Some aspects of conformal ${\cal N}=4$ SYM four point function

The four point functions of chiral primary BPS operators in ${\cal N}=4$ superconformal Yang Mills are expressed in a form manifestly satisfying the superconformal Ward identities. They are subsequently expanded in terms of conformal partial waves. Correlation functions of two pairs of identical chiral primaries, one pair having the lowest possible scale dimension, are considered. Crossing symmetries determine their free field value up to numeric constants. The contributions from different supermultiplets to the partial wave expansion is analysed, and determined in the case of the free fields and compared with established results at strong and weak coupling. In the large $N$, strong coupling limit, non-trivial cancellations are found between the free field values and results from supergravity. In the perturbative case values are obtained for the anomalous dimensions of lowest twist operators and the correction to the coupling by analysing the conformal wave expansions of certain hypergeometric and logarithmic functions. Next, we attempt to count shortened ${\cal N}=4$ SYM operators, beginning by constructing from fundamental fields the most general operators belonging to certain $SU(4)_R$ representations at low twists. The number of independent solutions to the conditions imposed on such operators is found via a combinatoric approach. Generating functions for the number of operators with spin $\ell=0,1,2,\dotsc$ are derived. Explicit values are obtained for specific $R$-symmetry representations at low twist in various sectors of the theory. The asymptotic behaviour at large twist is also considered. Finally the conformal field theory operator product expansion is analysed. Solutions in terms of series expansions are found, initially for scalar operators in two dimensions, and then more generally.