Math 2.0

Global analysis, analysis on manifolds Index theory Theorem

Atiyah–Patodi–Singer index theorem

Builds on Manifold with boundaryRiemannian manifoldElliptic operator on a compact manifoldAnalytic indexEta invariantHirzebruch signature theorem

Statement

Let MM be a compact oriented Riemannian manifold with boundary Y=∂MY = \partial M, whose metric is a product du2+gYdu^2 + g_Y on a collar [0,1)×Y[0, 1) \times Y of the boundary. Let D ⁣:C∞(M;E)→C∞(M;F)D \colon C^\infty(M; E) \to C^\infty(M; F) be a first-order elliptic operator. Assume that on the collar it has the form D=σ(∂∂u+A).D = \sigma \left( \frac{\partial}{\partial u} + A \right). Here σ ⁣:E∣Y→F∣Y\sigma \colon E|_Y \to F|_Y is a bundle isomorphism and AA is a self-adjoint elliptic operator on YY, both independent of uu. Impose the boundary condition P≥0(s∣Y)=0P_{\ge 0}(s|_Y) = 0, where P≥0P_{\ge 0} is the spectral projection of AA onto the span of its eigenvectors with eigenvalue ≥0\ge 0. Then DD with this boundary condition is Fredholm, and its index is ind⁡D=∫Mα0−h+ηA(0)2.\operatorname{ind} D = \int_M \alpha_0 - \frac{h + \eta_A(0)}{2}. Here α0\alpha_0 is the local index density of DD, the same density as for closed manifolds. Also h=dim⁡ker⁡Ah = \dim \ker A, and ηA(0)\eta_A(0) is the eta invariant of AA.

Remarks

Idea of the proof: glue the half-infinite cylinder (−∞,0]×Y(-\infty, 0] \times Y to MM along YY. The boundary condition corresponds to L2L^2 solutions on the elongated manifold, and on the cylinder the heat kernel is computed exactly by separation of variables, which produces the eta invariant. For operators such as the signature operator there is no elliptic local boundary condition, which is why a global spectral projection is used. Example: for the signature operator on a 4k4k-manifold with boundary, the theorem gives σ(M)=∫ML(p)−η(0).\sigma(M) = \int_M L(p) - \eta(0). Here L(p)L(p) is the Hirzebruch LL-form of the metric, and η(0)\eta(0) is the eta invariant of an operator on the even forms of YY. So the Hirzebruch signature theorem fails on manifolds with boundary, and the eta invariant measures the defect.

Going deeper

For a student

The theorem extends the Atiyah–Singer index theorem to manifolds with boundary. It introduced the eta invariant into geometry, with uses for the signature defects of 33-manifolds, for Chern–Simons invariants and for spectral flow.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

Show the path as a graph
Analytic index, concept, no evidence levelAnalytic indexconceptCobordism, concept, no evidence levelCobordismconceptElliptic operator on a compact manifold, concept, no evidence levelElliptic operator on …conceptEta invariant, concept, no evidence levelEta invariantconceptIntersection form, concept, no evidence levelIntersection formconceptManifold with boundary, concept, no evidence levelManifold with boundaryconceptOrientation of a manifold, concept, no evidence levelOrientation of a mani…conceptPontryagin class, concept, no evidence levelPontryagin classconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptSignature of a quadratic form, concept, no evidence levelSignature of a quadra…conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptHirzebruch signature theorem, statement, no evidence levelHirzebruch signature …statementAtiyah–Patodi–Singer index theorem, statement, no evidence levelAtiyah–Patodi–Singer …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Analytic index concept
  2. Cobordism concept
  3. Elliptic operator on a compact manifold concept
  4. Eta invariant concept
  5. Intersection form concept
  6. Manifold with boundary concept
  7. Orientation of a manifold concept
  8. Pontryagin class concept
  9. Riemannian manifold concept
  10. Signature of a quadratic form concept
  11. Smooth manifold concept
  12. Hirzebruch signature theorem statement
  13. Atiyah–Patodi–Singer index theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Eta_invariant
About this page
TagGGFAH1M80J
URIm2://GGFAH1M80J
Kindstatement, theorem
WritingW0 stub
MSC58J20, 58J28, 58J32
StatusThis copy of the site has no build data. Run m2 status GGFAH1M80J.
AboutAnalytic index, Eta invariant
UsesManifold with boundary, Riemannian manifold, Elliptic operator on a compact manifold, Analytic index, Eta invariant, Hirzebruch signature theorem

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