Fusing defect for the N = 2 super sinh-gordon model

Journal of Physics: Conference Series PAPER OPEN ACCESS Fusing defect for the N = super sinh-gordon model To cite this article: N I Spano et al 016 J. Phys.: Conf. Ser. 670 01049 View the article online for updates and enhancements. Related content - Exact Jacobian Elliptic Function Solutions to sinh-gordonequation Fu Zun-Tao, Liu Shi-Kuo and Liu Shi-Da - Solving mkdv sinh-gordon equation by a modified variable separated ordinary differential equation method Xie Yuan-Xi - Two-Particle States in the Sinh-Gordon and Sine-Gordon Models in 1+1 Dimensions * Wen-Fa Lu, Bo-Wei Xu and Yu-Mei Zhang This content was downloaded from IP address 148.51.3.83 on 15/06/018 at 04:1

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 Fusing defect for the N = super sinh-gordon model N I Spano, A R Aguirre, J F Gomes and A H Zimerman Rua Dr. Bento Teobaldo Ferraz 71, Block, 01140-070, São Paulo, Brazil E-mail: natyspano@ift.unesp.br Abstract. In this paper we derive the type-ii integrable defect for the N = supersymmetric sinh-gordon sshg model by using the fusing procedure. In particular, we show explicitly the conservation of the modified energy, momentum and supercharges. 1. Introduction An interesting topic in the study of integrable systems is the analysis of their integrability properties in the presence of impurities or defects. Accordingly, defects are introduced in twodimensional integrable field theories as internal boundary conditions located at a fixed point in the x-axis, which connect two different field theories of both sides of it. In particular, after the introduction of the defect, the spatial translation invariance is broken since some constraints are imposed to be satisfied at a particular space point, and hence it would be expected a violation of momentum conservation. However, it was verified in 1-4, that in order to preserve the integrability, the fields of the theory must satisfy a kind of Bäcklund transformation frozen at the defect point. This kind of integrable defects can be classified into two classes: type-i, if the fields on both sides only interact with each other at the defect point, and type-ii if they interact through additional degrees of freedom present only at the defect point 5. The type-ii formulation proved to be suitable not only for describing defects within the Tzitzéica-Bullough-Dodd a -Toda model, which had been excluded from the type-i setting, but it also provided additional types of defects for the sine-gordon sg and others affine Toda field theories ATFT 6. Interestingly, for the sg model 5, 7, and in general for a 1 r -ATFT 8 and the N = 1 sshg models 9, the type-ii defects can be regarded as fused pairs of type-i defects previously placed at different points in space. However, the type-ii defects can be allowed in models that cannot support type-i defects, as it was shown for the a -Toda model 5. On the other hand, the presence of integrable defects in the N = 1 sshg model has been already discussed in 10, 11. However, the kind of defect introduced in those papers can be regarded as a partial type-ii defect since only auxiliary fermionic fields appear in the defect Lagrangian, and consequently it reduces to type-i defect for sinh-gordon model in the bosonic limit. The proper supersymmetric extension of the type-ii defect for the N = 1 sshg model was recently proposed in 9, by using two methods: the