aa r X i v : . [ h e p - t h ] A ug A new class of integrable defects
E. Corrigan a and C. Zambon b Department of Mathematical SciencesUniversity of Durham, Durham DH1 3LE, U.K.
ABSTRACT
An alternative Lagrangian definition of an integrable defect is provided and analyzed. Thenew approach is sufficiently broad to allow a description of defects within the Tzitz´eica model,which was not possible in previous approaches, and may be generalizable. New, two-parameter,sine-Gordon defects are also described, which have characteristics resembling a pair of ‘fused’defects of a previously considered type. The relationship between these defects and B¨acklundtransformations is described and a Hamiltonian description of integrable defects is proposed. a E-mail: [email protected] b E-mail: [email protected]
Introduction
It was noticed some years ago [1, 2] that an integrable field theory in two-dimensional space-timecan accommodate discontinuities yet remain integrable. The fields on either side of a discon-tinuity are related to each other by a set of ‘defect’ conditions, including the influence of a‘defect’ potential whose form is required by integrability. The defect conditions themselves areinteresting since they are related, at least in the examples investigated so far, to B¨acklund trans-formations frozen at the location of the defect. It has been found, possibly owing ultimately tothe latter observation, that defects can be supported within the a (1) n series of affine Toda models[3, 4], of which the sine-Gordon model is the first member. Intriguingly, and despite translationinvariance being explicitly broken by the prescribed location, defect conditions compatible withintegrability are determined simply by demanding that the defect itself be able to contributeconsistently to ensure the whole system supports a conserved energy and momentum. The de-fect may be located anywhere (or even move at a constant speed [5]), but the defect conditionsapparently compensate for the evident lack of translation invariance. One might regard thedefect as a state within the model whose presence is indicated by a set of defect conditions de-scribed by an additional term in the Lagrangian description rather than being a field excitationor smooth field configuration. Typically, an integrable defect will be purely transmitting and itseffect does not depend upon its location, meaning it is essentially ‘topological’. At a classicallevel this is exemplified by the passage of a sine-Gordon soliton through a defect where thesoliton will be delayed (or advanced), but might alternatively, according to circumstances, beabsorbed by the defect or flipped to an anti-soliton [5]. Similar types of behaviour are observedfor the complex solitons of the a (1) n models [6]. At a quantum level, defects also appear to play arole though again they are purely transmitting and described by a transmission matrix that iscompatible with the bulk scattering matrix. The purely transmitting aspect of the setup was tobe expected from observations by Delfino, Mussardo and Simonetti [7], but it is still of interestto see exactly how this transpires in detail. In the sine-Gordon case, the transmission matrixwas anticipated by Konik and LeClair [8] but rederived and its properties explored in detailin [5]; for other members of the a (1) n series, the transmission matrices have been provided morerecently [9]. There are a number of related ideas and calculations, including perturbative checksof transmission factors for breathers, and an analysis of the interesting relationship betweenintegrable boundary conditions and defects; some of these are explored in the article by Bajnokand Simon [10].The sine-Gordon B¨acklund transformation was generalised to a (1) n affine Toda models by Fordyand Gibbons [11] and it seems surprising there appear to be no similarly explicit B¨acklundtransformations for the other series of Toda models. However, that fact is at least consistentwith the apparent absence of defects in most of these models, at least of the kind previouslyconsidered [9]. On the other hand, there are several types of B¨acklund transformation availablein the literature for the Tzitz´eica model [12, 13, 14, 15] c and, therefore, one might suppose thereshould be a generalisation of the defect, at least for this model, and possibly for others. Thepurpose of this article is to propose a generalisation by allowing a defect to have its own degreeof freedom in a certain well-defined manner, which is just general enough to encompass the c Note: the model introduced by Tzitz´eica is the a (2)2 member of the affine Toda collection of field theoriesand is also known as the Bullough-Dodd or Zhiber-Mikhailov-Shabat equation. Consider a defect located at the origin x = 0 and let u and v be the fields on either side of it inthe regions x < x >
0, respectively. Typically, a defect defined by B¨acklund conditionswill have a discontinuity, in the sense that while the conditions sewing the two fields at theorigin constrain their derivatives the fields themselves are not prescribed. In other words, it isexpected that the values of the fields approaching x = 0 from their respective domains neednot match and it should be expected that u (0 , t ) − v (0 , t ) = 0 . The basic idea to be exploredhere introduces a new variable λ ( t ) associated with the defect itself. The simplest setup onemight envisage does not directly associate dynamics to λ but is linear in λ t having a Lagrangiandescription of the form: L = θ ( − x ) L u + θ ( x ) L v + δ ( x ) (cid:18) uv t − vu t λ ( u − v ) t − λ t ( u − v ) − D ( u, v, λ ) (cid:19) , (2.1)where the Heaviside step function θ ( x ) and the Dirac delta function have been inserted to ensurethe fields u , v are restricted to their respected domains with the defect located at x = 0. In asense, λ ( t ) plays the role of a Lagrange multiplier: if the potential were absent, integrating over λ would require the discontinuity to be time-independent. However, because the potential also2epends on λ it has a more interesting effect. As we shall see, this is the case even if the potentialis quadratic and the defect links two free massive fields. For the purposes of distinguishing thecases with and without the extra degree of freedom, defects of the original type ( λ ≡
