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

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 $\eta$. In particular, we show that WDVV equations can be interpreted as a set of linear line congruences in suitable Pl\"ucker embeddings. This form leads to their representation as Hamiltonian systems of conservation laws. Moreover, we show that in low dimensions and for an arbitrary $\eta$ WDVV equations can be reduced to passive orthonomic form. This leads to the commutativity of the Hamiltonian systems of conservation laws. We conjecture that such a result holds in all dimensions.