A SSG = dtdx L SSG (x, t)

Transcription

1 ITP Budapest Report 583 Spectrum of oundary states in N= SUSY sine-gordon theory Z. Bajnok, L. Palla and G. Takács Institute for Theoretical Physics Eötvös University H-7 Budapest, Pázmány Péter sétány /A 5 June 00 Astract We consider N= supersymmetric sine-gordon theory SSG with supersymmetric integrale oundary conditions oundary SSG=BSSG. We find two possile ways to close the oundary ootstrap for this model, corresponding to two different choices for the oundary supercharge. We argue that these two ootstrap solutions should correspond to the two integrale Lagrangian oundary theories considered recently y Nepomechie. PACS numers:.55.ds,.30.p,.0.kk Keywords: supersymmetry, sine-gordon model, integrale quantum field theory, oundary scattering, ootstrap Introduction In this paper we consider the N= supersymmetric sine-gordon theory with supersymmetric integrale oundary conditions BSSG. Our aim is to find a closure of the oundary ootstrap for this model. The N= supersymmetric sine-gordon model is the natural supersymmetric extension of the ordinary sine-gordon model. It is an integrale field theory with infinitely many conserved charges []. The S matrix of the theory was otained in [], while the integrale and supersymmetric oundary conditions were considered in [3], ut it took a while until the most general integrale supersymmetric oundary interaction was found [4]. ajnok@afavant.elte.hu palla@ludens.elte.hu takacs@ludens.elte.hu

2 As a result of integraility, the oundary scattering factorizes, and the general solution of the oundary Yang-Baxter equation was found in [5], ut the constraint of supersymmetry was not imposed. Nepomechie was the first to consider supersymmetric oundary scattering [4], uilding on previous results otained in the case of supersymmetric sinh- Gordon theory [6]. However, no one has exposed the full structure of the solitonic reflection amplitude, although the results otained in the case of the tricritical Ising model [7] are closely related to this prolem. Besides that, the closure of the ootstrap and the spectrum of oundary states have not een even touched efore. Therefore our aim is to clear up the issue of supersymmetric oundary scattering in the BSSG model and to find the complete spectrum of oundary states and their associated reflection factors. The main idea motivated y the successful description of the ulk scattering is to look for the reflection amplitudes in a form where there is no mixing etween the supersymmetric and other internal quantum numers. This means an Ansatz for the reflection amplitudes as a product of two terms one of which is the ordinary osonic sine-gordon reflection amplitude, while the other descries the scattering of the SUSY degrees of freedom. Within the ootstrap procedure, we consider first the solitonic reflection amplitudes on the ground state and on the first two excited oundaries. The SUSY factors in these solutions have no poles in the physical strip, thus the masses of oundary states emerging are the same as in the osonic theory, however, SUSY introduces a nontrivial degeneracy. The spectrum of general higher excited oundaries is easily extracted from these results. We determine also the various reflection amplitudes on these excited oundaries. There are two ways to close the ootstrap, starting from two different ground state reflection amplitudes, corresponding to two possile choices of the oundary supercharge. Both solutions lead to the same spectrum of oundary states, ut the reflection amplitudes are different. In one case the reflections conserve fermionic parity, while in the other they do not. The layout of the paper is as follows. Section recalls riefly some important facts aout supersymmetric sine-gordon theory. In Section 3 the reader is reminded of the supersymmetric and integrale oundary interactions that can e added to the theory, and we also discuss of the oundary supercharge and derive a formula relating it to the oundary Hamiltonian. Section 4 gives a quick review containing only the most necessary facts of the spectrum and reflection factors of the ordinary non supersymmetric sine- Gordon model with integrale oundary conditions. In Section 5 we present the main results of the paper, which is a conjecture for the spectrum and the full set of reflection factors of the BSSG model. We consider the reather reflection amplitudes in Section 6 and then give our conclusions in Section 7. Bulk SSG theory The ulk supersymmetric sine-gordon theory is defined y the classical action A SSG = dtdx L SSG x, t L SSG = µφ µ Φ+i Ψγ µ µ Ψ+m ΨΨ cos β Φ+m cos βφ β where Φ is a real scalar, Ψ is a Majorana fermion field, m is a mass parameter and β is the coupling constant. The theory is invariant under an N =supersymmetry algera and

3 has infinitely many commuting local conserved charges []. These charges survive at the quantum level and render the theory integrale, which makes it possile to descrie the exact spectrum and the S matrix.. Spectrum and ulk scattering amplitudes The spectrum consists of the soliton/antisoliton multiplet, realizing supersymmetry in a nonlocal way, and reathers that are ound states of a soliton with an antisoliton. The uilding locks of supersymmetric factorized scattering theory were first descried in [0], using an Ansatz in which the full scattering amplitude is a direct product of a part carrying the SUSY structures and a part descriing all the rest of the dynamics. The full SSG S matrix was constructed in []. The supersymmetric solitons are descried y RSOS kinks Ka ɛ θ of mass M and rapidity θ, where a, take the values 0, and with a =/, and descrie the supersymmetric structure, while ɛ = ± corresponds to topological charge ± soliton/antisoliton. Multi-particle asymptotic states are uilt as follows K ɛ a 0 a θ K ɛ a a θ...k ɛ N an a N θ N where θ >θ >... > θ N for an in state and θ <θ <... < θ N for an out state. The two-particle scattering process K ɛ a θ +K ɛ c θ K ɛ ad θ +K ɛ dc θ has an amplitude of the form a S SUSY d c θ θ S SG θ θ ɛ ɛ ɛ ɛ 3 i.e. the tensor structure of the scattering amplitude factorizes into a part descriing the SUSY structure which we call the SUSY factor and another part corresponding to the topological charge the osonic factor. The osonic factor coincides with the usual sine-gordon S matrix S SG u = S SG u = S SG u + + = S SG u + + = S SG u + + = S SG u + + = [ l= Γl λ λu λu Γlλ + π Γl λ λu π π Γl λ + λu π /u u sinλu sinλπ u S SGu ; λ = π β sinλπ sinλπ u S SGu ; u = iθ, 4 while the SUSY factor is identical to the S matrix of the tricritical Ising model pertured y the primary field of dimension 3 5 [8]: case. S SUSY 0 θ = S SUSY θ = iπ θ/πi cos θ 4i π Kθ 4 0 Note that the relation etween the parameter λ and the coupling β is different from the sine-gordon 3 ]