generalization of the super-bäcklund transformations, and the fusing procedure. The purpose of this paper is to derive type-ii defects for the N = sshg equation by fusing defects of the kind already known in literature 1. The explicit form of the type-ii Bäcklund transformations for the N = sshg model will be presented. We will also compute its Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the authors and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 modified conserved energy, momentum and supercharges. Finally, by introducing appropriate field transformations, the PT symmetry of the bulk and the defect theories will be discussed.. N = super sinh-gordon model The action for the bulk N = sshg model is given by, with the bulk Lagrangian density, S bulk = dt dx L bulk, 1 L bulk = 1 xφ 1 tφ 1 xϕ + 1 tϕ iψ x t ψ + i ψ x + t ψ +iχ x t χ i χ x + t χ + m φ ϕ +4im ψψ + χχ φ ϕ 4im ψχ + χψ sinh φ sinh ϕ, where φ, ϕ are bosonic fields, ψ, ψ, χ, χ are fermionic fields. Then, the bulk field equations are, x t φ = m sinhφ + 4im ψψ + χχ sinh φ ϕ 4im ψχ + χψ φ sinh ϕ, x t ϕ = m sinhϕ 4im ψψ + χχ φ sinh ϕ +4im ψχ + χψ sinh φ ϕ, x t ψ = m ψ φ ϕ χ sinh φ sinh ϕ, 3 x + t ψ = m ψ φ ϕ χ sinh φ sinh ϕ, x t χ = m χ φ ϕ ψ sinh φ sinh ϕ, x + t χ = m χ φ ϕ ψ sinh φ sinh ϕ, The bulk action and the equation of motion have on-shell N = supersymmetry susy. The susy transformation is given by, δφ = δϕ = iɛ 1 ψ + ɛ 1 ψ iɛ χ + ɛ χ, iɛ ψ + ɛ ψ iɛ1 χ + ɛ 1 χ, δψ = ɛ 1 + φ + ɛ 1 m sinh φ ϕ ɛ + ϕ + ɛ m sinh ϕ φ, δχ = ɛ + φ ɛ m sinh φ ϕ ɛ 1 + ϕ ɛ 1 m sinh ϕ φ, δ ψ = ɛ ϕ + ɛ m sinh ϕ φ ɛ 1 φ + ɛ 1 m sinh φ ϕ, δ χ = ɛ 1 ϕ ɛ 1 m sinh ϕ φ ɛ φ ɛ m sinh φ ϕ, where ɛ k and ɛ k, with k = 1,, are fermionic parameters, and the light-cone notation x ± = x±t, and ± = 1 x ± t has been used. It can be easily verified that the equations of motions are invariant under these transformations. For simplicity, we will focus on the ɛ 1 -projection of the susy transformation 4, which will be denoted δ 1, and then we will compute the associated supercharge Q ɛ1 Q 1. Under a not-rigid susy transformation, i.e with parameters ɛx, t and ɛx, t, L bulk changes by a total derivative δ 1 L bulk = x iɛ 1 ψ φ + + φ + χ ϕ + + ϕ + m ψ sinh φ ϕ m χ sinh ϕ φ + t iɛ 1 ψ φ + φ + χ ϕ + + ϕ + m ψ sinh φ ϕ m χ sinh ϕ φ +ɛ 1 t iψ + φ + iχ + ϕ im ψ sinh φ ϕ + im χ sinh ϕ φ x iψ + φ + iχ + ϕ + im ψ sinh φ ϕ im χ sinh ϕ φ, 5 4

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 if the conservation law inside the last square-bracket in 5 is hold. Then, the associated bulk supercharges Q 1 is given by an integral of the fermionic density, namely Q 1 = dx iψ x + t φ + iχ x + t ϕ im ψ sinh φ ϕ + im χ sinh ϕ φ. 6 The derivation of the remaining supercharges follows the same line of reasoning. Their explicit form is given by the following expressions, Q 1 = Q = Q = dx i ψ x t φ + i χ x t ϕ imψ sinh φ ϕ + imχ sinh ϕ φ, 7 dx iχ x + t φ + iψ x + t ϕ + im χ sinh φ ϕ im ψ sinh ϕ φ, 8 dx i ψ x t ϕ + i χ x t φ + imχ sinh φ ϕ imψ sinh ϕ φ. 9 In next section we introduce the Lagrangian description of type-i defects in N = sshg model. 