A Geometry for multidimensional integrable systems

A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one obtains a geometric description of the operators. A dual theory is also possible, based on a deformation of differential forms. This calculus is applied to a number of multidimensional integrable systems, such as the KP hierarchy, thus obtaining a geometrical description of these systems. The limit in which the deformation disappears corresponds to taking the dispersionless limit in these hierarchies.

Note:

LaTeX, 29 pages. To be published in J.Geom.Phys

differential equations: nonlinear
integrability
hierarchy
differential forms
algebra: deformation

References(41)

Figures(0)

- Commun.Math.Phys. 89 77

Deformation Theory and Quantization. 1. Deformations of Symplectic Structures

F. Bayen
(
- Paris U., VI-VII
)
,
M. Flato
(
- UCLA
)
,
C. Fronsdal
(
- UCLA
)
,
A. Lichnerowicz
(
- College de France
)
,
D. Sternheimer
(
- College de France
)

- Annals Phys. 111 (1978) 61
•
DOI:
- 10.1016/0003-4916(78)90224-5

- Annals Phys. 111 151

Integrable discretizations of chiral models via deformation of the differential calculus

- J.Phys.A 29 (1996) 5007-5018
•
e-Print:
- hep-th/9512007
•
DOI:
- 10.1088/0305-4470/29/16/022

- J.Diff.Geom. 140 213

The Uniqueness of the Moyal algebra

Paul Fletcher
(
- Durham U.
)

- Phys.Lett.B 248 (1990) 323-328
•
DOI:
- 10.1016/0370-2693(90)90300-U

A One parameter family of Hamiltonian structures for the KP hierarchy and a continuous deformation of the nonlinear W(KP) algebra

Jose M. Figueroa-O'Farrill
(
- Bonn U.
)
,
Javier Mas
(
- Santiago de Compostela U.
)
,
Eduardo Ramos
(
- Leuven U.
)

- Commun.Math.Phys. 158 (1993) 17-44
•
e-Print:
- hep-th/9207092
•
DOI:
- 10.1007/BF02097230

- Physica D 90 1

From principal chiral model to selfdual gravity

Jerzy F. Plebanski
(
- CINVESTAV, IPN
)
,
Maciej Przanowski
(
- CINVESTAV, IPN
)
,
H. Garcia-Compean
(
- CINVESTAV, IPN
)

- Mod.Phys.Lett.A 11 (1996) 663-674
•
e-Print:
- hep-th/9509092
•
DOI:
- 10.1142/S0217732396000680

Further remarks on the chiral model approach to selfdual gravity

Hugo Garcia-Compean
(
- CINVESTAV, IPN
)
,
Jerzy F. Plebanski
(
- CINVESTAV, IPN
)
,
Maciej Przanowski
(
- CINVESTAV, IPN
)

- Phys.Lett.A 219 (1996) 249
•
e-Print:
- hep-th/9512013
•
DOI:
- 10.1016/0375-9601(96)00477-X

- Asterisque 123 1

- Lett.Math.Phys. 20 19

Phase space discretization and moyal quantization

e-Print:
- hep-th/9510077

- J.Sov.Math. 11 1

A Connection Between the Einstein and {Yang-Mills} Equations

L.J. Mason
(
- Pittsburgh U.
)
,
E.T. Newman
(
- Pittsburgh U.
)

- Commun.Math.Phys. 121 (1989) 659-668
•
DOI:
- 10.1007/BF01218161

Quantum mechanics as a statistical theory

J.E. Moyal
(
- Queen's U., Belfast
)

- Math.Proc.Cambridge Phil.Soc. 45 (2008) 1, 99-124,
- Proc.Cambridge Phil.Soc. 45 (1949) 99-124,
- In *Li, M. (ed.) et al.: Physics in non-commutative world* 9-33
•
DOI:
- 10.1017/S0305004100000487