Non-perturbative double scaling limits

Recently, the author has proposed a generalization of the matrix and vector models approach to the theory of random surfaces and polymers. The idea is to replace the simple matrix or vector (path) integrals by gauge theory or non-linear sigma model (path) integrals. We explain how this solves one of the most fundamental limitation of the classic approach: we automatically obtain non-perturbative definitions in non-Borel summable cases. This is exemplified on the simplest possible examples involving O(N) symmetric non-linear sigma models with N-dimensional target spaces, for which we construct (multi)critical metrics. The non-perturbative definitions of the double scaled, manifestly positive, partition functions rely on remarkable identities involving (path) integrals.Comment: 18 pages, one figur

Super Yang-Mills, Matrix Models and Geometric Transitions

I explain two applications of the relationship between four dimensional N=1 supersymmetric gauge theories, zero dimensional gauged matrix models, and geometric transitions in string theory. The first is related to the spectrum of BPS domain walls or BPS branes. It is shown that one can smoothly interpolate between a D-brane state, whose weak coupling tension scales as Nc or 1/gs, and a closed string solitonic state, whose weak coupling tension scales as Nc^2 or 1/gs^2. This is part of a larger theory of N=1 quantum parameter spaces. The second is a new purely geometric approach to sum exactly over planar diagrams in zero dimension. It is an example of open/closed string duality.Comment: 11 pages, 2 figures, .cls files included; to appear in the proceedings of the Strings 2004 conference in Pari

On a basic mean value Theorem with explicit exponents

In this paper we follow a paper from A. Sedunova (2017) regarding R. C. Vaughan's basic mean value Theorem (Acta Arith. 1980) to improve and complete a more general demonstration for a suitable class of arithmetic functions as started by A. C. Cojocaru and M. R. Murty (London Math. Soc. Stud. Texts 2006). As an application we derive a basic mean value Theorem for the von Mangoldt generalized functions.Comment: 13 page

A model for gauge theories with Higgs fields

We discuss in details a simple, purely bosonic, quantum field theory belonging to larger class of models with the following properties: a) They are asymptotically free, with a dynamically generated mass scale. b) They have a space of parameters which gets quantum corrections drastically modifying the classical singularity structure. The quantum theory can have massless solitons, Argyres-Douglas-like CFTs, exhibit confinement, etc... c) The physics can, to a large extent, be worked out in models with a large number of supersymmetries as well as in purely bosonic ones. In the former case, exact BPS mass formulas can be derived, brane constructions and embedding in M theory do exist. d) The models have an interesting 1/N expansion, and it is possible to define a double scaling limit in the sense of the ``old'' matrix models when approaching the singularities in parameter space. These properties make these theories very good toy models for four dimensional gauge theories with Higgs fields, and provide a framework where the effects of breaking supersymmetry can be explicitly studied. In our model, we work out in details the quantum space of parameters. We obtain the non-local lagrangian description of the Argyres-Douglas-like CFT, and show that it admits a strongly coupled fixed point. We also explicitly demonstrate property d). The possibility of defining such double scaling limits was not anticipated on the gauge theory side, and could be of interest to understand the gauge theory/string theory correspondence.Comment: 74 pages, 8 figures, 3 appendice

Bogoliubov theory of interacting bosons: new insights from an old problem

In a gas of $N$ interacting bosons, the Hamiltonian $H_c$, obtained by dropping all the interaction terms between free bosons with moment $\hbar\mathbf{k}\ne\mathbf{0}$, is diagonalized exactly. The resulting eigenstates $|\:S,\:\mathbf{k},\:\eta\:\rangle$ depend on two discrete indices $S,\:\eta=0,\:1,\:\dots$, where $\eta$ numerates the \emph{quasiphonons} carrying a moment $\hbar\mathbf{k}$, responsible for transport or dissipation processes. $S$, in turn, numerates a ladder of \textquoteleft vacua\textquoteright$\:|\:S,\:\mathbf{k},\:0\:\rangle$, with increasing equispaced energies, formed by boson pairs with opposite moment. Passing from one vacuum to another ($S\rightarrow S\pm1$), results from creation/annihilation of new momentless collective excitations, that we call \emph{vacuons}. Exact quasiphonons originate from one of the vacua by \textquoteleft creating\textquoteright$\:$an asymmetry in the number of opposite moment bosons. The well known Bogoliubov collective excitations (CEs) are shown to coincide with the exact eigenstates $|\:0,\:\mathbf{k},\:\eta\:\rangle$, i.e. with the quasiphonons created from the lowest-level vacuum ($S=0$). All this is discussed, in view of existing or future experimental observations of the vacuons (PBs), a sort of bosonic Cooper pairs, which are the main factor of novelty beyond Bogoliubov theory.Comment: 13 pages, 1 figur

Searches for the Higgs boson in Minimal Supersymmetric CP-conserving and CP-violating Standard Model scenarios at LEP

It is important to study extended models containing more than one physical Higgs boson in the spectrum. In particular, Two Higgs Doublet Models (2HDMs) are attractive extensions of the SM, predicting new phenomena with the fewest new parameters. The Higgs sector in the Minimal Supersymmetric extension of the SM (MSSM) is a 2HDM itself. The neutral Higgs searches performed at LEP are showing no evidence of the presence of a signal and have therefore been interpreted in the context of 2HDMs. Depending on the model considered exclusion of large regions of the parameter space can be obtained, but the existence of the lightest Higgs boson with masses lower than 90 GeV is not ruled out in all models by LEP. In the MSSM at least one of the neutral Higgs bosons is predicted to have its mass close to the electroweak energy scale; when radiative corrections are included, this mass should be less than about 140 GeV. This prediction provides a strong motivation for searches at present and future colliders.Comment: 4 pages, 4 figure