nLab
eta invariant

Context

Functional analysis

Arithmetic geometry

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Definition

Definition

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

η(s) n=sgn(λ n)1(λ n) s \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 DD is related to the zeta function of an elliptic differential operator H=D 2H = D^2 (regarding DD as a Dirac operator/supersymmetric quantum mechanics-like square root of HH) see below.

Definition

The eta invariant of DD is the special value η(0)\eta(0).

(e.g. Richardson, first page)

Remark

Def. 2 means that η 0\eta_0 is the regularized number of poistive minus negative eigenvalues of DD.

(Notice that if DD 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 DD be a self-adjoint operator such that

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

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

then

ddcη D+c(s)=sζ (D+c) 2((s+1)/2), \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(D+c)^2.

(e.g. Richardson prop. 2).

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

det(D 2)=exp(scη D(0)). 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()sgn(-) playing the role of a Dirichlet character.

context/function field analogytheta functionzeta functioneta function
physics/2d CFTpartition function θ(z,τ)=Tr(exp(τ(D z) 2))\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2)) as function of complex structure τ\mathbf{\tau} of worldsheet Σ\Sigma (hence polarization of phase space) and background gauge field/source z\mathbf{z}regularized Feynman propagator ζ(s)=Tr reg((D z) 2) s= 0 τ s1θ(0,τ)dτ\zeta(s) = Tr_{reg}((D_{\mathbf{z}})^2)^{-s} = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tautrace of signed power of Dirac operator/supercharge η(s)=Tr regsgn(D z)(D z) s\eta(s) = Tr_{reg} sgn(D_{\mathbf{z}}) (D_{\mathbf{z}})^{-s}
complex analytic geometry/analysissection θ(z,τ)\theta(\mathbf{z},\mathbf{\tau}) in terms of covering coordinates z\mathbf{z} on gJ(Σ τ)\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}}) of line bundle over Jacobian variety J(Σ τ)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 η\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)