3. Type-I defect for N = sshg model We consider a defect placed in x = 0 connecting two field theories Φ 1 in the region x < 0 and Φ in the region x > 0, First of all, let us consider a Lagrangian density for the region x < 0 Figure 1. Defect Representation. describing the set of fields Φ 1 φ 1, ψ 1, ψ 1, ϕ 1, χ 1, χ 1 and correspondingly Φ φ, ψ, ψ, ϕ, χ, χ in the region x > 0, and a defect located at x = 0, in the following way L = θxl 1 + θxl + δxl D, 10 where L 1 and L are the bulk Lagrangian densities corresponding to x < 0 and x > 0 regions, respectively, and the defect Lagrangian density L D is given by L D = 1 φ t φ 1 φ 1 t φ 1 ϕ t ϕ 1 ϕ 1 t ϕ + B 0 φ 1, φ, ϕ 1, ϕ i ψ 1 ψ + ψ 1 ψ + i χ 1 χ + χ 1 χ + i f tg + g t f +B 1 φ 1, φ, ϕ 1, ϕ, ψ 1, ψ, ψ 1, ψ, χ 1, χ, χ 1, χ, f, g 11 where f, g are fermionic auxiliary fields. For x = 0, we obtain the following defect conditions, x φ 1 t φ = φ1 B 0 + B 1, x ϕ 1 t ϕ = ϕ1 B 0 + B 1, x φ t φ 1 = φ B 0 + B 1, x ϕ t ϕ 1 = ϕ B 0 + B 1, iψ 1 ψ = ψ1 B 1 = ψ B 1, iχ 1 χ = χ1 B 1 = χ B 1, i ψ 1 + ψ = ψ1 B 1 = ψ B 1, i χ 1 + χ = χ1 B 1 = χ B 1, i t f = g B 1, i t g = f B 1, 1 3

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 where the defect potentials B 0 and B 1 are given given by 1, B 0 = mσ φ + ϕ + + m σ φ ϕ, 13 mσ B 1 = i m i σ φ+ + ϕ + φ ϕ fψ + χ + + f ψ + χ + φ+ ϕ + φ + ϕ gψ + + χ + g ψ χ, 14 where we have denoted φ ± = φ 1 ± φ, ψ ± = ψ 1 ± ψ, χ ± = χ 1 ± χ for the others fields the notation is similar, and σ is a free parameter associated with the defect. In next section, we will perform the fusing of two type-i defects placed at different points in order to construct a type-ii defect for the N = sshg model. 4. Fusing Defects Let us introduce two type-i defects in the N = sshg model, one located at x = 0, and a second one located at x = x 0 where Φ 1 φ 1, ψ 1, ψ 1, ϕ 1, χ 1, χ 1 is a set of fields in the region Figure. Fusing defects. x < 0, Φ 0 φ 0, ψ 0, ψ 0, ϕ 0, χ 0, χ 0 is the correspondingly set of fields for the region 0 < x < x 0 and Φ φ, ψ, ψ, ϕ, χ, χ in the x > 0. Then, the Lagrangian density describing this system can be written as, L = θxl 1 + δxl D1 + θxθx 0 xl 0 δx x 0 L D + θx x 0 L, 15 where the two type-i defect Lagrangian densities L Dk given by at x = 0 k = 1, and x = x 0 k =, are L Dk = 1 φ 0 t φ k φ k t φ 0 1 ϕ 0 t ϕ k ϕ k t ϕ 0 + i χ k χ 0 + χ k χ 0 i ψ i k ψ0 + ψ k ψ 0 1 k f k t g k + i g k t f k + B k 1 + B k 0, 16 with the defect potentials B k 0 = mσ k φ 0 + φ k ϕ 0 + ϕ k + m φ 0 φ k ϕ 0 ϕ k, 17 σ k B k mσk 1 = i φk + φ 0 + ϕ 0 + ϕ k f k ψ 0 + ψ k χ 0 χ k mσk +i φk + φ 0 ϕ k ϕ 0 g k ψ k + ψ 0 + χ 0 + χ k m i1 k φ0 φ k + ϕ k ϕ 0 f k σ k ψ 0 ψ k + χ 0 χ k m i1 k φ0 φ k + ϕ 0 ϕ k g k σ k ψ 0 ψ k χ 0 + χ k, 18 4

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 where σ k, with k = 1, are two free parameters associated two each defect. Thus for each defect we can write the following equations of motion for x = 0 : x φ 1 t φ 0 = φ1 B 1 0 + B 1 1, xϕ 1 t ϕ 0 = ϕ1 B 1 0 + B 1 1, x φ 0 t φ 1 = φ0 B 1 0 + B 1 1, xϕ 0 t ϕ 1 = ϕ0 B 1 0 + B 1 1, iψ 1 ψ 0 = ψ1 B 1 1 = ψ0 B 1 1, iχ 1 χ 0 = χ1 B 1 1 = χ0 B 1 1, i ψ 1 + ψ 0 = ψ1 B 1 1 = ψ0 B 1 1, i χ 1 + χ 0 = χ1 B 1 1 = χ0 B 1 1, and for x = x 0 : i t f 1 = g1 B 1 1, i tg 1 = f1 B 1 1, 19 x φ 0 t φ = φ0 B 0 + B 1, xϕ 0 t ϕ = ϕ0 B 0 + B 1, x φ t φ 0 = φ B 0 + B 1, xϕ t ϕ 0 = ϕ B 0 + B 1, iψ 0 ψ = ψ0 B 1 = ψ B 1, iχ 0 χ = χ0 B 1 = χ B 1, i ψ + ψ 0 = ψ0 B 1 = ψ B 1, i χ + χ 0 = χ0 B 1 = χ B 1, i t f = g B 1, i tg = f B 1, 0 Now taking the limit x 0 0 in the Lagrangian density 15, the bulk Langrangian term L 0 vanishes, and then the resulting Lagrangian density for fused defect becomes of the form of eq. 10 with L D = L D1 L D, namely, L D = 1 φ 0 t φ φ t φ 0 i ψ ψ0 + ψ ψ 0 + i f 1 t g 1 + f t g + g 1 t f 1 + g t f 1 ϕ 0 t ϕ ϕ t ϕ 0 + i χ χ 0 + χ χ 0 + B 1 0 + B 0 + B 1 1 + B 1. 