0) will becalled type I and those where λ plays a role will be called type II.The defect conditions at x = 0 implied by (2.1) are: u x = v t − λ t − ∂ D ∂u (2.2) v x = u t − λ t + ∂ D ∂v (2.3) u t = v t + 12 ∂ D ∂λ . (2.4)Then, it is not difficult to show directly that E + D is conserved, where E is the combined bulkcontributions to the total energy from the fields u and v . This was to be expected since timetranslation invariance has not been violated.On the other hand, as usual the contribution from the fields u and v to the total momentum isnot conserved and the requirement of being able to construct a compensating contribution fromthe defect is highly constraining. Defining P = Z −∞ dx u x u t + Z ∞ dx v x v t , (2.5)differentiating with respect to time, and using the bulk equations of motion, gives˙ P = 12 (cid:0) u t + u x − U ( u ) (cid:1) x =0 − (cid:0) v t + v x − V ( v ) (cid:1) x =0 . (2.6)Using the defect conditions (and simplifying the notation on the understanding all field quantitiesare evaluated at x = 0), the latter can be rewritten as − v t ∂ D ∂u − u t ∂ D ∂v + 2 λ t (cid:18) ∂ D ∂u + ∂ D ∂v + 12 ∂ D ∂λ (cid:19) + 12 (cid:18) ∂ D ∂u (cid:19) − (cid:18) ∂ D ∂v (cid:19) ! − U + V. (2.7)For type I defects it would be natural to require the last piece (without any time-derivatives)to vanish and the first two pieces should be a total time derivative leading to equations for thepotential D : ∂ D ∂u = ∂ D ∂v , (cid:18) ∂ D ∂u (cid:19) − (cid:18) ∂ D ∂v (cid:19) ! = U − V. (2.8)This was the setup originally considered in [1]. In fact, as was recalled in the introduction, theconditions (2.8) are highly constraining, effectively limiting U, V (and D ) to the set of sine/sinh-Gordon, Liouville, massive or massless, free fields. In particular, the Tzitz´eica equation isexplicitly excluded. It is also worth recalling the well-known fact that the same selection offields follows from insisting on the conservation of a spin three charge in the bulk (and that acareful analysis of the energy-like spin three charge is enough to provide the full set of integrableboundary conditions for the sine/sinh-Gordon model [17]). The Tzitz´eica equation does not3llow the conservation of a spin three charge but is the one additional possibility that arises ifone instead examines a bulk conserved charge of spin five.However, for type II defects, where λ = 0, the condition on the part of (2.7) containing noexplicit derivatives is weaker because it need not be zero as was assumed in (2.8). Rather, itshould be equated with 12 F ( u, v, λ ) ∂ D ∂λ ≡ ( u − v ) t F ( u, v, λ ) , for some function F depending on u, v and λ , but not their derivatives. In turn, this observationmodifies the impact of the other terms. Taking it into account and assuming the result is a totaltime derivative of − Ω, designed to be a functional of u (0 , t ) , v (0 , t ) and λ ( t ), requires: ∂ Ω ∂u = ∂ D ∂v − F∂ Ω ∂v = ∂ D ∂u + F∂ Ω ∂λ = − (cid:18) ∂ D ∂u + ∂ D ∂v + 12 ∂ D ∂λ (cid:19) , (cid:18) ∂ D ∂u (cid:19) − (cid:18) ∂ D ∂v (cid:19) = 2( U − V ) + F D λ . (2.9)This set of equations entails a number of compatibility relations and to examine these it isconvenient to use new field coordinates defined at the defect location: p = u (0 , t ) + v (0 , t )2 , q = u (0 , t ) − v (0 , t )2 . Then, after a few manipulations the conditions become (and hereafter subscripts will be usedto denote partial derivatives): Ω p = D p Ω q = −D q − F Ω λ = −D λ − D p . (2.10)Eliminating Ω leads to D pq = − F p D λp = −D pp F λ = − F p , (2.11)and, from these it follows that: D = f + g, F = − f q , Ω = f − g, where g depends only on λ and q , and f depends on q and p − λ . Under these circumstances,the last, nonlinear, relation becomes D p D q = 2( U − V ) + ( f λ + g λ ) F, f and g :12 ( f q g λ − f λ g q ) = U − V. (2.12)Interestingly, the left hand side of (2.12) is equal to the Poisson bracket of f and g regarded asfunctions of λ and its conjugate momentum π λ = − ( u − v ) = − q . In terms of the defect energyand momentum, D and Ω, the relationship (2.12) is {D , Ω } = − U − V ) , (2.13)an intriguing equation that relates the Poisson bracket of the energy and momentum contributedby the defect, which is non-zero because of the lack of translation invariance, to the potentialdifference across the defect.Finally, it is worth noting that the equation (2.12) is powerful because all the dependence on λ contained in the left hand side of the equation must cancel out; this significantly constrains notonly f and g but also the potentials U ( u ) and V ( v ). As will be seen below the list of possibilitieswill now include the Tzitz´eica model that had been excluded previously. In this section, using natural ans¨atze, a number of possible solutions to (2.12) are given. Be-sides the Tzitz´eica equation these solutions provide generalisations of already known integrabledefects. However, it is not clear that the examples given exhaust all possible solutions to (2.12).
For the sine-Gordon model, given the form of the potentials U ( u ) = e p + q + e − p − q ≡ e u + e − u , V ( v ) = e p − q + e − p + q ≡ e v + e − v , and bearing in mind the form of the constraint (2.12), the most general ansatz for f and g is f = Ae p − λ + Be − p + λ , g = Ce − λ + De λ , (3.1)where the coefficients A, B, C, D are functions only of q . In detail the constraint (2.12) requires( AD ) q = 2( e q − e − q ) , ( BC ) q = 2( e q − e − q ) , A q C = AC q , B q D = BD q , and hence C = αA, D = αB, αAB = 2( e q + e − q ) + 2 γ, where α and γ are constants. Since λ can be shifted by a function of q without causing anessential change, there is a family of equivalent solutions to these constraints and it is a matter5f convenience which choice is most suitable. For future purposes, it also turns out to be usefulto define γ = ( e τ + e − τ ) . A representative choice for f and g that will be used below is f = 1 σ (cid:0) e p − λ + e − p + λ (cid:0) e q + e − q + γ (cid:1)(cid:1) ,g = σ (cid:0) e λ (cid:0) e q + e − q + γ (cid:1) + 2 e − λ (cid:1) . (3.2)Using these, the defect conditions (2.2) can be rewritten in terms of p, q and λ as follows: p x − p t + 2 λ t = − σ e λ ( e q − e − q ) − σ e − p + λ ( e q − e − q ) ,q x − q t = − σ (cid:0) e λ ( e q + e − q + γ ) − e − λ (cid:1) ,q x + q t = 12 σ (cid:0) e − p + λ ( e q + e − q + γ ) − e p − λ (cid:1) . (3.3)For the sinh-Gordon model, the static solution in the bulk is u = v = 0 and this satisfies thedefect conditions (3.3) provided e λ = 12 cosh τ . On the other hand, purely imaginary solutions to the sinh-Gordon model are the solutions tothe sine-Gordon model, the least energy static solutions in the bulk correspond to u = 2 πia and v = 2 πib where a and b are integers, and the defect conditions permit a = b provided λ is chosensuitably. In fact, the conditions imply: e λ = (cid:26) / τ if a − b is even1 / τ if a − b is odd . (3.4) The Liouville field theory fits into the same scheme by truncating the previous choices for f and g in the sinh/sine-Gordon model found in (3.2). Thus, for example, f = 2 e p − λ g = e λ (cid:0) e q + e − q + γ (cid:1) , (3.5)is an adequate choice since 12 ( f q g λ − f λ g q ) = e p + q − e p − q . In this case, there is no place for an arbitrary parameter to correspond to σ since any such couldbe removed by a translation of λ . On the other hand, the parameter γ can be chosen freely.Further, dropping one or other of the exponential pieces e q (or e − q ) in g leads to a defect thatcouples the Liouville model for u (or v ) to free massless field theory for v (or u ).6 .3 The Tzitz´eica equation For the Tzitz´eica model the bulk potentials are, U = e p +2 q + 2 e − p − q = e u + 2 e − u , V = e p − q + 2 e − p + q = e v + 2 e − v , and the most general ansatz is f = Ae p − λ + Be − p + λ , g = Ce λ + De − λ , (3.6)with the coefficients A, B, C, D being functions only of q . The constraints following from (2.12)are A q D = 2 AD q , B q C = BC q , ( AC ) q = ( e q − e − q ) , ( BD ) q = 4( e q − e − q ) , for which the general solution is BD = 4( e q − e − q ) , A = αD , C = B α . It is always possible to shift λ by a function of q and, for example, A (and therefore D ) canbe chosen to be constants. Using a further shift one of these constants may be removed and aconvenient expression for the most general solution up to these translations of λ is: f = 1 σ (cid:0) e p − λ + e − p + λ (cid:0) e q + e − q (cid:1)(cid:1) ,g = σ (cid:16) e − λ + e λ (cid:0) e q + e − q (cid:1) (cid:17) . (3.7)This contains one free parameter σ . It is also instructive to consider the case where the fields to either side of the defect are free(and massive with mass parameter m ). In this situation, similar considerations lead to f = m (cid:18) ( p − λ ) β + αq (cid:19) ,g = m (cid:18) λ α + βq (cid:19) , (3.8)where α and β are undetermined parameters.One question is whether both of these parameters are effective after λ is eliminated (or, equiva-lently, integrated out in a functional integral). After some algebra, the result for the defect partof the Lagrangian (after removing a total time derivative) is the following: L D = δ ( x ) (cid:20) αβm ( α + β ) q t − (cid:18) α − βα + β (cid:19) ( uv t − vu t ) − m (cid:18) p α + β + ( α + β ) q (cid:19)(cid:21) . (3.9)This still depends upon two parameters, yet in an interesting manner. For example, the limit α → →
0, apart from an inessential sign change in the term linear in time derivatives. From thisobservation it is clear that the new framework does indeed engender an alternative type of defectto those considered previously. However, it is not straightforward to eliminate λ in the other,nonlinear, examples.The expressions for f and g in the sinh/sine-Gordon model given in (3.2) also contain two freeparameters and it is to be expected these survive in the quadratic limit regarded as an expansionabout a classical constant configuration. One way to facilitate the limit is to put σ = e η , andnote an alternative but quite symmetrical expression for D : D = 4 √ (cid:18) e − λ + p/ cosh p − η q + 2 τ e λ − p/ cosh p + 2 η q − τ (cid:19) , which may be expanded about the point p = q = λ = 0. After shifting λ → λ + q tanh τ , the quadratic form is diagonal and resembles (3.8); putting m = √ α and β are given by α = σ τ , β = 12 σ cosh τ . These parameters lie on the set of curves αβ = 14 cosh τ . On the other hand, the quadratic limit of the expression (3.7) giving the functions f and g for the Tzitz´eica equation is a particular one parameter set within the general two parameterfamily. Thus, for the Tzitz´eica equation ( m = √
6) one finds: α = 1 √ σ , β = 2 σ √ , (3.10)corresponding to points on the curve αβ = .If a plane travelling wave, u = e − iωt ( e ikx + Re − ikx ) , v = e − iωt T e ikx , ω = m cosh θ, k = m sinh θ, encounters a defect with the potential (3.9) then there is no reflection ( R = 0), and the trans-mission factor T is given by: T = i (cid:0) αe θ − βe − θ (cid:1) + 1 i ( αe θ − βe − θ ) − . (3.11)One difference from the previously considered cases (with α = 0 or β = 0) is the possibility ofa ‘bound state’ when α = β , for example of the form u = u cos ωt e mζx , x < v = 0 , x > , ζ = − / α, α < − /
2. The contributions to the energy of this solution from the bulkand defect exactly cancel, though both are time-dependent, leading to a zero energy excitationdegenerate with the constant ‘vacuum’ (in which all fields are zero everywhere).Since the present scheme can accommodate all the known single field integrable Toda systemsone might be optimistic that a generalisation of the scheme will encompass all Toda models, con-formal or affine, irrespective of the choice of root data. At this time, however, this generalisation,if it exists, is not known.
So far, nothing has been said about integrability. Nevertheless, this new class of defect is thoughtto be integrable on the basis of some indirect evidence. For example, if this is the case, at thevery least single solitons for both the sine-Gordon model and the complex Tzitz´eica model areexpected to pass safely through a defect suffering at most a delay. In this section, the behaviourof single soliton solutions for these two models will be explored. In addition, in appendix Aan energy-like spin 3 charge for the sine-Gordon model is calculated and found to be conservedon using the defect conditions (3.3). Ideally, a Lax pair formulation is needed to generalise theideas presented in [1].
