Fractional-Spin Integrals of Motion for the Boundary Sine-Gordon Model at the Free Fermion Point

Transcription

1 UMTG 200 Fractional-Spin Integrals of Motion for the Boundary Sine-Gordon Model at the Free Fermion Point Luca Mezincescu and Rafael I. Nepomechie Physics Department, P.O. Box , University of Miami Coral Gables, FL USA Abstract We construct integrals of motion IM for the sine-gordon model with boundary at the free Fermion point β 2 =4π which correctly determine the boundary S matrix. The algebra of these IM boundary quantum group at q = 1 is a one-parameter family of infinite-dimensional subalgebras of twisted sl2. We also propose the structure of the fractional-spin IM away from the free Fermion point β 2 4π.

2 1 Introduction Much is known about trigonometric solutions of the Yang-Baxter equations, primarily because the algebraic structure underlying these equations has been identified and elucidated namely, quantum groups 1]. In contrast, much less is known about corresponding solutions of the boundary Yang-Baxter equation 2], because the relevant algebraic structure some sort of boundary quantum group has not yet been formulated. Motivated in part by such considerations, we have undertaken together with A. B. Zamolodchikov a project 3] to construct fractional-spin integrals of motion 4] - 6] of the sine-gordon model with boundary 7], which should generate precisely such an algebraic structure. A further motivation for this work is to determine the exact relation between the parameters of the action and the parameters of the boundary S matrix given in 7]. We focus here on the special case of the free Fermion point β 2 =4π 8]. We construct Fermionic integrals of motion IM which correctly determine the boundary S matrix, and which flow to the topological charge as the boundary magnetic field h is varied from h =0toh. The algebra of these IM boundary quantum group at q =1isa one-parameter h family of infinite-dimensional subalgebras of twisted sl2. We also give an Ansatz for the structure of the fractional-spin IM away from the free Fermion point β 2 4π. A preliminary account of this work has appeared in Refs. 9] and 10]. We now outline the contents of the paper. In Sec. 2 we briefly review some basic properties of both bulk and boundary sine-gordon field theory for general values of β. In Sec. 3 we consider the special case of the free Fermion point. We first review the Fermionic description of the fractional-spin IM for the bulk theory. There follows the main part of the paper, in which we generalize these results to the boundary theory. In particular, we describe the Fermionic IM and their algebra, and we verify that these IM correctly determine the boundary S matrix. In Sec. 4 we propose the structure of the fractional-spin IM away from the free Fermion point. We conclude in Sec. 5 with a brief discussion of some open problems. 2 Sine-Gordon field theory We briefly review here some basic properties of the sine-gordon field theory for general values of β. We first consider the bulk theory. We describe the soliton/antisoliton S matrix 11] and the fractional-spin integrals of motion 6]. We then turn to the boundary sine-gordon theory, and describe the boundary S matrix 7]. 1

3 2.1 Bulk theory The Lagrangian density of the bulk sine-gordon field theory in Minkowski spacetime is given by L 0 = 1 2 µφ 2 m2 0 cosβφ, 1 β2 where Φx, t is a real scalar field, m 0 has dimensions of mass, and β is a dimensionless coupling constant. For 4π β 2 8π, the particle spectrum consists only of solitons and antisolitons, with equal masses m, and with topological charge T = β dx Φx, t 2 2π x equal to 1 or 1, respectively. The particles two-momenta p µ are conveniently parameterized in terms of their rapidities θ: p 0 = m cosh θ, p 1 =msinh θ. 3 This theory is integrable; i.e., it possesses an infinite number of integer-spin integrals of motion. Consequently, the scattering of solitons is factorizable. See 11] and references therein. For the case of two particles with rapidities θ 1 and θ 2, the two-particle S matrix Sθ whereθ=θ 1 θ 2 is defined by A a1 θ 1 A a2 θ 2 = S b 1 b 2 a 1 a 2 θ A b2 θ 2 A b1 θ 1, 4 where A ± θ are particle-creation operators. The S matrix obeys the Yang-Baxter equation and therefore has the form where S 12 θ S 13 θ θ S 23 θ =S 23 θ S 13 θ θ S 12 θ, 5 Sθ =ρθ aθ bθ cθ 0 0 cθ bθ aθ, 6 aθ = sinλπ u] bθ = sinλu cθ = sinλπ 7 2

