Equivariant index theorem

<h2 id="statement">Statement</h2>
Let 
 
 
 
 π
 :
 E
 →
 M
 
 
 {\displaystyle \pi :E\to M}
 
 be a <a href="/facts/Clifford_module_bundle/ESqf4ECI">clifford module bundle</a>. Assume a compact Lie group G acts on both E and M so that 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 is <a href="/facts/Equivariant_bundle/Dvfcso44">equivariant</a>. Let E be given a connection that is compatible with the action of G. Finally, let D be a <a href="/facts/Dirac_operator/1kdGnoTt">Dirac operator</a> on E associated to the given data. In particular, D commutes with G and thus the kernel of D is a finite-dimensional representation of G.
The equivariant index of E is a <a href="/facts/Brauer%2527s_theorem_on_induced_characters/jE0q973D">virtual character</a> given by taking the <a href="/facts/Supertrace/9IC2DHwu">supertrace</a>:

str
        ⁡
        (
        g
        ∣
        ker
        ⁡
        D
        )
        =
        tr
        ⁡
        (
        g
        ∣
        ker
        ⁡
        
          D
          
            +
          
        
        )
        −
        tr
        ⁡
        (
        g
        ∣
        ker
        ⁡
        
          D
          
            −
          
        
        )
        .
      
    
    {\displaystyle \operatorname {str} (g\mid \ker D)=\operatorname {tr} (g\mid \ker D^{+})-\operatorname {tr} (g\mid \ker D^{-}).}

<h2 id="see-also">See also</h2>
<ul><li>Equivariant topological K-theory</li>
<li><a href="/facts/Kawasaki%2527s_Riemann%25E2%2580%2593Roch_formula/TnfubmbI">Kawasaki's Riemann–Roch formula</a></li></ul>

<ul><li>Berline, Nicole; Getzler, E.; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, Berlin, New York: Springer-Verlag</li></ul>

Equivariant index theorem open-in-new

Equivariant index theorem