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I'm currently studying the superconformal algebra $psu(2,2|4)$, but I'm having trouble understanding its representation. Following arxiv:1012.4004

I know that the maximal compact subalgebra is su(2) $\oplus$ su(2) $\oplus$ su(4) (Lorentz & $R$-symmetry), and that a unitary rep is specified by the Dynkin labels [$s_1$, $s_2$] for the Lorentz algebra and with the conformal algebra su(2,2) we have [$s_1$, r, $s_2$]. The Dynkin labels of so(4) are $[q1,p,q2]$ such that they combine into [$s_1;r_1;q_1,p,q_2;r_2;s_2$] for su(2,2|4).

What exactly are Dynkin labels, and how are they different from Dynkin indices and weights? Why is it enough to specify a state by its Dynkin label?

I think my lack of knowledge about Lie algebras may be at the root of my problem. Can anyone recommend any good literature that covers these advanced topics? I've tried Brian C. Hall's text, but it hasn't been helpful.

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  • $\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$
    – Community Bot
    Apr 25 at 12:09

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Formally, Dynkin labels are the components of a highest weight vector in the basis of fundamental weights. Intuitively, they are just linear combinations of the weights and play the same role.

For a highest weight representation of a rank $r$ Lie algebra, you can compute either an $r$-tuple of eigenvalues under the Cartan generators or an $r$-tuple of Dynkin labels. The difference is that only the latter are guaranteed to be non-negative integers. Moreover, every $r$-tuple of non-negative integers is a set of Dynkin labels for some highest weight representation.

Dynkin indices on the other hand are norms of generators so they are convention dependent. For the applications you seem to be interested in, I think chapter 13 of the yellow CFT book would be a good reference.

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  • $\begingroup$ Thank you for your response and explanation! Which specific yellow book on CFT are you referring to? $\endgroup$
    – iron
    Jun 4 at 12:32
  • $\begingroup$ Di Francesco, Matthieu and Senechal. $\endgroup$
    – Connor Behan
    Jun 4 at 13:20

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