Integrable Structure Behind the WDVV Equations

An integrable structure behind the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations is identified with the reduction of the Riemann–Hilbert problem for the homogeneous loop group $\widehat{GL}(N,\mathbb C)$. The reduction requires the dressing matrices to be fixed points of an order-two loop group automorphism resulting in a subhierarchy of the $\widehat{gl}(N,\mathbb C)$ hierarchy containing only odd-symmetry flows. The model has Virasoro symmetry; imposing Virasoro constraints ensures the homogeneity property of the Darboux–Egoroff structure. Dressing matrices of the reduced model provide solutions of the WDVV equations.