Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Projective differential geometry of higher reductions of the
  two-dimensional Dirac equation   We investigate reductions of the two-dimensional Dirac equation imposed by
the requirement of the existence of a differential operator $D_n$ of order $n$
mapping its eigenfunctions to adjoint eigenfunctions. For first order operators
these reductions (and multi-component analogs thereof) lead to the Lame
equations descriptive of orthogonal coordinate systems. Our main observation is
that $n$-th order reductions coincide with the projective-geometric
`Gauss-Codazzi' equations governing special classes of line congruences in the
projective space $P^{2n-1}$, which is the projectivised kernel of $D_n$. In the
second order case this leads to the theory of $W$-congruences in $P^3$ which
belong to a linear complex, while the third order case corresponds to isotropic
congruences in $P^5$. Higher reductions are compatible with odd-order flows of
the Davey-Stewartson hierarchy. All these flows preserve the kernel $D_n$, thus
defining nontrivial geometric evolutions of line congruences. Multi-component
generalizations are also discussed. The correspondence between geometric
picture and the theory of integrable systems is established; the definition of
the class of reductions and all geometric objects in terms of the
multicomponent KP hierarchy is presented. Generating forms for reductions of
arbitrary order are constructed.

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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: eess
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: econ
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The categories of this paper are: hep-th
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: cs