In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 retur ...
on a
compact manifold
In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The onl ...
is formally the number of positive
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s minus the number of negative eigenvalues. In practice both numbers are often infinite so are defined using
zeta function regularization
In mathematics and theoretical physics, zeta function regularization is a type of regularization (physics), regularization or summability method that assigns finite values to Divergent series, divergent sums or products, and in particular can be ...
. 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
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 surface
In mathematics, a Hilbert ...
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 vari ...
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 their ...
.
Definition
The eta invariant of self-adjoint operator ''A'' is given by ''η''
''A''(0), where ''η'' is the analytic continuation of
:
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 t ...
, series=Second Series , issn=0003-486X , volume=118 , issue=1 , pages=131–177, jstor=2006957
Differential operators