Math 2.0

Global analysis, analysis on manifolds Index theory Theorem

Equivariant index theorem

Builds on Analytic indexTopological indexElliptic operator on a compact manifoldLie groupRepresentation of a Lie groupCharacter of a representationLefschetz fixed-point theoremAtiyah–Bott fixed-point theorem

Statement

Let GG be a compact Lie group acting smoothly on a closed smooth manifold MM. Let EE and FF be GG-equivariant complex vector bundles over MM, and let P ⁣:C∞(M;E)→C∞(M;F)P \colon C^\infty(M; E) \to C^\infty(M; F) be a GG-invariant elliptic operator. Then ker⁡P\ker P and coker⁡P\operatorname{coker} P are finite-dimensional representations of GG, and the equivariant analytic index ind⁡GP=[ker⁡P]−[coker⁡P]∈R(G)\operatorname{ind}_G P = [\ker P] - [\operatorname{coker} P] \in R(G) lies in the representation ring R(G)R(G). The equivariant analytic index equals the equivariant topological index: ind⁡GP=t-ind⁡GP.\operatorname{ind}_G P = \operatorname{t-ind}_G P. Here t-ind⁡GP\operatorname{t-ind}_G P is defined from the class of the principal symbol in the equivariant K-theory KG(T∗M)K_G(T^*M). The definition uses a GG-equivariant embedding of MM into a real representation of GG.

Remarks

The proof follows the K-theoretic proof of the non-equivariant theorem. Both sides are homomorphisms KG(T∗M)→R(G)K_G(T^*M) \to R(G) with the same formal properties, and they agree on the basic case of a representation space. Taking GG trivial gives the Atiyah–Singer index theorem, since R({1})=ℤR(\{1\}) = \Z and the class of a vector space is its dimension. Evaluating the character of ind⁡GP\operatorname{ind}_G P at g∈Gg \in G gives the Lefschetz number tr⁡(g∣ker⁡P)−tr⁡(g∣coker⁡P)\operatorname{tr}(g \mid \ker P) - \operatorname{tr}(g \mid \operatorname{coker} P). Atiyah, Segal and Singer computed this number by localization to the fixed point set MgM^g; when gg has no fixed points the number is 00. Example: the GG-signature theorem is the case of the signature operator.

Going deeper

For a student

The main use is the fixed point formula: the trace of a symmetry gg on the index is a sum of contributions of the fixed points of gg. For the de Rham complex this recovers the Lefschetz fixed-point theorem. For the Dolbeault complex it gives the holomorphic Lefschetz formula of Atiyah and Bott, from which they derived the Weyl character formula for compact connected Lie groups.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

Show the path as a graph
Analytic index, concept, no evidence levelAnalytic indexconceptCharacter of a representation, concept, no evidence levelCharacter of a repres…conceptCompact space, concept, no evidence levelCompact spaceconceptContinuous map, concept, no evidence levelContinuous mapconceptElliptic complex, concept, no evidence levelElliptic complexconceptElliptic operator on a compact manifold, concept, no evidence levelElliptic operator on …conceptEuler characteristic, concept, no evidence levelEuler characteristicconceptLefschetz number, concept, no evidence levelLefschetz numberconceptLie group, concept, no evidence levelLie groupconceptRepresentation of a Lie group, concept, no evidence levelRepresentation of a L…conceptRetract, concept, no evidence levelRetractconceptSimplicial complex, concept, no evidence levelSimplicial complexconceptSingular homology, concept, no evidence levelSingular homologyconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTopological index, concept, no evidence levelTopological indexconceptVector bundle, concept, no evidence levelVector bundleconceptLefschetz fixed-point theorem, statement, no evidence levelLefschetz fixed-point…statementAtiyah–Bott fixed-point theorem, statement, no evidence levelAtiyah–Bott fixed-poi…statementEquivariant index theorem, statement, no evidence levelEquivariant index the…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. Character of a representation concept
  3. Compact space concept
  4. Continuous map concept
  5. Elliptic complex concept
  6. Elliptic operator on a compact manifold concept
  7. Euler characteristic concept
  8. Lefschetz number concept
  9. Lie group concept
  10. Representation of a Lie group concept
  11. Retract concept
  12. Simplicial complex concept
  13. Singular homology concept
  14. Smooth manifold concept
  15. Topological index concept
  16. Vector bundle concept
  17. Lefschetz fixed-point theorem statement
  18. Atiyah–Bott fixed-point theorem statement
  19. Equivariant index theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Atiyah–Singer_index_theorem
About this page
TagT866256Y1W
URIm2://T866256Y1W
Kindstatement, theorem
WritingW0 stub
MSC58J20, 19L47, 57S15
StatusThis copy of the site has no build data. Run m2 status T866256Y1W.
AboutAnalytic index, Topological index
UsesAnalytic index, Topological index, Elliptic operator on a compact manifold, Lie group, Representation of a Lie group, Character of a representation, Lefschetz fixed-point theorem, Atiyah–Bott fixed-point theorem

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