We study the consistency of four-point functions of half-BPS chiral primary operators of weight p in four-dimensional N=4 superconformal field theories. The resulting conformal bootstrap equations impose non-trivial bounds for the scaling dimension o
f unprotected local operators transforming in various representations of the R-symmetry group. These bounds generalize recent bounds for operators in the singlet representation, arising from consistency of the four-point function of the stress-energy tensor multiplet.
We construct interpolating functions fully compatible with S-duality. We then consider the problem of resumming perturbative expansions for anomalous dimensions of low twist non-protected operators in N=4 super Yang-Mills theory. When the rank of the
gauge group is small, the interpolations suggest that anomalous dimensions of leading twist operators take their maximum value at the point $tau =exp(ipi/3)$. For fixed spin and large enough rank, there is a level-crossing region, where the anomalous dimension of the leading twist operator reaches its maximum and then bounces back.
We report on non-perturbative bounds for structure constants on N=4 SYM. Such bounds are obtained by applying the conformal bootstrap recently extended to superconformal theories. We compare our results with interpolating functions suitably restricte
d by the S-duality of the theory. Within numerical errors, these interpolations support the conjecture that the bounds found in this paper are saturated at duality invariant values of the coupling. This extends recent conjectures for the anomalous dimension of leading twist operators.
We analyze the properly normalized three-point correlator of two protected scalar operators and one higher spin twist-two operator in N=4 super Yang-Mills, in the limit of large spin j. The relevant structure constant can be extracted from the OPE of
the four-point correlator of protected scalar operators. We show that crossing symmetry of the four point correlator plus a judicious guess for the perturbative structure of the three-point correlator, allow to make a prediction for the structure constant at all loops in perturbation theory, up to terms that remain finite as the spin becomes large. Furthermore, the expression for the structure constant allows to propose an expression for the all loops four-point correlator G(u,v), in the limit u,v -> 0. Our predictions are in perfect agreement with the large j expansion of results available in the literature.
