# Conformal Affine Toda Soliton and Moduli of IIB Superstring on

Abstract

In this paper we construct the affine gauged WZW action of type IIB Green-Schwarz superstring in the coset space. From this gauged WZW action we obtain the super Lax connection. Further, in terms of torsion relation we get the pure Bose Lax connection and argue that the IIB Green-Schwarz superstring embedded in is the conformal affine Toda model. We review how the position of poles in the Riemann-Hilbert formulation of dressing transformation and how the value of loop parameters in the vertex operator of affine algebra determines the moduli space of the soliton solutions, which describes the moduli space of the Green-Schwarz superstring. We show also how this affine symmetry affinize the conformal symmetry in the twistor space, and how a soliton string corresponds to a Robinson congruence with twist and dilation spin coefficients of twistor.

## 1 Introduction

The AdS/CFT duality [1] set up the relations between the bulk classical theory and the boundary quantum theory. There exist many hints of the integrable structure on the both sides. Bena, Polchinski and Roiban [2, 3] find the infinite set of nonlocal conserved currents that is the hidden symmetry for the IIB Green-Schwarz superstring in . Dolan [4, 5] et. al. study the relation with quantum integrability in super Yang-Mills theory.

However, all the suggested integrable models of both sides was given after some approximation [6, 7]. In the classical bulk, the exact rotating string in after taking harmonic expansion around point string has been described by (confer the last reference in Ref.[6]) a Neumann-Rosochatius one-dimensional integrable system [8, 9]. For the quantum boundary CFT, one use the Feynmann diagram of tree level and one-loop to relate it with Bethe Ansatz of some spin chain [10, 11, 12, 13]. Then the classical and quantum anomalous dimensions and momentum are compared order by order.

Symmetry dictates interactions [14]. Metsaev and Tseylin [15] find that the global super-invariance and local symmetry dictates the unique IIB string action on . Roiban Siegel [16] bring the coset to in the more transparent symmetric gauge, give the simplest action with manifest conformal symmetry.

In this paper, our aim is to argue that the hidden symmetry has to dictate the classical action with the symmetry broken by anomaly, by the vacuum expectation value of chiral field. The transformation of the complex parameter , which labels the vacua (the moduli space), realizes as the opposite reparametrization of left and right moving string. This loop group is further central extended by the 2 cocycle of WZW term. In summary, the hidden affine symmetry should dictate a gauged WZW action. After shortly review Roiban-Siegel action [16], we suggest the axially gauged WZW action for super current with constraint label by then sketch the derivation of the pure bosonic EOM and its corresponding gauged WZW action. This EOM and action describe the dynamics of the chiral embedding of IIB string. The left and right evolution of moving frame is expressed as the Lax connection, which is uniquely determined by fields of diagonal Cartan element. It turns out to be the conformal affine Toda [17, 18, 19].

Once the bosonic EOM and (or) the gauged WZW action has been obtained, it is already obvious that it should dictate the conformal affine Toda, and to ascertain that the exact integrable model of both classical and quantum AdS/CFT are the affine Toda. In section 4, we only review the key and subtle point for the chain: gauged WZW to chiral embedding to conformal affine Toda, then review its soliton solution in section 5. All this is well know, so the readers better skip the section 4 and 5, go straightforward to the solitonic string picture and turn back in case of need. If something is unclear, then this part may serve as a directory to original papers. But we would like to stress: 1. The Riemann-Hilbert problem of dressing symmetry, description of the holomorphic and anti holomorphic behavior of left and right moving deserves attention. We will discuss the Riemann-Hilbert factorization in the next paper. 2. The vertex construction is not only effective to construct classical solution [17, 18, 19], but also implies the exact quantum version. 3. The geometry of pseudo-sphere [20, 21, 22] in connection with nonlinear model, with sine-Gordon, with Backlund transformation and R.H. lies in the heart of duality and dressing twist, but such kind of geometry seems not popular till now.

At last we find the ground states are stretched string. Its radius of deformation and correspondent energy-momentum are characterized by moduli parameter The hidden symmetry extend the conformal symmetry describes by twistor [23], to that of a “affine” conformal symmetry of twistor, the stretched string realize the Robinson congruence [23] of ray [24, 25].

##
2 Roiban-Siegel’s super vielbein construction and

quadratic WZW term of IIB string on

The Metsaev-Tseytlin action [15] of IIB Green-Schwarz superstring in is realized as the sigma model on coset .

Roiban and Siegel [16] extend the symmetry into by adding two factors

(1) |

Further, using a Wick rotation, it gives

(2) |

The coset representative vielbeins is given by

(3) |

The indices are introduced by

The currents are given by

(4) |

where are the connection.

