18 research outputs found
Hyper-complex four-manifolds from the Tzitz\'eica equation
It is shown how solutions to the Tzitz\'eica equation can be used to
construct a family of (pseudo) hyper-complex metrics in four dimensions.Comment: To be published in J.Math.Phy
Strominger--Yau--Zaslow geometry, Affine Spheres and Painlev\'e III
We give a gauge invariant characterisation of the elliptic affine sphere
equation and the closely related Tzitz\'eica equation as reductions of real
forms of SL(3, \C) anti--self--dual Yang--Mills equations by two
translations, or equivalently as a special case of the Hitchin equation.
We use the Loftin--Yau--Zaslow construction to give an explicit expression
for a six--real dimensional semi--flat Calabi--Yau metric in terms of a
solution to the affine-sphere equation and show how a subclass of such metrics
arises from 3rd Painlev\'e transcendents.Comment: 38 pages. Final version. To appear in Communications in Mathematical
Physic
Geometric discretization of the Koenigs nets
We introduce the Koenigs lattice, which is a new integrable reduction of the
quadrilateral lattice (discrete conjugate net) and provides natural integrable
discrete analogue of the Koenigs net. We construct the Darboux-type
transformations of the Koenigs lattice and we show permutability of
superpositions of such transformations, thus proving integrability of the
Koenigs lattice. We also investigate the geometry of the discrete Koenigs
transformation. In particular we characterize the Koenigs transformation in
terms of an involution determined by a congruence conjugate to the lattice.Comment: 17 pages, 2 figures; some spelling and typing errors correcte
Tzitz\'eica transformation is a dressing action
We classify the simplest rational elements in a twisted loop group, and prove
that dressing actions of them on proper indefinite affine spheres give the
classical Tzitz\'eica transformation and its dual. We also give the group point
of view of the Permutability Theorem, construct complex Tzitz\'eica
transformations, and discuss the group structure for these transformations
Desargues maps and the Hirota-Miwa equation
We study the Desargues maps \phi:\ZZ^N\to\PP^M, which generate lattices
whose points are collinear with all their nearest (in positive directions)
neighbours. The multidimensional compatibility of the map is equivalent to the
Desargues theorem and its higher-dimensional generalizations. The nonlinear
counterpart of the map is the non-commutative (in general) Hirota--Miwa system.
In the commutative case of the complex field we apply the nonlocal
-dressing method to construct Desargues maps and the
corresponding solutions of the equation. In particular, we identify the
Fredholm determinant of the integral equation inverting the nonlocal
-dressing problem with the -function. Finally, we establish
equivalence between the Desargues maps and quadrilateral lattices provided we
take into consideration also their Laplace transforms.Comment: 17 pages, 5 figures; v2 - presentation improve
Darboux transformations for linear operators on two dimensional regular lattices
Darboux transformations for linear operators on regular two dimensional
lattices are reviewed. The six point scheme is considered as the master linear
problem, whose various specifications, reductions, and their sublattice
combinations lead to other linear operators together with the corresponding
Darboux transformations. The second part of the review deals with
multidimensional aspects of (basic reductions of) the four point scheme, as
well as the three point scheme.Comment: 23 pages, 3 figures, presentation improve