# Contents

## Definition

###### Definition

Given a self-adjoint operator $D$ (usually first-order, such as a Dirac operator acting on sections of a vector bundle on a closed Riemannian manifold) with eigenvalues with multiplicities $\{\lambda_n\}$, then its eta function is given by the series

\begin{aligned} \eta(s) & \coloneqq \underoverset{n = -\infty}{^\infty }{\sum} sgn(\lambda_n) \frac{1}{ {\vert \lambda_n\vert}^s} \end{aligned}

expression wherever this converges, and extended by analytic continuation from there.

###### Remark

At the special value $s = 1$ the series in def. 1 does not converge, but if $D$ is indeed a Dirac operator then it is the expression of the Dirac propagator. Indeed the definition of $\eta$ by analytic continuation at $s = 1$ is the regularized Dirac propagator.

###### Remark

The eta function of $D$ is related to the zeta function of an elliptic differential operator $H = D^2$ (regarding $D$ as a Dirac operator/supersymmetric quantum mechanics-like square root of $H$) see below.

###### Definition

The eta invariant of $D$ is the special value $\eta(0)$.

(e.g. Richardson, first page)

###### Remark

Def. 2 means that $\eta_0$ is the regularized number of positive minus negative eigenvalues of $D$.

(Notice that if $D$ itself happens to have only positive eigenvalues, then its eta function already is on the notre the zeta function of an elliptic differential operator.)

## Properties

### Relation to the zeta function

Let $D$ be a self-adjoint operator such that

1. its eta function $\eta(s)$ is defined and analytic at $s= 0$;

2. for $c \in I \subset \mathbb{R}$ in an interval such that no $-c$ is an eigenvalue of $D$ such that both the eta series $\eta_{D+c}$ and the zeta function series $\zeta_{(D+c)^2}((s+1)/2)$ have a common lower bound $s \gt B$ for the values on wich the series converges

then

$\frac{d}{d c} \eta_{D+c}(s) = s \zeta_{(D + c)^2}((s+1)/2) \,,$

where on the left we have the zeta function of an elliptic differential operator for $(D+c)^2$.

(e.g. Richardson prop. 2).

In particular this means that under the above assumptions the functional determinant of $D^2$ is given by

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

### Relation with L-function

Relation of eta functions to Dirichlet L-functions includes (Atiyah-Donelly-Singer 83, Podesta 14)

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

## Examples

### For Dirac operator on Riemann surfaces

For the Dirac operator on a Riemann surface/complex curve the eta function was discussed in (Milson 78, Park 01), and related to the Selberg zeta function.

See at zeta function of a Riemann surface for more on this case.

## References

The $\eta$-invariant was introduced by Atiyah-Patodi-Singer in the series of articles

• Michael Atiyah, V. K. Patodi and Isadore Singer, Spectral asymmetry and Riemannian geometry I Proc. Cambridge Philos. Soc. 77 (1975), 43-69.

Spectral asymmetry and Riemannian geometry II. Proc. Cambridge Philos. Soc.

Spectral asymmetry and Riemannian geometry III, Proc. Cambridge Philos. Soc. 79 (1976), 71-99.

as the boundary correction term for the index formula on a manifold with boundary.

Introductions and surveys include

• Jean-Michel Bismut, Local index theory, eta invariants and holomorphic torsion: a survey, pp. 1-76, in: Surveys in diff. geom. (C. C Hsiang, S/T. Yau, eds.) 1998. International Press

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

• Xianzhe Dai, Eta invariant and holonomy Chern Centennial (2011) (pdf slides)

• Wikipedia, Eta invariant

Formulation in the broader context of bordism theory is in

Further discussion of the relation to holonomy is in

• Xianzhe Dai, Weiping Zhang, Eta invariant and holonomy, the even dimensional case, arXiv:1205.0562

• Eta invariant and Selberg zeta function of odd type over convex co-compact hyperbolic manifolds (pdf)

Discussion of relation to L-functions includes

• Michael Atiyah, H. Donnelly; , Isadore Singer, Eta invariants, signature defects of cusps, and values of L-functions, Annals of Mathematics. Second Series 118 (1): 131–177 (1983) doi:10.2307/2006957, ISSN 0003-486X, MR 707164

• Ricardo A. Podestá, The eta function and eta invariant of Z2r-manifolds (ariv:1407.7454)

Discussion of the case over Riemann surfaces includes

• John Milson, Closed geodesic and the $\eta$-invariant, Ann. of Math., 108, (1978) 1-39 ()

• Jinsung Park, Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps (arXiv:0111175)

Discussion in relation to analytic torsion and Chern-Simons theory includes

• John Lott, Eta and torsion, 1990 (pdf)

• Lisa C. Jeffrey, Symplectic quantum mechanics and Chern=Simons gauge theory I, arxiv/1210.6635

Revised on September 9, 2014 19:40:44 by Urs Schreiber (185.26.182.27)