Commutativity and orthonomic reducibility in all dimensions

Establish that for any dimension N and any nondegenerate symmetric metric η, the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations can be reduced to passive orthonomic form and, consequently, that the associated first-order Hamiltonian systems of conservation laws commute.

Background

The authors prove in low dimensions (N=4 and N=5) that choosing a distinguished independent variable allows them to rewrite parts of the WDVV equations as Hamiltonian first-order systems of conservation laws which commute, and that the reduced systems are in passive orthonomic form, implying compatibility. They then posit that this structural picture should extend to arbitrary dimension.

This conjecture connects the geometric interpretation of WDVV via line congruences and Monge metrics with a uniform Hamiltonian formalism in all dimensions, and aims to generalize the low-dimensional reducibility and commutativity results established in the paper.