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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The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions d. Calculations of the universal function of a conformal invariant ξ which appears in the two point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the ε = 4 − d expansion for the the operator φ 2 in φ 4 theory. The form for the associated functions of ξ for the two point functions for the basic field φ α and the auxiliary field λ in the the N → ∞ limit of the O(N ) non linear sigma model for any d in the range 2 < d < 4 are also rederived. These results are obtained by integrating the two point functions over planes parallel to the boundary, defining a restricted two point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two point function. Consistency of the results is checked by considering the limit d → 4 and also by analysis of the operator product expansions for φ α φ β and λλ. Using this method the form of the two point function for the energy momentum tensor in the conformal O(N ) model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two point functions are also derived making essential use of conformal invariance.

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The implications of conformal invariance, as relevant in quantum field theories at a renormalisation group fixed point, are analysed with particular reference to results for correlation functions involving conserved currents and the energy momentum tensor. Ward identities resulting from conformal invariance are discussed. Explicit expressions for two and three point functions, which are essentially determined by conformal invariance, are obtained. As special cases we consider the three point functions for two vector and an axial current in four dimensions, which realises the usual anomaly simply and unambiguously, and also for the energy momentum tensor in general dimension d. The latter is shown to have two linearly independent forms in which the Ward identities are realised trivially, except if d = 4, when the two forms become degenerate. This is necessary in order to accommodate the two independent forms present in the trace of the energy momentum tensor on curved space backgrounds for conformal field theories in four dimensions. The coefficients of the two trace anomaly terms are related to the three parameters describing the general energy momentum tensor three point function. The connections with gravitational effective actions depending on a background metric are described. A particular form due to Riegert is shown to be unacceptable. Conformally invariant expressions for the effective action in four dimensions are obtained using the Green function for a differential operator which has simple properties under local rescalings of the metric.* email: je10001@damtp.cam.ac.uk and ho@damtp.cam.ac.uk * This is entirely equivalent to the result given in [13] in (2.13) for a convenient choice of the arbitrary parameter q there.