Claude-Michel Viallet

Hamiltonian structures and Lax equations

Abstract We show that any hamiltonian system, which is integrable in the sense of Liouville, admits a Lax representation, at least locally at generic points in phase space. We introduce the most general Poisson bracket ensuring the involution property of the integrals of motion and existence of a Lax pair. We give examples of the structure we describe.

{BRS} Algebras: Analysis of the Consistency Equations in Gauge Theory

We compute all possible anomalous terms in quantum gauge theory in the natural class of polynomials of differential forms. By using the appropriate cohomological and algebraic methods, we do it for all dimensions of spacetime and all structure groups with reductive Lie algebras.

General solution of the consistency equation

Abstract We produce the general solution of the Wess-Zumino consistency condition for gauge theories of the Yang-Mills type, for any ghost number and form degree. We resolve the problem of the cohomological independence of these solutions. In other words we fully describe the local version of the cohomology of the BRS operator, modulo the differential on space-time. This in particular includes the presence of external fields and non-trivial topologies of space-time.

Some results on local cohomologies in field theory

Abstract We prove that the local cohomology of the exterior differential is trivial. This is done through homotopy and spectral sequence arguments. We then give relations between various cohomologies (especially cohomologies of one differential modulo another one). Using these results we give a direct proof of the locality of the gauge fixed action in the BRST formalism.

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.

Results on BRS cohomologies in gauge theory

Abstract We compute the cohomology of the Becchi-Rouet-Stora operator in gauge theory over general space-time M without boundary, with structure group G, for class of polynomial functions of the field. We show that the problem reduces to a standard problem for the finite dimensional group G. As a consequence, we prove that, within this class of polynomials, all anomalies and Schwinger terms are obtained from invariants of G.

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.

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.

Mappings in higher dimensions

We introduce non trivial two‐dimensional and three‐dimensional mappings. These mappings are birational transformations, the iterates of which give a nonlinear representation of infinite Coxeter groups. These mappings originate from integrable models in statistical mechanics. They exhibit a number of remarkable properties. Some have algebraic invariants allowing for orbits lying on some smooth manifolds. Others give examples of mappings the iterates of which are Hofstadte‐like patterns and exemplify onset of order from disorder (Saturn’s ring).

A note on the gauge symmetries of pure Chern-Simons theories with p-form gauge fields

The gauge symmetries of pure Chern - Simons theories with p-form gauge fields are analysed. It is shown that the number of independent gauge symmetries depends crucially on the parity of p. The case where p is odd appears to be a direct generalization of the p = 1 case and presents the remarkable feature that the timelike diffeomorphisms can be expressed in terms of the spatial diffeomorphisms and the internal gauge symmetries. In contrast, the timelike diffeomorphisms may be independent gauge symmetries when p is even. This happens when the number of fields and the spacetime dimension fulfil an algebraic condition which is written explicitly.