dorsal/arxiv
View SchemaIntegrable Discrete Geometry: the Quadrilateral Lattice, its Transformations and Reductions
| Authors | Adam Doliwa, Paolo Maria Santini |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9907014 |
| URL | https://arxiv.org/abs/solv-int/9907014 |
Abstract
We review recent results on Integrable Discrete Geometry. It turns out that most of the known (continuous and/or discrete) integrable systems are particular symmetries of the quadrilateral lattice, a multidimensional lattice characterized by the planarity of its elementary quadrilaterals. Therefore the linear property of planarity seems to be a basic geometric property underlying integrability. We present the geometric meaning of its tau-function, as the potential connecting its forward and backward data. We present the theory of transformations of the quadrilateral lattice, which is based on the discrete analogue of the theory of rectilinear congruences. In particular, we discuss the discrete analogues of the Laplace, Combescure, Levy, radial and fundamental transformations and their interrelations. We also show how the sequence of Laplace transformations of a quadrilateral surface is described by the discrete Toda system. We finally show that these classical transformations are strictly related to the basic operators associated with the quantum field theoretical formulation of the multicomponent Kadomtsev-Petviashvilii hierarchy. We review the properties of quadrilateral hyperplane lattices, which play an interesting role in the reduction theory, when the introduction of additional geometric structures allows to establish a connection between point and hyperplane lattices. We present and fully characterize some geometrically distinguished reductions of the quadrilateral lattice, like the symmetric, circular and Egorov lattices; we review also basic geometric results of the theory of quadrilateral lattices in quadrics, and the corresponding analogue of the Ribaucour reduction of the fundamental transformation.
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"abstract": "We review recent results on Integrable Discrete Geometry. It turns out that\nmost of the known (continuous and/or discrete) integrable systems are\nparticular symmetries of the quadrilateral lattice, a multidimensional lattice\ncharacterized by the planarity of its elementary quadrilaterals. Therefore the\nlinear property of planarity seems to be a basic geometric property underlying\nintegrability. We present the geometric meaning of its tau-function, as the\npotential connecting its forward and backward data. We present the theory of\ntransformations of the quadrilateral lattice, which is based on the discrete\nanalogue of the theory of rectilinear congruences. In particular, we discuss\nthe discrete analogues of the Laplace, Combescure, Levy, radial and fundamental\ntransformations and their interrelations. We also show how the sequence of\nLaplace transformations of a quadrilateral surface is described by the discrete\nToda system. We finally show that these classical transformations are strictly\nrelated to the basic operators associated with the quantum field theoretical\nformulation of the multicomponent Kadomtsev-Petviashvilii hierarchy. We review\nthe properties of quadrilateral hyperplane lattices, which play an interesting\nrole in the reduction theory, when the introduction of additional geometric\nstructures allows to establish a connection between point and hyperplane\nlattices. We present and fully characterize some geometrically distinguished\nreductions of the quadrilateral lattice, like the symmetric, circular and\nEgorov lattices; we review also basic geometric results of the theory of\nquadrilateral lattices in quadrics, and the corresponding analogue of the\nRibaucour reduction of the fundamental transformation.",
"arxiv_id": "solv-int/9907014",
"authors": [
"Adam Doliwa",
"Paolo Maria Santini"
],
"categories": [
"solv-int",
"hep-lat",
"nlin.SI"
],
"title": "Integrable Discrete Geometry: the Quadrilateral Lattice, its Transformations and Reductions",
"url": "https://arxiv.org/abs/solv-int/9907014"
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