1 We note that the fields of the bulk Langrangian term L 0 only contribute to the total defect Lagrangian at x = 0, and become auxiliary fields. The fused bosonic potential for N = sshg model B 0 = B 1 0 +B 0 = B 0 + +B 0, is a combination of two N = 1 potentials previously obtained in 9, and it can be written explicitly as follows, B + 0 = m B 0 = m φ+ 0 e +φ σ 1 e φ + σ e φ e ϕ + +ϕ 0 σ 1 e ϕ + σ e ϕ φ+ e φ 0 1 e ϕ + ϕ 0 e φ + 1 e φ σ 1 σ 1 e ϕ + 1 e ϕ σ 1 σ + e φ+ +φ 0 σ 1 e φ + σ e φ e ϕ + +ϕ 0 σ 1 e ϕ + σ e ϕ + e φ+ φ 0 1 e σ 1 e ϕ + ϕ 0 φ 1 e ϕ σ 1 + 1 e φ σ, + 1 e ϕ. 3 σ 5

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 For the fermionic part we need to use the equations of motion 1 for each region, in order to eliminate the auxiliary fields ψ 0, ψ 0, χ 0, χ 0, we get ψ 0 = ψ + mσ1 φ1 u + 1 f 1 + φ1 u 1 g mσ 1 + φ u + f + φ u g χ 0 = χ + mσ1 φ1 u + 1 f 1 φ1 u 1 g mσ 1 + φ u + f φ u g ψ 0 = ψ + m + φ1 v1 σ f 1 + φ1 v 1 + g m 1 + φ v 1 σ f + φ v + g χ 0 = χ + m φ1 v1 σ f 1 φ1 v 1 + g m 1 φ v 1 σ f φ v + g where we define the functions u ± k = sinh φk + φ 0 ± ϕ k + ϕ 0, v ± k = sinh φk φ 0 ± ϕ k ϕ 0 4 5 6 7 8 Then noting that iχ χ 0 ψ ψ 0 = i χ χ + ψ ψ + im σ 1 σ φ1 u + 1 φ u f 1g + φ1 u 1 φ u + g 1f, i χ χ 0 ψ ψ0 = i χ χ + ψ ψ+ im σ1 σ φ1 v 1 φ v + f 1g + φ1 v + 1 φ v g 1f, we find that the fermionic part of the fused defect Lagrangian is given by, where L D fermion = i ψ 1 ψ ψ 1 ψ i χ 1 χ χ 1 χ + i f 1 t g 1 + f t g + g 1 t f 1 + g t f B + 1 = i + i m + im + im 9 +B + 1 + B 1, 30 e φ+ +ϕ + + φ 0 +ϕ 0 σ 4 e φ +ϕ 4 f + σ 1 e φ +ϕ 4 f 1 + e m φ+ +ϕ + + φ 0 +ϕ 0 4 σ e φ +ϕ 4 f + σ 1 e φ +ϕ 4 f 1 ψ + χ + e φ+ ϕ + + φ 0 ϕ 0 σ1 4 e φ ϕ 4 g 1 + σ e φ ϕ 4 g + e σ1 σ φ+ ϕ + σ1 σ + φ 0 ϕ 0 4 φ 0 + φ + σ1 e φ ϕ 4 g 1 + σ e φ ϕ 4 + ϕ φ 0 + φ + ϕ + + g ψ + + χ + ϕ 0 + ϕ + + φ f 1 g ϕ 0 + ϕ + φ g 1 f, 31 6

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 and B 1 = i i m m e φ+ +ϕ + φ 0 +ϕ 0 1 4 + e φ+ +ϕ + φ 0 +ϕ 0 4 e φ+ ϕ + φ 0 ϕ 0 1 4 + e φ+ ϕ + φ 0 ϕ 0 4 + im φ 0 φ + σ 1 σ + im σ 1 σ + ϕ φ 0 φ + ϕ e φ +ϕ 4 σ1 1 σ1 e φ +ϕ 4 g 1 1 e φ ϕ 4 σ1 1 σ1 e φ ϕ 4 f 1 1 g 1 1 e φ +ϕ 4 g σ e φ +ϕ 4 σ f 1 1 e φ ϕ 4 f σ e φ ϕ 4 σ + ϕ 0 ϕ + + ϕ 0 ϕ + φ Finally the type-ii defect Lagrangian density can be expressed as follows, g ψ + χ + f ψ + + χ + + φ f 1 g g 1 f. 3 L D = 1 φ 0 t φ φ t φ 0 1 ϕ 0 t ϕ ϕ t ϕ 0 + i ψ 1 ψ ψ 1 ψ i χ 1 χ χ 1 χ + i f 1 t g 1 + f t g + g 1 t f 1 + g t f + B + 0 + B 0 + B+ 1 + B 1. 33 Then, the corresponding type-ii defects conditions for the N = sshg model at x = 0 are, x φ 1 t φ 0 = φ1 B 0 + B 1, x φ t φ 0 = φ B 0 + B 1, x ϕ 1 t ϕ 0 = ϕ1 B 0 + B 1, x ϕ t ϕ 0 = ϕ B 0 + B 1, t φ 1 φ = φ0 B 0 + B 1, t ϕ 1 ϕ = ϕ0 B 0 + B 1, iψ 1 ψ = ψ1 B 1 = ψ B 1, i ψ 1 ψ = ψ1 B 1 = ψ B 1, iχ 1 χ = χ1 B 1 = χ B 1, i χ 1 χ = χ1 B 1 = χ B 1, i t g 1 = f1 B 1, i t f 1 = g1 B 1, i t g = f B 1, i t f = g B 1. The explicit form of the Bäcklund transformation for N = sshg model is presented in appendix A. 5. Conservation of the momentum and energy In this section, we will discuss the modified conserved momentum and energy. Let us consider first the total canonical momentum, which is given by the following contributions with P = 0 dx P 1 + + 0 34 dx P, 35 P p = t φ p x φ p t ϕ p x ϕ p iψ p x ψ p + ψ p x ψp + iχ p x χ p + χ p x χ p, p = 1,. 36 Using the bulk equations 3, we can write the time derivative of momentum as dp 1 = dt xφ 1 + 1 tφ 1 1 xϕ 1 1 tϕ 1 iψ 1 t ψ 1 + ψ 1 t ψ1 +iχ 1 t χ 1 + χ 1 t χ 1 1 xφ 1 tφ + 1 xϕ + 1 tϕ +iψ t ψ + ψ t ψ iχ t χ + χ t χ V 1 + V W 1 + W. 37 7