4 0 S SUSY 0 0 S SUSY S SUSY 0 θ θ θ = S SUSY = S SUSY 0 = S SUSY 0 θ θ = θ/πi cos Kθ 4i θ θ = iπ θ/πi cos 4i + π 4 θ θ = θ/πi cos 4i π Kθ = Γk /+θ/πiγk θ/πi π Γk +/ θ/πiγk + θ/πi k= Kθ Kθ As the SUSY factor has no poles in the physical strip, the solitonic amplitudes 3 have poles at exactly the same locations as the sine-gordon soliton S matrix. These correspond to ound states reathers B n of mass m n =Msin πn,n=,... [λ]. λ The S matrix of the reathers was first found in [9]. The reathers form a particle multiplet composed of a oson and a fermion, on which supersymmetry is represented in a standard way [0, ]. For the ordinary sine-gordon theory, the correspondence etween the Lagrangian theory and the ootstrap S matrix 4 is very well estalished. There is much less evidence for the correctness of the S matrix 3 as the scattering amplitude of SUSY sine-gordon theory. Besides the original construction [] ased on arguments related to N =supersymmetric minimal models, another indication is that at a particular value of the coupling β where it is expected to have a restriction to the SUSY version of Lee-Yang theory superconformal minimal model SM/8 pertured y the relevant superconformal primary field Φ, 3, which is equivalent to Virasoro minimal model M 3/8 pertured y the primary field Φ, 5, the first reather supermultiplet has the same scattering amplitude as predicted from RSOS restriction of imaginary coupled a Toda theory in [] see also [3].. Bulk SUSY charges The ulk theory has two supersymmetry charges of opposite chirality Q and Q, which together form a Majorana spinor. They act on one-particle states A i θ in the following way [8, 0, ]: Q A i θ = m i e θ/ Q A i θ, Q Ai θ = m i e θ/ Q Ai θ where m i are the particle masses and Q, Q are matrices satisfying Q =, Q =. { } In the one-particle asis K 0,K,K 0,K we omit the upper index ɛ, as the SUSY action does not depend on the topological charge, the supersymmetry algera is represented y the matrices 0 i i 0 0 Q = i , Q = i

5 The SUSY algera of the sine-gordon theory has a central charge and the matrix Z = { Q, Q} = descries the SUSY central charge in the aove asis. This is not to e confused with the topological charge T of the sine-gordon solitons, which is represented y the upper indices ɛ. Z can take the values 0 or ±, and it distinguishes etween solitons/antisolitons mediating from odd to even and from even to odd vacua of the osonic potential [4] this means that using the terminology of [5] the theory is -folded. The aove representation of SUSY descries BPS saturated ojects. There was a controversy in the literature whether the solitons in SSG are BPS saturated, since N = SUSY does not protect their mass from acquiring radiative corrections. However, it was shown in [6] that they remain BPS saturated at one-loop and proaly to all orders, due to anomalous quantum corrections to the classical formula for the central charge Z. The fermionic parity operator Γ= F is given y the matrix Γ= Using the definition of Γ it is possile to specify a asis of pure osonic and pure fermionic states for any given fixed numer of particles. However, the composition coproduct rules of the kink states as given in are not free and therefore in the oson-fermion internal space supersymmetry acts nonlocally. The action of supersymmetry on multiparticle states involves raiding factors depending on Γ that are defined y the coproduct : Q = Q I +Γ Q Q = Q I +Γ Q Γ = Γ Γ The action on reather states can e derived using the ootstrap, ut can also e otained from the representation theory of the SUSY algera. It turns out that the central charge Z as well as the topological charge T vanishes identically for the reathers. For further details we refer to [0, ]. 3 Boundary SSG theory 3. Integrale and supersymmetric oundary interactions Boundary SSG BSSG theory can e descried y an action of the form 0 A BSSG = dt dxl SSG x, t+ L B tdt 5

