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Construct RHF for ATFT using higher Hamiltonian structures

Construct real Hamiltonian forms for affine two-dimensional Toda field theories using higher Hamiltonian structures of the affine Toda field theory hierarchy rather than the canonical Hamiltonian structure.

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Background

The paper develops real Hamiltonian forms (RHF) for affine Toda field theories (ATFT) associated with Dr+1 and demonstrates reductions via external automorphisms to Br and G2 models. The constructions and examples presented focus on the canonical Hamiltonian structure.

The authors highlight that ATFT possess hierarchies of Hamiltonian structures. Extending the RHF construction to these higher (non-canonical) Poisson structures remains unresolved and is posed as an explicit open problem.

References

It is also an open problem to construct the RHF for ATFT using some of its higher Hamiltonian structures.

— On affine Toda field theories related to ${\bf D}_r$ algebras and their real Hamiltonian forms (2401.18027 - Gerdjikov et al., 31 Jan 2024) in Section 5 (Conclusions)