The (Fang-)Fronsdal formulation for free fully symmetric (spinor-) tensors rests on (γ-)trace constraints on gauge fields and parameters. When these are relaxed, glimpses of the underlying geometry emerge: the field equations extend to non-local expressions involving the higher-spin curvatures, and with only a pair of additional fields an equivalent "minimal" local formulation is also possible. In this paper we complete the discussion of the "minimal" formulation for fully symmetric (spinor-) tensors, constructing one-parameter families of Lagrangians and extending them to (A)dS backgrounds. We then turn on external currents, that in this setting are subject to conventional conservation laws and, by a close scrutiny of current exchanges in the various formulations, we clarify the precise link between the local and non-local versions of the theory. To this end, we first show the equivalence of the constrained and unconstrained local formulations, and then identify a unique set of non-local Lagrangian equations which behave in the same fashion in current exchanges.where, as in the rest of the present paper, "primes" denote traces, was first considered by Schwinger long ago [17] 3 . More recently, a "minimal" Lagrangian formulation for the compensator equations was also obtained in [18]: rather than O(s) fields as in [10], it involves only a Lagrange multiplier β µ 1 ...µ s−4 , that first emerges for spin s = 4, aside from the basic field ϕ µ 1 ...µs and the compensator α µ 1 ...µ s−3 . In contrast with the non-local case, the generalization of this 2 The web site http://www.ulb.ac.be/sciences/ptm/pmif/Solvay1proc.pdf contains the Proceedings of the First Solvay Workshop on Higher-Spin Gauge Theories [2], with some contributions closely related to the present work [3,4,5] and many references to the original literature. 3 We are grateful to G. Savvidy for calling Schwinger's result to our attention.

The tachyon-free nonsupersymmetric string theories in ten dimensions have dilaton tadpoles which forbid a Minkowski vacuum. We determine the maximally symmetric backgrounds for the U Sp(32) Type I string and the SO(16) × SO(16) heterotic string. The static solutions exhibit nine dimensional Poincaré symmetry and have finite 9D Planck and Yang-Mills constants. The low energy geometry is given by a ten dimensional manifold with two boundaries separated by a finite distance which

We consider a free massless scalar field coupled to an infinite tower of background higher-spin gauge fields via minimal coupling to the traceless conserved currents. The set of Abelian gauge transformations is deformed to the non-Abelian group of unitary operators acting on the scalar field. The gauge invariant effective action is computed perturbatively in the external fields. The structure of the various (divergent or finite) terms is determined. In particular, the quadratic part of the logarithmically divergent (or of the finite) term is expressed in terms of curvatures and related to conformal higher-spin gravity. The generalized higher-spin Weyl anomalies are also determined. The relation with the theory of interacting higher-spin gauge fields on anti de Sitter spacetime via the holographic correspondence is discussed.

We show that the Wigner equations describing the continuous spin representations can be obtained as a limit of massive higher-spin field equations. The limit involves a suitable scaling of the wave function, the mass going to zero and the spin to infinity with their product being fixed. The result allows to transform the Wigner equations to a gauge invariant Fronsdal-like form. We also give the generalisation of the Wigner equations to higher dimensions with fields belonging to arbitrary representations of the massless little group.