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
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.)
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||eta function|
|physics/2d CFT||partition function as function of complex structure of worldsheet (hence polarization of phase space) and background gauge field/source||regularized Feynman propagator||trace of signed power of Dirac operator/supercharge|
|complex analytic geometry/analysis||section in terms of covering coordinates on of line bundle over Jacobian variety||zeta function of an elliptic differential operator||eta function of a self-adjoint operator|
|arithmetic geometry for a function field||Weil zeta function|
|arithmetic geometry for a number field||Hecke theta function||Dedekind zeta function||Hecke L-function|
|arithmetic geometry for||Jacobi theta function||Riemann zeta function||Dirichlet L-function|
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
Xianzhe Dai, Eta invariant and holonomy Chern Centennial (2011) (pdf slides)
Wikipedia, Eta invariant
Further discussion of the relation to holonomy is in
Eta invariants play role in