In the previous section the sinh/sine-Gordon model were considered together but solitons are realsolutions of sine-Gordon or purely imaginary solutions of the sinh-Gordon equation. For easeof notation, and compatibility with earlier sections, the fields u and v will be pure imaginary.Then the defect conditions (3.3) will determine how a soliton scatters with the defect. Thedefect parameters will be taken to be real.In a situation where the intial defect has either no discontinuity, or a discontinuity proportionalto 4 π , a single soliton solution can be written as follows: e u/ = 1 + E − E , E = e ax + bt + c , a = √ θ, b = −√ θ, e v/ = 1 + zE − zE , where z represents the delay, the rapidity θ > x -axis, and e c is purely imaginary. Replacing E → − E (or equivalently shifting c → c + iπ ) provides an expression for an anti-soliton.The final pair of defect conditions (3.3) do not involve λ t and can be used to obtain two expres-sions for λ , e λ = − σ σ ( q x + q t ) + e p ( q x − q t )( e p − e − p )( e q + e − q + γ ) (4.1) e − λ = − σ σ ( q x + q t ) + e − p ( q x − q t )( e p − e − p ) . (4.2)9hese two expressions must be consistent and will determine both z and λ . In fact, there willbe two possible choices for z corresponding to the iπ ambiguity in the possible static solutionsfor λ given by (3.4). Explicitly, the two possibilities for the delay are given by z = z or z = z ,where z = tanh (cid:18) θ − η + τ (cid:19) tanh (cid:18) θ − η − τ (cid:19) , z = 1 /z , σ = e η . (4.3)For z = z , the companion expression for λ is given by e λ = 1 √ τ (1 + E )(1 + zE )(1 + ρE )(1 + ˜ ρE ) , ρ = tanh (cid:18) θ − η + τ (cid:19) , ˜ ρ = tanh (cid:18) θ − η − τ (cid:19) , (4.4)where E = E (0 , t ), and there is a similar expression for λ .Interestingly, the expression (4.3) indicates that the delay is identical to a delay that wouldbe experienced by a soliton passing through two defects of type I (see for example [5]) withparameters η ± τ . Because E is purely imaginary the expression (4.4) for λ indicates that λ is complex and nowhere singular as a function of real t . In order to decide which of the twosolutions should be chosen the starting value of λ (that is, the value λ has when the soliton isfar away but approaching the defect) needs to be specified - effectively, the defect has two statesassociated with it even when the static field configurations to either side of it are u = 0 and v = 0. The modulus of e λ , with 0 < | z | ≤
1, grows to a maximum at E = z − then falls to itsinitial value. On the other hand, the phase of e λ is more interesting since it is the product offour terms, each having a single soliton (or anti-soliton) form (but is a function only of time): e i Im λ = (cid:18) E − E (cid:19) (cid:18) z E − z E (cid:19) (cid:18) − ρE ρE (cid:19) (cid:18) − ˜ ρE ρE (cid:19) . (4.5)The first factor (provided E , which is pure imaginary, has a positive imaginary part) has aphase whose angle monotonically decreases by π as t runs over its range ( −∞ , ∞ ). On the otherhand, if the imaginary part of E had been negative, the phase angle would have increased by π . So, the total effect of the four terms will be either zero (if not more than one of ρ or ˜ ρ isnegative), or − π (if both ρ and ˜ ρ are negative). Thus the phase angle of e λ either shifts by 0or − π . The case where the imaginary part of λ shifts by − π is quite interesting. There, theingoing soliton emerges as a soliton but only after flipping to an anti-soliton and back again, ina virtual sense, since that is what would have happened had the soliton passed two separateddefects with the chosen parameters. In other words, keeping track of λ distinguishes the twopossible cases ( z >
0) where a soliton emerges as a soliton. In the other two cases (one of ρ or˜ ρ is negative), the soliton emerges as an antisoliton.As was the case with type I defects, and as indicated above, the delay z can indicate a changein the character of the soliton as it passes (if η − τ < θ < η + τ , then z <
0, and an approachingsoliton will emerge as an anti-soliton), or the soliton may be absorbed (if θ = η − τ or θ = η + τ ,meaning z = 0). In the latter case, the expression for λ interpolates ‘even’ and ‘odd’ staticsolutions given by (3.4), as it should since the defect stores the topological charge (and theenergy-momentum and other charges) transported by the soliton. The limit τ → η . This lends a little more credibility to the idea(already mentioned in [5]) that a pair of defects with the same parameter behaves like a soliton.10hese results are very suggestive of the idea that at least for the sine-Gordon model the type IIdefects are ‘squeezed’, or ‘fused’, pairs of type I defects.Finally, it is not difficult to check directly that the first of the three defect conditions (3.3) issatisfied by the soliton solution without any further constraints on λ or z .A question that will not be addressed here is how the type II defect should be described by atransmission matrix in the quantum sine-Gordon field theory. Presumably, a generalisation ofthe Konik-LeClair transmission matrix (see [5]) will need to be found and this will be postponedfor a future investigation. The solitons for the Tzitz´eica equation can be analysed similarly although in this case the solitonis complex (though its energy and momentum are real). Using the same conventions as beforewith the potential associated with the choice (3.7), the defect conditions are: p x − p t + 2 λ t = − σ e λ ( e q − e − q ) − σ e − p + λ ( e q − e − q ) ,q x − q t = − σ (cid:0) e λ ( e q + e − q ) − e − λ (cid:1) ,q x + q t = 12 σ (cid:0) e − p + λ ( e q + e − q ) − e p − λ (cid:1) . (4.6)Single soliton solutions in the bulk are given by the expressions [18, 19, 20] e u = (1 + E ) (1 − E + E ) , e v = (1 + zE ) (1 − zE + z E ) , with E = e ax + bt + c , a = √ θ, b = −√ θ, where z represents the delay of the outgoing soliton. The constant e c is chosen so that theexpressions for u and v are nonsingular for all real choices of t and x . The last two of the defectconditions (4.6) can be regarded as a pair of cubic equations for Λ ≡ e λ of the form α Λ + β Λ + γ = 0 , α Λ + β Λ + γ = 0 , (4.7)where the coefficients depend upon p, σ , q and the derivatives of q . Together, these may besolved to giveΛ = α β γ + β γ ( α γ − α γ ) α β β γ − ( α γ − α γ ) ,
