Zero-curvature subconformal structures and dispersionless integrability in dimension five

Abstract

We extend the recent paradigm "Integrability via Geometry" from dimensions 3 and 4 to higher dimensions, relating dispersionless integrability of partial differential equations to curvature constraints of the background geometry. We observe that in higher dimensions on any solution manifold the symbol defines a vector distribution equipped with a subconformal structure, and the integrability imposes a certain compatibility between them. In dimension 5 we express dispersionless integrability via the vanishing of a certain curvature of this subconformal structure. We also obtain a "master equation" governing all second order dispersionless integrable equations in 5D, and count their functional dimension. It turns out that the obtained background geometry is parabolic of the type $(A_3,P_{13})$. We provide its Cartan theoretic description and compute the harmonic curvature components via the Kostant theorem. Then we relate it to 3D projective and 4D conformal geometries via twistor theory, discuss symmetry reductions and nested Lax sequences, as well as give another interpretation of dispersionless integrability in 5D through Levi-degenerate CR structures~in~7D.