SDiff(2) KP hierarchy

TL;DR

This paper introduces the SDiff(2) KP hierarchy, an extension of the KP hierarchy related to area-preserving diffeomorphisms, with new potentials, a twistor description, special solutions, and symmetry structures.

Contribution

It proposes the SDiff(2) KP hierarchy, develops its Lax formalism, introduces potentials and tau functions, and explores its solutions and symmetries, extending the KP framework.

Findings

A new hierarchy related to area-preserving diffeomorphisms is formulated.

A twistor description of solutions is provided.

Infinitesimal symmetries form a centrally extended algebra.

Abstract

An analogue of the KP hierarchy, the SDiff(2) KP hierarchy, related to the group of area-preserving diffeomorphisms on a cylinder is proposed. An improved Lax formalism of the KP hierarchy is shown to give a prototype of this new hierarchy. Two important potentials, $S$ and $\tau$, are introduced. The latter is a counterpart of the tau function of the ordinary KP hierarchy. A Riemann-Hilbert problem relative to the group of area-diffeomorphisms gives a twistor theoretical description (nonlinear graviton construction) of general solutions. A special family of solutions related to topological minimal models are identified in the framework of the Riemann-Hilbert problem. Further, infinitesimal symmetries of the hierarchy are constructed. At the level of the tau function, these symmetries obey anomalous commutation relations, hence leads to a central extension of the algebra of infinitesimal area-preserving diffeomorphisms (or of the associated Poisson algebra).