Global compatibility of WDVV equations

Establish the compatibility (absence of hidden integrability conditions) of the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) associativity equations S_{αβγν}(F) = η^{λμ}(F_{λαβ}F_{μγν} − F_{λαν}F_{μγβ}) = 0 for a potential F in arbitrary dimension N and for any nondegenerate symmetric metric η; specifically, show that all cross-derivative conditions generated by orthonomic reduction are identically satisfied.

Background

Compatibility is central for ensuring that the reduced first-order Hamiltonian systems derived from WDVV commute. The authors demonstrate passivity (compatibility) for N=4 and N=5 with arbitrary η, and show that this implies commutativity of the associated Hamiltonian flows.

However, beyond these cases, it remains unknown whether the full WDVV system is compatible in general. Resolving this problem would complete the structural program proposed in the paper.