Constructive Matrix Theory for Higher Order Interaction

This paper provides an extension of the constructive loop vertex expansion to stable matrix models with interactions of arbitrarily high order. We introduce a new representation for such models, then perform a forest expansion on this representation. It allows to prove that the perturbation series of the free energy for such models is analytic in a domain uniform in the size N of the matrix. Our method applies to complex (rectangular) matrices. The extension to Hermitian square matrices, which was claimed wrongly in the first arXiv version of this paper, is postponed to a future study.Comment: 44 pages, 9 figure

Localic Metric spaces and the localic Gelfand duality

In this paper we prove, as conjectured by B.Banachewski and C.J.Mulvey, that the constructive Gelfand duality can be extended into a duality between compact regular locales and unital abelian localic C*-algebras. In order to do so we develop a constructive theory of localic metric spaces and localic Banach spaces, we study the notion of localic completion of such objects and the behaviour of these constructions with respect to pull-back along geometric morphisms.Comment: 57 page

Non-perturbative \lambda\Phi^4 in D=1+1: an example of the constructive quantum field theory approach in a schematic way

During the '70, several relativistic quantum field theory models in $D=1+1$ and also in $D=2+1$ have been constructed in a non-perturbative way. That was done in the so-called {\it constructive quantum field theory} approach, whose main results have been obtained by a clever use of Euclidean functional methods. Although in the construction of a single model there are several technical steps, some of them involving long proofs, the constructive quantum field theory approach contains conceptual insights about relativistic quantum field theory that deserved to be known and which are accessible without entering in technical details. The purpose of this note is to illustrate such insights by providing an oversimplified schematic exposition of the simple case of $\lambda\Phi^4$ (with $m>0$) in $D=1+1$. Because of the absence of ultraviolet divergences in its perturbative version, this simple example -although does not capture all the difficulties in the constructive quantum field theory approach- allows to stress those difficulties inherent to the non-perturbative definition. We have made an effort in order to avoid several of the long technical intermediate steps without missing the main ideas and making contact with the usual language of the perturbative approach.Comment: 63 pages. Typos correcte

Internalising modified realisability in constructive type theory

A modified realisability interpretation of infinitary logic is formalised and proved sound in constructive type theory (CTT). The logic considered subsumes first order logic. The interpretation makes it possible to extract programs with simplified types and to incorporate and reason about them in CTT.Comment: 7 page

Scaling behaviour of three-dimensional group field theory

Group field theory is a generalization of matrix models, with triangulated pseudomanifolds as Feynman diagrams and state sum invariants as Feynman amplitudes. In this paper, we consider Boulatov's three-dimensional model and its Freidel-Louapre positive regularization (hereafter the BFL model) with a ?ultraviolet' cutoff, and study rigorously their scaling behavior in the large cutoff limit. We prove an optimal bound on large order Feynman amplitudes, which shows that the BFL model is perturbatively more divergent than the former. We then upgrade this result to the constructive level, using, in a self-contained way, the modern tools of constructive field theory: we construct the Borel sum of the BFL perturbative series via a convergent ?cactus' expansion, and establish the ?ultraviolet' scaling of its Borel radius. Our method shows how the ?sum over trian- gulations' in quantum gravity can be tamed rigorously, and paves the way for the renormalization program in group field theory

Constructive Field Theory in Zero Dimension

In this pedagogical note we propose to wander through five different methods to compute the number of connected graphs of the zero-dimensional $\phi^4$ field theory,in increasing order of sophistication. The note does not contain any new result but may be helpful to summarize the heart of constructive resummations, namely a replica trick and a forest formula.Comment: 12 pages,one figur