Translationally invariant symmetric polynomials as coordinates for N-body problems with identical particles are proposed. It is shown that in those coordinates the Calogero and Sutherland Nbody Hamiltonians, after appropriate gauge transformations, can be presented as a quadratic polynomial in the generators of the algebra sl N in finite-dimensional degenerate representation. The exact solvability of these models follows from the existence of the infinite flag of such representation spaces, preserved by the above Hamiltonians. A connection with Jack polynomials is discussed. a

Using Noether's procedure we present a complete solution for the trilinear interactions of arbitrary spins s 1 , s 2 , s 3 in a flat background, and discuss the possibility to enlarge this construction to higher order interactions in the gauge field. Some classification theorems of the cubic (self)interaction with different numbers of derivatives and depending on relations between the spins are presented. Finally the expansion of a general spin s gauge transformation into powers of the field and the related closure of the gauge algebra in the general case are discussed.

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We present an off-shell generating function for all cubic interactions of Higher Spin gauge fields constructed in [1]. It is a generalization of the on-shell generating function proposed in [2], is written in a very compact way, and turns out to have a remarkable structure.

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We present a detailed analysis of a scalar conformal four-point function obtained from AdS/CFT correspondence. We study the scalar exchange graphs on AdS d+1 and discuss their analytic properties. Using methods of conformal partial wave analysis, we present a general procedure to study conformal four-point functions in terms of exchanges of scalar and tensor fields. The logarithmic terms in the four-point function are connected to the anomalous dimensions of the exchanged fields. Comparison of the results from AdS d+1 graphs with the conformal partial wave analysis suggests a possible general form for the operator product expansion of scalar fields in the boundary CFT d . 1 1 The duality between string or M -theory compactifications on AdS d+1 and d-dimensional superconformal gauge theories suggested by AdS/CFT correspondence [1] has been the subject of intensive research over the past couple of years (for a recent review see [2]). Gradually, the emerging picture takes the form of the long-sought string/gauge theory relationship [3]. Recently, in a minkowsian version of the correspondence the d-dimensional conformal field theory (CFT) has been discussed in the context of local quantum field theory [4] defined on a standard (flat) compactified Minkowski space M c 1,d−1 . This space arises as the boundary of the AdS 1,d spacetime. The isometry group of both spaces is SO(d, 2) and the state space of the boundary CFT is related to the state space of the bulk theory [5].Such a view of the AdS/CFT correspondence implies that the known local structure of conformal field theory, (see for example [6] and references therein), is connected to the local structure of the the field (or string) theory living on AdS. In particular, harmonic analysis on the isometry group SO(d, 2) ("conformal partial wave analysis" CPWA), of n-point functions of the boundary CFT should be valid. This is equivalent to the existence of an operator product expansion (OPE) for the boundary CFT. Such expansions are convergent in a topology defined by the n-point functions on which they are applied (CPWA), or into which they are inserted (OPE). Perhaps the most well-known application ground for CPWA and OPEs is the (Euclidean) case d = 4, when the boundary CFT is the N = 4 SYM theory with gauge group SU (N ). In that case, the large-N , large-λ expansion (λ = g 2 Y M N with g Y M being the gauge coupling), corresponds to a perturbative form of the AdS theory in terms of the so-called "Witten graphs"[1]. Technical exploitations of the AdS/CFT correspondence are mainly based on this graphical expansion [8,9].Our aim in this work is to make a thorough investigation of a four-point function of scalar fields in the boundary CFT obtained from a graphical expansion in AdS. We choose to work in general dimensions d to ensure a broad applicability of our results. In Section 2 we set the stage for our study by considering a theory on AdS with a single cubic local interaction term.This may be viewed as the minimal AdS theory leading to a non-trivial four-point fu...

The higher spin interaction currents for the conformally coupled scalar in AdS 4 space for both regular and irregular boundary condition corresponding to the free and interacting critical point of the boundary O(N) sigma model are constructed. The explicit form of the linearized interaction of the scalar and spin two and four gauge fields in the AdS D space using Noether's procedure for the corresponding spin two and four linearized gauge and generalized Weyl transformations are obtained.