The antisymmetrization, tracelessness, index contraction and inverse of are defined as[16]

(5) |

Here and is the symplectic metric of . Only the antisymmetrical traceless current is dynamical. Thus the kinetic part is

(6) |

To keep the symmetry, Roiban and Siegel find the topological should be

(7) |

where , this sign is used instead of in Ref.[15] and for the etc..

The total action is

(8) |

The simplest action is given by choosing the symmetric gauge

(9) |

In this gauge, only 16 survives. The explicit form of the currents is given by

(10) |

where

(11) |

## 3 The Gauged WZW Action of IIB string on

The symmetry and symmetry have dictated Roiban Siegel’s action. Now the reparametrization symmetry will dictate gauged WZW action.

### 3.1 Fermionic action

In order to obtain the spontaneously broken ground state of the IIB string
in , we will add the D and FI terms which are
determined by the vacuum expectation values. Thus we must gauging the WZW
action (8) i.e. find the Hamiltonian reduction of (8)^{1}^{1}1Remark: we choose the RS action instead of the beautiful Berkovits formalism
of superstring quantization on background[27],
since there the symmetry is not manifest. The kinetic part includes the
Cartan form of Lorentz generators [28, 29], thus no dressing
symmetry. The WZW is gauged by vector gauge [29], not the axial
one, thus the chiral symmetry seems not manifest also.. The IIB string is
chirally embedded [26] in , such that the
left and right moving tangent vectors of the world sheet is mapped to the
two co-moving tangents of respectively. But meanwhile,
by SUSY, the fermionic beins should be rotated by axial
in accompany. The Cartan forms which describes the string embedded
in the should be corresponded to the components of a
subalgebra in and this and axial are generated by the primary generator of the Here the embedding
obviously is excluded. The principal embedding of in fundamental
representation can not give the decoupled diagonal bosonic block with a subgroup in upper left block for and in lower right block
the extended supersymmetry axial Only the adjoint
embedding works this way.

The super covariant derivatives on are

(12) |

(13) |

where the target and x are functions of the world sheet variable . In the super covariant symmetrical killing gauge

(14) |

Surviving becomes

(15) |

(16) |

They commute with the supercharge

The super currents is defined similar with (4)

(17) |

here

we adapt similar notation for later. But now the even derivative on the world sheet is replaced by the odd and pulled back by the static map to the world-sheet.

The embedding in is given by the generators

(18) |

(19) |

Remark: The lies not in the upper right (lower left) off diagonal block as usually (e.g. [30] appendix) in distinguished basis. It is, as in WZW term (8) of RS, the antisymmetric combination of conjugate terms in upper right and lower left, and has different phase for components in the of in [15].

### 3.2 Bosonic EOM and gauged WZW action of IIB string in

In the Killing gauge, we find the pure bosonic integrable condition^{2}^{2}2We will give it in the next paper.:

(20) |

Using the one can obtain the pure bosonic gauged WZW action

(21) | |||||

One can show that for coset the vierbein and the nonlinear representative G can be identified in some gauge. So here and latter, we will use the (i.e. the in [33], the in [17]) instead of .

It is well known [33, 34, 35, 36] that gauged WZW model given by the action (21) or EOM(22-25) are equivalent to the chiral embedding (Gervais and Matsuo [26]). The Green-Schwarz IIB superstring propagate on the background can be regard as the following chiral embedding . The IIB superstring embed in only the of the . Consider the image on tangent plane of of the two left and right moving tangent vectors respectively of world-sheet as the moving frame. The rotation of the moving frame is given by the Frenet-Serret equations [26] in the manifold (actually the vector representation space of affine if we include the twisted dilation and phase variation of ).

## 4 Chiral embedding and conformal affine Toda model

The chiral embedding dictates the Toda model [26]. Later these has been generalized to the affine case [19]. In this paper we will apply these well known result for the chiral embedding of IIB string in to find stretched string soliton.

At first let us formulate the conclusion. The U in EOM (22-25) and action (21) is the transfer matrix of conformal affine Toda model written in principal curvature coordinate [22],[17] as

(26) |

But, here the Lax connection is in different gauge with in (21-25),

(27) |

Here is the affine Toda field (35).

Now we turn to chiral embedding, to obtain the conformal Affine Toda after (35).

Let be the rotation of moving frame of left and right chiral embedding [26]. We will show that they are the transfer matrix (the in [17]) in the triangular gauge or in the asymptotic line gauge. The transfer matrices “U” in different gauge satisfy

(28) |

where the element will be given later in (35).

Now let’s gauss decompose

(29) |

Here is the diagonal Cartan element, is upper triangular and is lower triangular.

Acting on the highest (lowest) weight vector of the level one representation of affine algebra. The upper(lower) triangular factor is annihilated. One can show as [17] that the remain left moving are “holomorphic”, right moving are “anti-holomorphic” respectively

(30) |

(31) |

Then the various combination of the minors of Wronski determinant [33, 26] of will be expressed uniquely by the Cartan field

(35) |

where is the finite (level zero) part of and we will see that is the affine Toda field and are the fields corresponding to the grade derivative and the center in Cartan subalgebra of respectively.

