On perturbations of Hilbert spaces and probability algebras with a generic automorphism

International audienceWe prove that $IHS_A$, the theory of infinite dimensional Hilbert spaces equipped with a generic automorphism, is $\aleph_0$-stable up to perturbation of the automorphism, and admits prime models up to perturbation over any set. Similarly, $APr_A$, the theory of atomless probability algebras equipped with a generic automorphism is $\aleph_0$-stable up to perturbation. However, not allowing perturbation it is not even superstable

Structures of SU-rank omega with a dense independentsubset of generics

Extending the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory of SU-rank $\omega$ with a "dense codense" independent collection$H$ of element of rank $\omega$, where density of $H$ means it intersectsany definable set of $SU$-rank omega. We show that under some technical conditions, the class of such structures is first order.We prove that the expansion is supersimple and characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one-basedness and CM-triviality. In the one-based case, we describe a natural "geometry of generics modulo $H$" associated with such expansions and show it is modular

Polish topometric groups

International audienceWe define and study the notion of \emph{ample metric generics} for a Polish topological group, which is a weakening of the notion of ample generics introduced by Kechris and Rosendal in \cite{Kechris-Rosendal:Turbulence}. Our work is based on the concept of a \emph{Polish topometric group}, defined in this article. Using Kechris and Rosendal's work as a guide, we explore consequences of ample metric generics (or, more generally, ample generics for Polish topometric groups). Then we provide examples of Polish groups with ample metric generics, such as the isometry group \Iso(\bU_1) of the bounded Urysohn space, the unitary group ${\mathcal U}(\ell_2)$ of a separable Hilbert space, and the automorphism group \Aut([0,1],\lambda) of the Lebesgue measure algebra on $[0,1]$. We deduce from this and earlier work of Kittrell and Tsankov that this last group has the automatic continuity property, i.e., any morphism from \Aut([0,1],\lambda) into a separable topological group is continuous

Bounces/Dyons in the Plane Wave Matrix Model and SU(N) Yang-Mills Theory

We consider SU(N) Yang-Mills theory on the space R^1\times S^3 with Minkowski signature (-+++). The condition of SO(4)-invariance imposed on gauge fields yields a bosonic matrix model which is a consistent truncation of the plane wave matrix model. For matrices parametrized by a scalar \phi, the Yang-Mills equations are reduced to the equation of a particle moving in the double-well potential. The classical solution is a bounce, i.e. a particle which begins at the saddle point \phi=0 of the potential, bounces off the potential wall and returns to \phi=0. The gauge field tensor components parametrized by \phi are smooth and for finite time both electric and magnetic fields are nonvanishing. The energy density of this non-Abelian dyon configuration does not depend on coordinates of R^1\times S^3 and the total energy is proportional to the inverse radius of S^3. We also describe similar bounce dyon solutions in SU(N) Yang-Mills theory on the space R^1\times S^2 with signature (-++). Their energy is proportional to the square of the inverse radius of S^2. From the viewpoint of Yang-Mills theory on R^{1,1}\times S^2 these solutions describe non-Abelian (dyonic) flux tubes extended along the x^3-axis.Comment: 11 pages; v2: one formula added, some coefficients correcte

The Spectrum of Yang Mills on a Sphere

In this note, we determine the representation content of the free, large N, SU(N) Yang Mills theory on a sphere by decomposing its thermal partition function into characters of the irreducible representations of the conformal group SO(4,2). We also discuss the generalization of this procedure to finding the representation content of N=4 Super Yang Mills.Comment: 18 pages v2. references added. typos fixe