Grassmannian Approach to Super KP Hierarchies

TL;DR

This paper develops a comprehensive super-KP hierarchy framework that unifies existing hierarchies, describes them via a field theoretic approach, and constructs vertex operators to generate wave functions, enhancing the understanding of super integrable systems.

Contribution

It introduces a 'maximal' super-KP hierarchy unifying known SKP hierarchies and constructs vertex operators using superbosonization, providing a complete mbda-function description.

Findings

Unified super-KP hierarchy including MRSKP and JSKP

Constructed vertex operators acting on mbda-functions

Clarified relations among different SKP hierarchies

Abstract

We present a theory of 'maximal' super-KP(SKP) hierarchy whose flows are maximally extended to include all those of known SKP hierarchies, including, for example, the MRSKP hierarchy of Manin and Radul and the Jacobian SKP(JSKP) introduced by Mulase and Rabin. It is shown that SKP hierarchies has a natural field theoretic description in terms of the B-C system, in analogous way as the ordinary KP hierarchy. For this SKP hierarchy, we construct the vertex operators by using Kac-van de Leur superbosonization. The vertex operators act on the \(\tau\)-function and then produce the wave function and the dual wave function of the hierarchy. Thereby we achieve the description of the 'maximal' SKP hierarchy in terms of the \(\tau\)-function, which seemed to be lacking till now. Mutual relations among the SKP hierarchies are clarified. The MRSKP and the JSKP hierarchies are obtained as special cases when the time variables are appropriately restricted.