4 and u = iθ. The relation between the coupling constant β and the parameter λ of the S matrix λ = 8π β can be inferred 11] from semiclassical results for the S matrix and the mass spectrum. The unitarization factor ρθ can be found in Ref 11]. In addition to having an infinite number of integrals of motion IM of integer spin, the sine-gordon model also has fractional-spin IM. Within the framework of Euclidean-space perturbed conformal field theory 12], the IM Q ± and Q ± with Lorentz spin λ and λ, respectively are given by 6] Q ± = 1 dzj ± d zh ±, 2πi Q ± = 1 d z 2πi J ± dz H ±, 9 where, up to a normalization factor, J ± x,t = exp ± 2iˆβ ] φx,t, J± x,t=exp 2iˆβ ] φx,t, ] ˆβ2 2 H ± x,t = m 0 ˆβ 2 2 exp ±i ˆβ ˆβ φx,t iˆβ φx,t, H ± x,t = m 0 ˆβ2 ˆβ 2 2 exp i 2 ˆβ ˆβ φx,t±iˆβφx,t ], 10 with ˆβ = β/ 4π and where φx, t and φx, t are given by 4π x φx, t = Φx, t dy t Φy, t, 2 4π x φx, t = Φx, t dy t Φy, t, 11 2 respectively. An appropriate regularization prescription is implicit in the above expressions for the currents. These IM obey the so-called q-deformed twisted affine sl2 algebra sl q 2 with zero center, where q = e iπλ

5 Moreover, these IM have nontrivial commutation relations with the particle-creation operators, which in terms of the matrix notation Aθ A θ = A θ are of the form Q ± Aθ = q ±σ 3 Aθ Q ± αe λθ σ Aθ, Q ± Aθ = q σ 3 Aθ Q± ᾱe λθ σ Aθ, TAθ = Aθ T σ 3 Aθ, 13 where the values of α and ᾱ depend on the normalization and regularization of the currents, with α 1, ᾱ 1 for q Associativity of the products in QA a1 θ 1 A a2 θ 2 0 and invariance of the vacuum Q 0 =0 where Q = Q ±,T or Q = Q ±,T ; or, equivalently, Š, Q ] = 0 15 where Š is the S operator and is the comultiplication leads to the S matrix 6, up to the unitarization factor. Note that this derivation of the S matrix yields the relation 8 between the coupling constant β and the parameter λ of the S matrix. Also note that only half of the fractional-spin IM are needed to determine the S matrix. This approach of computing S matrices is a realization in quantum field theory of the general program 1] of linearizing the problem of finding trigonometric solutions of the Yang-Baxter equations. 2.2 Boundary theory Following Ghoshal and Zamolodchikov 7], we consider the sine-gordon field theory on the negative half-line x 0, with action { 0 S = dt dx L 0 M cos β 2 Φ Φ 0 } x=0, 16 where L 0 is given by Eq. 1, and M and Φ 0 are parameters which specify the boundary conditions. There is compelling evidence that this choice of boundary conditions preserves the integrability of the bulk theory, and hence, the scattering of the solitons remains factorizable. The boundary S matrix Rθ,which is defined by7] Aθ B=RθAθ B, 17 4

6 where B is the boundary operator, describes the scattering of solitons and antisolitons off the boundary. The boundary S matrix must satisfy the boundary Yang-Baxter equation 2] S 12 θ 1 θ 2 R 1 θ 1 S 21 θ 1 θ 2 R 2 θ 2 =R 2 θ 2 S 12 θ 1 θ 2 R 1 θ 1 S 21 θ 1 θ 2, 18 and therefore has the form 7], 13] coshλθ iξ i Rθ =rθ k 2 sinh2λθ i k sinh2λθ coshλθ iξ Evidently, this matrix depends on the three boundary parameters ξ, k, k ; however, by a suitable gauge transformation, the off-diagonal elements can be made to coincide. Thus, the boundary S matrix in fact depends on only two boundary parameters, as does the action 16. The relation between these two sets of parameters is not known for general values of β. 3 Free Fermion point As discussed in the Introduction, we would like to construct fractional-spin IM for the sine- Gordon field theory with boundary for general values of β. We consider in this Section the special case β 2 =4πwhich corresponds to q = 1 i.e., undeformed algebra. We begin by reviewing the Fermionic description of the fractional-spin IM for the bulk theory 14]. We then generalize these results to the boundary theory. In particular, we describe the Fermionic IM and their algebra, and we verify that these IM correctly determine the boundary S matrix. 3.1 Bulk theory As is well known 8], the bulk sine-gordon field theory 1 with β 2 =4πis equivalent to a free massive Dirac field theory. The corresponding Lagrangian density is 1 L FF 0 = i 2 Ψ / Ψim ΨΨ = i ψ ψ ψ ψ ψ ψ ψ ψ m ψ ψ ψ ψ ], 20 ψ ψ where Ψ =,Ψ ψ =. Evidently, the Lagrangian has a U1 symmetry; ψ ψ, ψ, have charge 1 and ψ, ψ have charge 1. 1 Our conventions are η 00 = 1 =η 11 ; Ψ =Ψ T γ 0 ;γ 0 =iσ 2 ; γ 1 = σ 1,whereσ i are the Pauli matrices; ± = 1 2 ± 0 1 5

