Integrability vs Supersymmetry: Poisson Structures of The Pohlmeyer Reduction

Understand

We construct recursively an infinite number of Poisson structures for the supersymmetric integrable hierarchy governing the Pohlmeyer reduction of superstring sigma models on the target spaces AdS_{n}\times S^n, n=2,3,5.

• These Poisson structures are all non-local and not relativistic except one, which is the canonical Poisson structure of the semi-symmetric space sine-Gordon model (SSSSG).
• We verify that the superposition of the first three Poisson structures corresponds to the canonical Poisson structure of the reduced sigma model.
• Using the recursion relations we construct commuting charges on the reduced sigma model out of those of the SSSSG model and in the process we explain the integrable origin of the Zukhovsky map and the twisted inner product used in the sigma model side.