1Λ = α β γ + α β ( α γ − α γ ) α β β γ − ( α γ − α γ ) . (4.8)Demanding these two expressions are compatible and inserting the soliton solutions reveals,after some algebra, three possibilities for z : z = ( e − θ + η + e iπ/ )( e − θ + η + e − iπ/ )( e − θ + η − e iπ/ )( e − θ + η − e − iπ/ ) , e η = √ σz = ¯ z = ( e − θ + η − e iπ/ )( e − θ + η + e − iπ/ )( e − θ + η + e iπ/ )( e − θ + η − e − iπ/ ) . (4.9)11hese may also be rewritten more suggestively: z = coth (cid:18) θ − η − iπ (cid:19) coth (cid:18) θ − η iπ (cid:19) , (4.10) z = ¯ z = coth (cid:18) θ − η iπ (cid:19) tanh (cid:18) θ − η − iπ (cid:19) , (4.11)and z z z = 1 . Finally, as examples, for the two cases z = z or z = z expressions for the field λ are e λ = (1 + E )(1 + z E )(1 + 2 ρ E + z E ) , ρ = ( e − θ + η − √ e − θ + η + √ e − θ + η − e iπ/ )( e − θ + η − e − iπ/ ) , (4.12) e λ = (1 + E )(1 + z E )(1 + 2 ρ E + z E ) e − iπ/ , ρ = ( e − θ + η − √ e iπ/ )( e − θ + η + √ e iπ/ )( e − θ + η + e iπ/ )( e − θ + η − e − iπ/ ) . (4.13)For z = z the corresponding formulae are the complex conjugates of the expressions in (4.13).The possible asymptotic values of u and v for soliton solutions are u = 2 πia , v = 2 πib , andthe corresponding asymptotic values of λ required by the defect conditions are λ = 2 πic or λ = ± πi/ πic with a, b, c integers. The formulae (4.12) and (4.13) for λ provide examples ofthis. Once again, as was found to be the case for the sine-Gordon model, part of the specificationof the defect must be the initial choice of λ (essentially, for the soliton, one of three). In [1] it was pointed out using several examples that integrable defect conditions for type Idefects coincide with B¨acklund transformations ‘frozen’ at the defect location. This impressionwas strongly reinforced by subsequent analysis of the a (1) n affine Toda models [2, 6]. However, itwas also found that while the a (2)2 Toda model has B¨acklund transformations these cannot beused directly to construct integrable defects within the type I scheme. At first sight this seemedpuzzling and the purpose of this section is to show how a ‘folding’ procedure [3, 21] may beused to obtain a B¨acklund transformation for the Tzitz´eica model, making use of two similar,yet different, sets of defect conditions obtained in [2] for the a (1)2 Toda model.First a little background is necessary. The equation of motion for an a (1)2 Toda field u is ∂ u = − X j =0 α j e α j · u , (5.1)where, with respect to a basis of orthonormal vectors { e , e , e } in a three dimensional Euclideanspace, the a (1)2 roots are: α = e − e , α = e − e , α = e − e . (5.2)12he projections of the field u onto the orthonormal basis are u , u , u satisfying the constraint u + u + u = 0. From (5.1) it follows that the corresponding equations for the projections read ∂ u j = − e u j − u j +1 − e − u j + u j − ) , j = 0 , , , (5.3)where the subscripts on the right hand side are to be understood modulo 3. Then, the foldingprocedure consists of setting one of the fields to zero, for instance u = 0 (i.e. u = − u ),to obtain the Tzitz´eica equation of motion with the same normalisations as had been assumedwhen writing down the Tzitz´eica potential in section 4.2. Note, the alternative choices u = 0or u = 0 would lead to the same conclusion. The defect conditions that must hold at the defect( x = x ) when sewing together two a (1)2 Toda fields u and λ are ∂ x u − A∂ t u − B∂ t λ + D u = 0 , ∂ x λ − B T ∂ t u + A∂ t λ − D λ = 0 , B = (1 − A ) , (5.4)with D = √ X j =0 (cid:18) σ e α j · ( B T u + Bλ ) / + 1 σ e α j · B ( u − λ ) / (cid:19) , B = 2 X a =0 w a ( w a − w a +1 ) T , (5.5)where w , w ( w ≡ w = 0) are the fundamental highest weights of the Lie algebra a (1)2 . By usingsimilar notation as in (5.3) for the two fields u and λ and light-cone coordinates x ± = ( t ± x ) / ∂ + ( u − u ) − ∂ + ( λ − λ ) = √ σ ( e u − λ − e u − λ + e u − λ ) ,∂ + ( u − u ) − ∂ + ( λ − λ ) = √ σ ( e u − λ − e u − λ + e u − λ ) ,∂ + ( u − u ) − ∂ + ( λ − λ ) = √ σ ( e u − λ − e u − λ + e u − λ ) ,∂ − ( u − u ) − ∂ − ( λ − λ ) = √ σ − (2 e − u + λ − e − u + λ − e − u + λ ) ,∂ − ( u − u ) − ∂ − ( λ − λ ) = √ σ − (2 e − u + λ − e − u + λ − e − u + λ ) ,∂ − ( u − u ) − ∂ − ( λ − λ ) = √ σ − (2 e − u + λ − e − u + λ − e − u + λ ) . (5.6)In the bulk, the expression (5.6) would be the B¨acklund transformation discovered by Fordyand Gibbons [11].Unfortunately, the folding procedure cannot be applied directly to the defect conditions becausethey are simply incompatible with folding. This fact can be expressed heuristically by notingthat the defect conditions (5.6) do not treat solitons and antisolitons identically (a featurealready pointed out in [2, 6] and expected since solitons and anti-solitons are associated withdifferent representations of the a algebra ), because each type of soliton experiences a differentdelay on transmission through the defect. The soliton solution of the Tzitz´eica model can bethought of as a particular soliton-antisoliton solution of the a (1)2 affine Toda model, and, sincethe components of a multi-soliton are delayed independently by the defect, its components willbe treated differently by (5.6). Therefore, the Tzitz´eica soliton cannot survive intact. A remedyis provided by observing that an alternative defect setting is available if the matrix B in (5.5) isreplaced by its transpose. The resulting set of defect conditions describes a system, which is stillintegrable yet interchanges the delays experienced by a soliton or antisoliton when compared13ith the previous case. This suggests that two different types of defect, one built using B (at x = x ) and the other with B T (at x = x ), then ‘squeezed’ together ( x → x ), might allow thefolding procedure to be applied successfully. The second set of defect conditions matches two a (1)2 fields λ and v and would be written in a similar manner to (5.6) but using B T instead of B .Since the incoming ( u ) and outgoing ( v ) solitons are required to satisfy the Tzitz´eica equationof motion, the projections u , v can be set equal to zero. Consequently, the field λ is forced tosatisfy the following constraint (at x = x ):2 e − λ = e − p + λ ( e q + e − q ) , p = u + v , q = u − v . (5.7)Setting u ≡ u, v ≡ v, λ ≡ − λ and sending σ → / ( √ σ ) the two sets of defect conditionslead to ∂ − ( p − λ ) = σ e λ ( e q − e − q ) (5.8) ∂ + λ = − σ e − p + λ ( e q − e − q ) , (5.9) ∂ − q = σ e λ ( e q + e − q ) − e − λ ) , (5.10) ∂ + q = 12 σ ( e − p + λ ( e q + e − q ) − e p − λ ) . (5.11)If, instead of being ‘frozen’ at x = x , equations (5.8)-(5.11) were required to hold in the bulk,they do, in fact, represent a B¨acklund transformation for the Tzitz´eica equation. This can beseen by cross-differentiating the expressions (5.10) and (5.11) to eliminate λ , to find that if thefield u satisfies the Tzitz´eica equation then the field v also satisfies it. Also, cross-differentiatingexpressions (5.8) and (5.9) an equation of motion satisfied by the field λ emerges: ∂ λ = − ( e q + e − q ) e λ − p ( e λ − e − λ ) . (5.12)Inevitably, this depends on the fields u and v . The B¨acklund transformation (5.8)-(5.11) seemsnot to have been reported elsewhere in the literature [13, 14, 15].On the other hand, since equations (5.8-5.11) are supposed to hold only at x = x , and becausethe quantity λ is confined at x = x and depends only on t , the sum of the pair (5.8) and (5.9),together with (5.10) and (5.11) are precisely the three defect conditions (4.6). Hence, for thetype II defect, the number of defect conditions following from the Lagrangian (2.1) is one lessthan the number of equations specifying the B¨acklund transformation described above. Thisis quite different to the previous situation where the Lagrangian description of a type I defectled directly to the frozen B¨acklund transformation (and hence to the B¨acklund transformationitself).Clearly, using the same idea, the defect conditions (3.3) can be augmented to obtain an alternate14¨acklund transformation for the sine-Gordon model that depends on two parameters: ∂ − ( p − λ ) = σ e λ ( e q − e − q ) ∂ + λ = − σ e − p + λ ( e q − e − q ) ,∂ − q = σ e λ ( e q + e − q + γ ) − e − λ ) ,∂ + q = 12 σ ( e − p + λ ( e q + e − q + γ ) − e ( p − λ ) ) . (5.13)From these relations, in a similar manner as previously, the equations of motion for the sine-Gordon fields u and v are recovered and the field λ satisfies, ∂ λ = − e − p (4 e λ − ( e q + e − q )(2 − γ e λ )) . (5.14)The fact that there appear to be generalisations of the defect conditions, which are only indi-rectly related to B¨acklund transformations, and yet likely to be integrable, generates a sense ofoptimism that the framework will generalise to encompass all affine Toda models. So far, properties of defects, and the relationship of the defect conditions to the conservation of asuitably defined momentum, have been derived from first principles from a Lagrangian startingpoint. It is interesting to ask if the framework can be formulated within a Hamiltonian setting.In this section this will be attempted, at least at a formal level, by explaining the main ideas,albeit sketchily. The setup demonstrates explicitly that the presence of a defect reduces theindependent degrees of freedom of the system in the sense of providing defect conditions thatcan be regarded as a set of constraints on the fields u and v (for type I defects), or u , v and λ (for type II defects). This fact is highlighted by the emergence of second class constraints in theHamiltonian (for a detailed description of these, see for example [22]).The discussion can begin by considering a system with a type I defect. In this case, the startingpoint is the following Lagrangian density L = θ ( − x ) L u + θ ( x ) L v + δ ( x ) (cid:18) u v t − v u t − D ( u, v ) (cid:19) , (6.1)with L u = 12 ∂ µ u ∂ µ u − U ( u ) , L v = 12 ∂ µ v ∂ µ v − V ( v ) . (6.2)According to the usual definitions, and treating the theta and delta functions formally, thecanonical momenta conjugate to the fields u and v are, π u = ∂ L ∂u t = θ ( − x ) u t − δ ( x ) v ,π v = ∂ L ∂v t = θ ( x ) v t + δ ( x ) u . (6.3)15y comparison with what happens within each half line, x < x >
0, the canonical momentaare not well-defined at the defect location. In other words, at x = 0 it is not possible to write thetime derivatives of the fields (Lagrangian variables) in terms of the canonical momenta (Hamil-tonian variables). At x = 0 the canonical momenta are not independent, and the definitions(6.3) provide constraints amongst the canonical variables. These are χ = π u + v , χ = π v − u H = Z ∞−∞ dx H (6.5)with H = θ ( − x ) (cid:18) π u + u x U (cid:19) + θ ( x ) (cid:18) π v + v x V (cid:19) + δ ( x ) ( D + µ χ + µ χ ) , (6.6)where µ and µ are functions of the fields u, v together with their momenta. They can bedetermined by using the fact that the constraints χ and χ must be preserved in time. In otherwords, the relations χ t = { χ , H } = 0 , χ t = { χ , H } = 0 , (6.7)must hold. The Poisson bracket of two functionals F = R ∞−∞ dx F and G = R ∞−∞ dx G is definedformally as follows { F, G } = Z ∞−∞ dx (cid:18) δFδu δGδπ u − δFδπ u δGδu (cid:19) + Z ∞−∞ dx (cid:18) δFδv δGδπ v − δFδπ v δGδv (cid:19) . (6.8)Using this definition and the Hamiltonian (6.5), for which, δHδπ u ≡ u t = ∂ H ∂π u = θ ( − x ) π u + δ ( x ) µ , δHδπ v ≡ v t = ∂ H ∂π v = θ ( x ) π v + δ ( x ) µ ,δHδu ≡ − π u t = ∂ H ∂u − ∂∂x ∂ H ∂u x = θ ( − x )( − u xx + U ′ ) + δ ( x ) (cid:16) D u − µ u x (cid:17) ,δHδv ≡ − π v t = ∂ H ∂v − ∂∂x ∂ H ∂v x = θ ( x )( − v xx + V ′ ) + δ ( x ) (cid:16) D v + µ − v x (cid:17) , (6.9)where U ′ = U u and V ′ = V v , the Poisson brackets (6.7) can be calculated. In consequence, (6.7)leads to explicit expressions for the functions µ and µ , which are µ = −D v + v x µ = D u + u x . Assembling all these ingredients, the Hamiltonian density (6.6) becomes H = θ ( − x ) (cid:18) π u + u x U (cid:19) + θ ( x ) (cid:18) π v + v x U (cid:19) + δ ( x ) h(cid:16) π u + v (cid:17) ( v x − D v ) + (cid:16) π v − u (cid:17) ( u x + D u ) + D i . (6.10)16xpressions (6.9) are the canonical Hamilton equations and using the definitions of the canonicalmomenta they coincide with the defect conditions and equations of motion of the type I defectproblem (the latter by performing a differentiation with respect to time).In principle, the conservation of any charge can be verified by calculating its Poisson bracketwith the Hamiltonian. For example, consider the total momentum of the system, which isdefined by P = Z ∞−∞ dx P with P = θ ( − x ) π u u x + θ ( x ) π v v x + δ ( x )Ω( u, v ) . (6.11)It is straightforward to calculate the time derivative of P using its Poisson bracket with theHamiltonian to obtain,˙ P = δ ( x ) (cid:20) (cid:0) D u − D v (cid:1) − U + V + u t (Ω u − D v ) + v t (Ω v − D u ) (cid:21) = 0 . (6.12)The final step follows from the facts that ( D u − D v ) / U − V ), D = ( f + g ) and Ω = ( f − g ),with f = f ( p ) and g = g ( q ), as was described previously in [1].It should be noticed that the constraints (6.4) are second class. Hence, as mentioned at thebeginning of this section, they indicate that not all degrees of freedom are independent. Bydefinition, a constraint is first class if its Poisson brackets with all other constraints are zero- the constraints themselves can be imposed, if needed - otherwise, it is second class. In thepresent case, it is straightforward to check that the Poisson brackets of the constraints areconstant. In fact, C ij ≡ { χ i , χ j } , C = (cid:18) − (cid:19) . The matrix C can be used to construct the Dirac brackets, the standard tool for dealing withsecond class constraints.Next, consider the type II defect and suppose the Lagrangian density is given by (2.1). Then,there are three fields u , v and λ , whose canonical momenta are π u = ∂ L ∂u t = θ ( − x ) u t − δ ( x ) (cid:16) v − λ (cid:17) , π v = ∂ L ∂v t = θ ( x ) v t + δ ( x ) (cid:16) u − λ (cid:17) ,π λ = ∂ L ∂λ t = − δ ( x )( u − v ) . Consequently, the primary constraints are χ = π u + v − λ = 0 , χ = π v − u λ = 0 , χ = π λ + ( u − v ) = 0 , (6.13)and the Hamiltonian density reads H = θ ( − x ) (cid:18) π u + u x U (cid:19) + θ ( x ) (cid:18) π v + v x V (cid:19) + δ ( x )( D + µ χ + µ χ + µ χ ) . (6.14)17ince these constraints must be consistent with the evolution equations, their time derivativemust vanish. By using the following Poisson bracket { F, G } = Z ∞−∞ dx (cid:18) δFδu δGδπ u − δFδπ u δGδu (cid:19) + Z ∞−∞ dx (cid:18) δFδv δGδπ v − δFδπ v δGδv (cid:19) + (cid:18) δFδλ δGδπ λ − δFδπ λ δGδλ (cid:19) x =0 , (6.15)it is possible to verify that χ t = −D u − u x + µ − µ = 0 , χ t = −D v + v x − µ + 2 µ = 0 , χ t = −D λ + 2( µ − µ ) = 0 . Unlike the previous case, this system of equations does not determine completely the functions µ j . In fact, requiring the constraints to be preserved with time forces µ = −D v + v x + 2 µ , µ = D u + u x + 2 µ (6.16)( u − v ) x + D u + D v + 12 D λ = 0 . (6.17)Expression (6.17) is a secondary constraint. However, it is not genuinely new since it coincideswith an algebraic sum of some of the canonical Hamiltonian equations, as can be verified byusing the following Hamiltonian density H = θ ( − x ) (cid:18) π u + u x U (cid:19) + θ ( x ) (cid:18) π v + v x V (cid:19) + δ ( x ) D + δ ( x ) h(cid:16) π u + v − λ (cid:17) ( v x − D v ) + (cid:16) π v − u λ (cid:17) ( u x + D u ) + µ (2 π u + 2 π v + π λ ) i . (6.18)In fact, 0 = ( π λ + 2 π u + 2 π v ) t ≡ − ( u − v ) x − D u − D v − D λ , which coincides with (6.17). As was shown in the previous case, all Hamilton equations can beobtained and they lead to the equations of motion and defect conditions (note that λ t ≡ µ ).Finally, as mentioned before, the Poisson brackets (6.15) may be used to verify the conservationof charges. For example, given the total momentum (6.11), it can be checked that˙ P = δ ( x ) (cid:20) (cid:0) D u − D v (cid:1) − U + V + λ t ( D λ + 2 D u + 2 D v + Ω λ ) + u t (Ω u − D v ) + v t (Ω v − D u ) (cid:21) . Since D = ( f + g ) and Ω = ( f − g ), with f = f ( p − λ, q ) and g = g ( λ, q ), the above expressionbecomes ˙ P = δ ( x ) (cid:20) (cid:0) D u − D v (cid:1) − U + V + 12 f q D λ (cid:21) ≡ . In summary, from the Hamiltonian density (6.18), it is possible to read off the final constraints,which are χ , χ , γ = 2 π u + 2 π v + π λ , where χ , χ are second class, while γ is first class. In fact, it can be checked that { χ , γ } = { χ , γ } = { γ , γ } = 0. The first class constraints are usually related to the presence of a gaugefreedom. In the type II defect framework, the existence of a first class constraint indicates thefreedom to translate the field λ by any function of q , as was pointed out in section 3.18 Comments and conclusions
The main result of this paper has been to extend the framework within which an integrabledefect may be described. The previous framework (referred to as type I in this article) seemedfairly natural yet even for a single scalar field was unable to accommodate all possible relativisticintegrable models because the Tzitz´eica, or a (2)2 affine Toda, model was conspicuously absent.For multiple scalar fields the possible type I defects are restricted to the a (1) n series of affine Todamodels. In all cases, the type I defects are intimately related to B¨acklund transformations, in thesense that the conditions relating the fields on either side of an integrable defect take the formof a B¨acklund transformation frozen at the location of the defect. At first sight, this relationshipseemed attractive since it provided a use for B¨acklund transformations that had not been noticedbefore. On the other hand, the Tzitz´eica equation has several B¨acklund transformations associ-ated with it and none of them emerged naturally from within the type I framework. Moreover,the integrability of the type I defects is intimately related to momentum conservation, in thesense that insisting there should be a total momentum including a contribution from the defectitself leads to restrictions that would be associated normally with the requirements of havinghigher spin conserved quantities. It is a curious situation: certain integrable systems (thosewith type I defects) can violate translational invariance yet preserve momentum. The questionis: can this phenomenon be extended to other integrable systems by changing the framework?It appears the answer is yes, and one particular different framework (referred to as type II) isdescribed in this paper. In fact, only a slight change appears to be necessary, the Tzitz´eicamodel is incorporated, and the relationship with frozen B¨acklund transformations is modified.The trick is to introduce a new degree of freedom located on the defect and couple it in aminimal manner to the discontinuity across the defect. In the absence of a generalised Lax pairfor the type II system, momentum conservation becomes a tool for identifying the possibilities,backed up by other less direct evidence. Turning the argument around and starting from thedefect conditions allows an apparently new B¨acklund transformation to be established for theTzitz´eica equation. The type II framework certainly contain all single field integrable systemsof Toda type (or free fields) but it is not yet demonstrated these are the only