In John Terning's book, on pages 14 and 15, there are lists of $\mathcal{N} = 2$ and $\mathcal{N} = 4$ supermultiplets, labeled in terms of the dimensions of the corresponding R-symmetry $d_R$ and spin-symmetry $2j+1$. I want to figure out a way to get all these numbers by Young Tableaus in a systematic way.

Of course, the $\mathcal{N}=2$ case is relatively straightforward. Its clear that the numbers in the labels for $\mathcal{N}=4$ can separately be recovered using

$4_{R} \otimes 4_{R} = 10_R \oplus 6_R$

for $SU(4)_R$ and tensor products of this basic identity with $4_R$. Of course, one can also do the same thing for the $SU(2)$ spin symmetry

$2_{SU(2)} \otimes 2_{SU(2)} = 3_{SU(2)} + 1_{SU(2)}$

But if one writes $(\textbf{R}, 2j+1)$ as the label, then how does one justify

$(\textbf{4}_R, 2)\otimes(\textbf{4}_R, 2) = (\textbf{10}_R,1) \oplus (\textbf{6}_R,3)$


$(\textbf{4}_R, 2)\otimes((\textbf{10}_R,1) \oplus (\textbf{6}_R,3)) = (\bar{\textbf{20}}, 2) + (\bar{\textbf{4}},4)$

What I'm asking is: how do you get this particular grouping?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.