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.

Various aspects of the four point function for scalar fields in conformally invariant theories are analysed. This depends on an arbitrary function of two conformal invariants u, v. A recurrence relation for the function corresponding to the contribution of an arbitrary spin field in the operator product expansion to the four point function is derived. This is solved explicitly in two and four dimensions in terms of ordinary hypergeometric functions of variables z, x which are simply related to u, v.

By solving the two variable differential equations which arise from finding the eigenfunctions for the Casimir operator for O(d, 2) succinct expressions are found for the functions, conformal partial waves, representing the contribution of an operator of arbitrary scale dimension ∆ and spin ℓ together with its descendants to conformal four point functions for d = 4, recovering old results, and also for d = 6. The results are expressed in terms of ordinary hypergeometric functions of variables x, z which are simply related to the usual conformal invariants. An expression for the conformal partial wave amplitude valid for any dimension is also found in terms of a sum over two variable symmetric Jack polynomials which is used to derive relations for the conformal partial waves.

Possible short and semi-short representations for N = 2 and N = 4 superconformal symmetry in four dimensions are discussed. For N = 4 the well known short supermultiplets whose lowest dimension conformal primary operators correspond to -BPS states and are scalar fields belonging to the SU(4) r symmetry representations [0, p, 0] and [q, p, q] and having scale dimension ∆ = p and ∆ = 2q+p respectively are recovered. The representation content of semi-short multiplets, which arise at the unitarity threshold for long multiplets, is discussed. It is shown how, at the unitarity threshold, a long multiplet can be decomposed into four semishort multiplets. If the conformal primary state is spinless one of these becomes a short multiplet. For N = 4 a

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