Supersymmetric Integrable Hierarchies and String Theory

TL;DR

This thesis develops the supersymmetric Lax formalism, introduces new supersymmetric hierarchies, and explores their algebraic structures and connections to superstring theory, advancing the understanding of integrable systems in supersymmetric contexts.

Contribution

It presents the first comprehensive development of supersymmetric Lax formalism, introduces three supersymmetric KP hierarchies, and links an integrable hierarchy to noncritical superstring theory.

Findings

Supersymmetric extensions of the KP hierarchy are constructed and analyzed.

The algebra of additional symmetries is shown to be isomorphic to superdifferential operators.

The sKdV-B hierarchy is identified, proven bihamiltonian integrable, and extended by odd flows.

Abstract

This thesis is roughly organized into two parts. The first one (the first three chapters), expository in nature, attempts to place the current work in context: at first historically, but then focusing on the Lax formalism and the Adler--Gel'fand--Dickey scheme for hierarchies of the KdV type. The second part (the last four chapters) comprises the main body of this work. It begins by developing the supersymmetric Lax formalism, introducing the ring of formal superpseudodifferential operators and the associated Poisson structures. We discuss three supersymmetric extensions of the KP hierarchy (MRSKP, \SKP2, and JSKP). We define and compute their additional symmetries and we find that the algebra of additional symmetries are in all three cases isomorphic to the Lie algebra of superdifferential operators. We discuss a new reduction of \SKP2 and the relation between MRSKP and \SKP2 is clarified. Finally we consider the (so far) only integrable hierarchy to have played a role in noncritical superstring theory (sKdV-B). We identify it, prove its bihamiltonian integrability, and extend it by odd flows. We close with a discussion of new integrable supersymmetrizations of the KdV-like hierarchies suggested by the study of sKdV-B. (This is the author's PhD Thesis from the Physics Deparment of the University of Bonn, July 1994.)