KP Hierarchies, Polynomial and Rational W Algebras on Riemann Surfaces: a Global Approach

TL;DR

This paper develops a covariant pseudodifferential calculus on Riemann surfaces, enabling the definition of KP operators and W algebras in higher genus, with detailed analysis of the higher genus NLS hierarchy.

Contribution

It introduces a global covariant calculus on Riemann surfaces that allows defining KP operators, Miura maps, and W algebras in higher genus settings.

Findings

Global pseudodifferential calculus on Riemann surfaces established

Construction of polynomial and rational W algebras associated with reductions

Detailed analysis of the higher genus NLS hierarchy

Abstract

A covariant pseudodifferential calculus on Riemann surfaces, based on the Krichever-Novikov global picture, is presented. It allows defining scalar and matrix KP operators, together with their reductions, in higher genus. Globally defined Miura maps are considered and the arising of polynomial or rational ${\cal W}$ algebras on R.S. associated to each reduction are pointed out. The higher genus NLS hierarchy is analyzed in detail.