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2.2 N = 4 Super Yang-Mills

The conjectured dual gauge theory of the AdS5 × S5 superstring is the maximally supersymmetric (N = 4) Yang-Mills theory in four dimensions [6642]. Its field content is given by a gluon field Am(x), six scalars fi(x) (i = 1,...,6) as well as 4 Majorana gluinos, which we choose to write as a 16 component 10d Majorana-Weyl spinor xa(x) (a = 1,...16). All fields are in the adjoint representation of SU(N). The action of N = 4 Super Yang-Mills is uniquely determined by two parameters, the coupling constant gYM and the rank of the gauge group SU(N)
integral { } --2- 4 1- 2 1- 2 1- 1- i- S = g2 d x Tr 4 (Fmn) + 2 (Dmfi) - 4 [fi,fj][fi,fj] + 2 xD/ x - 2 x Gi[fi,x] (13) YM
with the covariant derivative Dm = @m - i[Am, ]. Furthermore, (Gm,Gi) are the ten-dimensional Dirac matrices.

Due to the large amount of supersymmetry present, the conformal invariance of the classical field theory survives the quantization procedure: The coupling constant g YM is not renormalized and its b-function vanishes to all orders in perturbation theory [1087241]. This is why one often refers to N = 4 Super Yang-Mills as a “finite” quantum field theory. Nevertheless, composite gauge invariant operators, i.e. traces of products of fundamental fields and their covariant derivatives at the same space-point, e.g. Oi ...i (x) = Tr[fi (x) fi (x)...fi (x)] 1 k 1 2 k, are renormalized and acquire anomalous dimensions. These may be read off from the two point functions (stated here for the case of scalar operators)

----dAB----- <OA(x) OB(y) > = (x- y)2DOA (14)
where DOA is the scaling dimension of the composite operator OA. Classically, these scaling dimensions are simply the sum of the individual dimensions of the constituent fields ([fi] = [Am] = 1 and [x] = 3/2). In quantum theory the scaling dimensions receive anomalous corrections, organized in a double expansion in 2 c = gYM N (loops) and 2 1/N (genera)
oo oo D = D + sum cl sum -1--D . (15) 0 N 2g l,g l=1 g=0
Determining the scaling dimensions in perturbation theory is a difficult task due to the phenomenon of operator mixing: One has to identify the correct basis of (classically) degenerate gauge theory operators OA in which (14View Equation) indeed becomes diagonal. This task is greatly facilitated through the use of the gauge theory dilatation operator, to be discussed in Section 4.

The core statement of the AdS/CFT correspondence is that the scaling dimensions D(c, 1/N 2) are equal to the energies E of the AdS5 × S5 string excitations. A central problem, next to actually computing these quantities on either side of the correspondence, is to establish a “dictionary” between states in the string theory and their dual gauge theory operators. Here the underlying symmetry structure of SU(2,2|4) is of help, whose bosonic factors are SO(2,4)×SO(6). SO(2,4) corresponds to the isometry group of AdS5 or the conformal group in four dimensions respectively. SO(6), on the other hand, emerges from the isometries of the five sphere and the R-symmetry group of internal rotations of the six scalars and four gluinos in SYM. Clearly, then any state or operator can be labeled by the eigenvalues of the six Cartan generators of SO(2,4)×SO(6)

(E; S1,S2;J1,-J2,J3) (16) S3 S5
where we shall denote the Si as the commuting “spins” on the three sphere within AdS5, the Jk as the commuting angular momenta on the 5 S and E is the total energy. These are the conserved quantities of the string discussed above (see Equation (8View Equation)).

The strategy is now to search for string solutions with energies

E = E(S ,S ,J ,J ,J ) 1 2 1 2 3
and to identify these with scaling dimensions of Super Yang-Mills operators carrying the identical set of charges (S1,S2,J1,J2,J3). Generically the string energies will depend on additional data, such as winding numbers or oscillator levels, which we have suppressed in the above.

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