6 where L B t is a term local at x =0. We consider oundary interactions which are oth supersymmetric and integrale. The case when L B t depends only on the value of Φ and Ψ at x = 0was considered in [3], where it was found that supersymmetry and integraility restricts the oundary interaction to the form L B = ± 4m β cos β Φ ± ψψ 5 x=0 which gives four discrete choices. Here we write the Majorana spinor in component form ψ Ψ=. ψ However, this is not the whole story. The Majorana fermion in the ultraviolet limit is descried y the c = Ising conformal field theory. It was noted in [7] that in order for the Majorana fermion to descrie correctly the oundary states of the conformal field theory, one has to include a fermionic oundary degree of freedom at that is related to the oundary value of the Ising spin. Using this fact, a two-parameter set of integrale supersymmetric oundary conditions was derived in [4]. The oundary interaction term is of the form L ± B = ± ψψ + ia t a f Φ aψ ψ+b Φ x=0 6 The functions f and B are fixed y the requirement of oundary integraility and supersymmetry cf. [4]. We denote the theories otained y adding L ± B to the ulk action as BSSG ±. The most important property of this action is that it depends on two continuously varying oundary parameters, exactly as in the case of the non-supersymmetric oundary sine-gordon theory [8]. This is important for consistency with the ootstrap since the reflection factors we find depend on two parameters as well. The oundary Lagrangian 5 can e otained as a special case of 6 when the parameters are tuned so that f =0, and therefore the oundary fermion a decouples. 3. The oundary SUSY charge and the Hamiltonian The ulk SUSY charges Q and Q can e written as integrals of local fermionic densities q and q: Q = qx, tdx, Q = qx, tdx They have the anticommutation relation {Q, Q} =MZ and satisfy the following relation among others QQ + Q Q =H 7 where H = hx, tdx is the Hamiltonian, Z is the SUSY central charge and M is the soliton mass. In a oundary theory with supersymmetric integrale oundary condition, the conserved supercharge can e written as follows: Q ± = 0 qx, t ± qx, t dx + Q B x =0,t 6

7 where Q B is a oundary contriution, localized at x =0. There are two possile choices ± corresponding to the two possile conformal fermionic oundary conditions in the ultraviolet limit. As shown in [4], these correspond to the two choices of sign in 6 and therefore to BSSG ±. Similarly, the Hamiltonian takes the form H = 0 hx, tdx + H B x =0,t where H B is the oundary interaction. Let us for the moment neglect the contriution from the central charge Z this is possile e.g. in a sector containing only reathers. Then, using eqn. 7 it is easy to see that 0 Q ± = hx, tdx +H Bx =0,t where H B is again some term localized at x =0. However, Q± is conserved in time, and therefore d 0 hx, tdx +H dt Bx =0,t =0. In the supersymmetric oundary sine-gordon theory, this property uniquely determines H B as the oundary interaction H B and therefore Q ± = H Including Z, it is natural to expect that this relation extends as follows: Q ± = H ± M Z 8 where Z is an appropriate extension of Z to the oundary situation. We shall see that the two ootstrap solutions we propose correctly reproduce this formula. 4 Boundary sine-gordon model 4. Ground state reflection factors The most general reflection factor modulo CDD-type factors ofthe soliton antisoliton multiplet s, s on the ground state oundary, denoted y, satisfying the oundary versions of the Yang-Baxter, unitarity and crossing equations was found y Ghoshal and Zamolodchikov [8]: P R SG η, ϑ, u = + η, ϑ, u Qη, ϑ, u = Qη, ϑ, u P η, ϑ, u P + 0 η, ϑ, u Q 0 u Q 0 u P0 η, ϑ, u R 0 u ση, u σiϑ, u cosη coshϑ P 0 ± η, ϑ, u = cosλucosηcoshϑ sinλusinη sinhϑ, Q 0 u = sinλucosλu, 9 7,

8 where η and ϑ are the two real parameters characterizing the solution, [ Γ ] 4lλ λu π Γ 4λl + λu π R 0 u = Γ /u u 4l 3λ λu π Γ 4l λ + λu π l= is the oundary condition independent part and [ cos x Γ σx, u = + x π cosx + λu l= Γ x π λu +l λ π Γ x π +l λ λu π ] λu +l λ π Γ + /u u x λu +lλ π π descries the oundary condition dependence. The reflection factors of the reathers can e otained y the ulk ootstrap procedure [9]. 4. The general spectrum and the associated reflection factors The spectrum of oundary excited states was determined in [0, ]. It can e parametrized y a sequence of integers n,n,...,n k, whenever the π ν n >w n >... 0 condition holds, where ν n = η λ The mass of such a state is πn + λ and w k = π η λ πk λ m n,n,...,n k = M cosν ni +M cosw ni. 0 i odd i even The reflection factors depend on which sectors we are considering. In the even sector, i.e. when k is even, we have Q n,n,...,n k η, ϑ, u =Qη, ϑ, u a ni η, u a ni η, u, i odd i even and P ± n,n,...,n k η, ϑ, u =P ± η, ϑ, u a ni η, u a ni η, u, i odd i even for the solitonic processes, where and a n η, u = {y} = y+ λ n l= y+ λ { η } π l y λ ; η = πλ + η u, x =sin + xπ + y sin u xπ λ when k is odd, the same formulae apply if in the ground state In the odd sector, i.e. reflection factors the η η and s s changes are made. The reather sector can e otained again y ulk fusion.. 8

