Past exam of the groups and representations course of the University of Oxford / 2022 / 2

Normalize the simple roots to $({\alpha }_i,{\alpha }_i)=2$. The A2 root system has $({\alpha }_1,{\alpha }_2)=-1$, so its Dynkin diagram is two nodes joined by one bond:The fundamental weights satisfy $({\omega }_i,{\alpha }_j)={\delta }_{ij}$. Hence the Gram matrix in the fundamental weight basis, the metric for Dynkin label coordinates, isFor weights of a representation with Dynkin labels $a$ and $b$, their inner product is $a^{T}Gb$. In the simple root basis the metric is instead $A$ itself; stating the basis removes this apparent convention ambiguity.

Write $a=a_1$, $b=a_2$ with $a,b\ge 0$, and let $\lambda =a{\omega }_1+b{\omega }_2$ be the highest weight. The positive roots are ${\alpha }_1$, ${\alpha }_2$ and ${\alpha }_1+{\alpha }_2$, and the Weyl vector is $\rho ={\omega }_1+{\omega }_2$. The Weyl dimension formula givesFor the quadratic Casimir operator, use the root-length normalization $C(\lambda )=(\lambda ,\lambda +2\rho )$. Substituting the Gram matrix in solution yieldsTaking the trace of the defining quadratic Casimir operator identity gives $d(a,b)C(a,b)=\dim (\mathfrak{su}(3))c(a,b)=8c(a,b)$, where $c$ is the Dynkin index in the matching normalization. ThereforeIn particular $c(3)=1$ and $c(8)=6$. The common particle-physics convention uses $C_2=C/2$ and $T=c/2$, giving $C_2(3)=4/3$ and $T(3)=1/2$. The subsequent discrete gauge anomaly congruences use the integer Dynkin index $c$, while the sign test for the supplied beta function is independent of a common rescaling of all indices.

The Weyl dimension formula is increasing in each nonnegative Dynkin label. If $a\ge 3$ or $b\ge 3$, the dimension is at least $d(3,0)=10$. Among $0\le a,b\le 2$, the cases with both labels nonzero, other than $(1,1)$, have dimension at least $d(2,1)=15$. Thus the following is the complete list of irreducible representations of dimension strictly less than $10$:Here $6={\mathrm{Sym}}^{2}3$ is the two-index symmetric representation and $8$ is the Adjoint representation. Interchanging $a$ and $b$ gives the conjugate representation, explaining equal dimensions, Casimir eigenvalues and Dynkin indices in each conjugate pair. In the alternative convention of solution, divide the $C$ and $c$ columns by two.

The Dynkin index is additive over a direct sum of group representations, so $n$ copies of the fundamental representation have $c(r_S)=n$. With $c(\mathrm{ad})=6$, the supplied beta function becomesFor positive gauge coupling $g$, asymptotic freedom at this order requires $\beta (g)<0$, hence $n<132$. For nonnegative integer $n$,At $n=132$ the one-loop coefficient vanishes, so this formula alone does not establish asymptotic freedom. This count follows the scalar multiplicity convention of the printed beta function; changing how a real or complex scalar field is counted requires changing its coefficient too.

The two-index symmetric representation of SU(3) is $6$, with Dynkin index $5$. Thus $n$ copies have $c(r_S)=5n$, and the supplied beta function isThe strict sign condition for asymptotic freedom is $n<132/5=26.4$, givingIndeed $132-5(26)=2>0$, while $132-5(27)=-3<0$.

Use the integer normalization $c(3)=1$ from solution. For two fundamental representations, the specified discrete gauge anomaly coefficient is $\mathcal{A}=q_1+q_2$. Consequently all solutions in the finite cyclic group of charges areThere are $n$ ordered charge pairs. The trivial pair is included: the problem does not require a faithful action. If a faithful action on these Weyl fermions were additionally required, one would retain precisely the $q$ coprime to $n$.

This cancels the discrete gauge anomaly defined in the paper. Two left-handed fundamental representations by themselves have a nonzero pure cubic SU(3) gauge anomaly; a complete gauge theory would need additional fermions to cancel it. That separate condition does not alter the requested discrete congruence.

For the fundamental representation and the two-index symmetric representation, the Dynkin indices are $1$ and $5$. Therefore the specified discrete gauge anomaly condition for ${\mathbb{Z}}_4$ isAll ordered solutions areOnly $(1,3)$ and $(3,1)$ give a faithful ${\mathbb{Z}}_4$ action on this pair of Weyl fermions; faithfulness was not imposed. As in solution, cancellation here concerns the stipulated discrete gauge anomaly, not cancellation of the pure cubic SU(3) gauge anomaly. The integer normalization matters: one must not replace $c$ by a half-integer trace index inside an unchanged modular congruence.