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arXiv · 2509.13757

On the geometry of WDVV equations and their Hamiltonian formalism in arbitrary dimension

S. Opanasenko·R. Vitolo
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Abstract

It is known that in low dimensions WDVV equations can be rewritten as commuting quasilinear bi-Hamiltonian systems. We extend some of these results to arbitrary dimension $N$ and arbitrary scalar product $η$. In particular, we show that suitable subsets of WDVV equations can be interpreted as a set of linear line congruences in suitable Plücker embeddings. This form leads to their representation as $N-2$ Hamiltonian systems of conservation laws. Moreover, we show that WDVV equations can be reduced to an orthonomic form, which is also passive in low dimensions $N\leq 5$. This also leads to the commutativity of the Hamiltonian systems of conservation laws ($N\leq 5$), after which we can find a solution of the WDVV equations from a joint solution of the Hamiltonian systems. Finally, we conjecture that passivity holds in all dimensions.

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S. Opanasenko, R. Vitolo. 2026-09-14. On the geometry of WDVV equations and their Hamiltonian formalism in arbitrary dimension. https://arxiv.org/abs/2509.13757

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