Feb. 1994

KHTP-94-01/ SNUCTP 94-09

Deformed Coset Models From Gauged WZW Actions

[.1in]
Q-Han Park^{1}^{1}1 E-mail address;
Department of Physics, Kyunghee University

Seoul, 130-701, Korea

[.7in] ABSTRACT

[.2in]

A general Lagrangian formulation of integrably deformed G/H-coset models is given. We consider the G/H-coset model in terms of the gauged Wess-Zumino-Witten action and obtain an integrable deformation by adding a potential energy term , where algebra elements belong to the center of the algebra h associated with the subgroup H. We show that the classical equation of motion of the deformed coset model can be identified with the integrability condition of certain linear equations which makes the use of the inverse scattering method possible. Using the linear equation, we give a systematic way to construct infinitely many conserved currents as well as soliton solutions. In the case of the parafermionic SU(2)/U(1)-coset model, we derive -solitons and conserved currents explicitly.

Integrable field theories in two dimensions (2-d IFT) have been successfully developed in the past decades. Recently, Zamolodchikov has shown that there exist deformations of conformal field theories (2-d CFT) with relevant operators which preserve integrability[1]. In the case of certain rational conformal field theories, these deformations have been explained in terms of the affine extension of the Toda field theory[2][3]. However, a general Lagrangian framework for integrably deformed 2-d CFT’s has not been known so far.

The purpose of this letter is to provide a Lagrangian formulation of general G/H-coset models and their integrable deformations in the context of the gauged Wess-Zumino-Witten (WZW) model[4][5]. We show that an integrable deformation of general G/H-coset models is possible when the gauged WZW action for the G/H-coset model is added by a potential energy term , where algebra elements belong to the center of the algebra h associated with the subgroup H. In the case of SU(2)/U(1), this reduces to the Lagrangian, given recently by Bakas[6], of the parafermion model deformed by the energy operator . The main observation of this work is that the classical equation of motion of the deformed coset model takes the form of a zero curvature which can be identified with the integrability condition of the associated linear equations with a spectral parameter. This allows us to apply the inverse scattering method to the problem and using this method, we give a systematic way to construct infinitely many conserved currents and -soliton solutions. As an example, we construct explicitly conserved currents and -soliton solutions of the deformed parafermionic SU(2)/U(1)-coset model.

We first recall that a Lagrangian of the G/H-coset model is given in terms of the gauged WZW functional[4][5], which in light-cone variables is

(1) |

where is the usual WZW action [5] for a map G on two-dimensional Minkowski space . The connection gauge the anomaly free subgroup H of G. In this letter, we take the diagonal embedding of H in , where and denote left and right group actions by multiplication , so that Eq.(1) becomes invariant under the vector gauge transformation with H). The key observation to be made is that the equation of motion of the gauged WZW action takes the form of a zero curvature,

(2) |

which, together with the constraint equation

(3) |

describes the coset model at the classical level.

Now, we consider a deformation of the G/H-coset model by adding a primary field to the gauged WZW action,

(4) |

where is a coupling constant and are elements of the Lie algebra associated with the Lie group G. In the following, we assume that belong to the center of the algebra h so that , . In such a case, the equation of motion once again takes the form of a zero cuvature but with a spectral parameter

(5) |

Since the term is invariant under the vector gauge transformation, so is the action and the constraint equation remains unchanged which we solve explicitly for and ,

(6) |

where and are generators of normalized by . By taking the trace of Eq.(5) multiplied with any , we may easily see that and also satisfy the zero curvature condition, i.e. which reflects the vector gauge invariance of the action. In the following, we solve the zero curvature condition by and for some and rewrite and in terms of whenever necessary. Note that Eq.(5), with given as in Eq.(6), is precisely the integrability condition of a couple of linear equations,

(7) |

This linear equation and also the integrability condition generalize those of the affine Toda equation which can be solved by applying the inverse scattering method.[7] For example, consider a reduction of the GL(N,R)/U(1)-coset model by setting and where if mod(N) and zero otherwise. If we fix the gauge by , then the integrability condition becomes precisely the affine SL(N,R)-Toda equation. Therefore, the deformed G/H-coset model constitute a more general integrable system than the affine Toda theories and as we will show below, a similar inverse method can be applied to the deformed G/H-coset model in obtaining infinitely many conserved currents as well as soliton solutions.

In order to understand the symmetry of deformed coset models, we transform the linear equation in a different but equivalent form which makes the equation solvable by iteration. Let so that the linear equation changes into

(8) |

and

(9) | |||||

where the subscript denotes the orthogonal decomposition; according to the decomposition of the Lie algebra and the constraint equation (3) has been used. Eq.(9) may be solved for iteratively by assuming which, combined with Eq.(8), leads to an infinite number of conservation laws, , where

(10) |

For in particular, this becomes the conservation of the stress-energy tensor . Also, note that the integrability condition Eq.(5) is invariant under the exchange;

(11) |

Repeating Eq.(8) through Eq.(10) with the above exchange, we obtain another set of infinite number of conserved currents, which together with Eq.(10), constitute the conserved currents of the deformed coset model.

