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The parametric representation has been used since a long time for the evaluation of Feynman diagrams. As a dimension independent intermediate representation, it allows a clear description of singularities. Recently, it has become a choice tool for the investigation of the type of transcendent numbers appearing in the evaluation of Feynman diagrams. The inclusion of numerators has however stagnated since the ground work of Nakanishi. I here show how to greatly simplify the formulas through the use of Dodgson identities., Comment: 23 pages, TikZ pictures. Two additional result files, best read with Maple. v2: expanded explanations on some parts. Results and additional files unchanged

Building on our recent derivation of the Ward-Schwinger-Dyson equations for the cubic interaction model, we present here the first steps of their resurgent analysis. In our derivation of the WSD equations, we made sure that they had the properties of compatibility with the renormalisation group equations and independence from a regularisation procedure which was known to allow for the comparable studies in the Wess-Zumino model. The interactions between the transseries terms for the anomalous dimensions of the field and the vertex is at the origin of unexpected features, for which the effect of higher order corrections is not precisely known at this stage: we are only at the beginning of the journey to use resurgent methods to decipher non-perturbative effects in quantum field theory.

We develop a system of equations for the propagators and three point functions of the $\phi^3$ quantum field theory in six dimensions. Inspired from a refinement by Ward on the Schwinger--Dyson equations, the main characteristics of this system are to be formulated purely in terms of renormalized quantities and to give solutions satisfying renormalisation group equations. These properties were difficult to get together, due to the overlapping divergences in the propagator. The renormalisation group equations are an integral part of any efficient resolution scheme of this system and will be instrumental in the study of the resurgent properties of the solutions. It is our belief that this method can be generalized to the case of gauge fields, shedding some light on their quantum properties., Comment: 22 pages, tikZ figures. v3: Precisions added from suggestions of the referees. v2: some clarifications in section 1.3

From 't Hooft's argument, one expects that the analyticity domain of an asymptotically free quantum field theory is horned shaped. In the usual Borel summation, the function is obtained through a Laplace transform and thus has a much larger analyticity domain. However, if the summation process goes through the process called acceleration by Ecalle, one obtains such a horn shaped analyticity domain. We therefore argue that acceleration, which allows to go beyond standard Borel summation, must be an integral part of the toolkit for the study of exactly renormalisable quantum field theories. We sketch how this procedure is working and what are its consequences., Comment: 6 pages

In many domains of physics, methods are needed to deal with non-perturbative aspects. I want here to argue that a good approach is to work on the Borel transforms of the quantities of interest, the singularities of which give non-perturbative contributions. These singularities in many cases can be largely determined by using the alien calculus developed by Jean \'Ecalle. My main example will be the two point function of a massless theory given as a solution of a renormalization group equation., Comment: 4 pages, double-column

Starting from the Schwinger--Dyson equation and the renormalization group equation for the massless Wess--Zumino model, we compute the dominant nonperturbative contributions to the anomalous dimension of the theory, which are related by alien calculus to singularities of the Borel transform on integer points. The sum of these dominant contributions has an analytic expression. When applied to the two-point function, this analysis gives a tame evolution in the deep euclidean domain at this approximation level, making doubtful the arguments on the triviality of the quantum field theory with positive \(\beta\)-function. On the other side, we have a singularity of the propagator for time like momenta of the order of the renormalization group invariant scale of the theory, which has a nonperturbative relationship with the renormalization point of the theory. All these results do not seem to have an interpretation in terms of semiclassical analysis of a Feynman path integral., Comment: 15 pages, one embedded tikz figure. v2: prose polishing and misprints corrections, additional references v3: stronger case for acceleration. Generally matches the published version

We map the Schwinger--Dyson equation and the renormalization group equation for the massless Wess--Zumino model in the Borel plane, where the product of functions get mapped to a convolution product. The two-point function can be expressed as a superposition of general powers of the external momentum. The singularities of the anomalous dimension are shown to lie on the real line in the Borel plane and to be linked to the singularities of the Mellin transform of the one-loop graph. This new approach allows us to enlarge the reach of previous studies on the expansions around those singularities. The asymptotic behavior at infinity of the Borel transform of the solution is beyond the reach of analytical methods and we do a preliminary numerical study, aiming to show that it should remain bounded., Comment: 21 pages, 2 figures, use Tikz New version includes corrections asked by referee

Building on our previous works on perturbative solutions to a Schwinger-Dyson for the massless Wess-Zumino model, we show how to compute 1/n corrections to its asymptotic behavior. The coefficients are analytically determined through a sum on all the poles of the Mellin transform of the one loop diagram. We present results up to the fourth order in 1/n as well as a comparison with numerical results. Unexpected cancellations of zetas are observed in the solution, so that no even zetas appear and the weight of the coefficients is lower than expected, which suggests the existence of more structure in the theory., Comment: 16 pages, 2 figures. Some points clarified, typos corrected, matches the version to be published in Lett. Math. Phys

We display the construction of a twisted superalgebra for the N=1 Euclidian supergravity on 4-manifolds with an almost complex structure. It acts on a representation of twisted supersymmetry made of forms with odd and even statistics and it is covariant under a SU(2)\subset SO(4) Lorentz invariance of the manifold's tangent-space. It contains 4 twisted supersymmetry generators, one nilpotent scalar, one vector and one pseudo-scalar. The superalgebra closes on the twisted fields of supergravity in its new minimal set of auxiliary fields. Its couplings to the twisted Wess and Zumino and vector multiplets are also determined., Comment: Added references and corrected misprints

