etainvariants

Etainvariants, more commonly called eta invariants, are spectral invariants associated with elliptic self-adjoint operators on manifolds. They measure the asymmetry between positive and negative eigenvalues of such an operator, encoded in a regularized spectral sum. The concept arose in global analysis and geometric topology as part of tools to relate analysis to topology.

For a self-adjoint elliptic operator D with eigenvalues λi, the eta function is defined as η(s) = sumi sign(λi) |λi|^{-s} for values of s where the sum converges. This function extends meromorphically to the complex plane, and the eta invariant is defined as η(0). In general, η(0) is a real number that depends on the operator and the chosen Riemannian metric; it is not a purely topological invariant.

Eta invariants play a central role in the Atiyah-Patodi-Singer index theorem for manifolds with boundary. In that framework, the index of a Dirac-type operator with APS boundary conditions includes a correction term that involves the eta invariant of the boundary operator and the dimension of its kernel. They also appear in theoretical physics, notably in the study of global anomalies and in constructions of certain Chern-Simons-type terms.

The eta invariant is sensitive to geometric data, changing with smooth variations of the metric or operator. In families, its variation is related to spectral flow and is typically defined modulo integers in many applications. The plural term etainvariants is sometimes used to refer to the collection of eta invariants associated with a family of operators or manifolds.