The Constrained KP Hierarchy and the Generalised Miura Transformation

Recently much attention has been paid to the restriction of KP to the submanifold of operators which can be represented as a ratio of two purely differential operators L = AB−1. Whereas most of the aspects concerning this reduced hierarchy, like the Lax flows and the Hamiltonians, are by now well understood, there still lacks a clear and conclusive statement about the associated Poisson structure. We fill this gap by placing the problem in a more general framework and then showing how the required result follows from an interesting property of the second Gelfand-Dickey brackets under multiplication and inversion of Lax operators. As a byproduct we give an elegant and simple proof of the generalised Kupershmidt-Wilson theorem. ♯ e-mail: [email protected] ♭ e-mail: [email protected] W algebras provide a link between two dimensional conformal field theories and integrable systems. At the heart of their relation with the former lies the so called “free field realization”, which in the context of the latter receives the name of (quantized) Miura transformation [1]. On the other hand, the approach based on integrable systems has provided a unifying framework for those algebras via the formalism of pseudo-differential operators (ΨDO); i.e. there exists a basic common ingredient, the so called Adler map, from which the W algebras can be readily constructed as Poisson bracket algebras. The data that specify a particular W algebra is encoded in the particular form of the associated Lax operator. This scheme has raised the hope of establishing a full classification or atlas. Whereas only partial results have been achieved, the increasing size of this atlas, with new examples being constructed everyday, calls for a deeper understanding of the way in which different W algebras can be related. This letter is a modest step towards this direction. Although many of the results presented here were already known, our approach makes emphasis on the astonishing simplicity that lies behind a number of important and, up to now, disperse results, and whose individual proof required a considerable amount of insight and calculational thrust. Our original motivation stemmed from the construction of free (multi) boson realization for non-linear W∞ type of algebras[2]. From the point of view of integrable systems, the associated piece of data is a KP-Lax operator of the form L = AB−1, where A = (∂+φ1) · · · (∂+φm) and B = (∂+φ1) · · · (∂+φn). This kind of Lax operator arises naturally in the context of matrix models [3] and the associated hierarchy and Hamiltonian structure has been the subject of recent intense research [4]. The possibility of inducing a W∞ type of algebra from the free boson Poisson brackets for φi and φj is the content of what we call the generalized Kupershmidt-Wilson theorem. As the reader will see, this theorem follows as a trivial corollary from the property of “self-similarity” of the second Gelfand-Dickey brackets against product and inversion of Lax operators. The natural geometrical arena for KP is the space of pseudodifferential operators (ΨDO) of the form