Dynkin index ( $I (R)$ )

The Dynkin index of a representation

R

fixes the trace normalization

tr_{R} (T^{a} T^{b}) = I (R) δ^{ab}

up to the convention for the generators. It controls matter contributions to gauge beta functions and mixed anomalies.

Table of contents
- Cubic anomaly coefficient Dynkin index

Cubic anomaly coefficient ( $A (R)$ )

 0  0

For a representation

R

S U (N)

with

N \geq 3

, choose the Hermitian quantum generator convention and normalize the cubic anomaly coefficient by

tr_{R} (T^{a} {T^{b}, T^{c}}) = A (R) tr_{N} (T^{a} {T^{b}, T^{c}}), A (N) = 1.

(1)

Here the anticommutator symmetrizes the last two generators; cyclic property of the trace makes the whole tensor symmetric. This coefficient multiplies the local four-dimensional gauge anomaly from a chiral fermion in

R

. It adds under direct sums.

The conjugate representation has generators

- (T^{a})^{T}

, so transposition and the three minus signs give

A (\overline{R}) = - A (R)

. Hence a real representation or pseudoreal representation has zero cubic anomaly coefficient. For

S U (3)

, the two-index antisymmetric representation is the conjugate triplet and has coefficient

- 1

. For

S U (2)

the symmetric cubic invariant itself vanishes, so normalization to a nonzero fundamental cubic coefficient is unavailable; this statement concerns the local cubic anomaly and does not settle global anomalies.^[ref]

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The Dynkin index, also known as the Dynkin index of a representation, is a concept that arises in the study of Lie algebras and Lie groups, particularly in the context of representation theory. It provides a way to quantify the degree of "mixing" or "interaction" of a representation with the structure of the algebra, especially when considering the space of invariant functions or the geometry associated with the representation.

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Dynkin index (I(R))

Cubic anomaly coefficient (A(R))