Next, we present a systematic way to derive soliton solutions. First, by setting we transform the linear equation into

(12) |

where

(13) |

A trivial solution of Eq.(12) is given by and . As we show in the following, nontrivial soliton solutions can be obtained from the trivial one by employing the Riemann problem technique with zeros[8]. Let be a closed contour or a contour extending to infinity on the complex plane of the parameter . Consider the matrix function which is analytic with simple zeros inside and analytic with simple zeros outside . We assume that none of these zeros lies on the contour and for . We normalize by . Differentiating with respect to and , one can easily see that

(14) |

Since is analytic inside (outside) , we find that the matrix functions and , defined by

(15) |

where or depending on the region, become independent of . Also, satisfies the linear equation;

(16) |

If we make an identification with as in Eq.(13), and in general provide nontrivial -soliton solutions of deformed coset models. In order to prove the existence of matrix functions and with the properties as stated above, we construct them explicitly in the following; since , the zeros of are the poles of and vice versa. Thus we consider the ansätze;

(17) |

where the matrix functions are to be determined. The fact that requires and to satisfy algebraic conditions which may be obtained through the evaluation of residues of the equation at ,

(18) |

Also, the residues at of Eq.(15) require

(19) |

as well as

(20) |

In order to solve Eqs.(18)-(20), we assume that Then Eq.(18) becomes

(21) |

and Eqs.(19) and (20) become respectively

(22) |

Note that and can be solved in terms of arbitrary constant vectors and ,

(23) |

while and can be obtained in terms of and by solving the linear algebraic equation (21),

(24) |

where . Having determined and as above, we finally obtain -soliton solutions by evaluating Eq.(15) at ,

(25) |

Of course, instead of taking the value at , we may evaluate at which results in leading to the same result. In general, further restrictions are required for and depending on the specific coset structure. For example, if G and H are unitary, in the linear equation become anti-hermitian which in turn impose further restrictions on ,

(26) |

where is the complex conjugation of . This requires that and , or so that . Then the matrix becomes

(27) |

Beside of unitarity, we may impose other types of restrictions according to the specific group structure of G as well as reductions by discrete subgroups of G and H. These cases will be considered elsewhere.

Having demonstrated a systematic way to construct conserved currents and soliton solutions, we take the deformed parafermionic SU(2)/U(1)-coset model as an example and compute explicitly conserved currents and -soliton solutions. We fix the gauge in such a way that the group manifold is parametrized by

(28) |

We choose the center elements . Then, the connections are

(29) |

The linear equation (7) now becomes

(30) |

and

(31) |

whose integrability gives rise to the classical equation of motion of and ;

(32) |

and the U(1)-current conservation law which arises, after the gauge fixing, from the remaining global U(1)-invariance of the action under and . In order to obtain higher conservation laws, we consider the recursive equation (9) which in the present case is given by

(33) |

With

(34) |

With , the first iterative solution , for example, is

(35) |

Higher conservation laws; also result from Eq.(33) with currents defined by

(36) |

For , the conservation law becomes

(37) |

which is the conservation of the stress energy tensor, .
Higher order conserved currents () in general become
non-local.^{2}^{2}2
I would like to thank I.Bakas for raising questions at this point.
For example,

(38) |

This however can be made local if we redefine currents and which satisfy .

Finally, we calculate explicitly -soliton solutions of Eq.(32). Eq.(25) for the SU(2)/U(1)-case becomes

(39) |

where and

(40) |

If we write , -soliton solution can be obtained directly from Eq.(39) such that

(41) |

In particular, if we choose and write , we obtain the one soliton solution,

(42) |

and

(43) |

In this letter, we have shown that, to each coset G/H, there exist a corresponding integrable field theory which arises from a deformation of coset conformal field theory. At the classical level, these models are shown to generalize the affine Toda field theories and possess infinite dimensional symmetries as well as soliton solutions. The quantum aspect of these models are of great interest and will be considered elsewhere. ACKNOWLEDGEMENT I am grateful to Professor H.J.Shin for many useful discussions and constructive criticism, and to Professors I.Bakas, B.K.Chung, S.Nam, D.Kim and C.Lee for their help. This work was supported in part by the program of Basic Science Research, Ministry of Education, and by Korea Science and Engineering Foundation.

REFERENCES

[1] A.Zamolodchikov, Sov.Phys.JETP Lett. 46 (1987); Int.J.Mod. Phys. A3 (1988) 743.

[2] T.Eguchi and S.Yang, Phys.Lett.B224 (1989) 373.

[3] T.Hollowood and P.Mansfield, Phys.Lett.B226 (1989) 73.

[4] D.Karabali, Q-H.Park, H.J.Schnitzer and Z.Yang, Phys.Lett.B216 (1989) 307; D.Karabali and H.J.Schnitzer, Nucl.Phys.B329 (1990) 649; K.Gawedski and A.Kupiainen, Phys.Lett.B215(1988) 119, Nucl.Phys.B320 (1989) 625.

[5] E.Witten, Commun.Math.Phys.92 (1984) 455.

[6] I.Bakas, ‘Conservation laws and geometry of perturbed coset models’, CERN-TH.7047/93, hep-th/9310122

[7] A.V.Mikhailov, Physica 3D (1981) 73.

[8] V.E.Zakharov, S.V.Manakov, S.P.Novikov and L.P.Pitaievski, Theory of Solitons, Moscow, Nauka 1980 (Russian); English transl.:New York, Plenum 1984.