Then it turns out [17] that the in (28) turns to be the transfer matrix satisfied (26) with Lax connection (27) expressed solely by

(36) |

(37) |

Here we fix , and set .

In terms of (36)(37) the self consistence condition of

(38) |

becomes the EOM of conformal affine Toda .

For simplicity, let us restrict to the case , the EOM becomes that of the conformal Sine-Gordon

(39) |

(40) |

(41) |

Conformal invariance

This EOM is invariant under left ( and right independent reparametrization [17]

(42) |

(43) |

(44) |

The contribution of the two fields and in the improved energy-momentum tensor is [17, 19]

so the improved energy momentum tensor will be traceless [19]

(45) |

The integral of improve energy momentum tensor in 2-dim, will give a surface term which is the topological term (the first Chern-class) and equals the number of soliton. The traceless of energy momentum tensor implies that the conformal invariance is recovered. The total energy and momentum of one soliton equals

(46) |

## 5 Soliton solution of affine conformal Toda

Its soliton solutions can be obtained in three way:

A. Solve the equations of motion (39,40,41) to obtain one soliton solution, then use the Bcklund transformation.

B. By Riemann-Hilbert method in connection with dressing transformation [38].

The dressing transformation of by to is solved by the following Riemann-Hilbert method.

Factorize

(47) |

into

(48) |

with analytic respectively in two region, e.g. upper and lower complex plane.

Then set

(49) |

where

(50) |

It is easy to check

(51) |

Consequently the equations

(52) |

is transformed into

(53) |

Thus R.H transformations induce different “gauge transformation” on the Lax connection of to get the same

(54) |

actually it is shown that [39]

(55) |

(56) |

(57) |

The dressing transformation form a dressing group with the following multiplication rule

(58) |

Here is decomposed as

(59) |

where are

(60) |

are the factor of the Gaussian decomposition of .

C. By using vertex operator (e.g. principal realization) of Kac-Moody algebra and factorized it into in connection with dressing transformation.

### 5.1 One soliton solution

I. The Riemann-Hilbert method with one zero(pole) at [38]

(61) |

can be solved by using the dressing transformation from the vacuum solution as followings [18].

The vacuum solution

(62) |

(63) |

If we set the

(67) |

as the direction of the common tangent [20, 21, 22], define

(68) |

Then we can show that is the projection operator in the eq.(61).

1. Solve the affine Toda equation (41) with to get

(69) |

2. Further solve the factorized dressing eq.(65) by substitute the (63) into it to obtain

(70) |

Then solve this eq. as in 5.1 of ref.[18];

3. The most elegant method is to use the factorized vertex operators as in II.

II. The g written in exponential forms becomes

(71) | |||||

(72) |

Here

(73) |

are the positive and negative frequency respectively of

(74) |

(75) |

where is the position of pole [40] and is the spectral parameter. The the vertex operators principal construction is defined by

(76) |

where oscillators satisfy

(77) |

(78) |

### 5.2 N-solitons solution

Similarly the n-soliton solutions of conformal affine model as the orbit of the vacuum solution under the dressing group may be construct from dressing product of one soliton solution. It is easy to see from (69),(71) that

may be expressed by Then as shown by Ref.[18] their dressing product equals

(81) |

where will be the zero mode position and momentum of the -th solitons. So from (70) the -solitons solutions may be expressed by

Further by factorize each normal ordered into and shift to left or right, then by Wick theorem, the usual function will be obtained.

### 5.3 Soliton of chiral embedded string in

To find it from the known solution, we simply embed the in the , the generator map to the cyclic elements of the . Since this cyclic element is invariant under the action of the Coxeter element which is the maximal Weyl reflection for the automorphism of the Dynkin diagram [40], then it is easy to find that the one soliton solution of the affine is given by [19]

(82) |

where

(83) |

here coxeter number equals . The n-soliton solutions of are similar with 5.2.

## 6 The stretched string soliton

Since the space has the Poincare metric in conformal flat coordinate. We illustrate the motion of string in by the following figures

The metric of in the pure bosonic part of action by (11), becomes the Poincare metric

(84) |

for the pseudo-sphere. The world-sheet and maps to the two principal curvature coordinates. The normal image of the motion on pseudo-sphere will describe the dynamics of nonlinear model [22], now on .

The ground states of solitons is the twistly stretched string on . The two end points of string approach the horizon and the left (right) moving string at fixed time is described by a geodesic line on the pseudo-sphere, which is the semicircle in fig.1 and fig.2. The geodesic radius of the semicircle equals the exponential rapidity and it evolves with constant velocity in time counterclockwise (clockwise) around with period .