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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e1201802.png" />-invariant''
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e1201803.png" /> be an unbounded [[Self-adjoint operator|self-adjoint operator]] with only pure point spectrum (cf. also [[Spectrum of an operator|Spectrum of an operator]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e1201804.png" /> be the eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e1201805.png" />, counted with multiplicity. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e1201806.png" /> is a first-order elliptic [[Differential operator|differential operator]] on a compact [[Manifold|manifold]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e1201807.png" /> and the series
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e1201808.png" /></td> </tr></table>
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''$ \eta $-invariant''
  
is convergent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e1201809.png" /> large enough. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018010.png" /> has a meromorphic continuation to the complex plane, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018011.png" /> a regular value (cf. also [[Analytic continuation|Analytic continuation]]). The value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018012.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018013.png" /> is called the eta-invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018014.png" />, and was introduced by M.F. Atiyah, V.K. Patodi and I.M. Singer in the foundational paper [[#References|[a1]]] as a correction term for an index theorem on manifolds with boundary (cf. also [[Index formulas|Index formulas]]). For example, in that paper, they prove that the signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018015.png" /> of a compact, oriented, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018016.png" />-dimensional [[Riemannian manifold|Riemannian manifold]] with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018017.png" /> whose [[Metric|metric]] is a product metric near the boundary is
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Let $A$ be an unbounded [[Self-adjoint operator|self-adjoint operator]] with only pure point spectrum (cf. also [[Spectrum of an operator|Spectrum of an operator]]). Let $a _ { n }$ be the eigenvalues of $A$, counted with multiplicity. If $A$ is a first-order elliptic [[Differential operator|differential operator]] on a compact [[Manifold|manifold]], then $| a _ { n } | \rightarrow \infty$ and the series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018018.png" /></td> </tr></table>
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\begin{equation*} \eta ( s ) = \sum _ { a _ { n } \neq 0 } \frac { a _ { n } } { | a _ { n } | } | a _ { n } | ^ { - s } \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018019.png" /> is the signature operator on the boundary and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018020.png" /> the Hirzebruch <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018022.png" />-polynomial associated to the Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120180/e12018023.png" />.
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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|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 [[#References|[a1]]] as a correction term for an index theorem on manifolds with boundary (cf. also [[Index formulas|Index formulas]]). For example, in that paper, they prove that the signature $\operatorname{sign}( M )$ of a compact, oriented, $4 k$-dimensional [[Riemannian manifold|Riemannian manifold]] with boundary $M$ whose [[Metric|metric]] is a product metric near the boundary is
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\begin{equation*} \operatorname { sign } ( M ) = \int _ { M } \mathcal{L} ( M , g ) - \eta _ { D } ( 0 ), \end{equation*}
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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 [[#References|[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.
 
The definition of the eta-invariant was generalized by J.-M. Bismut and J. Cheeger in [[#References|[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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.F. Atiyah,  V.K. Patodi,  I.M. Singer,  "Spectral asymmetry and Riemannian Geometry"  ''Math. Proc. Cambridge Philos. Soc.'' , '''77'''  (1975)  pp. 43–69</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.-M. Bismut,  J. Cheeger,  "Eta invariants and their adiabatic limits"  ''J. Amer. Math. Soc.'' , '''2''' :  1  (1989)  pp. 33–77</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  M.F. Atiyah,  V.K. Patodi,  I.M. Singer,  "Spectral asymmetry and Riemannian Geometry"  ''Math. Proc. Cambridge Philos. Soc.'' , '''77'''  (1975)  pp. 43–69</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J.-M. Bismut,  J. Cheeger,  "Eta invariants and their adiabatic limits"  ''J. Amer. Math. Soc.'' , '''2''' :  1  (1989)  pp. 33–77</td></tr></table>

Latest revision as of 16:58, 1 July 2020

$ \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
How to Cite This Entry:
Eta-invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Eta-invariant&oldid=17359
This article was adapted from an original article by V. Nistor (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article