possibilities. Thelatter appears reasonable since (2.12) is highly constraining but a complete proof of integrabilityneeds to be found in order to be sure.It is already known that the a (1) n affine Toda models can support type I defects of several kindsand that defects are able to relate different a n conformal Toda models to each other (therebygeneralising the relationship between the Liouville model and free fields [9]). However, otheraffine Toda models based on the root data of the b, c, d, e, f, g series of Lie algebras do notappear to fit in to the type I framework. This is surprising: in most respects, the affine Todafield theories at least in the bulk, have similar features, though it does appear from the literaturethat the a (1) n series is special in having a B¨acklund transformation of a simple type. It remainsto be seen if the type II framework can be adapted to all Toda models. The folding processcannot explain the apparent difficulties with the d, e series. However, once these are understoodthe folding process might be an essential part of the story for the remaining cases. For thatreason it would be natural to examine the d, e series next.At this stage it is worth outlining a possible direction for a generalisation containing multi-component fields. Using the same notation as previously, taking as a starting point the defect19ontribution L D = δ ( x ) ( q · Aq t + 2 λ · q t − D ( λ, q, p )) , (7.1)where A is an antisymmetric matrix, then insisting on overall momentum conservation, leads tothe following constraints on D and Ω: D = f ( p − λ, q ) + g ( p + λ, q ) , Ω = f ( p − λ, q ) − g ( p + λ, q ) . (7.2)Further, the two functions f and g are constrained by a generalisation of the Poisson bracketrelation (2.12) that reads, ∇ q f · ∇ λ g − ∇ q g · ∇ λ f + ∇ λ f · A ∇ λ g = U ( u ) − V ( v ) . (7.3)Here A is the antisymmetric matrix occurring in (7.1) and U, V are the bulk potentials for thefields to either side of the defect. The left hand side of (7.3) is a bona fide Poisson bracket sinceit is antisymmetric and satisfies the Jacobi relation, yet, as before, all dependence on λ mustcancel out. This provides severe constraints on U and V , which will be explored elsewhere.At the quantum level, it was demonstrated in [5, 6] that type I defects within the a (1) n seriesare described by infinite-dimensional transmission matrices, which are determined up to a singleparameter by a set of ‘triangle relations’ ensuring their compatibility with the bulk S-matrix.Moreover, arguments have been provided to demonstrate that the free parameter is essentiallythe same, though possibly renormalised, as the free parameter in the type I Lagrangian. Clearly,the next question concerns the transmission matrix in the context of type II defects. For the sine-Gordon model, the transmission matrix in this framework should depend on two independentparameters and there should be some evidence or influence of the confined field λ , at leastrecognising the iπ ambiguity mentioned in section 4.1. At a quantum level, the Tzitz´eica modelcontains a triplet of equal mass states, reflecting its origin in a (1)2 affine Toda field theory underthe folding process, only two of which correspond to classical solitons, and its S-matrix is known[23]. It is to be hoped there will be a transmission matrix based on an ansatz that takes intoaccount the mysterious role of λ (this time the ambiguity is threefold - see section 4.2). Acknowledgements
We are grateful for conversations with colleagues in Durham, especially Peter Bowcock. Inparticular, we wish to thank him for discussions on the content of section 5, much of which hedeveloped independently.We also wish to express our gratitude to the UK Engineering and Physical Sciences ResearchCouncil for its support under grant reference EP/F026498/1.
A Energy-like spin three charge for the sine-Gordon model
In this appendix it is shown that an energy-like spin three charge for the sine-Gordon modelwith a defect of type II is conserved. The bulk charge, which is not expected to be conserved in20he presence of a defect, conveniently normalised, reads E = Z −∞ dx (cid:18) u t + u x u x u t + 4 u tx + ( u tt + u xx ) + ( u t + u x ) U ′′ (cid:19) + Z ∞ dx (cid:18) v t + v x v x v t + 4 v tx + ( v tt + v xx ) + ( v t + v x ) V ′′ (cid:19) , and its time derivative is˙ E = h ( u t u x + u t u x ) − ( v t v x + v t v x ) + 4(2 u tt + U ′ ) u tx − v tt + V ′ ) v tx − u t u x U ′′ − v t v x V ′′ ) i x =0 , (A.1)where U ′ = U u and V ′ = V v . This is not expected to be zero but the right hand side may turnout to be the total time derivative of a functional −D that depends only on the defect variables p, q and λ . In that case, E + D will be conserved. Since the expression (A.1) is calculatedat x = 0, it is convenient to rewrite it by using the variables p and q . Then, using the defectconditions (2.2)-(2.4) with the functions f and g given by (3.2), the expression (A.1) becomesa total time derivative˙ E = 4 ddt (cid:0) p t − λ t ) q t f λq − ( p t − λ t ) f − q t ( f + g ) qq − λ t g − q t λ t g λq (cid:1) +4 ddt (cid:16) ( p t − λ t )( U ′ − V ′ ) − λ t ( U ′ − V ′ ) − q t ( U ′ + V ′ ) (cid:17) − ddt Ω ( p, q, λ ) , (A.2)(where again on the right hand side all field quantities are evaluated at x = 0), with ∂ Ω ∂q = 3 f λ ( U − V ) − f q ( f + g ) λ + 14 ( f + g ) q (cid:0) f λ + ( f + g ) q − U + V ) (cid:1) ,∂ Ω ∂p = − f + g ) q ( U − V ) + 34 f qq ( f + g ) λ − f λ (cid:0) f λ + 3( f + g ) q − U + V ) (cid:1) ,∂ Ω ∂λ = 14 ( f + g ) λ (cid:0) f λ + g λ − f λ g λ − f + g ) λ ( f + g ) qq + 3( f + g ) q − U + V ) (cid:1) . The formula (A.2) has been obtain by making use of the following properties of the defectpotential for the sine-Gordon model f p = − f λ , f λλ = f, g λλ = g, f qqq = f q , g qqq = g q , f λq = f q , g λq = g q . (A.3)Finally, it has been verified that the cross derivatives of the function Ω are consistent, that is ∂ Ω ∂q∂p = ∂ Ω ∂p∂q , ∂ Ω ∂q∂λ = ∂ Ω ∂λ∂q , ∂ Ω ∂p∂λ = ∂ Ω ∂λ∂p . For this task, in addition to (A.3), the following relations have been used( U ± V ) p = ( U ∓ V ) q , f qq ( f + g ) q = f q ( f + g ) qq , f q ( g + g λ ) = g q ( f + f λ ) (A.4)where ( U − V ) = 12 ( f q g λ − f λ g q ) = ( U + V ) pq . eferences [1] P. Bowcock, E. Corrigan and C. Zambon, Classically integrable field theories with defects ,Int. J. Mod. Physics
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