ at 0:

$$det_{reg} H \coloneqq \exp(- \zeta_H^\prime(0)) \,.$$

Via the analytic continuation involved in defining $\zeta_H(0)$ in the first place, this may be thought of as a regularization of the ill-defined naive definition “$\prod_n \lambda_n$” of the determinant of $H$. As such functional determinants often appear in quantum field theory as what is called zeta function regularization.

Conversely, the logarithm

$$Z \coloneqq - \frac{1}{2}\zeta_H^\prime(0) = \tfrac{1}{2} log\,det_{reg} H$$

is what is called the vacuum energy in quantum field theory (for $H^{-1}$ the Feynman propagator).

If $H = D^2$ has a square root $D$ (a Dirac operator-type square root as in supersymmetric quantum mechanics) then under some conditions on the growth of the eigenvalues, then the functional determinant may also be expressed in terms of the eta function of $D$ as

$$det H = det (D^2) = \exp( \frac{\partial}{\partial s}\frac{\partial}{\partial c} \eta_{D}(0)) \,.$$

See at eta invariant – Relation to zeta function for more on this.

Relation to partition functions and number-theoretic zeta/theta functions

By basic integration identities we have that

Proposition

The series expression in def. is equal to the Mellin transform of the partition function

$$\zeta_H(s) = \int_{(0,\infty)} t^{s-1} \left( \underset{\lambda_k \neq 0}{\sum} \exp(-t \lambda_k) \right) d t \,.$$

(see e.g. Quine-Heydari-Song 93 (8), Richardson, pages 8-9, BCEMZ 03, section A.2, Connes-Marcolli 06, theorem 13.11).

Remark

If one thinks of the operation $H$ as a Hamiltonian of a quantum mechanical system, then the term

$$Tr(\exp(-\beta H)) \coloneqq \underset{\lambda_k \neq 0}{\sum} \exp(-\beta \lambda_k) + 1$$

is the partition function of this system. Accordingly, prop. says that the zeta function of $H$ is obtained from its partition function by

$$\zeta_H(s) = \int_{(0,\infty)} \beta^{s-1} \; \left(Tr(\exp(-\beta H)) - 1 \right) \; d \beta \,.$$

Further, by a change of integration variable $t\coloneqq x^2$ in the expression in prop. one obtains

Proposition

The series expression in def. is equal to

$$\begin{aligned} \zeta_H(s) & = 2 \int_{(0,\infty)} x^{2s-1} \left( \underset{\lambda_k \neq 0}{\sum} \exp(- x^2 \lambda_k) \right) d x \end{aligned} \,.$$

In particular if $H = D^2$ is the square of a Dirac operator/supersymmetric quantum mechanics-type square root operator $D$ with eigenvalues $\pm \alpha_k$,then $\lambda_k = \alpha_k^2$ and hence in this case the series is

$$\begin{aligned} \zeta_H(s) & = 2 \int_{(0,\infty)} x^{2s-1} \left( \underset{\alpha_k \neq 0}{\sum} \exp(- (x \alpha_k)^2) \right) d x \end{aligned} \,.$$

By comparison one observes:

Remark

The integral expression in prop. is analogous to the expression of zeta functions in number theory/arithmetic geometry as integrals of a theta function (for instance discussed here for the Riemann zeta function)

$$\hat\zeta_f(2 s) = \int_{(0,\infty)} (\theta(x^2) - 1) x^{2s-1} d x \,.$$

Under this analogy the theta function in the case of the differential operator $H$ is

$$\theta_H(x) \coloneqq \underset{\lambda_k \neq 0}{\sum} \exp(- x \lambda_l) \,.$$

This is formally the same definition as that of adelic theta functions (e.g.Garrett 11, section 1.8)

Remark

The determinant line bundle of the functional determinant of the Dirac operator on a complex torus is a complex-analytic theta function as above, quotiented by the Dedekind eta function.

Early references explaining this include Alvarez-Gaumé & Moore & Vafa 86, Alvarez-Gaumé & Bost & Moore & Nelson & Vafa 87. In a bigger perspective, this relation plays a central role in the general discussion of self-dual higher gauge theory (Witten 96).

