Marc P. Bellon

Integrable Coxeter groups

Abstract We describe the construction of a class of mappings in projective space C PN for any N. These mappings are non-linear representations of Coxeter groups by birational and therefore almost everywhere defined and invertible transformations. We give specific examples of the construction and exhibit algebraic invariants. The class of mappings we consider has a variety of behaviours according to the number of independent invariants. We introduce the notion of integrability of a group of mappings. The concept is related to the nation of integrability in the realm of statistical physics and field theory as will appear elsewhere. There is a natural set of deformation parameters of our mappings, allowing for a study of their stability. We comment on the algebraic structures we are handling.

Rational mappings, arborescent iterations, and the symmetries of integrability

We describe a class of nonlinear birational representations of groups generated by a finite number of involutions. These groups are symmetries of the Yang-Baxter equations and their higher-dimensional generalizations. They provide discrete dynamical systems with a variety of behaviors, from chaotic to integrable, according to the number of invariants of the representation.

Infinite discrete symmetry group for the Yang-Baxter equations. Vertex models

Abstract We show that the Yang-Baxter equations for two-dimensional vertex models admit as a group of symmetry the infinite discrete group A 2 (1) . The existence of this symmetry explains the presence of a spectral parameter in solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetries. Although generalizing very naturally the previous one, this is a much bigger hyperbolic Coxeter group. We indicate how this symmetry should be used to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of a family of three-dimensional vertex models.

Quasi integrability of the sixteen-vertex model

Abstract We analyze the symmetries of the sixteen-vertex model. We prove the existence of a natural parametrization of the parameter space of the model by elliptic curves, grounding the inversion trick for the exact calculation of the partition function. We proceed with a “pre-Bethe-ansatz” system of equations whose analysis produces an algebraic modular invariant and yields candidates for criticality and disorder conditions.

Higher dimensional mappings

Abstract We present a new class of mappings acting on many variables, and depending on many parameters. These mappings are nonlinear (birational) representations of discrete groups generated by involutions, having their origin in the theory of integrable models in statistical mechanics. Various quantities of statistical mechanics present automorphy properties under the action of these groups, which appear thus as a generalization to several complex variables of the fundamental group for Riemann surface. They enjoy many remarkable properties, and we give a preliminary study of these mappings.

Infinite discrete symmetry group for the Yang-Baxter equations : spin models

Abstract We show that the star-triangle equation possesses an infinite discrete group of symmetry. This group is the Coxeter group A 2 (1) . It explains the presence of the spectral parameter in solutions of the equations. We describe a strategy for the resolution of the equations, and apply it to specific examples.

D = 4 supersymmetry for gauge fields of any spin

Abstract We construct a covariant formulation valid in any space-time dimensions for free massless fermionic gauge field of any spin and any permutation symmetry of their Lorentz indices. These fields can then be ranged in an off-shell N = 1, D = 4 global supersymmetry supermultiplet. The supersymmetry transformations acting on this supermultiplet generalize the usual supermatter case.

A Schwinger–Dyson Equation in the Borel Plane: Singularities of the Solution

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 gets 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.

Higher Order Corrections to the Asymptotic Perturbative Solution of a Schwinger-Dyson Equation

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.

Towards three-dimensional Bethe ansatz

We introduce a “pre-Bethe-Ansatz” system of equations for three dimensional vertex models. We bring to the light various algebraic curves of high genus and discuss some situations where these curves simplify. As a result we describe remarkable subvarieties of the space of parameters.