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Extra resources for Physics Reports vol.405
85) where we have deﬁned the hypercharge B = 0 and length L = 1. The vacuum state |0 is invariant under su(4), but it is not physical. Conversely, the highest-weight state |Z is physical, but superﬁcially breaks su(4) to su(2) × su(2). When dealing with physical states it is convenient to employ a notation suited for su(2) × su(2) invariance. We deﬁne the oscillator da˙ with index a˙ = 1, 2 d1† = c4 , d2† = c3 , d1 = c4† , d2 = c3† . 84), is annihilated by a1,2 , b1,2 , c1,2 , d1,2 . The drawback is that the notation breaks the su(4) invariant notation and the expressions for the u(2, 2|2 + 2) generators thus complicate.
53) Note that we chose a basis of ﬁxed classical dimension D0 , therefore D0M N = M N D0 . The equation for l = 0 naturally requires D0,ev = D0 . The equation at leading non-trivial order (l = 2) simpliﬁes to M DlM N ON 0 = Dev,l O0 . 54) In general, the eigenvalue problem is an algebraic equation which can only be solved numerically. 53) for any value of l 42 The generators L, L ˙ , R do not depend on g and commute with D(g). Therefore one can also restrict to deﬁnite representations of su(2) × su(2) × su(4).
In perturbation theory multiplets close to the unitarity bound have some special features, see for example [12,14,140]. 72) and whose anomalous dimension D is non-zero. 75). Nevertheless, the remaining states of the interacting long multiplet cannot disappear, instead they form an additional short highest-weight multiplet, see Fig. 7. 5, +1) . 76) 31 Alternatively r = s = 0. i i 32 These are usually called semi-shortening conditions. Here we shall distinguish between short and BPS multiplets. 33 Alternatively r − s = 1.
Physics Reports vol.405