Contents

complex geometry

# Contents

## Idea

In the context of regularization in physics, zeta function regularization is a method/prescription for extracing finite values for traces of powers of Laplace operators/Dirac operators by

1. considering $s$-powers for all values of $s$ in the complex plane where the naive trace does make sense and then

2. using analytic continuation to obtain the desired special value at $s = 1$ – as for zeta functions.

### Analytic regularization of propagators

One speaks of analytic regularization (Speer 71) or zeta function regularization (e.g. M 99, BCEMZ 03, section 2) if a Feynman propagator/Green's function for a bosonic field, which is naively given by the expression “$Tr\left(\frac{1}{H}\right)$” (for $H$ the given wave operator/Laplace operator) is made well defined by interpreting it as the principal value of the special value at $s= 1$

$Tr_{reg} \left(\frac{1}{H}\right) \coloneqq pv\, \zeta_H(1)$

of the zeta function which is given by the expression

$\zeta_H(s) \coloneqq Tr\left( \frac{1}{H} \right)^s$

for all values of $s \in \mathbb{C}$ for which the right hand side exists, and is defined by analytic continuation elsewhere.

Analogously the zeta function regularization of the Dirac propagator for a fermion field with Dirac operator $D$ is defined by

$Tr_{reg} \left(\frac{D}{D^2} \right) \coloneqq pv\, \eta_D(1)$

where $\eta$ is the eta function of $D$.

### Functional determinants

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 the functional determinant of $H$ (Ray-Singer 71) is the exponential of the 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.

### Higher amplitudes

Accordingly, more general scattering amplitudes are controled by multiple zeta functions (…).

## 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)

### Zeta regularization for divergent integrals

the zeta regularizatio method can be extended to include also a regularization for the divergent integrals $\int_{a}^{\infty}x^{m}dx$ which appears in QFT, this is made by means of the identity

$\begin{array}{l} \int_{a}^{\infty }x^{m-s} dx =\frac{m-s}{2} \int_{a}^{\infty }x^{m-1-s} dx +\zeta (s-m)-\sum_{i=1}^{a}i^{m-s} +a^{m-s} \\ -\sum_{r=1}^{\infty }\frac{B_{2r} \Gamma (m-s+1)}{(2r)!\Gamma (m-2r+2-s)} (m-2r+1-s)\int_{a}^{\infty }x^{m-2r-s} dx \end{array}$

for the case of $m=-1$ although the harmonic series has a pole we can regularize by the 2 possibilities

$\sum_{n=0}^{\infty} \frac{1}{n+a} = -\Psi (a)$ or $\sum_{n=0}^{\infty} \frac{1}{n+a} = -\Psi (a)+log(a)$ in particular
$\sum_{n=1}^{\infty} \frac{1}{n} = \gamma$ Euler-Mascheroni constant, and $\Psi(a)= -\frac{\Gamma '(a)}{\Gamma (a)}$

So within this reuglarization there wouldn’t be any UV ultraviolet divergence

### 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.

context/function field analogytheta 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 CFTpartition 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 operatorzeta function of an elliptic differential operatoreta function of a self-adjoint operatorfunctional determinant, analytic torsion
complex analytic geometrysection $\theta(\mathbf{z},\mathbf{\tau})$ 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 surfaceSelberg zeta functionDedekind eta function
arithmetic geometry for a function fieldGoss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number fieldHecke theta function, automorphic formDedekind 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

Original articles include

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

• 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

Modern accounts and reviews include

• Daniel Freed, page 8 of On determinant line bundles, Math. aspects of string theory, ed. S. T. Yau, World Sci. Publ. 1987, (revised pdf, dg-ga/9505002)

• Emilio Elizalde, Ten Physical Applications of Spectral Zeta Functions (1995)

• Valter Moretti, Local z-function techniques vs point-splitting procedures: a few rigorous results

Commun. Math. Phys. 201, 327 (1999).

• 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

• Nicolas Robles, Zeta function regularization, 2009 (pdf)