four eigenvalues $\lambda = 32g_{YM}^2$ and so on.

If $a_{\rm max}$ is a multiple of 4, the larger eigenvalues are given by $\lambda=4a_{\rm max}g_{YM}^2 +8g_{YM}^2$, $\lambda=4a_{\rm max}g_{YM}^2 + 16g_{YM}^2$ and $$\lambda_i=4a_{\rm max}g_{YM}^2 + 16g_{YM}^2 +16ig_{YM}^2\qquad i=1,2,\cdots,{a_{\rm max}\over 4}\, ;$$ all of these eigenvalues on the last line above are non-degenerate.

If $a_{\rm max}$ (which by assumption is even) is not a multiple of 4, the larger eigenvalues are given by $\lambda=4a_{\rm max}g_{YM}^2 +8g_{YM}^2$ with a degeneracy of 2 and $$\lambda_i=4a_{\rm max}g_{YM}^2 + 16g_{YM}^2 +16ig_{YM}^2\qquad i=1,2,\cdots,{a_{\rm max}+2\over 4}\, ;$$ all of these eigenvalues on the last line above are non-degenerate.

Once again, the degeneracies observed almost certainly indicate a symmetry enhancement in the large $N$ limit.

Discussion
==========

In this article we have computed the one loop anomalous dimension of an operator built from $O(N)$ $Z$s and 3 or 4 $Y$ “impurities”. What lessons can be learnt from these results, together with the results of [@Koch:2010gp], which dealt with the case of 2 impurities? Before we start the discussion, it is useful to recall the structure of the operators which participate in the case of two impurities $$\chi_A(b_0,b_1)=\chi_{\tiny \yng(2,2,2,2,2,1,1,1,1)\,\yng(2,2,2,2,2,1,1)\,\yng(1,1)}(Z,Y)
\qquad
\chi_B(b_0,b_1)=\chi_{\tiny \yng(2,2,2,2,2,1,1,1,1)\,\yng(2,2,2,1,1,1,1,1,1)\,\yng(1,1)}(Z,Y)$$ $$\chi_D(b_0,b_1)=\chi_{\tiny \yng(2,2,2,2,2,1,1,1,1)\,\yng(2,2,2,2,1,1,1,1)\,\yng(2)}(Z,Y)
\qquad
\chi_E(b_0,b_1)=\chi_{\tiny \yng(2,2,2,2,2,1,1,1,1)\,\yng(2,2,2,2,1,1,1,1)\,\yng(1,1)}(Z,Y)$$ Firstly, the actual result 