Examples

Of Laplace operator on complex torus and Dedekind eta function

For $\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$ a complex torus (complex elliptic curve) equipped with its standard flat Riemannian metric, then the zeta function of the corresponding Laplace operator $\Delta$ is

$$\zeta_{\Delta} = (2\pi)^{-2 s} E(s) \coloneqq (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum} \frac{1}{{\vert k +\tau l\vert}^{2s}} \,.$$

The corresponding functional determinant is

$$\exp( E^\prime_{\Delta}(0) ) = (Im \tau)^2 {\vert \eta(\tau)\vert}^4 \,,$$

where $\eta$ is the Dedekind eta function.

(recalled e.g. in Todorov 03, page 3)

For more see also at zeta function of a Riemann surface.

Analytic torsion

The functional determinant of a Laplace operator of a Riemannian manifold acting on differential n-forms is up to a sign in the exponent a factor in what is called the analytic torsion of the manifold.

Related concepts

• partition function

context/function field analogy	theta function $\theta$	zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$)	L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$)	eta function $\eta$	special values of L-functions
physics/2d CFT	partition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$	analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$	analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$	analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$	regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
Riemannian geometry (analysis)		zeta function of an elliptic differential operator	zeta function of an elliptic differential operator	eta function of a self-adjoint operator	functional determinant, analytic torsion
complex analytic geometry	section $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$	zeta function of a Riemann surface	Selberg zeta function		Dedekind eta function
arithmetic geometry for a function field		Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number field	Hecke theta function, automorphic form	Dedekind zeta function (being the Artin L-function $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation)	Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)		class number $\cdot$ regulator
arithmetic geometry for $\mathbb{Q}$	Jacobi theta function ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character)	Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$)	Artin L-function of a Galois representation $\mathbf{z}$ , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

References

General

An early reference is

• Hans Duistermaat, Victor Guillemin, The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics,Inventiones mathematicae (1975) Volume: 29, page 39-80 (EuDML)

(see also at Duistermaat-Guillemin trace formula)

Textbook accounts include

• Nicole Berline, Ezra Getzler, Michèle Vergne, section 9.6 of Heat Kernels and Dirac Operators

Review includes

• Ken Richardson, section 3 of Introduction to the Eta invariant (pdf)

• J. R. Quine, S. H. Heydari, R. Y. Song, Zeta regularized products, Transactions of the AMS volume 338, number 1, 1993 (pdf)

• Wikipedia, Functional determinant – Zeta function version

• Wikipedia, Zeta function regularization

• Alain Connes, Matilde Marcolli, A walk in the noncommutative garden (arXiv:0601054)

Zeta function regularization

• Eugene Speer, On the structure of Analytic Renormalization, Comm. math. Phys. 23, 23-36 (1971) (Euclid)

• A. Bytsenko, G. Cognola, Emilio Elizalde, Valter Moretti, S. Zerbini, section 2 of Analytic Aspects of Quantum Fields, World Scientific Publishing, 2003, ISBN 981-238-364-6

Functional determinant

The definition of a functional determinant via the exponential of the derivative of the zeta function at 0 originates in

• D. Ray, Isadore Singer, R-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7: 145–210, (1971) doi:10.1016/0001-8708(71)90045-4, MR 0295381

Discussion in the special case of 2d CFT (worldsheet string theory) is in

• Luis Alvarez-Gaumé, Gregory Moore, Cumrun Vafa, Theta functions, modular invariance, and strings, Communications in Mathematical Physics Volume 106, Number 1 (1986), 1-4 (Euclid)

• Luis Alvarez-Gaumé, Jean-Benoit Bost, Gregory Moore, Philip Nelson, Cumrun Vafa, Bosonization on higher genus Riemann surfaces, Communications in Mathematical Physics, Volume 112, Number 3 (1987), 503-552 (Euclid)

• Andrey Todorov, The analogue of the Dedekind eta function for CY threefolds, 2003 pdf