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 Now, from the explicit form of the defect potentials B 0 = B 0 + + B 0, B 1 = B 1 + + B 1 eqs. 31 3, and the defect conditions 34, we find the following set of relations, given in and ψ B 1 = ψ B 1 = χ B 1 = χ B 1 = 0, ψ+ B 1 = ψ+ B + 1 = χ + B 1 = χ + B + 1 = 0, 38 φ0 B 0 + = φ+ B 0 +, φ 0 B 1 + = φ+ B 1 +, φ0 B0 = φ+ B0, φ 0 B1 = φ+ B1, ϕ0 B 0 + = ϕ+ B 0 +, ϕ 0 B 1 + = ϕ+ B 1 +, 39 ϕ0 B0 = ϕ+ B0, ϕ 0 B1 = ϕ+ B1. Then, by using the above relations and the defect conditions 34, the equation 37 takes the following form, dp dt = φ+ B 0 φ B 0 ϕ+ B 0 ϕ B 0 + φ+ B 1 φ B 1 ϕ+ B 1 ϕ B 1 + φ B 0 φ+ B 1 + φ+ B 0 φ B 1 ϕ B 0 ϕ+ B 1 ϕ+ B 0 ϕ B 1 t φ 0 φ+ B 0 + B 1 t ϕ 0 ϕ+ B 0 + B 1 1 tφ + φ0 B 0 + B 1 1 tϕ + ϕ0 B 0 + B 1 t ψ + ψ+ B 1 + t ψ+ ψ+ B 1 t χ + χ+ B 1 + t χ + χ+ B 1 V 1 + V W 1 + W +i t ψ 1 ψ + ψ 1 ψ χ 1 χ χ 1 χ, 40 where the right-hand-side of the above equation becomes a total time derivative since the defect potentials B 0 ± and B± 1 satisfy the following Poisson-bracket-like relations, V 1 V = φ0 B 0 + φ B0 φ 0 B0 φ B 0 + ϕ 0 B 0 + ϕ B0 + ϕ 0 B0 ϕ B 0 +, 41 W 1 W = φ0 B + 1 φ B 0 φ 0 B 1 φ B + 0 + φ 0 B + 0 φ B 1 φ 0 B 0 φ B + 1 together with the constraint, ϕ0 B + 1 ϕ B 0 ϕ 0 B 1 ϕ B + 0 + ϕ 0 B + 0 ϕ B 1 ϕ 0 B 0 ϕ B + 1 +i f1 B1 g 1 B 1 + f 1 B 1 + g 1 B1 + f B1 g B 1 + f B 1 + g B1, 4 φ0 B + 1 φ B 1 φ 0 B 1 φ B + 1 ϕ 0 B + 1 ϕ B 1 + ϕ 0 B 1 ϕ B + 1 = 0. 43 Then, we get that the modified conserved momentum can be written in a simple form, P = P + B 1 + + B+ 0 B 1 B 0 + iψ 1ψ ψ 1 ψ + χ 1 χ + χ 1 χ. 44 Now, let us consider the total energy where E = 0 dx E 1 + + 0 dx E, 45 E p = 1 xφ p + 1 tφ p 1 xϕ p 1 tϕ p iψ p x ψ p ψ p x ψp +iχ p x χ p χ p x χ p + V p + W p. 46 8

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 We can find its time-derivative in the same way as before by using the bulk equations 3. The result reads de dt = t φ 1 x φ 1 t ϕ 1 x ϕ 1 iψ 1 t ψ 1 ψ 1 t ψ1 + iχ 1 t χ 1 χ 1 t χ 1 t φ x φ + t ϕ x ϕ + iψ t ψ ψ t ψ iχ t χ χ t χ. 47 Then using the defect conditions 34 and the defect potentials 31 3, we find that the modified conserved energy is given by E = E + B 0 + B 1 + i ψ 1 ψ ψ 1 ψ + χ 1 χ χ 1 χ. 48 6. Modified conserved supercharges We have seen that the bulk theory action is invariant under susy transformation 4, and it was explicitly shown for δ 1 projection. However, this is not necessarily true for the defect theory, and therefore we should show that the presence of the defect will not destroy the supersymmetry of the bulk theory. Let us compute the defect contribution for Q 1. By introducing the defect at x = 0, we have Q 1 = 0 + 0 dx iψ 1 + φ 1 + iχ 1 + ϕ 1 im ψ 1 sinh φ 1 ϕ 1 + im χ 1 sinh ϕ 1 φ 1 dx iψ + φ + iχ + ϕ im ψ sinh φ ϕ + im χ sinh ϕ φ.49 Now, by taking the time-derivative respectively, we get dq 1 dt = iψ 1 + φ 1 + iχ 1 + ϕ 1 + im ψ 1 sinh φ 1 ϕ 1 im χ 1 sinh ϕ 1 φ 1 iψ + φ + iχ + ϕ + im ψ sinh φ ϕ im χ sinh ϕ φ. 50 Using the defect conditions 34, we get dq 1 dt = φ+ iψ t + φ 0 iψ φ B 0 + B 1 iψ + φ0 B 0 + + B+ 1 ϕ+ +iχ t + ϕ 0 + iχ ϕ B 0 + B 1 + iχ + ϕ0 B 0 + + B+ 1 im ψ + + ψ φ+ + φ ϕ+ + ϕ sinh +im ψ + ψ φ+ φ ϕ+ ϕ sinh ϕ+ + ϕ φ+ + φ +im χ + + χ sinh im χ + χ sinh ϕ+ ϕ φ+ φ. 51 Now, by making use of the defect conditions intensively, we find after some algebra that the right-hand-side of the equation becomes a total time-derivative, and then the modified conserved 9

