Jean Michel Maillet

Integrability for multidimensional lattice models

Abstract The generating principle for the algebraic construction of the hierarchy of the d -simplex equations generalizing the Yang-Baxter equation in any dimension d is given. Following this principle, we construct the generalization of the Lax equations for multidimensional integrable systems. We show that this contribution leads, as in d =2, to the existence of an infinite series of conserved charges for such models. In the d =3 case a reinterpretation of these results is given in terms of functionals of loops, leading to the notion of extended gauge symmetry and gauge fields associated to quantum groups.

Lax equations and quantum groups

Abstract We construct Lax equations associated to quantum groups. In particular we derive quantum trace formulas for the Lax matrix L giving the quantum analogue of its usual classical invariants generated by tr( L n ). These quantities generate an abelian subalgebra of the quantum group and define hamiltonian flows that admit quantum Lax representations. This hierarchy of quantum Lax equations is shown to verify zero-curvature relations. In the field theory case we give the lattice construction of these quantum structures.

The tetrahedron equation and the four-simplex equation

Abstract The tetrahedron equation and the four-simplex equation are multidimensional generalizations of the Yang-Baxter or triangle equations. We discuss common features of these members of the family of “simplex equations”. Zamolodchikovs solution of the tetrahedron equation is rewritten in an algebraic form and a generalization of it to the four-simplex case is proposed. Relevance of the simplex equation for the understanding of multidimensional integrability is briefly discussed.

Gauging the quantum groups

Abstract A notion of local Hopf algebras is introduced, in which the co-algebra structure is intrinsically connected to the dynamics described on a phase-space manifold. The concept is motivated by the introduction of an integrable dynamics in dimension d = 3. This leads to the notion of gauging of a quantum group. The connection of these results with gauge theories defined on a space of loops is also discussed.

Topological Yang-Mills theory is Abelian chiral anomaly in superspace

Abstract Ghost moments of the BRST operator in the four-dimensional topological Yang-Mills theory determine the Gauss law generator of the Parisi-Sourlas extended Yang-Mills theory, with the superspace curvature two-form reproducing BRST transformations of the component fields. The superspace generalization of the abelian chiral anomaly contains Donaldsons invariants when expanded in the supercoordinates, and its cohomological nontriviality follows from standard arguments: The chiral anomaly is the superspace exterior differential of the Parisi-Sourlas generalized Chern-Simons three-form which fails to remain invariant under superspace gauge transformations.

On a generalization of BRS and gauge transformations

Abstract Generalized non-linear BRS and gauge transformations containing non-trivial Lie algebra cocycles and acting on differential p-form gauge fields (p ≥ 1) are constructed in the context of free minimal differential algebras. The associated gauge algebra is analyzed and a method for constructing BRS gauge-invariant homogeneous polynomials of the gauge fields and their curvatures is given by quotienting the associated BRS algebra with an appropriate ideal.