9 5 Supersymmetric oundary sine-gordon model 5. Ground state reflection factors 5.. The general solution for the reflection factor Following the ulk case, we suppose that the reflection matrix factorizes as R SUSY θ R SG θ. In this special form the constraints as unitarity, oundary Yang-Baxter equation and crossing-unitarity relation [8] can e satisfied separately for the two factors. Since the sine-gordon part already fulfills these requirements, we concentrate on the supersymmetric part. From the RSOS nature of the ulk S-matrix 3 it is clear that the oundary must also have RSOS laels and the adjacency conditions etween the nearest kink and the oundary must also hold. Thus the following reflections are possile: K a θ B a = c R ac θk c θ B c or in detail and K 0 θ B = R 0 θk 0 θ B ; K θ B = R θk θ B. K aθ B a = R aa θk a θ B a + R a θk θ B, a, a, =0, In the second process the lael of the oundary state has changed, which shows that B 0 and B form a doulet. All of the constraints mentioned aove factorize in the sense that they give independent equations for the reflections on the oundary B / and on the doulet B 0,. Since the ground state oundary is expected to e nondegenerate we first concentrate on reflection factors off the singlet oundary B /. The most general solution of the oundary Yang-Baxter equation is of the form [5] R 0 θ =+A sinhθ/mθ ; R θ = A sinhθ/mθ while unitarity and crossing symmetry give the following restrictions MθM θ A sinh θ/ =, iπ θ M θ =cosh iπ 4 Kθ iπ+θ/πi M iπ + θ We suppose that the oundary states B a,a=0, can e otained y oundary ootstrap from the ground state B/ [, 7]. Therefore we do not consider the Yang-Baxter equation and the other constraints for the amplitudes R / a θ as these will e guaranteed to e fulfilled y the ootstrap. We shall see later that in general the oundary states B a,a=0, come in multiple copies, each of which forms a doulet of states with the same energy. 9.

10 5.. Action of the oundary supersymmetry charges Q ± We need to construct the action of the oundary supercharge on the asymptotic states. We expect that the action is given y Q ± = Q ± Q + Q B where Q, Q act on the particles as in the ulk theory Section.. The reason for this is that they are given y integrals of local fermionic densities, and asymptotic particles are localized far away from the wall, so the action of these charges is not affected y the presence of the oundary. Q B is the action of the oundary contriution, which we take to e Q B = γγ, where γ is some unknown parameter related to the energy of the oundary ground state see later. The reason for this choice is that we expect the oundary supercharge to commute with the ulk S-matrix, which is symmetric under the action of Q, Q and Γ y construction[0, ]. is also supported y classical considerations in [4], showing also that the classical version of γ is a function of the parameters in the oundary Lagrangian 6. Next we need to give the action of Q, Q and Γ on the oundary ground state B/. Following [7], we choose Q B =0, Q B =0, Γ B = B. 3 The first two relations express that the oundary ground state is supersymmetric, while the last one shows that it is an eigenvector of Γ. We expect that ecause the ground state is nondegenerate. The choice of the eigenvalue ± is not important, as it could e compensated y a redefinition of γ. It is a consequence of 8 and 3 that the ground state energy is γ /, which will e shown later to e consistent with the action of Q ± on the excited oundary states Supersymmetric reflection amplitudes Now wewould like to impose the supersymmetry constraints on the ground state reflection amplitudes. The two choices Q ± will give different solutions [7, ]. If the oundary supercharge Q + commutes with the reflections theory BSSG + then we otain [ R 0 θ =R θ = θ/πi Γk θ Γk θ ] πi πi Γk θ Γk + {θ θ} = θ/πi P θ θ / k= 4 πi 4 πi 4 If, however, it is Q that commutes with the reflections BSSG then the result is R 0 θ = Kθ iξkiπ θ iξ θ/πi P θ R θ = cos ξ + i sinh θ cos ξ i sinh θ Kθ iξkiπ θ iξ θ/πi P θ 5 Note that our charge Q differs y a sign from that used in [7], while agrees with the convention in []. This is important when comparing our results with those in [7]. 0

11 where ξ is related to γ as γ = M cos ξ. 6 Note that symmetry of the reflection under Γ requires R 0 θ =R θ, thus in the first case BSSG + the reflections also commute with the operator Γ, while in the other case BSSG they do not. We remark that there are no poles in the physical strip in any of the reflection factors aove. In the case BSSG +, the supersymmetry constraints do not determine the value of γ in contrast to the results of [7, ]. The reason is that it is the supersymmetry of the reflections on B a,a=0, which connects γ with a parameter in the reflection matrix itself. However, we construct these reflections y the ootstrap, which determines them completely, and γ is left as a free parameter. As it was argued aove and as will also e seen later γ is connected to the vacuum energy, so it is not a new parameter of the theory in principle it is expressile in terms of the Lagrangian parameters. The only independent parameters introduced y the oundary are η and ϑ which are present in the osonic sine-gordon reflection factors R SG. 5. The general spectrum and the associated reflection factors 5.. The Γ symmetric case BSSG + We start with the analysis of the ground state reflection factors R a θ R SG θ where the SUSY component has the form 4. Since the only poles of these reflection factors are due to the sine-gordon part their explanation has to e similar to that in the osonic theory. However, we have to supplement the formulae for the osonic theory with RSOS indices in a consistent way. The sine-gordon reflection factor has oundary independent poles at i nπ for n =,,..., which can e descried y diagram a. This λ is identical to the non supersymmetric diagram except that it is decorated with RSOS indices, which are displayed inside circles. Clearly the dashed line denotes the full reather supermultiplet now consisting of a oson and a fermion. The oundary dependent poles of R SG are located at iθ = ν n = η λ n +π λ ; n =0,,.... At the position of these poles we associate oundary ound states to the reflection amplitudes R a,a=0, a, / n = K g / a a,/ n iν n, where B, 7 where the g-factor is the SUSY part of the oundary coupling, coming from the SUSY component of the reflection factor for definitions of oundary couplings, see [8]. The

