Bidifferential calculi and integrable models

The existence of an infinite set of conserved currents in completely integrable classical models, including chiral and Toda models as well as the KP and self-dual Yang-Mills equations, is traced back to a simple construction of an infinite chain of closed (respectively, covariantly constant) 1-forms in a (gauged) bi-differential calculus. The latter consists of a differential algebra on which two differential maps act. In a gauged bi-differential calculus these maps are extended to flat covariant derivatives.

integrability
current: conservation law
differential forms
model: chiral
field theory: Toda
Kadomtsev-Petviashvili equation
sine-Gordon equation
field equations: Liouville
dimension: 2
gauge field theory: Yang-Mills

References(30)

Figures(0)

Scattering of Massless Lumps and Nonlocal Charges in the Two-Dimensional Classical Nonlinear Sigma Model

M. Luscher
(
- Hamburg U.
)
,
K. Pohlmeyer
(
- Heidelberg U.
)

- Nucl.Phys.B 137 (1978) 46-54
•
DOI:
- 10.1016/0550-3213(78)90049-4

Remarks About the Existence of Nonlocal Charges in Two-Dimensional Models

- Phys.Lett.B 82 (1979) 442-444
•
DOI:
- 10.1016/0370-2693(79)90263-6

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

Noncommutative geometry and integrable models

- Lett.Math.Phys. 39 (1997) 69-79
•
e-Print:
- hep-th/9601024
•
DOI:
- 10.1023/A:1007376001478

Noncommutative geometry and a class of completely integrable models

- Czech.J.Phys. 48 (1998) 1319-1324
•
e-Print:
- math-ph/9809023
•
DOI:
- 10.1023/A:1021632517435

- Phys.Rev.B 9 1921

Local cohomology in field theory (with applications to the Einstein equations)

C.G. Torre
(
- Utah State U.
)

e-Print:
- hep-th/9706092

Some solutions of complex Einstein equations

J.F. Plebanski
(
- CINVESTAV, IPN
)

- J.Math.Phys. 16 (1975) 2395-2402
•
DOI:
- 10.1063/1.522505

An Infinite Number of Hidden Variables in Hyperkahler Metrics

Kanehisa Takasaki
(
- Kyoto U., RIMS
)

- J.Math.Phys. 30 (1989) 1515
•
DOI:
- 10.1063/1.528283

Selfdual gravity as a two-dimensional theory and conservation laws

Viqar Husain
(
- Alberta U.
)

- Class.Quant.Grav. 11 (1994) 927-938
•
e-Print:
- gr-qc/9310003
•
DOI:
- 10.1088/0264-9381/11/4/011

The Affine symmetry of selfdual gravity

Viqar Husain
(
- Alberta U.
)

- J.Math.Phys. 36 (1995) 6897-6906,
- J. Math. Phys. 36 (1995) 6897-6906, and Edmonton Alberta U. - Thy-94-26 (94,rec.Oct.) 17 p
•
e-Print:
- hep-th/9410072
•
DOI:
- 10.1063/1.531197

The Symmetry structure of the antiselfdual Einstein hierarchy

I.A.B. Strachan
(
- Newcastle upon Tyne U.
)

- J.Math.Phys. 36 (1995) 3566-3573
•
e-Print:
- hep-th/9410047
•
DOI:
- 10.1063/1.530982

Hypercomplex Integrable Systems

J.D.E. Grant
(
- Hull U.
)
,
I.A.B. Strachan
(
- Hull U.
)

- Nonlinearity 12 (1999) 5, 1247
•
e-Print:
- solv-int/9808019
•
DOI:
- 10.1088/0951-7715/12/5/302

Currents, charges, and canonical structure of pseudodual chiral models

Thomas Curtright
(
- Miami U.
)
,
Cosmas K. Zachos
(
- Argonne
)

- Phys.Rev.D 49 (1994) 5408-5421
•
e-Print:
- hep-th/9401006
•
DOI:
- 10.1103/PhysRevD.49.5408

Relativistically Invariant Two-Dimensional Models in Field Theory Integrable by the Inverse Problem Technique. (In Russian)

V.E. Zakharov
(
- Landau Inst.
)
,
A.V. Mikhailov
(
- Landau Inst.
)

- Sov.Phys.JETP 47 (1978) 1017-1027,
- Zh.Eksp.Teor.Fiz. 74 (1978) 1953-1973

Some Properties of an Analog of the Nonlinear $\sigma$ Model

Chiara R. Nappi
(
- Harvard U.
)

- Phys.Rev.D 21 (1980) 418
•
DOI:
- 10.1103/PhysRevD.21.418

Noncommutative differential calculus and lattice gauge theory

A. Dimakis
(
- Gottingen U.
)
,
Folkert Mueller-Hoissen
(
- Gottingen U.
)
,
T. Striker
(
- Gottingen U.
)

- J.Phys.A 26 (1993) 1927-1949,
- J. Phys. A26 (1993) 1927-1949 and Goettingen Univ. - GOET-TP 92-44 (92/07,rec.Jul.) 24 p

Quantum mechanics on a lattice and q deformations

A. Dimakis
(
- Gottingen U.
)
,
Folkert Mueller-Hoissen
(
- Gottingen U.
)

- Phys.Lett.B 295 (1992) 242-248
•
DOI:
- 10.1016/0370-2693(92)91561-M

- Prog.Theor.Phys.Suppl. 94 42

- Lett.Math.Phys. 46 189

- J.Math.Phys. 31 1426

The Soliton: A New concept in applied science

Alwyn C. Scott
(
- Wisconsin U., Madison
)
,
f.Y.F. Chu
(
- Wisconsin U., Madison
)
,
David W. McLaughlin
(
- Iowa State U.
)

- IEEE Proc. 61 (1973) 1443-1483
•
DOI:
- 10.1109/PROC.1973.9296

Integrable Hamiltonian Systems and Interactions Through Quadratic Constraints

K. Pohlmeyer
(
- Hamburg U.
)

- Commun.Math.Phys. 46 (1976) 207-221,
- Commun.Math.Phys.46 (1976) 207-221 and Hamburg Desy - DESY 75-28 (75,REC.AUG) 29p
•
DOI:
- 10.1007/BF01609119

- Nucl.Phys.A 236 381

Condition of Selfduality for SU(2) Gauge Fields on Euclidean Four-Dimensional Space

Chen Ning Yang
(
- SUNY, Stony Brook
)

- Phys.Rev.Lett. 38 (1977) 1377
•
DOI:
- 10.1103/PhysRevLett.38.1377

1-25 of 30
1
2
25 / page