Types of quantum field thories
Given a self-adjoint operator (usually first-order, such as a Dirac operator acting on sections of a vector bundle on a closed Riemannian manifold) with eigenvalues with multiplicities , then its eta function is given by the series
At the special value the series in def. 1 does not converge, but if is indeed a Dirac operator then it is the expression of the Dirac propagator. Indeed the definition of by analytic continuation at is the regularized Dirac propagator.
The eta function of is related to the zeta function of an elliptic differential operator (regarding as a Dirac operator/supersymmetric quantum mechanics-like square root of ) see below.
The eta invariant of is the special value .
(e.g. Richardson, first page)
(Notice that if itself happens to have only positive eigenvalues, then its eta function already is on the notre the zeta function of an elliptic differential operator.)
e.g. (Müller 94 (0.2)).
Let be a self-adjoint operator such that
its eta function is defined and analytic at ;
where on the left we have the zeta function of an elliptic differential operator for .
(e.g. Richardson prop. 2).
In particular this means that under the above assumptions the functional determinant of is given by
|context/function field analogy||theta function||zeta function (= Mellin transform of )||L-function (= Mellin transform of )||eta function||special values of L-functions|
|physics/2d CFT||partition function as function of complex structure of worldsheet (hence polarization of phase space) and background gauge field/source||analytically continued trace of Feynman propagator||analytically continued trace of Feynman propagator in background gauge field :||analytically continued trace of Dirac propagator in background gauge field||regularized 1-loop vacuum amplitude / regularized fermionic 1-loop vacuum amplitude / vacuum energy|
|Riemannian geometry (analysis)||zeta function of an elliptic differential operator||zeta function of an elliptic differential operator||eta function of a self-adjoint operator||functional determinant, analytic torsion|
|complex analytic geometry||section of line bundle over Jacobian variety in terms of covering coordinates on||zeta function of a Riemann surface||Selberg zeta function||Dedekind eta function|
|arithmetic geometry for a function field||Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)|
|arithmetic geometry for a number field||Hecke theta function, automorphic form||Dedekind zeta function (being the Artin L-function for the trivial Galois representation)||Artin L-function of a Galois representation , 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 regulator|
|arithmetic geometry for||Jacobi theta function ()/ Dirichlet theta function ( a Dirichlet character)||Riemann zeta function (being the Dirichlet L-function for Dirichlet character )||Artin L-function of a Galois representation , 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|
e.g. (Müller 94, below (0.3))
Define then the exponentiated eta-invariant to be the element
in the inverse of the determinant line
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).
Prop. 1 means that the eta-invariant satisfies something like the Atiyah-axioms for TQFT (but of course 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.
(and hopefully it coincides with the section given by the exponentiated under suitable conditions?)
See at zeta function of a Riemann surface for more on this case.
The -invariant was introduced by Atiyah-Patodi-Singer in the series of articles
Spectral asymmetry and Riemannian geometry II. Proc. Cambridge Philos. Soc.
Spectral asymmetry and Riemannian geometry III, Proc. Cambridge Philos. Soc. 79 (1976), 71-99.
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
Wikipedia, Eta invariant
Formulation in the broader context of bordism theory is in
Further discussion of the relation to holonomy is in
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
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)
with more in
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
Werner Müller, Eta invariants and manifolds with boundary, J. Diff. Geom. 40 (1994) 311-377 (pdf)
and regarding the result as taking values in the determinant line over the boundary is due to
with review and streamlined results in