12 a u n / /> /> a / /> a,/ n> Σ x a / / /> x / /> / /,/ n>,/ n> a Soliton ulk pole ootstrap I. c ootstrap II. two states a =0, for a given n form a doulet which realizes the structure, that is the K a kinks can scatter on it. The action of the oundary supercharge on these states can e calculated using the coproduct rules in [0, ], taking into account the action of the charges on the oundary ground state: Q + 0, / n = r γ i M cos ν n, / n Q +, / n = r γ +i M cos ν n 0, / n, r = g /,/ n g / 0,/ n The oundary supercharge satisfies γ Q + a, / n = + M cos ν n + M a, / n which is exactly the relation Q + = H + M Z, since the central charge of this state is Z =we take the ground state / to have Z =0, while the ulk soliton K a/ has Z =, the ground state has energy γ / y virtue of the relations,3 and M cos ν n is the energy that the excited state has relative totheground state 0. The SUSY reflection factors of K a off a, / n can e computed from the ootstrap principle diagrams and c: { g a R a θ = g x S x=0, a where g a g / a,/ n θ iν n S x θ + iν n R x θ } 8 9 The result turns out to e R a θ =P θkθ + iν n Kθ iν n g g a νn θ δ a cos + δ a, sin i 0

13 a a,/ n> a,/ n> ν n ν n wm / > a ν n k / > / / / / > / >,/ n>,/ n> d w type poles e ν type poles Note the appearance of the g factors in the result. They are the SUSY parts of the oundary couplings and come in two types: one corresponds to the asorption of the particle while creating a higher excited oundary state, the other descriing the emission of the particle and transition to some lower excited oundary state. The ones aove are of the asorption type. The residues of the full reflection factor are descried y the product of an emission and an asorption type full oundary coupling for a definition of oundary couplings and their relation to the residue of the reflection factor see [8]. Due to the tensor product structure the full oundary coupling is given y the osonic part multiplied with the SUSY g-factor, as in the case of ulk scattering []. The product of the appropriate emission and asorption SUSY g-factors is constrained to coincide with the value of the SUSY part of the reflection factor at the position of the pole in the osonic factor. It can e seen in general that this does not give enough constraints to determine their value unamiguously due to the degeneracy introduced y the RSOS indices a,, and as no physical quantity should explicitly depend on their value see the example of the relation Q + = H + M Z discussed aove we do not present any solution for them. Being constructed y the ootstrap, the reflection factors 0 necessarily satisfy the constraints of oundary factorization and crossing-unitarity; in addition, they commute with Q + which is guaranteed y the fact that the action of the oundary supercharge is also derived from the ootstrap as in 8. The full reflection factor on the a, / n excited oundary can e otained y multiplying this result with the appropriate excited osonic reflection factor: R a θ Q n η, ϑ, θ or R a θ P ± n η, ϑ, θ. Clearly 0 has neither pole nor zero in the physical strip. So the poles of the reflection factors on the first excited wall are exactly the same as in the non supersymmetric theory: that is they are at iν k or at iw m. The decoration of the non supersymmetric diagrams shows, that diagram e explains the ν type of poles, while diagram d explains the w type, ut only for w m >ν n. For 3

14 w m <ν n we have a oundary ound state which we denote y,a, m, n = K g a,/ n a iw m a, / n,,a, m,n so this is also a doulet, ut now it is the K a type kinks that are ale to reflect on it. It can e checked easily that the relation Q + = H + M Z holds for these states as well, consistently with the previous interpretation of γ for these states Z =0. At this point the question emerges whether the two states a =0, forming the doulet,a, m, n are physically different or there is a possiility for some identification so that a single state can explain the pole in the reflection matrix. This can e decided y examining whether one can descrie the residue of the reflection factor with a single intermediate state, which implies a relation etween the R / 00,R/,R/ 0 and R/ 0 components of the reflection factor at the pole. This relation is violated for generic values of the parameters and so one must really introduce the two states aove. Following the same analysis we performed in [], ut now using a decorated version of the Coleman-Thun diagrams it can e seen that the poles in the reflection matrix on the aove oundary excited state, which can not e explained y Coleman-Thun diagrams are located at iν k. Since the poles appear in association with oth reflection factors R θ, the corresponding oundary states, which are denoted y,,a, k, m, n have a fourfold degeneracy. It is clear that the general oundary ound state has the structure a k...,a, n k...,m,n or,a k...,a, m k,n k...,m,n From this we see that in the supersymmetric case the oundary excited states have a nontrivial degeneracy in contrast to the osonic theory. The degeneracy is laeled y RSOS sequences starting from /. In oth states in the laels a i can freely take the values 0 and, and, as a result, the degeneracy of the states is k. The associated reflection factors can e computed from successive application of the ootstrap procedure, which is illustrated on the figure w m k a k νn w ν w ν m n m n / ak / a / a k k x k / xk / / x / k / k / / 4