7 By solving the field equations ψ ± = m 2 ψ ±, ψ± = m 2 ψ ±, 21 we are led following the conventions of 7] to the mode expansions m ψ x, t = dθ e θ 2 ωa θe ip x ω A θ e ip x], 4π m ψ x, t = dθ e θ 2 ω A θe ip x ωa θ e ip x], 22 4π where p x = p 0 t p 1 x with p µ given by Eq. 3, and ω = e iπ/4,ω = e iπ/4. Canonical quantization implies that the only nonvanishing anticommutation relations for the modes are { A± θ,a ± θ } =δθθ. 23 We regard A ± θ as creation operators. The fractional-spin integrals of motion 9 and the topological charge 2 have at the free Fermion point the form 14] Q ± = i m Q ± = i m T = dx ψ ± ψ ± = dθ e θ A ± θ A θ, dx ψ ± ψ± = dθ e θ A ± θ A θ, dx ψ ψ ψ ψ = dθ A θ A θ A θ A θ ], 24 where the dot denotes differentiation with respect to time. Here we normalize the IM so that their commutation relations with the particle-creation operators are given by Eq. 13 with α =ᾱ=1. Let us set Q ± 1 Q ±, Q ± 1 Q ±, T 0 T. By closing the algebra of these IM, we obtain the infinite-dimensional twisted affine Lie algebra sl2 with zero center 14]: Q n,qm] = T nm, Tn,Q m] ± = ±2Q ± nm, 25 all other commutators vanish, where Q n = dθ enθ A θ A θ Q n = dθ enθ A θ A θ }, n odd, 26 6

8 T n = dθ e nθ A θ A θ A θ A θ ], n even. 27 Note that Q n = Q n, T n = T n. 28 The expressions in coordinate space for Q ± n and T n have derivatives of highest order n. 3.2 Boundary theory The sine-gordon field theory with boundary 16 for β 2 =4πis equivalent to 9], 15] { 0 } S = dt dx L FF 0 L boundary, 29 where L FF 0 is given by Eq. 20, and L boundary is given by L boundary = i 2 e iφϕ ψ ψ e iφϕ ψ ψ aȧ h e iϕ ψ e iϕ ψ e iφ ψ e iφ ψ a ] x=0, 30 where at is a Fermionic boundary degree of freedom, and h, φ, ϕ are real parameters which specify the boundary conditions. Varying the action gives upon eliminating at through its equations of motion the following boundary conditions e iϕ ψ e iϕ ψ e iφ ψ e iφ ψ =0, 31 x=0 d e iϕ ψ e iϕ ψ e iφ ψ e iφ ψ dt x=0 h 2 e iϕ ψ e iϕ ψ e iφ ψ e iφ ψ x=0 =0. 32 This action is a generalization of the one for the off-critical Ising field theory free massive Majorana field with a boundary magnetic field h 7]. The action given in 15] differs slightly from 30; specifically, it involves two Fermionic boundary degrees of freedom instead of one. However, it gives the same boundary conditions 31, 32, apart from minor differences in notation. The relation between the parameters of the Bosonic and Fermionic actions is discussed in 15]. 7

9 Computing the boundary S matrix Rθ according to the definition 17 and the mode expansions 22 along the lines of Ref. 7], we obtain the result 19, with λ =1and e 2iξ = eiφϕ m h 2 e iφϕ m h 2, 33 k ± = h 1 2 2m rθ = me iφϕ h 2 cosφ ϕ m2 h 4, m cosφ ϕ m2 h 2 h 4 coshφ ϕ i sinh θ m cosh 2 θ. 35 h 2 Indeed, the action 29, 30 was formulated specifically to reproduce this result of 7]. We turn now to the problem of constructing IM. In analogy with the bulk IM 24, we define the following charges on the half-line: Q ± = i m 0 0 dx ψ ± ψ ±, Q ± = i dx m ψ ± ψ±, 0 T = dx ψ ψ ψ ψ. 36 Evidently, these charges are of the form 0 dx J 0,whereJ 0 is the time component of a two-component current J µ which is conserved µ J µ = 0. Nevertheless, in general these charges unlike dx J 0 arenot separately conserved, since J 1 x=0 0. For future convenience, we also define in analogy with 26, 27 the quantities Q n = 0 dθ e nθ A θ } A θ Q n = 0 dθ e nθ A θ, n odd, 37 A θ T n = 0 dθ e nθ A θ A θ A θ A θ ], n even, 38 which also obey the twisted sl2 algebra Q n,q m] Tn,Q ± m] = T nm, = ±2Q ± nm. 39 8

