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Integrable systems in 4D associated with sixfolds in Gr(4,6)
Exactly Solvable and Integrable Systems · Submitted 18 May 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our main result is a complete description of integrable systems $\Sigma(X)$.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $Gr(d,n)$ be the Grassmannian of $d$-dimensional linear subspaces of an $n$-dimensional vector space $V$. | background 1.00 | 0.05 | 0.01 | asserted 0.96 |
| s1 | 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 $X$. | background 0.99 | 0.12 | 0.02 | asserted 1.00 |
| s2 | We investigate a special case of this construction where $X$ is a sixfold in $Gr(4, 6)$. | contribution 0.77 | 0.52 | 0.02 | asserted 0.97 |
| s3 | The corresponding system $\Sigma(X)$ reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. | result 0.55 | 0.70 | 0.04 | asserted 1.00 |
| s4 | Equations of this type arise in self-dual Ricci-flat geometry. | background 0.91 | 0.06 | 0.02 | asserted 1.00 |
| s5 | Our main result is a complete description of integrable systems $\Sigma(X)$. | result 0.62 | 0.94 | 0.94 | asserted 1.00 |
| s6 | These naturally fall into two subclasses. | result 0.93 | 0.74 | 0.05 | asserted 1.00 |
| s7 | (1) Systems of Monge-Amp\`ere type. | result 0.96 | 0.67 | 0.05 | asserted 0.82 |
| s8 | The corresponding sixfolds $X$ are codimension 2 linear sections of the Pl\"ucker embedding $Gr(4,6)\subset\mathbb{P}^{14}$. | result 0.97 | 0.67 | 0.04 | asserted 1.00 |
| s9 | (2) General linearly degenerate systems. | result 0.94 | 0.57 | 0.03 | asserted 0.83 |
| s10 | 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. | result 0.97 | 0.46 | 0.03 | asserted 1.00 |
| s11 | 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. | result 0.98 | 0.93 | 0.38 | asserted 1.00 |
| s12 | In fact, all solutions carry hyper-Hermitian geometry. | result 0.75 | 0.79 | 0.06 | asserted 1.00 |
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The paper
Original 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.
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References
How the literature uses it
8 arXiv papers cite this in 23 sentences
Papers citing it · 18 sentences without a confident label
Cites it as background 3 papers
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We recall that the rYME [ref] was first introduced in [ref] , subsequenty studied e.g. in [ref] , [ref] , [ref] , [ref] , and it also arises in the context of the classification [ref] of the integrable four-dimensional systems associated with sixfolds in Gr(4,6).
A straightforward construction of $\Bbb Z$-graded Lie algebras of full-fledged nonlocal symmetries via recursion operators · Introduction · 1.00
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As it has already been mentioned in the introductory part of this paper, the rYME [ref] was introduced for the first time in [ref] , including its one-parameter isospectral Lax representation [equation] cf. also [ref] .
On recursion operators for full-fledged nonlocal symmetries of the reduced quasi-classical self-dual Yang-Mills equation · Coverings and the recursion operator for full-fledged nonlocal symmetries of the rYME · 0.98
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The problem of classification multidimensional equations is actively studied by many authors, using different algebraic and geometry approaches [ref] .
Lax Pair for a Novel Two-Dimensional Lattice · Conclusion · 1.00
What it cites · 11 sentences without a confident label
Builds on 1 paper
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This example can be viewed as a 4D generalisation of the Bäcklund transformation for the Veronese web equation constructed in [ref] .
Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese webs · Formulation of the problem · 0.91
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