15 The result depends on the Z charge of the scattering particles. In the Z =0case the result can e written in the following form R k... θ,, n k k...,m,n = R a k...,a, n k...,m,n a θ f a ia i+ i i+ w mi,ν ni+,θ 3 where f a a w m,ν n,θ is the contriution of the dotted square summing over x =0, that is f a a w m,ν n,θ = S x =0, S x a Collecting the common factors we have where x a i= x θ iw m S θ iν n S x θ + iν n θ + iw m f a a w m,ν n,θ=kθ iw m Kθ + iw m Kθ iν n Kθ + iν n h a a w m,ν n,θ= x=0, g a, n a, m,n g a, m,n θ iwm cos 4i θ iνn cos 4i a, a, n,m,n {a } h a a w m,ν n,θ + π θ + 4 δ iwm x,a cos 4i π δ x,a cos θ + iνn 4i + π 4 δ x, π δ x, In the Z =case, which is indicated with dotted lines on the diagram, the result contains an extra factor where R x k θ, k...,, m k...,m,n,a k...,a, m k...,m,n = hx k a k, k R θ k...,, n k...,m,n a k...,a, n k...,m,n g a k... h x k a k, k = θ a k... θ iπ g k... sin i π δ wmk x k,a k + δ xk, k +cos + π δ x k,a k δ xk, k k The Γ non symmetric BSSG case The discussion of the Γ non symmetric BSSG solution runs entirely parallel to the previous Γ symmetric case, the only difference eing that in the input of the ootstrap procedure, i.e. in the ground state reflection amplitude the supersymmetry factors, R a R a θ R SG θ θ, a =0,, are taken now from 5. 4 These factors depend explicitly on γ, and this dependence pertains in the SUSY reflection amplitudes 5

16 on exited oundaries. Nevertheless, since none of these amplitudes has a pole in the physical strip, following the steps of the previous considerations leads to the same conclusion regarding the indexing and degeneracies of the oundary states. Therefore we concentrate here mainly on the differences etween the two solutions. At the position of the ν n poles in the ground state reflection amplitude we again associate oundary ound states a, / n to R a a =0, as in 7 though of course the present values of oundary couplings may differ from the previous ones. The action of the present oundary supercharge, Q, on these states is Q 0, / n = r γ + M sin ν n, / n, 0, / n. Q, / n = r γ M sin ν n The action of Q on these states is compatile with the relation Q = H M Z, provided we keep the interpretation of γ / as the ground state energy. Using the SUSY factors eqn. 5 in the ootstrap equation 9 for the reflections of the K a kinks on these oundary states gives R a θ = Z θ g [δ a cos ξ cos ν n g + a i sinh θ cosh θ a iδ a, sinh θ cos ξ + sin ν ] n, where ξ is expressed in terms of γ in eqn. 6, and Z θ =P θkθ + iξkθ iξkθ + iν n Kθ iν n. Although this reflection amplitude has a slightly more complicated form than the one in eqn. 0, it also solves the oundary Yang-Baxter equation and the other constraints y construction. The difference etween the two comes from the fact that 0 commutes with Q +, while the present reflection factor is invariant under Q. Finally we point out that the expressions 3,4 for the kink reflection amplitudes on the general higher excited oundary states remain valid in the case BSSG as well, if the ground state reflection factor 4 is replaced y 5. 6 Breather reflection factors The reathers have vertex type scattering matrices in contrast to the RSOS type ones of the kinks. These scattering matrices enter into the equations determining the reflection factors of the reathers, nevertheless there is no need for their explicit form as the reather reflection factors on the various oundaries can e otained from that of the soliton kinks y using the ulk fusion and the ootstrap [9]; the procedure is summarized schematically on diagram f. If two ulk kinks form a ound state at a rapidity difference iρ 0 <ρ<π the ound state is identified with a supermultiplet φ, ψ of mass M cosρ/. In case of the kth reather ρ = ρ k = π kπ. The fusing coefficients of these processes are defined via []: λ K a θ + iρ/k c θ iρ/ = f φ ac φθ + f ψ ac ψθ 6

17 a / / a = Σ x / x / / f The ootstrap procedure for the reather reflection factors on the oundary ground state with the non vanishing coefficients eing: and f φ 0 0 = f φ =π ρ/4π f φ 0 f ψ 0 = f ψ 0 =π ρ/4π if ψ 0 = π ρ/4π f φ = π ρ/4π if ψ ρ π = Kiρ π ρ/π cos 4 ρ + π = Kiρ π ρ/π cos 4 To descrie the ground state reflection amplitudes of the osonic φ and fermionic ψ components we represent them as φθ = K f φ 0θ + iρ/k 0 θ iρ/ + K θ + iρ/k θ iρ/ 0, ψθ = f ψ 0 K 0θ + iρ/k 0 θ iρ/ K θ + iρ/k θ iρ/. These expressions show that they also provide an ordinary doulet representation of the oundary supercharge Q ± and that the fermionic parity Γ act on them in the standard way. The actual reflection factors are otained from the ootstrap equation on diagram f, where the dashed lines represent either φ or ψ. The osonic and fermionic reflection factors are qualitatively different in the Γ symmetric and Γ non symmetric cases, since the ootstrap equations contain oth the R 0 and the R ground state kink reflection amplitudes, and these are significantly different in the two cases. Writing the reather reflection factors on the ground state oundary as φ θ ψ = A+ B φ θ B A ψ 7

