Conserved charges and supersymmetry in principal chiral and WZW models

Abstract

Conserved and commuting charges are investigated in both bosonic and supersymmetric classical chiral models, with and without Wess-Zumino terms. In the bosonic theories, there are conserved currents based on symmetric invariant tensors of the underlying algebra, and the construction of infinitely many commuting charges, with spins equal to the exponents of the algebra modulo its Coxeter number, can be carried out irrespective of the coefficient of the Wess-Zumino term. In the supersymmetric models, a different pattern of conserved quantities emerges, based on antisymmetric invariant tensors. The current algebra is much more complicated than in the bosonic case, and it is analysed in some detail. Two families of commuting charges can be constructed, each with finitely many members whose spins are exactly the exponents of the algebra (with no repetition modulo the Coxeter number). The conserved quantities in the bosonic and supersymmetric theories are only indirectly related, except for the special case of the WZW model and its supersymmetric extension.

Pseudoduality in sigma models

“…For a recent discussion and references to the older literature on local and non-local conservation laws for sigma models based on groups see [27,28]. The connection between the method described here and other methods is not clear.…”

Integrable Hierarchies and Modern Physical Theories

“…These findings were subsequently extended to incorporate the effects of WZ terms and supersymmetry [2] and analogous results were also established for sigma-models whose target manifolds are symmetric spaces G/H [3], but again only for G and H classical groups. The general case, including exceptional groups, has been treated quite recently [4] by making use of a direct link between the k-tensors and the Drinfeld-Sokolov modified KdV (DS/mKdV) hierarchies [6].…”

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This is a summary of progress made [1][2][3][4] in understanding the occurrence and properties of local, conserved, commuting charges in non-linear sigma-models, including principal chiral models (PCMs) and WZW models. Initial investigations [1] focussed on PCMs based on classical groups G and established that currents and symmetric tensors with the required properties could be defined by a formula

where the generators t a belong to the defining representation of g. Commutation of the resulting charges depends upon some intricate algebraic identities satisfied by the k-tensors which we shall refer to as the commutation conditions.

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“…This is a summary of progress made [1][2][3][4] in understanding the occurrence and properties of local, conserved, commuting charges in non-linear sigma-models, including principal chiral models (PCMs) and WZW models. Initial investigations [1] focussed on PCMs based on classical groups G and established that currents and symmetric tensors with the required properties could be defined by a formula…”

Zero-curvature formalism of supersymmetric principal chiral model

“…A fermionic and bosonic Lax type representation is obtained which is associated with the monodromy operator of SPCM. The linear system in superspace is then used to generate bosonic non-local superfield currents of SPCM, which, when expressed in terms of components, coincide with those obtained in [19,20,26].…”