Eta invariant

Summary

The eta invariant of a self-adjoint elliptic operator on a closed manifold measures the asymmetry of its spectrum between positive and negative eigenvalues. It is the value at $s = 0$ of the eta function $\sum \operatorname{sign}(\lambda) \abs{\lambda}^{-s}$, defined by analytic continuation. Atiyah, Patodi and Singer introduced it as the boundary correction in their index theorem for manifolds with boundary.