Contents

# 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. 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. 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 theta-function / Mellin transform

The eta function is a kind of odd version of the Mellin transform of an odd version of the theta function:

$\eta_D(s) = \frac{1}{\Gamma((s+1)/2)} \underoverset{0}{\infty}{\int} t^{(s-1)/2} Tr(D\,\exp(-t D^2)) \, d t \,.$

e.g. (Müller 94 (0.2)).

### Relation to the zeta function

#### Generally

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

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

#### On odd-dimensional manifolds

Under suitable conditions the exponentiated $\eta$-invariant $\exp(\pi i \eta(0))$ equals the Selberg zeta function of odd type. (Millson 78, Park 01, theorem 1.2, Guillarmou-Moroianu-Park 09).

### 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

### Boundaries, determinant line bundles and perturbative Chern-Simons

Let $\pi \colon X \to Z$ be a $Z$-parameterized collection of spin Riemannian manifold of odd dimension with boundary.

Equip the corresponding collection of Dirac operators $D_X$ with the boundary condition given by a choice of isometry

$Ker^+ D_{\partial X} \simeq Ker^- D_{\partial X} \,.$

e.g. (Müller 94, below (0.3))

Define then the exponentiated eta-invariant to be the element

$\tau_X \coloneqq \exp\left( \pi i \left( \eta_X(0) + dim\left(ker D_X\right) \right) \right) \in Det^{-1}_{\partial X}$

in the inverse of the determinant line

$Det_{\partial X} \coloneqq \left( det\left(ker^- D_{\partial X}\right) \right) \otimes \left( det\left(ker^+ D_{\partial X}\right) \right)^{-1} \,.$

(Here it is maybe noteworthy that, by the above, $\zeta_S = \exp(i \pi\, \eta_X(0))$ is the Selberg zeta function.)

In fact this is a smooth section of the determinant line bundle as $X$ varies.

###### Proposition

These sections given by the exponentiated eta invariant satisfy the sewing law.

This is due to (Dai-Freed 94), reviewed in (Freed 95a). See also (Witten 15) for discussion in relation to anomaly cancellation of fermions (specifically for the eta invariant in the Green-Schwarz mechanism see Witten 99, section 2.2).

###### Remark

Prop. means that the eta-invariant satisfies something like the Atiyah-axioms for TQFT (but of course $\eta$ depends on a metric), a point of view highlighted in (Bunke 94).
Indeed, this exponentiated eta invariant is one factor (together with analytic torsion and the classical CS invariant) of the perturbative path integral quantization of Chern-Simons theory (Witten 89 (2.17) (2.23)), see at Chern-Simons theory – Perturbative path integral quantization.

###### Remark

Also the theta function is a section of, up to isomorphism, this determinant line bundle (or maybe its inverse) (Freed 95b, p. 31).

(and hopefully it coincides with the section given by the exponentiated $\eta$ under suitable conditions?)

## Examples

### For Dirac operator on Riemann surfaces

For the Dirac operator on a Riemann surface/complex curve the eta function was discussed in (Millson 78).

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

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. 3 (C.-C. Hsiung, S.-T. Yau, eds.) 1996. 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

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

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

• Colin Guillarmou, Sergiu Moroianu, Jinsung Park, Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds (arXiv:0901.4082)

Discussion in relation to analytic torsion and perturbative quantum Chern-Simons theory goes back to

with more in

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

• Lisa Jeffrey, Symplectic quantum mechanics and Chern-Simons gauge theory I, (arxiv/1210.6635)

• Lisa Jeffrey, Brendan McLellan, Eta-Invariants and Anomalies in U(1) Chern-Simons Theory (pdf)

• M. B. Young, section 2 of Chern-Simons theory, knots and moduli spaces of connections (pdf)

Discussion of the eta-invariant on manifolds with boundary is in

and regarding the result as taking values in the determinant line over the boundary is due to

with review and streamlined results in