Additional Symmetries of Supersymmetric KP Hierarchies

TL;DR

This paper explores the additional symmetries of various supersymmetric KP hierarchies, revealing they form an algebra isomorphic to superdifferential operators, indicating their kinematical nature.

Contribution

It demonstrates that the algebra of additional symmetries in several supersymmetric KP hierarchies is isomorphic to superdifferential operators, highlighting their universal kinematical character.

Findings

Symmetry algebra is isomorphic to superdifferential operators.

Additional symmetries are kinematical, not dynamical.

Results apply to multiple supersymmetric KP hierarchies.

Abstract

We investigate the additional symmetries of several supersymmetric KP hierarchies: the SKP hierarchy of Manin and Radul, the $\hbox{SKP}_2$ hierarchy, and the Jacobian SKP hierarchy. In all three cases we find that the algebra of symmetries is isomorphic to the algebra of superdifferential operators, or equivalently $\SW_{1+\infty}$. These results seem to suggest that despite their realization depending on the dynamics, the additional symmetries are kinematical in nature.