Poisson Structures for Dispersionless Integrable Systems and Associated W-Algebras

TL;DR

This paper explores Poisson structures for dispersionless integrable systems, focusing on their reductions to w-algebras, and provides explicit free field representations for these algebras, expanding understanding of their algebraic properties.

Contribution

It introduces new classes of w-algebras derived from dispersionless KP hierarchies and establishes their isomorphisms and free field representations.

Findings

w-algebras are isomorphic to direct sums of simpler algebras

Explicit free field representations are constructed for these w-algebras

Analysis of Poisson structures and their reductions for specific cases

Abstract

In analogy to the KP theory, the second Poisson structure for the dispersionless KP hierarchy can be defined on the space of commutative pseudodifferential operators $L=p^n+\sum_{j=-\infty}^{n-1}u_j p^j$. The reduction of the Poisson structure to the symplectic submanifold $u_{n -1}=0$ gives rise to the w-algebras. In this paper, we discuss properties of this Poisson structure, its Miura transformation and reductions. We are particularly interested in the following two cases: a) L is pure polynomial in p with multiple roots and b) L has multiple poles at finite distance. The w-algebra corresponding to the case a) is defined as $w_ {[m_1,m_2, ... ,m_r]}$, where m_i means the multiplicity of roots and to the case b) is defined by $w(n,[m_1,m_2, ... ,m_r])$ where m_i is the multiplicity of poles. We prove that w(n,[m_1, m_2, ... , m_r])$-algebra is isomorphic via a transformation to $w_{[m_1,m_2, ... ,m_r]} \bigoplus w_{n+m} \bigoplus U(1) with $m=\sum m_i$. We also give the explicit free fields representations for these w-algebras.