18 in the Γ symmetric BSSG + case one otains B = B θ =0, A + = Zθcos i π 4 while in the Γ non symmetric BSSG case we get with A ± = Zθ, A = Zθcos θ i + π 4 Zθ =P θ + iρ/p θ iρ/ Kθ θ/iπ, B = B = Zθ γ cosρ/ sinhθ, M θ γ cosh 4M θ γ i sinh 4M + [ ρ sin 4 [ sin ρ 4 +sinh θ +sinh θ ] ], Zθ =Kθ θ/iπ F θ iρ/f θ + iρ/, Fθ =P θkθ + iξkθ iξ. The two cases are indeed qualitatively different: in the BSSG + solution the conservation of fermionic parity forids the φ ψ reflection on the ground state oundary, while in the BSSG solution this reflection is possile. The form of the BSSG + solution is the same as the one otained in [4] y imposing fermion numer conservation. The structural form of the reflection factors in the case BSSG are identical to the ones otained in [3] from the eight vertex free fermion model with oundary. In [4] it was proposed that γ, which appears explicitly in the reflection matrix, could e fixed in terms of η and ϑ y the oundary ootstrap. Here we see that this is not the case, as the ootstrap gives no constraint for γ, due to degeneracies appearing in the oundary excited states. We recall however that γ is not a free parameter, as it is determined y the ground state oundary energy or equivalently, y the oundary supercharge, thus it can e expressed in terms of the Lagrangian parameters in principle. A more explicit description of the reflection factor of the first reather on the ground state oundary in the BSSG case was given in oundary sinh-gordon model studied in [6], ut the precise connection etween the parameters used in that paper and the present one is yet to e determined. In the osonic theory the ground state reflection amplitude of the kth reather has poles at iθ = η λ π +k l π [ ] k λ, l =0,...,. In the supersymmetric theory, these poles signal the presence of the excited oundary states,a, l, k l as intermediate states in the reather reflection process. Since there are two intermediate states a =0,, the determinant of the reflection matrix should not vanish at the position of these poles as the residue of the reflection matrix at the pole is proportional 8,

19 to the projector on the suspace of on-shell intermediate states. It is straightforward to verify that this is indeed the case for oth the BSSG + and the BSSG solutions. Using the ootstrap procedure it is also possile to otain the reather reflection factors on excited oundaries. If the RSOS sequence characterizing the oundary state ends with a lael a a =0or a =, then e.g. the osonic reather can e represented y φθa,,... n,... = K f φ a θ + iρ/k aθ iρ/ a, a a,... n,... in the ootstrap procedure. To emphasize that even in the BSSG + case there are φ ψ type reflections on excited oundaries we give here the reflection matrix of reathers on the a/ n states. In the asis of φθ0, / n, φθ, / n, ψθ0, / n ψθ, / n it can e written as Ẑθ where D = cos ρ cos νn sin θ and i C ± =cos ν n θ cos i π 4 C D/r 0 C + rd 0 0 D/r C 0 rd 0 0 C + 5 cos ρ cos θ θ cos i i ± π. 4 It is easy to show that in spite of the non trivial oson fermion reflection the operator Γ commutes with this reflection matrix. A nontrivial check on the consistency of the ootstrap solution can e otained y considering the pole structure of the full reflection amplitude containing the SUSY factor 5. The osonic reflection factor of B k on the osonic oundary excited state n has a pole at iθ = π η + π k +n +[]. In the supersymmetric case, this means that a λ λ oundary excited state of the form a, n + k 6 enters as an on-shell intermediate state in the scattering of B k on a, / n. However, due to the doulet oson/fermion structure of the reather naively one would expect 4 states to explain the residue of the 4 4 reflection factor. In the conjectured spectrum, on the other hand, the only possile process goes via 6 and it allows for only two intermediate states a =0,. Therefore one expects that the determinant of the matrix 5 should have a doule zero there. It can e verified y direct calculation that this doule zero is indeed there without imposing any restriction on the parameters. In the reflection of the kth reather on the nth excited oundary, there is another family of poles at iθ = η π k l +, l =0,...,n [] that in the supersymmetric λ λ case should correspond to intermediate states of the form,,a, l, k l, n. At these poles, the numer of intermediate states is 4 a, =0, and so we expect that the determinant of the SUSY factor does not vanish, which indeed turns out to e the case. 9

20 7 Conclusions To start we summarize the results of this paper. We considered the oundary scattering amplitudes in oundary supersymmetric sine-gordon theory BSSG. Imposing the constraint of supersymmetry on solutions of the oundary Yang-Baxter equation, we found two consistent sets of amplitudes that descrie the reflection of solitons off the oundary in its ground state. Then we considered the two ootstrap systems uilt from these fundamental amplitudes and conjectured the closure of this ootstrap, i.e. the set of oundary states and the reflection factors on them. We also derived a relation etween the oundary supercharge and the Hamiltonian and checked that this relation holds for the ootstrap solutions. Although the reflection amplitudes are different, the spectrum of states is the same in the two ootstrap solutions. This common spectrum is characterized partly y a sequence of integers, just like in the case of the ordinary sine-gordon model [], ut also y an RSOS sequence of length k + if the length of the integer sequence is k starting from /. The energy of the state depends only on the integer laels, the different RSOS sequences correspond to degenerate states. It is interesting to note that the non supersymmetric oundary spectrum allows for a tensor product type supersymmetrization, and no further constraints are otained in accord with the ulk case []. In the case of the BSSG + theory, the reflection amplitudes depend on two parameters η and ϑ, that are inherited from the osonic reflection factors and were originally introduced in [8]. In the osonic case it is known how these parameters are related to the parameters of the oundary Lagrangian in the pertured CFT formalism [5]. Besides that, the SUSY algera introduces a further parameter γ, which is related to the energy of the oundary ground state and so must e a function of the parameters of the BSSG Lagrangian. In the BSSG theory the difference is that γ appears also in the expression for the reflection factors themselves. In the osonic case the expression for the oundary energy in terms of Lagrangian parameters is also known [5]. The existence of two different families of solutions and the numer of parameters are in accordance with the expectations that they descrie the scattering in the Lagrangian theories corresponding to the oundary interaction 6 [4]. It is a very interesting and important issue to connect the ootstrap parameters η, θ and the vacuum energy parameter γ to the parameters of the Lagrangian description for the supersymmetric case as well. In the case of the non supersymmetric oundary sine- Gordon theory this was achieved y considering it as a comined ulk and oundary perturation of a c =free massless oson with Neumann oundary condition. However, even the interpretation of the ulk SSG theory as a pertured CFT is nontrivial, and we are investigating this prolem. We are also working on getting more evidence to link the ulk S matrix and the reflection factors to the Lagrangian theory. Work is in progress in these directions and we hope to report on the results in the very near future. Acknowledgments The authors would like to thank G.M.T. Watts for very useful discussions and R.I. Nepomechie for comments on the manuscript. G.T. thanks the Hungarian Ministry of Education for a Magyary Postdoctoral Fellowship, while B.Z. acknowledges partial support from a Bolyai János Research Fellowship. This research was supported in part y the Hungarian Ministry of Education under FKFP 0043/00 and the Hungarian National Science Fund OTKA 0

