Integrable systems in 4D associated with sixfolds in Gr(4,6)

Abstract

Let $Gr(d,n)$ be the Grassmannian of $d$-dimensional linear subspaces of an $n$-dimensional vector space $V$. A submanifold $X\subset Gr(d, n)$ gives rise to a differential system $\Sigma(X)$ that governs $d$-dimensional submanifolds of $V$ whose Gaussian image is contained in $X$. We investigate a special case of this construction where $X$ is a sixfold in $Gr(4, 6)$. The corresponding system $\Sigma(X)$ reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. Equations of this type arise in self-dual Ricci-flat geometry. Our main result is a complete description of integrable systems $\Sigma(X)$. These naturally fall into two subclasses. (1) Systems of Monge-Amp\`ere type. The corresponding sixfolds $X$ are codimension 2 linear sections of the Pl\"ucker embedding $Gr(4,6)\subset\mathbb{P}^{14}$. (2) General linearly degenerate systems. The corresponding sixfolds $X$ are the images of quadratic maps $\mathbb{P}^6- Gr(4, 6)$ given by a version of the classical construction of Chasles. We prove that integrability is equivalent to the requirement that the characteristic variety of system $\Sigma(X)$ gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.