mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the eta invariant of a self-adjoint elliptic

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...

on a compact manifold is formally the number of positive eigenvalues minus the number of negative eigenvalues. In practice both numbers are often infinite so are defined using

zeta function regularization Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label=Demotic Greek, classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived ...

. It was introduced by who used it to extend the Hirzebruch signature theorem to manifolds with boundary. The name comes from the fact that it is a generalization of the Dirichlet eta function. They also later used the eta invariant of a self-adjoint operator to define the eta invariant of a compact odd-dimensional smooth manifold. defined the

signature defect In mathematics, the signature defect of a singularity measures the correction that a singularity contributes to the signature theorem. introduced the signature defect for the cusp singularities of Hilbert modular surfaces. defined the signature d ...

of the boundary of a manifold as the eta invariant, and used this to show that Hirzebruch's signature defect of a cusp of a

Hilbert modular surface In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular variet ...

can be expressed in terms of the value at ''s''=0 or 1 of a

Shimizu L-function In mathematics, the Shimizu ''L''-function, introduced by , is a Dirichlet series associated to a totally real algebraic number field. defined the signature defect of the boundary of a manifold as the eta invariant, the value as ''s''=0 of the ...

Definition

The eta invariant of self-adjoint operator ''A'' is given by ''η''_''A''(0), where ''η'' is the analytic continuation of :

\eta(s)=\sum_ \frac

and the sum is over the nonzero eigenvalues λ of ''A''.

References

* * *{{Citation , last1=Atiyah , first1=Michael Francis , author1-link=Michael Atiyah , last2=Donnelly , first2=H. , last3=Singer , first3=I. M. , title=Eta invariants, signature defects of cusps, and values of L-functions , doi=10.2307/2006957 , mr=707164 , year=1983 , journal= Annals of Mathematics , series=Second Series , issn=0003-486X , volume=118 , issue=1 , pages=131–177, jstor=2006957 Differential operators