ZERO CURVATURE CONDITION AND 2D GRAVITY THEORIES

1992 ◽  
Vol 07 (15) ◽  
pp. 3447-3472 ◽  
Author(s):  
A. DAS ◽  
W.-J. HUANG ◽  
S. ROY
Keyword(s):  
Current Algebra ◽  
2D Gravity ◽  
Integrable Model ◽  
Integrable Models ◽  
Operator Product ◽  
Curvature Condition ◽  
Kdv Hierarchy ◽  
Zero Curvature ◽  
Anomaly Equation ◽  
Gravity Theories

We propose interpreting the zero curvature condition associated with an integrable model as an anomaly equation. This can lead to the WZWN action and the associated current algebra quite readily and clarifies further the connections found between the integrable models and 2D gravity theories. We analyze, in detail, the cases SL (2, R) (KdV hierarchy), OSp (2/1) (sKdV hierarchy) and SL (3, R) (Boussinesq hierarchy) and obtain the operator product expansions of the appropriate fields. We also make some observations on the generalization of our method to SL (n, R).

ZERO CURVATURE CONDITION OF OSp(2/2) AND THE ASSOCIATED SUPERGRAVITY THEORY

1992 ◽  
Vol 07 (18) ◽  
pp. 4293-4311 ◽  
Author(s):  
ASHOK DAS ◽  
WEN-JUI HUANG ◽  
SHIBAJI ROY
Keyword(s):  
Current Algebra ◽  
Hamiltonian Structure ◽  
Operator Product ◽  
Curvature Condition ◽  
Supergravity Theory ◽  
Kdv Equations ◽  
Zero Curvature ◽  
Anomaly Equation ◽  
Graded Lie Algebra ◽  
The One

The N=2 fermionic extensions of the KdV equations are derived from the zero curvature condition associated with the graded Lie algebra of OSp(2/2). These equations lead to two bi-Hamiltonian systems, one of which is supersymmetric. We also derive the one-parameter family of N=2 supersymmetric KdV equations without a bi-Hamiltonian structure in this approach. Following our earlier proposal, we interpret the zero curvature condition as a gauge anomaly equation which brings out the underlying current algebra for the corresponding 2D supergravity theory. This current algebra is then used to obtain the operator product expansions of various fields of this theory.

scholarly journals LAX-PAIR FORMULATION OF THE W-GRAVITY THEORIES IN TWO DIMENSIONS

1992 ◽  
Vol 07 (27) ◽  
pp. 6907-6932 ◽  
Author(s):  
LINA JEAN-MARC ◽  
PRASANTA K. PANIGRAHI
Keyword(s):  
Light Cone ◽  
2D Gravity ◽  
Lax Pair ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Curvature Condition ◽  
Induced Gravity ◽  
Light Cone Gauge ◽  
Zero Curvature ◽  
Cosmological Constants

The Lax-pair formulation of the two-dimensional induced gravity in the light-cone gauge is extended to the more general wN theories. After presenting the w2 and w3 gravities, we give a general prescription for an arbitrary wN case. This is further illustrated with the w4 gravity to point out some peculiarities. The constraints and the possible presence of the cosmological constants are systematically exhibited in the zero-curvature condition, which also yields the relevant Ward identities. The restrictions on the gauge parameters in the presence of the constraints are pointed out too, and are contrasted with those of the ordinary 2D gravity.

scholarly journals SELECTED TOPICS IN CLASSICAL INTEGRABILITY

2012 ◽  
Vol 27 (05) ◽  
pp. 1230003 ◽  
Author(s):  
ANASTASIA DOIKOU
Keyword(s):  
Integrable Systems ◽  
Toda Chain ◽  
Lax Pair ◽  
Integrable Models ◽  
Integrals Of Motion ◽  
Curvature Condition ◽  
R Matrix ◽  
Zero Curvature ◽  
Algebraic Description ◽  
Classical Integrable

Basic notions regarding classical integrable systems are reviewed. An algebraic description of the classical integrable models together with the zero curvature condition description is presented. The classical r-matrix approach for discrete and continuum classical integrable models is introduced. Using this framework the associated classical integrals of motion and the corresponding Lax pair are extracted based on algebraic considerations. Our attention is restricted to classical discrete and continuum integrable systems with periodic boundary conditions. Typical examples of discrete (Toda chain, discrete NLS model) and continuum integrable models (NLS, sine–Gordon models and affine Toda field theories) are also discussed.

scholarly journals INTEGRABLE MODELS AND THE HIGHER DIMENSIONAL REPRESENTATIONS OF GRADED LIE ALGEBRAS

1998 ◽  
Vol 13 (02) ◽  
pp. 133-144
Author(s):  
J. C. BRUNELLI ◽  
ASHOK DAS
Keyword(s):  
Lie Algebras ◽  
Integrable Models ◽  
Fundamental Representation ◽  
Graded Lie Algebras ◽  
Systematic Method ◽  
Curvature Condition ◽  
Zero Curvature ◽  
Higher Dimensional

We construct a zero curvature formulation, in superspace, for the sTB-B hierarchy which naturally reduces to the zero curvature condition in terms of components, thus solving one of the puzzling features of this model. This analysis, further, suggests a systematic method of constructing higher dimensional representations for the zero curvature condition starting with the fundamental representation. We illustrate this with the examples of the sTB hierarchy and the sKdV hierarchy. This would be particularly useful in constructing explicit higher dimensional representations of graded Lie algebras.

scholarly journals CLASSICAL N=1 and N=2 SUPER W ALGEBRAS FROM A ZERO-CURVATURE CONDITION

1994 ◽  
Vol 09 (03) ◽  
pp. 383-398 ◽  
Author(s):  
FRANÇOIS GIERES ◽  
STEFAN THEISEN
Keyword(s):  
Superconformal Algebra ◽  
Order System ◽  
Curvature Condition ◽  
First Order ◽  
Zero Curvature

Starting from superdifferential operators in an N=1 superfield formulation, we present a systematic prescription for the derivation of classical N=1 and N=2 super W algebras by imposing a zero-curvature condition on the connection of the corresponding first-order system. We illustrate the procedure on the first nontrivial example (beyond the N=1 superconformal algebra) and also comment on the relation with the Gelfand-Dickey construction of W algebras.

