Supersymmetry of Affine Toda Models as Fermionic Symmetry Flows of the Extended mKdV Hierarchy

TL;DR

This paper develops a framework linking supersymmetric affine Toda models with fermionic symmetry flows of the extended mKdV hierarchy, revealing new supercharges and applying the theory to specific sinh-Gordon models.

Contribution

It introduces a novel algebraic dressing approach to connect supersymmetric Toda models with fermionic symmetry flows, expanding understanding of their symmetries and supercharges.

Findings

Derived supersymmetry transformations as fermionic symmetry flows.

Established a generalized set of relativistic-like fermionic current identities.

Applied the framework to detailed N=(1,1) and N=(2,2) sinh-Gordon models.

Abstract

We couple two copies of the supersymmetric mKdV hierarchy by means of the algebraic dressing technique. This allows to deduce the whole set of $(N,N)$ supersymmetry transformations of the relativistic sector of the extended mKdV hierarchy and to interpret them as fermionic symmetry flows. The construction is based on an extended Riemann-Hilbert problem for affine Kac-Moody superalgebras with a half-integer gradation. A generalized set of relativistic-like fermionic local current identities is introduced and it is shown that the simplest one, corresponding to the lowest isospectral times $t_{\pm 1}$ provides the supercharges generating rigid supersymmetry transformations in 2D superspace. The number of supercharges is equal to the dimension of the fermionic kernel of a given semisimple element $E \in \widehat{\mathfrak{g}}$ which defines both, the physical degrees of freedom and the symmetries of the model. The general construction is applied to the $N=(1,1)$ and $N=(2,2)$ sinh-Gordon models which are worked out in detail.