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 supercharge can be written in Q 1 = Q 1 +Q D1, with the defect contribution given by the following expression, Q D1 = i mσ k u + k f k + u k g k. 5 k=1 where we have introduced the function, u ± k = sinh φk + φ 0 ± ϕ k + ϕ 0 Analogously, we can find the remaining modified conserved supercharges,. 53 Q = Q + Q D, Q 1 = Q 1 + Q D1, Q = Q + Q D, 54 with the corresponding defect contributions given by, Q D = Q D1 = Q D = i mσ k u + k f k u k g k, 55 k=1 k=1 k=1 i m1 k v σk k f k + v + k g k, 56 i m1 k1 v σk k f k v + k g k, 57 where the functions v ± k are defined to be, v ± k = sinh φk φ 0 ± ϕ k ϕ 0. 58 The derivation of the exact form of the all modified conserved, together with the modified conserved energy and momentum, provides a strong evidence indicating the classical integrability of the fused defect for the N = sshg model. A more rigorous analysis should require the derivation of the generating function of an infinite set of modified conserved quantities. That can be done following the on-shell Lax approach to derive the corresponding type-ii defect K- matrix for the model. From its explicit form is possible to derive an infinite set of modified conserved quantities. Alternative approaches can also be used in order to prove the involutivity of the charges, for instance the off-shell r-matrix and the multisymplectic approach as well. Some of these issues will be considered in future investigations. 7. PT symmetry First of all, it can be shown that the bulk Lagrangian density and fields equations are invariant under the simultaneous transformations of parity transformation P, and time reversal T, namely x, t x, t, if the fields transform as follows, φx, t φx, t, ϕx, t ϕx, t, ψx, t χx, t, ψx, t χx, t, In addition, we notice that applying this PT transformation, the ɛ 1 -projection of the susy transformation maps to the ɛ -projection. Then, as it can be verified, by applying the PT 10

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 transformation over Q 1 we get the second supercharge, namely Q = PTQ 1. Analogously, it happens with Q = PTQ 1. Now, in the presence of a type-i defect the PT transformation relates the fields on the lefthand side to the ones on the right-hand side, and conversely. Then, to preserve the invariance under PT symmetry the fields in the respective bulk Lagrangian densities L p should transform in the following way, φ 1 x, t φ x, t, ϕ 1 x, t ϕ x, t, ψ 1 x, t χ x, t, ψ1 x, t χ x, t. Consequently, the auxiliary fermionic fields f, g in the defect Lagrangian should transform as, ft ft, gt gt. Under these field transformations it can be shown that the type-i defect equations are invariant. On the other hand, the type-ii defect Lagrangian density for the N = sshg model is invariant under PT transformation, if the corresponding auxiliary fields transform in the following way, φ 0 t φ 0 t, ϕ 0 t ϕ 0 t, f 1 t f t, g 1 t g t. 59 In this case, we can verified that if this PT transformation is applied over Q D1 we obtain the defect contribution to the second supercharge, namely Q D = PTQ D1. The same is valid for Q D = PTQ D1. The invariance of the N = sshg model under PT symmetry is strongly related with the description of the equation of motion in the superspace formalism. In such language, the fields appear as