eta invariant


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Arithmetic geometry


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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}{ {\vert \lambda_n\vert}^s} \end{aligned}

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


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


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.


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

(e.g. Richardson, first page)


Def. 2 means that η 0\eta_0 is the regularized number of positive 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.)


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


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)) \,.

Relation with L-function

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

context/function field analogytheta function θ\thetazeta function ζ\zeta (= Mellin transform of θ(0,)\theta(0,-))L-function L zL_{\mathbf{z}} (= Mellin transform of θ(z,)\theta(\mathbf{z},-))eta function η\etaspecial values of L-functions
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}analytically continued trace of Feynman propagator ζ(s)=Tr reg(1(D 0) 2) s= 0 τ s1θ(0,τ)dτ\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tauanalytically continued trace of Feynman propagator in background gauge field z\mathbf{z}: L z(s)Tr reg(1(D z) 2) s= 0 τ s1θ(z,τ)dτL_{\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\tauanalytically continued trace of Dirac propagator in background gauge field z\mathbf{z} η z(s)=Tr reg(sgn(D z)|D z|) s\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s regularized 1-loop vacuum amplitude pvL z(1)=Tr reg(1(D z) 2)pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right) / regularized fermionic 1-loop vacuum amplitude pvη z(1)=Tr reg(D z(D z) 2)pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right) / vacuum energy 12L z (0)=Z H=12lndet reg(D z 2)-\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 θ(z,τ)\theta(\mathbf{z},\mathbf{\tau}) of line bundle over Jacobian variety J(Σ τ)J(\Sigma_{\mathbf{\tau}}) in terms of covering coordinates z\mathbf{z} on gJ(Σ τ)\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 zL_{\mathbf{z}} for z=0\mathbf{z} = 0 the trivial Galois representation)Artin L-function L zL_{\mathbf{z}} of a Galois representation z\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 (z=0\mathbf{z} = 0)/ Dirichlet theta function (z=χ\mathbf{z} = \chi a Dirichlet character)Riemann zeta function (being the Dirichlet L-function L zL_{\mathbf{z}} for Dirichlet character z=0\mathbf{z} = 0)Artin L-function of a Galois representation z\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


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.


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

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 η\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 (