Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integrable systems in 4D associated with sixfolds in Gr(4,6)   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.

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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: econ
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The categories of this paper are: q-fin
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The categories of this paper are: q-alg
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The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat