Conserved charges and supersymmetry in principal chiral and WZW models

TL;DR

This paper explores conserved charges and supersymmetry in classical principal chiral and WZW models, revealing different patterns of conserved quantities in bosonic and supersymmetric cases, with implications for integrability.

Contribution

It introduces a detailed analysis of conserved charges based on invariant tensors in both bosonic and supersymmetric models, highlighting new structures and differences.

Findings

Bosonic models have infinitely many commuting charges based on symmetric tensors.

Supersymmetric models exhibit a different pattern with antisymmetric invariant tensors.

Conserved quantities are related to algebra exponents and Coxeter numbers, with specific differences in bosonic and supersymmetric cases.

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.