Noncommutative negative order AKNS equation and its soliton solutions

2018 ◽  
Vol 33 (35) ◽  
pp. 1850209 ◽  
Author(s):  
H. Wajahat A. Riaz ◽  
Mahmood ul Hassan
Keyword(s):  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Curvature Condition ◽  
Integrable Structure ◽  
Zero Curvature ◽  
Negative Order ◽  
Akns Equation

A noncommutative negative order AKNS (NC-AKNS(-1)) equation is studied. To show the integrability of the system, we present explicitly the underlying integrable structure such as Lax pair, zero-curvature condition, an infinite sequence of conserved densities, Darboux transformation (DT) and quasideterminant soliton solutions. Moreover, the NC-AKNS(-1) equation is compared with its commutative counterpart not only on the level of nonlinear evolution equation but also for the explicit solutions.

scholarly journals Zero Curvature Formalism for Supersymmetric Integrable Hierarchies in Superspace

1997 ◽  
Vol 12 (34) ◽  
pp. 2623-2630 ◽  
Author(s):  
H. Aratyn ◽  
C. Rasinariu ◽  
A. Das
Keyword(s):  
Symmetric Space ◽  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Space Structure ◽  
Hermitian Symmetric Space ◽  
Curvature Condition ◽  
Lax Equation ◽  
Abelian Gauge ◽  
Graded Algebras ◽  
Zero Curvature

We generalize the Drinfeld–Sokolov formalism of bosonic integrable hierarchies to superspace, in a way which systematically leads to the zero curvature formulation for the supersymmetric integrable systems starting from the Lax equation in superspace. We use the method of symmetric space as well as the non-Abelian gauge technique to obtain the supersymmetric integrable hierarchies of the AKNS type from the zero curvature condition in superspace with the graded algebras, sl (n+1,n), providing the Hermitian symmetric space structure.

2D SUPERGRAVITY WITH APPLICATIONS TO INTEGRABLE MODELS

1993 ◽  
Vol 08 (31) ◽  
pp. 2943-2954
Author(s):  
L. N. ARNAUDOV ◽  
E. M. PRODANOV ◽  
R. CH. RASHKOV
Keyword(s):  
Recent Work ◽  
Ward Identity ◽  
Degrees Of Freedom ◽  
Hamiltonian Reduction ◽  
Simons Theory ◽  
Curvature Condition ◽  
Zero Curvature ◽  
Covariant Action ◽  
Chern Simons ◽  
Chern Simons Theory

In recent work two different approaches for obtaining the covariant action of 2D quantum supergravity are developed. The first one is based on Hamiltonian reduction of flat OSP (2|1) connection in holomorphic polarization. Adding extra degrees of freedom with the help of gauging procedure, the action and the superconformal Ward identity are obtained. It is shown that super-Virasoro transformations preserve the form of the Lax connection and therefore are symmetries of the sKdV equations. In the second approach starting with Chern-Simons theory and using non-canonical polarization the zero-curvature condition entails the same results.

scholarly journals Two Expanding Integrable Models of the Geng-Cao Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Xiurong Guo ◽  
Yufeng Zhang ◽  
Xuping Zhang
Keyword(s):  
Heat Equation ◽  
Lie Algebras ◽  
Burgers Equation ◽  
Integrable Model ◽  
Integrable Models ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Generalized Burgers Equation ◽  
Integrable Couplings ◽  
Integrable Coupling

As far as linear integrable couplings are concerned, one has obtained some rich and interesting results. In the paper, we will deduce two kinds of expanding integrable models of the Geng-Cao (GC) hierarchy by constructing different 6-dimensional Lie algebras. One expanding integrable model (actually, it is a nonlinear integrable coupling) reduces to a generalized Burgers equation and further reduces to the heat equation whose expanding nonlinear integrable model is generated. Another one is an expanding integrable model which is different from the first one. Finally, the Hamiltonian structures of the two expanding integrable models are obtained by employing the variational identity and the trace identity, respectively.

scholarly journals NEW SPECIAL OPERATORS IN W-GRAVITY THEORIES

1991 ◽  
Vol 06 (38) ◽  
pp. 3531-3541 ◽  
Author(s):  
S. KALYANA RAMA
Keyword(s):  
Ising Model ◽  
Minimal Model ◽  
2D Gravity ◽  
Selection Rule ◽  
Energy Operator ◽  
Two Dimensional ◽  
Gravity Theories ◽  
The Way

We find new special physical operators of W3-gravity having non-trivial ghost sectors. Some of these operators may be viewed as the Liouville dressings of the energy operator of the Ising model coupled to two-dimensional (2D) gravity and this fills in the gap in the connection between pure W3-gravity and Ising model coupled to 2D gravity found in our previous work. We formulate a selection rule required for the calculation of correlators in W-gravity theories. Using this rule, we construct the non-ghost part of the new operators of WN-gravity and find that they represent the (N, N + 1) minimal model operators from both inside and outside the minimal table. Along the way we obtain the canonical spectrum of WN-gravity for all N.

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