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2. Perturbation theory In this section we will investigate the corrections to the generators of the symmetry algebra in perturbation theory. Attention is payed to the dilatation operator which will take a special role. 1. Quantum representations The superconformal symmetry algebra psu(2, 2|4) is spanned by the generators J. 27) where FCAB are the structure constants of psu(2, 2|4). The generators can act on the set of states, or, more precisely, there is a representation which we shall also denote by J.

5 4 2. The dilatation operator The dilatation generator is a means to investigate scaling dimensions in a conformal field theory. 1 by comparing different methods of obtaining scaling dimensions. We will then go on to study aspects of the dilatation operator which will be useful in the following chapters. 2 we will consider the symmetry algebra and states in perturbation theory. 4 we will compute the one-loop dilatation operator within a subsector [50]. 5. 6 we will consider the planar limit and introduce the notation to be used in most parts of this work.

These have been classified [12] and the following two bounds have been found:29 D 2 + s1 + p + 23 q1 + 21 q2 or D = p + 23 q1 + 21 q2 , s1 = 0 , D 2 + s2 + p + 21 q1 + 23 q2 or D = p + 21 q1 + 23 q2 , s2 = 0 . 72) Unitary multiplets fall into different series, the first one is the ‘half-BPS’ series with highest weights given by30 w = (p; 0, 0; 0, p, 0; 0, p) . 73) These are the shortest physical multiplets. Multiplets which are of this type in the classical theory are protected, the scaling dimension of the classical theory is preserved exactly in the quantum theory, D = p.

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