W(sl(n)) : Existence, Cartan Basis and Infinite Abelian Subalgebras

Abstract

Classical integrable systems are amenable to a variety of different solution techniques. The two most prominent ones are the calculus of pseudo differential operators and τ-functions and the inverse scattering method centered around the classical Yang Baxter equation. The transition to the quantum regime, however, has so far only been successful for the case of the inverse scattering method. The classical field theories for which the resulting quantum inverse scattering method can be implemented have by now become paradigmatic examples of integrable quantum field theories. Supplemented by the Smatrix- and form factor bootstraps many of the physically relevant questions can be answered. There are, however, many classically integrable field theories for which even the first step, the construction of the quantum monodromy matrix solving a Yang Baxter equation, has not been achieved. This motivates the search for possible alternatives. In particular, there are indications that the quantum analogues of the first of the mentioned classical techniques should reveal some of the required mathematical structure. The seminal work of A.B. Zamolodchikov on the integrable deformations of 2-dimensional CFTs [?] suggests the existence of integrable structures in a large class of meromorphic CFTs. The existence of an infinite set of conserved charges in involution is then equivalent to the vanishing of a certain subset of the structure constants. This is a dynamical problem for which no general solution techniques are known. For the low members of the W(s1(n)) series it has been solved with reference to the classical case (involution) and a specific induction scheme visible in the free field realization (existence).