Masslessness in nn-dimensions

We determine the representations of the ``conformal'' group ${\bar{SO}}_0(2, n)$, the restriction of which on the ``Poincar\'e'' subgroup ${\bar{SO}}_0(1, n-1).T_n$ are unitary irreducible. We study their restrictions to the ``De Sitter'' subgroups ${\bar{SO}}_0(1, n)$ and ${\bar{SO}}_0(2, n-1)$ (they remain irreducible or decompose into a sum of two) and the contraction of the latter to ``Poincar\'e''. Then we discuss the notion of masslessness in $n$ dimensions and compare the situation for general $n$ with the well-known case of 4-dimensional space-time, showing the specificity of the latter.Comment: 34 pages, LaTeX2e, 1 figure. To be published in Reviews in Math. Phy

Massless Particles in Arbitrary Dimensions

Various properties of two kinds of massless representations of the n-conformal (or (n+1)-De Sitter) group $\tilde{G}_n=\widetilde{SO}_0(2,n)$ are investigated for $n\ge2$. It is found that, for space-time dimensions $n\ge3$, the situation is quite similar to the one of the n=4 case for $S_n$-massless representations of the n-De Sitter group $\widetilde{SO}_0(2,n-1)$. These representations are the restrictions of the singletons of $\tilde{G}_n$. The main difference is that they are not contained in the tensor product of two UIRs with the same sign of energy when n>4, whereas it is the case for another kind of massless representation. Finally some examples of Gupta-Bleuler triplets are given for arbitrary spin and $n\ge3$.Comment: 33 pages, LaTeX2e. To be published in Reviews in Math. Phy

Linear Form of 3-scale Relativity Algebra and the Relevance of Stability

We show that the algebra of the recently proposed Triply Special Relativity can be brought to a linear (ie, Lie) form by a correct identification of its generators. The resulting Lie algebra is the stable form proposed by Vilela Mendes a decade ago, itself a reapparition of Yang's algebra, dating from 1947. As a corollary we assure that, within the Lie algebra framework, there is no Quadruply Special Relativity.Comment: 5 page

Simple Non Linear Klein-Gordon Equations in 2 space dimensions, with long range scattering

We establish that solutions, to the most simple NLKG equations in 2 space dimensions with mass resonance, exhibits long range scattering phenomena. Modified wave operators and solutions are constructed for these equations. We also show that the modified wave operators can be chosen such that they linearize the non-linear representation of the Poincar\'e group defined by the NLKG.Comment: 19 pages, LaTeX, To appear in Lett. Math. Phy

Superconformal field theories from IIB spectroscopy on AdS5×T11AdS_5\times T^{11}

We report on tests of the AdS/CFT correspondence that are made possible by complete knowledge of the Kaluza-Klein mass spectrum of type IIB supergravity on $AdS_5 \times T^{11}$ with T^{11}=SU(2)^2/U(1). After briefly discussing general multiplet shortening conditions in SU(2,2|1) and PSU(2,2|4), we compare various types of short SU(2,2|1) supermultiplets on AdS_5 and different families of boundary operators with protected dimensions. The supergravity analysis predicts the occurrence in the SCFT at leading order in N and g_s N, of extra towers of long multiplets whose dimensions are rational but not protected by supersymmetry.Comment: 11 pages, To appear in the proceedings of the STRINGS '99 conference, Potsdam (Germany), 19-25 July 199

Improved Implementation of Point Location in General Two-Dimensional Subdivisions

We present a major revamp of the point-location data structure for general two-dimensional subdivisions via randomized incremental construction, implemented in CGAL, the Computational Geometry Algorithms Library. We can now guarantee that the constructed directed acyclic graph G is of linear size and provides logarithmic query time. Via the construction of the Voronoi diagram for a given point set S of size n, this also enables nearest-neighbor queries in guaranteed O(log n) time. Another major innovation is the support of general unbounded subdivisions as well as subdivisions of two-dimensional parametric surfaces such as spheres, tori, cylinders. The implementation is exact, complete, and general, i.e., it can also handle non-linear subdivisions. Like the previous version, the data structure supports modifications of the subdivision, such as insertions and deletions of edges, after the initial preprocessing. A major challenge is to retain the expected O(n log n) preprocessing time while providing the above (deterministic) space and query-time guarantees. We describe an efficient preprocessing algorithm, which explicitly verifies the length L of the longest query path in O(n log n) time. However, instead of using L, our implementation is based on the depth D of G. Although we prove that the worst case ratio of D and L is Theta(n/log n), we conjecture, based on our experimental results, that this solution achieves expected O(n log n) preprocessing time.Comment: 21 page

Anti de Sitter Holography via Sekiguchi Decomposition

In the present paper we start consideration of anti de Sitter holography in the general case of the (q+1)-dimensional anti de Sitter bulk with boundary q-dimensional Minkowski space-time. We present the group-theoretic foundations that are necessary in our approach. Comparing what is done for q=3 the new element in the present paper is the presentation of the bulk space as the homogeneous space G/H = SO(q,2)/SO(q,1), which homogeneous space was studied by Sekiguchi.Comment: 10 pages, to appear in the Proceedings of the XI International Workshop "Lie Theory and Its Applications in Physics", (Varna, Bulgaria, June 2015

Some interesting violations of the Breitenlohner-Freedman bound

We demonstrate that AdS_5 x T^{pq} is unstable, in the sense of Breitenlohner and Freedman, for unequal p and q. This settles, negatively, the long-standing question of whether the T^{pq} manifolds for unequal p and q might correspond to non-supersymmetric fixed points of the renormalization group. We also show that the AdS_3 x S^7 vacuum of Sugimoto's USp(32) open string theory is unstable. This explains, at a heuristic level, the apparent absence of a heterotic string dual.Comment: 16 pages, latex. v2: minor correction

Analysis of Higher Spin Field Equations in Four Dimensions

The minimal bosonic higher spin gauge theory in four dimensions contains massless particles of spin s=0,2,4,.. that arise in the symmetric product of two spin 0 singletons. It is based on an infinite dimensional extension of the AdS_4 algebra a la Vasiliev. We derive an expansion scheme in which the gravitational gauge fields are treated exactly and the gravitational curvatures and the higher spin gauge fields as weak perturbations. We also give the details of an explicit iteration procedure for obtaining the field equations to arbitrary order in curvatures. In particular, we highlight the structure of all the quadratic terms in the field equations.Comment: Latex, 30 pages, several clarifications and few references adde