Dynkin Index

1) Dynkin index of an irreducible representation . Let be a finite-dimensional simple complex Lie algebra . The Killing form κ is the invariant bilinear form on defined by the trace in the adjoint representation:

for x, y ∈ . For any arbitrary irreducible finite-dimensional representation R _Λ of the corresponding form is a multiple of the Killing form:

I _Λ is called the (second order) Dynkin index of R _Λ. It is related to the eigenvalue C _Λ of the quadratic Casimir operator by

In physics the second order Dynkin Index is often a more natural quantity than the quadratic Casimir. For example, in four-dimensional gauge theories, I _Λ gives (up to a representation independent constant) the contribution of elementary fields carrying the representation R _Λ to the one-loop renormalization group β-function . I _Λ is also equal to the number of zero modes of the Dirac equation for (spin-) fermions carrying the representation R _Λ in the background of an instanton .

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