Due to the large amount of supersymmetry present, the conformal invariance of the classical field theory
survives the quantization procedure: The coupling constant is not renormalized and its
-function
vanishes to all orders in perturbation theory [108, 72, 41]. This is why one often refers to
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.
, are renormalized and acquire anomalous dimensions. These
may be read off from the two point functions (stated here for the case of scalar operators)
The core statement of the AdS/CFT correspondence is that the scaling dimensions are
equal to the energies
of the
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
or the conformal group in four dimensions respectively.
SO(6), on the other hand, emerges from the isometries of the five sphere and the
-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)
The strategy is now to search for string solutions with energies
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