components of two N = superfields, one of them being a chiral superfield, while the other one is anti-chiral For more details see 1. The fact that the PT symmetry is preserved in the presence of the type-ii defect, somehow suggests the possibility of describing the defect conditions in terms of superfields. In other words, there should exists a type-ii Bäcklund transformation for the N = sshg equation consistent with the defect conditions of the fused defect. As it was shown for the N = 1 sshg equations, the type-ii defect conditions are equivalent to frozen type-ii Bäcklund transformation of the model see appendix A. 8. Final Remarks In this paper, we have derived a type-ii integrable defect for the N = sshg model by using the fusing procedure. At the Lagrangian level, we have shown that the type-ii defect for this supersymmetric model can be also obtained by fusing two type-i defects located initially at different points in the x-axis. We have shown the conservation of the modified quantities of the energy, momentum and supercharges. Moreover, the invariance under PT symmetry was verified. From the results obtained in this paper and those previously found in 9, it would be interesting to explore the possibility of finding new integrable boundary conditions for the N = 1 and N = sshg models, by performing a consistent half-line limit, specially following the reasoning of 13 15. There are many others algebraic aspects related to type-ii defects that have not been addressed in this work, like the Lax representation, the involutivity of the charges via the r-matrix approach, and the construction of the soliton solutions. These issues are expected to be addressed in future investigations. 11

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 Acknowledgments The authors are very grateful to the organisers of the XXIIIth International Conference on Integrable Systems and Quantum Symmetries ISQS-3 for the opportunity to present this work. NIS, JFG and AHZ would like to thank CNPq-Brasil for financial support. ARA is supported by FAPESP grant 01/13866-3. Appendix A. Type-II Bäcklund transformations for N = sshg model ψ = m σ1 φ0 u + 1 f 1 + φ0 u 1 g 1 + σ φ0 u + f + φ0 u g A.1 ψ = 1 m φ0 v 1 + g 1 + φ0 v1 f 1 1 φ0 v + g + φ0 v f A. σ1 σ χ = m σ1 φ0 u + 1 f 1 φ0 u 1 g 1 + σ φ0 u + f φ0 u g A.3 χ = 1 m φ0 v 1 + g 1 φ0 v1 f 1 1 φ0 v + g φ0 v f A.4 σ1 σ m g 1 = φ 0 φ + σ 1 σ + ϕ + ϕ 0 ϕ + + φ g m + φ0 v1 σ ψ + + χ + A.5 1 + g 1 = m σ1 σ φ 0 + φ + + ϕ + ϕ 0 + ϕ + + φ g mσ 1 φ0 u + 1 ψ + χ + A.6 m f 1 = φ 0 φ + σ 1 σ ϕ + ϕ 0 ϕ + φ f m + φ0 v 1 + σ ψ + χ + A.7 1 + f 1 = m σ1 σ φ 0 + φ + ϕ + ϕ 0 + ϕ + φ f mσ 1 φ0 u 1 ψ + + χ + A.8 g = m φ 0 φ + σ 1 σ ϕ + ϕ 0 ϕ + φ g 1 m φ0 v σ ψ + + χ + A.9 + g = m σ1 σ φ 0 + φ + ϕ + ϕ 0 + ϕ + φ g 1 mσ φ0 u + ψ + χ + A.10 f = m φ 0 φ + σ 1 σ + ϕ + ϕ 0 ϕ + + φ f 1 m φ0 v + σ ψ + χ + A.11 + f = m σ1 σ φ 0 + φ + + ϕ + ϕ 0 + ϕ + + φ f 1 mσ φ0 u ψ + + χ + A.1 1

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 + φ = m φ+ 0 e +φ σ 1 e φ + σ e φ e φ+ +φ 0 σ 1 e φ + σ e φ i m σ1 u + 1 f 1 + σ u + f ψ + χ + + σ 1 u 1 g 1 + σ u g ψ + + χ + im σ1 σ sinh φ 0 + φ + + ϕ f 1 g + sinh φ 0 + φ + ϕ g 1 f A.13 φ = m φ+ e φ 0 1 e φ + 1 e φ e φ+ φ 0 1 e φ + 1 e φ σ 1 σ σ 1 σ + i m 1 v 1 + g 1 1 v + g ψ 1 + χ + + v1 f 1 1 v f ψ + + χ + σ1 σ σ1 σ im + sinh φ 0 φ + σ 1 σ + ϕ f 1 g + sinh φ 0 φ + ϕ g 1 f A.14 φ + + φ 0 = m φ+ e φ 0 1 e φ 1 e φ e φ+ φ 0 1 e φ 1 e φ σ 1 σ σ 1 σ + i m 1 v 1 + g 1 + 1 v + g ψ 1 + χ + + v1 f 1 + 1 v f ψ + + χ + σ1 σ σ1 σ im sinh ϕ 0 ϕ + σ 1 σ + φ f 1 g sinh ϕ 0 ϕ + φ g 1 f A.15 + φ + φ 0 = m φ+ 0 e +φ σ 1 e φ σ e φ e φ+ +φ 0 σ 1 e φ σ e φ i m σ1 u + 1 f 1 σ u + f ψ + χ + + σ 1 u 1 g 1 σ u g ψ + + χ + im σ1 σ sinh ϕ 0 + ϕ + + φ f 1 g sinh ϕ 0 + ϕ + φ g 1 f A.16 + ϕ = m e ϕ + +ϕ 0 σ 1 e ϕ + σ e ϕ e ϕ + +ϕ 0 σ 1 e ϕ + σ e ϕ + i m σ1 u + 1 f 1 + σ u + f ψ + χ + σ 1 u 1 g 1 + σ u g ψ + + χ + + im σ1 σ sinh ϕ 0 + ϕ + + φ f 1 g + sinh ϕ 0 + ϕ + φ g 1 f A.17 ϕ = m e ϕ + ϕ 0 1 e ϕ + 1 e ϕ e ϕ + ϕ 0 1 e ϕ + 1 e ϕ σ 1 σ σ 1 σ i m 1 v 1 + g 1 1 v + g ψ 1 + χ + v1 f 1 1 v f ψ + + χ + σ1 σ σ1 σ im sinh ϕ 0 ϕ + σ 1 σ + φ f 1 g + sinh ϕ 0 ϕ + φ g 1 f A.18 ϕ + + ϕ 0 = m e ϕ + ϕ 0 1 e ϕ 1 e ϕ e ϕ + ϕ 0 1 e ϕ 1 e ϕ σ 1 σ σ 1 σ i m 1 v 1 + g 1 + 1 v + g ψ 1 + χ + v1 f 1 + 1 v f ψ + + χ + σ1 σ σ1 σ im + sinh φ 0 φ + σ 1 σ + ϕ f 1 g sinh φ 0 φ + ϕ g 1 f A.19 13

XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-3 IOP Publishing Journal of Physics: Conference Series 670 016 01049 doi:10.1088/174-6596/670/1/01049 + ϕ + ϕ 0 = m e ϕ + +ϕ 0 σ 1 e ϕ σ e ϕ e ϕ + +ϕ 0 σ 1 e ϕ σ e ϕ + i m σ1 u + 1 f 1 σ u + f ψ + χ + σ 1 u 1 g 1 σ u g ψ + + χ + + im σ1 σ sinh φ 0 + φ + + ϕ f 1 g sinh φ 0 + φ + ϕ g 1 f It was verified that these Bäcklund transformations correspond to the equations 3 for each fields. A.0 References 1 P. Bowcock, E. Corrigan and C. Zambon, Classically integrable field theories with defects, Int. J. Mod. Phys. A19 004 8 hep-th/03050. P. Bowcock, E. Corrigan and C. Zambon, Affine Toda field theories with defects, JHEP 01 004 056 hep-th/040100. 3 E. Corrigan and C. Zambon, Jump-defects in the nonlinear Schrödinger model and other non-relativistic field theories, Nonlinearity 19 006 1447 nlin/051038. 4 P. Bowcock and J. M. Umpleby, Quantum complex sine-gordon dressed boundaries, JHEP 38 008 0811 hep-th/0809.0661 5 E. Corrigan and C. Zambon, A new class of integrable defects, J. Phys. A4 009 47503 hep-th/0908.316. 6 E. Corrigan and C. Zambon, Integrable defects in affine Toda field theory and infinite dimensional representations of quantum groups, Nucl. Phys B 848 011 545 arxiv:101.4186. 7 E. Corrigan and C. Zambon, A transmission matrix for a fused pair of integrable defects in the sine-gordon model, J. Phys. A 43 010 34501 arxiv:1006.0939. 8 C. Robertson, Folding defect affine Toda field theories, J. Phys. A 47 014 18501 arxiv:1304.319. 9 A.R Aguirre, J.F Gomes, N.I. Spano and A.H. Zimerman, Type-II super Baäcklund transformation and integrable defects for the N = 1 super sinh-gordon model, JHEP 06 015 15 arxiv:1504.07978. 10 J.F. Gomes, L.H. Ymai, and A.H. Zimerman, Classical integrable super sinh-gordon equation with defects, J. Phys. A 39 006 7471 hep-th/0601014. 11 A.R. Aguirre, J.F. Gomes, N.I. Spano and A.H. Zimerman, N=1 super sinh-gordon model with defects revisited, JHEP 0 015 175 arxiv:141.579. 1 J.F. Gomes, L.H. Ymai and A.H. Zimerman, Integrability of a classical N = super sinh-gordon model with jump defects, JHEP 03 008 001 hep-th/0710.1391. 13 E. Corrigan and C. Zambon, Infinite dimension reflection matrices in the sine-gordon model with a boundary, JHEP 06 01 050 arxiv:10.6016. 14 A.R. Aguirre, J.F. Gomes, L.H. Ymai and A.H. Zimerman, N=1 super sinh-gordon model in the half line: Breather solutions, JHEP 04 013 136 arxiv:1304.458. 15 C. Zambon, The classical nonlinear Schrödinger model with a new integrable boundary, JHEP 08 014 036 arxiv:1405.0967. 14