Recently obtained results for two and three point functions for quasi-primary operators in conformally invariant theories in arbitrary dimensions d are described. As a consequence the three point function for the energy momentum tensor has three linearly independent forms for general d compatible with conformal invariance. The corresponding coefficients may be regarded as possible generalisations of the Virasoro central charge to d larger than 2. Ward identities which link two linear combinations of the coefficients to terms appearing in the energy momentum tensor trace anomaly on curved space are discussed. The requirement of positivity for expectation values of the energy density is also shown to lead to positivity conditions which are simple for a particular choice of the three coefficients. Renormalisation group like equations which express the constraints of broken conformal invariance for quantum field theories away from critical points are postulated and applied to two point functions. Talk presented at the XXVII Ahrenshoop International Symposium.

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We discuss conserved currents and operator product expansions (OPE's) in the context of a O(N) invariant conformal field theory. Using OPE's we find explicit expressions for the first few terms in suitable short-distance limits for various four-point functions involving the fundamental N-component scalar field ,: (x), :=1, 2, ..., N. We propose an alternative evaluation of these four-point functions based on graphical expansions. Requiring consistency of the algebraic and graphical treatments of the four-point functions we obtain the values of the dynamical parameters in either a free theory of N massless fields or a non-trivial conformally invariant O(N) vector model in 2

We present a detailed analysis of the 4-point functions of the lowest weight chiral primary operators O I ∼ trφ (i φ j) in N = 4 SYM 4 at strong coupling and show that their structure is compatible with the predictions of AdS/CFT correspondence. In particular, all powersingular terms in the 4-point functions exactly coincide with the contributions coming from the conformal blocks of the CPOs, the R-symmetry current and the stress tensor. Operators dual to string modes decouple at strong coupling. We compute the anomalous dimensions and the leading 1/N 2 corrections to the normalization constants of the 2-and 3-point functions of scalar and vector double-trace operators with approximate dimensions 4 and 5 respectively. We also find that the conformal dimensions of certain towers of double-trace operators in the 105, 84 and 175 irreps are non-renormalized. We show that, despite the absence of a non-renormalization theorem for the double-trace operator in the 20 irrep, its anomalous dimension vanishes. As by-products of our investigation, we derive explicit expressions for the conformal block of the stress tensor, and for the conformal partial wave amplitudes of a conserved current and of a stress tensor in d dimensions.1

We argue that the N = 1 higher-spin theory on AdS 4 is holographically dual to the N = 1 supersymmetric critical O(N) vector model in three dimensions. This appears to be a special form of the AdS/CFT correspondence in which both regular and irregular bulk modes have similar roles and their interplay leads simultaneously to both the free and the interacting phases of the boundary theory. We study various boundary conditions that correspond to boundary deformations connecting, for large-N, the free and interacting boundary theories. We point out the importance of parity in this holography and elucidate the Higgs mechanism responsible for the breaking of higher-spin symmetry for subleading N.