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 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 $N\le 5$. This also leads to the commutativity of the Hamiltonian systems of conservation laws ($N\le 5$), 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.

Accepted for publication in the Journal of the London Mathematical Society