# 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}{(\lambda_n)^s} \end{aligned}

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

###### 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 poistive 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

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

### Analogy with L-series

###### Remark

The series in def. 1 is analogous to that of a Dirichlet L-series, whith the signa $sgn(-)$ playing the role of a Dirichlet character.

context/function field analogytheta functionzeta functioneta function
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 $\mathbf{z}$regularized Feynman propagator $\zeta(s) = Tr_{reg}((D_{\mathbf{z}})^2)^{-s} = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$trace of signed power of Dirac operator/supercharge $\eta(s) = Tr_{reg} sgn(D_{\mathbf{z}}) (D_{\mathbf{z}})^{-s}$
complex analytic geometry/analysissection $\theta(\mathbf{z},\mathbf{\tau})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$zeta function of an elliptic differential operatoreta function of a self-adjoint operator
arithmetic geometry for a function fieldWeil zeta function
arithmetic geometry for a number fieldHecke theta functionDedekind zeta functionHecke L-function
arithmetic geometry for $\mathbb{Q}$Jacobi theta functionRiemann zeta functionDirichlet L-function

## 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

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 invariants play role in

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

Revised on August 27, 2014 06:10:27 by Urs Schreiber (188.200.54.65)