Eta-invariant

$ \eta $-invariant

Let $A$ be an unbounded self-adjoint operator with only pure point spectrum (cf. also Spectrum of an operator). Let $a _ { n }$ be the eigenvalues of $A$, counted with multiplicity. If $A$ is a first-order elliptic differential operator on a compact manifold, then $| a _ { n } | \rightarrow \infty$ and the series

\begin{equation*} \eta ( s ) = \sum _ { a _ { n } \neq 0 } \frac { a _ { n } } { | a _ { n } | } | a _ { n } | ^ { - s } \end{equation*}

is convergent for $\operatorname { Re } ( s )$ large enough. Moreover, $ \eta $ has a meromorphic continuation to the complex plane, with $s = 0$ a regular value (cf. also Analytic continuation). The value of $\eta _ { A }$ at $0$ is called the eta-invariant of $A$, and was introduced by M.F. Atiyah, V.K. Patodi and I.M. Singer in the foundational paper [a1] as a correction term for an index theorem on manifolds with boundary (cf. also Index formulas). For example, in that paper, they prove that the signature $\operatorname{sign}( M )$ of a compact, oriented, $4 k$-dimensional Riemannian manifold with boundary $M$ whose metric is a product metric near the boundary is

\begin{equation*} \operatorname { sign } ( M ) = \int _ { M } \mathcal{L} ( M , g ) - \eta _ { D } ( 0 ), \end{equation*}

where $D = \pm ( * d - d * )$ is the signature operator on the boundary and $\mathcal{L} ( M , g )$ the Hirzebruch $L$-polynomial associated to the Riemannian metric on $M$.

The definition of the eta-invariant was generalized by J.-M. Bismut and J. Cheeger in [a2], where they introduced the eta-form of a family of elliptic operators as above. It can be used to recover the eta-invariant of operators in the family.

References

[a1]	M.F. Atiyah, V.K. Patodi, I.M. Singer, "Spectral asymmetry and Riemannian Geometry" Math. Proc. Cambridge Philos. Soc. , 77 (1975) pp. 43–69
[a2]	J.-M. Bismut, J. Cheeger, "Eta invariants and their adiabatic limits" J. Amer. Math. Soc. , 2 : 1 (1989) pp. 33–77