21 grants T037674/0 and T3499/0. References [] S. Ferrara, L. Girardello and S. Sciuto: An infinite set of conservation laws of the supersymmetric sine-gordon theory, Phys. Lett. B [] C. Ahn: Complete S-matrices of supersymmetric sine-gordon theory and pertured superconformal minimal model, Nucl. Phys. B [3] T. Inami, S. Odake and Y.-Z. Zhang: Supersymmetric extension of the sine-gordon theory with integrale oundary interactions, Phys. Lett. B , hep-th/ [4] R.I. Nepomechie: The oundary supersymmetric sine-gordon model revisited, Phys. Lett. B , hep-th/ [5] C. Ahn and W.M. Koo: Exact oundary S matrices of the supersymmetric sine-gordon theory on a half line, J. Phys. A , hepth/ [6] C. Ahn and R.I. Nepomechie: Exact solution of the supersymmetric sinh-gordon model with oundary, Nucl. Phys. B , hepth/ [7] R.I. Nepomechie: Supersymmetry in the oundary tricritical Ising field theory, preprint UMTG-34, hep-th/0033 [8] A.B. Zamolodchikov: Fractional-spin integrals of motion in pertured conformal field theory, in Fields, Strings and Quantum Gravity, eds. H. Guo, Z. Qiu and H. Tye, Gordon and Breach, 989. [9] R. Shankar and E. Witten: The S matrix of the supersymmetric nonlinear sigma model, Phys. Rev. D [0] K. Schoutens: Supersymmetry and factorizing scattering, Nucl. Phys. B [] T.J. Hollowood and E. Mavrikis: The N= supersymmetric ootstrap and Lie algeras, Nucl. Phys. B , hep-th/ [] G. Takács: A new RSOS restriction of the Zhier-Mikhailov-Shaat model and Φ, 5 perturations of nonunitary minimal models, Nucl. Phys. B , hep-th/ [3] C. Ahn and R.I. Nepomechie: The scaling supersymmetric Yang-Lee model with oundary, Nucl. Phys. B , hep-th/ [4] E. Witten and D.I. Olive: Supersymmetry algeras that include topological charges, Phys. Lett. B

22 [5] Z. Bajnok, L. Palla, G. Takács and F. Wágner: The k-folded sine-gordon model in finite volume, Nucl. Phys. B , hep-th/ [6] N. Graham and R.L. Jaffe: Energy, central charge, and the BPS ound for +-dimensional supersymmetric solitons, Nucl. Phys. B , hep-th/ M.A. Shifman, A.I. Vainshtein and M.B. Voloshin: Anomaly and quantum corrections to solitons in two-dimensional theories with minimal supersymmetry, Phys. Rev. D , hep-th/ A. Litvintsev and P. van Nieuwenhuizen: Once more on the BPS ound for the SUSY kink, preprint YITP-00-8, hep-th/ [7] R. Chatterjee and A. B. Zamolodchikov: Local magnetization in critical Ising model with oundary magnetic field, preprint RU-93-54, hep-th/9365. [8] S. Ghoshal and A.B. Zamolodchikov: Boundary S matrix and oundary state in two-dimensional integrale quantum field theory, Int. J. Mod. Phys. A , Erratum-iid. A , hep-th/ [9] S. Ghoshal: Bound state oundary S matrix of the sine-gordon model, Int. J. Mod. Phys. A , hep-th/ A. Fring, R. Köerle: Factorized scattering in the presence of reflecting oundaries, Nucl. Phys. B , hep-th/ [0] P. Mattsson and P. Dorey: Boundary spectrum in the sine-gordon model with Dirichlet oundary conditions, J. Phys. A , hep-th/ P. Mattsson Integrale Quantum Field Theories, in the Bulk and with a Boundary, Ph.D. thesis, hep-th/06. [] Z. Bajnok, L. Palla, G. Takács and G.Z. Tóth: The spectrum of oundary states in sine-gordon model with integrale oundary conditions, Nucl. Phys. B , hep-th/ [] L. Chim: Boundary S matrix for the tricritical Ising model, Int. J. Mod. Phys. A , hep-th/ [3] C. Ahn and W.M. Koo: Supersymmetric Sine-Gordon Model and the Eight-Vertex Free Fermion Model with Boundary, Nucl. Phys. B , hep-th/ [4] M. Moriconi and K. Schoutens: Reflection matrices for integrale N= supersymmetric theories, Nucl. Phys. B , hep-th/ [5] Al.B. Zamolodchikov, unpulished. Z. Bajnok, L. Palla and G. Takács: Finite size effects in oundary sine-gordon theory, Nucl. Phys. B , hep-th/00857.