We consider the effects of higherloop corrections to a Schwinger--Dyson equations for propagators. This is made possible by the efficiency of the methods we developed in preceding works, still using the supersymmetric Wess--Zumino model as a laboratory. We obtain the dominant contributions of the three and four loop primitive divergences at high order in perturbation theory, without the need for their full evaluations. Our main conclusion is that the asymptotic behavior of the perturbative series of the renormalization function remains unchanged, and we conjecture that this will remain the case for all finite order corrections., Comment: 12 pages, 2 imbedded TiKZ pictures. A few clarifications matching the published version

Focusing on the connection between the Landau theory of second order phase transitions and the holographic approach to critical phenomena, we study diverse field theories in an anti-de Sitter black hole background. Through simple analytical approximations, solutions to the equations of motion can be obtained in closed form which give rather good approximations of the results obtained using more involved numerical methods. The agreement we find stems from rather elementary considerations on perturbation of Schr\"odinger equations., Comment: 14 pages, 2 figures. Version published in Advances in High Energy Physics

Efficient computation methods are devised for the perturbative solution of Schwinger--Dyson equations for propagators. We show how a simple computation allows to obtain the dominant contribution in the sum of many parts of previous computations. This allows for an easy study of the asymptotic behavior of the perturbative series. In the cases of the four-dimensional supersymmetric Wess--Zumino model and the $\phi_6^3$ complex scalar field, the singularities of the Borel transform for both positive and negative values of the parameter are obtained and compared., Comment: 9 pages, no figures. Match of the published version, with the corrections in proof

I introduce an approximation scheme that allows to deduce differential equations for the renormalization group $\beta$-function from a Schwinger--Dyson equation for the propagator. This approximation is proven to give the dominant asymptotic behavior of the perturbative solution. In the supersymmetric Wess--Zumino model and a $\phi^3_6$ scalar model which do not have divergent vertex functions, this simple Schwinger--Dyson equation for the propagator captures the main quantum corrections., Comment: Clarification of the presentation of results. Equations and results unchanged. Match the published version. 12 pages

We obtain the contributions to the renormalization group functions of all the diagrams containing the unique one-loop primitive divergence of a simple supersymmetric Wess--Zumino model, up to more than 200 loops. The asymptotic behavior of the coefficients in the expansion of the anomalous dimension is analyzed., Comment: Changed title, match the version to be published in Nucl. Phys. B 12 pages, 2 tables

These are expanded notes for a short course given at the Universidad Nacional de La Plata. They aim at giving a self-contained account of the results of Alain Connes and Michel Dubois--Violette., Comment: 17 pages

In his famous book, Felix Klein describes a complex variable for the quotients of the ordinary sphere by the finite groups of rotations and in particular for the most complex situation of the quotient by the symmetry group of the icosahedron. The purpose of this work and its sequels is to obtain similar results for the quotients of the three--dimensional sphere. Various properties of the group $SU(2)$ and of its representations are used to obtain explicit expressions for coordinates and the relations they satisfy., Comment: 8 pages

Using Dyson--Schwinger equations within an approach developed by Broadhurst and Kreimer and the renormalization group, we show how high loop order of the renormalization group coefficients can be efficiently computed in a supersymmetric model., Comment: 8 pages, 2 figures

A general method for the determination of the harmonics of quotients of the 3-sphere is given. They can all be deduced from three objects already known from Klein. We further show explicitly how these harmonics can be organized in irreducible representations of the holonomy group: this allows for the determination of the full correlation matrix of the Cosmic Microwave Background in the spherical harmonics basis., Comment: 18 pages, changed title and expanded explanations, to follow the published version. The results are unchanged

We extend the semiclassical study of the Neumann model down to the deep quantum regime. A detailed study of connection formulae at the turning points allows to get good matching with the exact results for the whole range of parameters., Comment: 10 pages, 5 figures Minor edits

We use semi--classical and perturbation methods to establish the quantum theory of the Neumann model, and explain the features observed in previous numerical computations., Comment: 14 pages, 3 figures

We study numerically the spectrum and eigenfunctions of the quantum Neumann model, illustrating some general properties of a non trivial integrable model., Comment: 12 pages, 5 figures. Expanded introduction and references to put our work in the proper historical context

Topological euclidean gravity is built in eight dimensions for manifolds with $Spin(7) \subset SO(8)$ holonomy. In a previous work, we considered the construction of an eight-dimensional topological theory describing the graviton and one graviphoton. Here we solve the question of determining a topological model for the combined system of a metric and a Kalb--Ramond two-form gauge field. We then recover the complete $N=1, D=8$ supergravity theory in a twisted form. We observe that the generalized self-duality conditions of our model correspond to the octonionic string equations., Comment: 17 pages

A topological theory for euclidean gravity in eight dimensions is built by enforcing octonionic self-duality conditions on the spin connection. The eight-dimensional manifold must be of a special type, with G_2 or Spin(7) holonomy. The resulting theory is related to a twisted version of N=1, D=8 supergravity. The situation is comparable to that of the topological Yang--Mills theory in eight dimensions, for which the SO(8) invariance is broken down to Spin(7), but is recovered after untwisting the topological theory., Comment: LaTeX file, 9 pages

The progress brought to the study of chiral fermions and gauge theories by quantization methods with a bulk time suggests their usefulness in supersymmetric theories. Using superspace methods, we show how an explicitly supersymmetric version of such quantization methods may be given., Comment: 6 pages