Dynamical conservation laws in supersymmetric theories of symmetric-space valued fields

Abstract

We define supersymmetric extensions of non-linear σ-models with fields taking values in riemannian symmetric spaces, and develop the associated classical theory of dynamical symmetry. A central role is played by a pair of dual spinorial superfields, in close analogy with the structure of dual currents identified in the purely bosonic case by Eichenherr and Forger. When the field variations vanish sufficiently rapidly at spatial infinity, there are infinitely many conserved non-local charges (non fermionic) over and above anything implied directly by translation, Lorentz, conformal, supersymmetry or isospin invariance. The charges in general conform to the pattern conjectured by Curtright and Zachos. Each of these charges is invariant under simple supersymmetry. When the underlying symmetric space is a complex manifold, the invariance group of such a model enlarges to include chiral and O(2) supersymmetry transformations; the non-local charges are also unchanged by these. The relations between this dynamical symmetry and instantons (when they exist), and a limit in which these models become purely fermionic while retaining their dynamical symmetry, are also described. The ordinary and chiral Gross-Neveu models are examples of this limit.