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Ferapontov-type operator conjecture for first-order WDVV systems

Prove that every first-order WDVV system admits a first-order Ferapontov-type Hamiltonian operator compatible with the third-order homogeneous Dubrovin–Novikov operator identified in this paper.

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Background

Beyond the established third-order Hamiltonian structure, the authors conjecture the existence of a compatible first-order operator of Ferapontov type for all first-order WDVV systems, extending computations verified in specific low-dimensional cases.

A proof would confirm a general bi-Hamiltonian framework for WDVV in arbitrary dimension and for arbitrary nondegenerate η, aligning with the projective-geometric picture developed in the paper.

References

It is natural to conjecture that all first-order WDVV systems admit a first-order operator of Ferapontov type that is compatible with the third-order operator found in this paper for all such systems.

— On the geometry of WDVV equations and their Hamiltonian formalism in arbitrary dimension (2509.13757 - Opanasenko et al., 17 Sep 2025) in Main results, paragraph “Hamiltonian formalism”