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 In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...

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 In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example is ...

is formally the number of positive

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

s 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 f ...

. It was introduced by who used it to extend the

Hirzebruch signature theorem In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem) is Friedrich Hirzebruch's 1954 result expressing the signature of a smooth closed oriented manifold by a linear combi ...

to manifolds with boundary. The name comes from the fact that it is a generalization of the

Dirichlet eta function In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: \eta(s) = \sum_^ = \frac - \frac + \frac - \frac + \cdo ...

. 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 varie ...

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 The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...

, series=Second Series , issn=0003-486X , volume=118 , issue=1 , pages=131–177, jstor=2006957 Differential operators