A New Description of the Jacobian Super KP Hierarchy

As for any other theory, a wide comprehension of the theory of soliton equations has been achieved only after the development of different techniques, which allowed the investigation of its various aspects. The efforts of the people involved in such investigations have furthermore produced a deeper understanding of many related branches of mathematics. Among the several concepts that have been used, either introduced from scratch or already available, we can for example cite, to name only a few, the theory of algebraic curves; the universal (and the Sato) Grassmannian; the Krichever map; pseudodifferential operators; Lax, Zakharov-Shabat equations and iso-spectral deformations; Backlund-Darboux-Lie transformations; bi-Hamiltonian structures; Faa di Bruno polynomials, recurrence relations and Riccati equations; and so on. On the other hand, stimulated both by physical research and mathematical interest, several attempts have been made to extend the theory to the super-symmetric domain. The first "surprise" in this direction has been the realization that such an extension is by no means unique. Indeed, two nonisomorphic super KP theories have been defined up to now: the MRSKP of Manin and Radul [28] and the JSKP of Mulase and Rabin [33, 37]. While constructing a generalisation of a theory, one has to choose the concepts and features that he considers characterising the theory, and then he has to develop the new theory trying to keep consistence with these. The starting point of Manin and Radul, in this context, was the pseudo-differential operator and Lax equation approach. Studying their theory, M ulase and (independently) Rabin discovered that on the algebraic geometric side that SKP theory is not good. The reason leading to this conclusion is the fact that the flows of MRSKP not only move a point in the super Jacobian of a super curve, but deform the curve itself. In the context of integrable systems this is surely a discouraging property. In fact, a fundamental requirement for such a system, that can not be put apart, is to evolve along straight lines on suitable tori (in this case the super Jacobian). Mulase and Rabin were mainly interested in the algebraic geometry of SKP, e.g. in extending the successful (even if not constructive) characterisation of Jacobians among Abelian varieties to the super-commutative domain, and so on. It is in studying these issues that they finally came up with their JSKP, where now only the super line bundle moves along the flows of the theory. Having at our disposal a certain number of different (but equivalent) approaches to the KP hierarchy, it is thus tempting to follow other ways in defining a super KP theory and look at the corresponding outcomes. The aim of the present investigation is exactly to provide, in the context of super KP hierarchies, some of the techniques that have proved to be successful for the investigation of the KP theory. Therefore, starting from the bi-Hamiltonian formulation of KP [10, 5], formalised by the Faa di Bruno polynomials and recurrence relations, we give a new definition of the JSKP hierarchy and study it by systematically adapting the corresponding methods. This will hopefully shed new light both on JSKP and on KP. We have divided the exposition of the subsequent material in two Parts. In Part I we aim at providing a general but nevertheless accurate view of the theory, so we give the definitions and present different descriptions of the Jacobian super KP hierarchy. Section 2 is motivational in character: we briefly review the KP theory emphasising the role of Faa di Bruno polynomials and the associated conservation equations, which are equivalent to KP. This allows us to introduce the main characters and to offer our definition of the JSKP hierarchy in Section 3. Then in Section 4 we provide the link to the theory of M ulase and Rabin, showing that our JSKP is isomorphic to their hierarchy. Finally, in Section 5 we give a Lax representation by means of two super pseudo-differential operators, while in Section 6 we present a hyper-cohomological interpretation of the theory along the same lines of [15]. Part II is devoted to the detailed study of JSKP, a tool of central importance being the theory of Darboux transformations (Section 7). In Section 8 we explain the connection with the DKP hierarchy introduced in [25] to give a geometric description of the Darboux transformation, while in Section 9 we use that technique to linearise the flows on the super universal Grassmannian. Finally, in Section 10 we move our attention to the problem of reductions, considering two simple examples. For the convenience of the reader we also include an Appendix where useful formulae have been collected. We wish to thank G. Falqui for having proposed to study this interesting subject and for many useful and enlightening discussions, C. Reina for the enjoyable and fruitful work we did together during the staying at SISSA for the Ph.D., and all the people whose help has contributed to the development of the present work.