10 We shall see that for the boundary theory 29, 30, the integrals of motion are linear combinations of 37-38, and the algebra of these IM is a subalgebra of 39. Before treating the general case, it is useful to first consider the special cases of fixed h and free h = 0 boundary conditions. Fixed boundary conditions For fixed boundary conditions h, the Fermion fields satisfy ψ e iφϕ ψ x=0 ψ e iφϕ ψ x=0 = 0, = These boundary conditions preserve the U1 symmetry of the bulk Lagrangian. One can show using the field equations that for fixed boundary conditions there are three linearly independent combinations of the charges 36 which are integrals of motion: ˆQ 1 fixed = 2 Q e 2iφϕ Q, ˆQ 1 fixed = 2 Q e 2iφϕ Q, ˆT 0 fixed = 2T. 41 In analogy with the bulk analysis, we wish to rewrite these IM in terms of Fourier modes. A direct but laborious approach is to substitute into the above expressions the mode expansions 22, to use the boundary S matrix to express modes with negative rapidity in terms of modes with positive rapidity see Eq. 17, and then to perform the x integrals and use certain identities which are obeyed by the boundary S matrix. The result is ˆQ ± 1 fixed = 2 Q ± 1 e ±i2φϕ Q 1 ±, ˆT 0 fixed = 2T 0, 42 where Q ± n and T n are given by Eqs. 37, 38. We now observe that the same result can be obtained much more readily by assuming that for the purpose of writing a conserved charge in terms of Fourier modes we can make the replacement 0 dx dx 43

11 in the coordinate-space expression for the charges 36; and then make in the final result the replacement dθ 2 dθ We shall use this device extensively in the work that follows. Moreover, we find higher-derivative integrals of motion which in terms of Fourier modes are given by ˆQ ± nfixed = Q ± n Q± 2n e±i2φϕ Q ± n2 n Q±, n odd 1, ˆT nfixed = T n T n, n even 0, 45 where Q ± n and T n are given by Eqs. 37, 38. See Appendix A for details. Notice that the expressions for ˆQ ± nfixed and ˆT nfixed are invariant under n 2n and n n, respectively. The IM for fixed boundary conditions obey the algebra we now drop the label fixed ˆQ n, ˆQ m] = ˆT nm ˆT nm4 ˆT 2nm ˆT 2nm ˆTn, ˆQ ± m] 2 cos2φ ϕ ˆTnm ˆT nm2, = ±2 ˆQ± mn ˆQ ± mn, 46 and all other commutators vanish. Free boundary conditions For free boundary conditions h = 0, the Fermion fields satisfy ψ e iφϕ ψ = 0, x=0 ψ e iφϕ ψ = x=0 These boundary conditions break the U1 symmetry of the bulk Lagrangian. For free boundary conditions, there are only two linearly independent combinations of the charges 36 which are integrals of motion: ˆQ1 free and ˆQ 1 free,where ˆQ 1 free =2 Q e 2iφϕ Q ie iφϕ T. 48 GoingtoFouriermodes,wehave ˆQ 1 free =2 Q 1 e2iφϕ Q 1 ieiφϕ T 0, 49 10

12 where Q ± n and T n are given by Eqs. 37, 38. Moreover, we find higher-derivative integrals of motion which in terms of Fourier modes are given by ˆQ nfree = Q n Q 2n e2iφϕ Q n2 Q n ie iφϕ T n1 T 1n, n odd 1, ˆT nfree = T n T n T n4 T n4 2ie iφϕ Q n1 Q n3 Q 1n Q 3n 2ie iφϕ Q n1 Q n3 Q 1n Q 3n, n even 2, 50 together with ˆQ nfree. Here Q ± n and T n are given by Eqs. 37, 38. See Appendix A for details. Notice that the expressions for ˆQ nfree and ˆT nfree are invariant under n 2 n and n 4n, respectively; and ˆT nfree=ˆt nfree. The algebra of the IM for free boundary conditions is given by dropping the label free ˆQn, ˆQ m] = ˆT nm ˆT nm2, ˆTn, ˆQ ] m = 2 ˆQmn 2ˆQ mn2 ˆQ mn4 ˆTn, ˆQ m] and all other commutators vanish. General boundary conditions ˆQ mn 2ˆQ mn2 ˆQ mn4, = 2 ˆQ mn 2ˆQ mn2 ˆQ mn4 ˆQ mn2ˆq mn2 ˆQ mn4, 51 A principal result 10] is that for general values of h, the quantities ˆQ 1 =2 Q e 2iφϕ Q ie iφϕ 1 h2 m eiφϕ T im ] eiφϕ ψ x=0 ψ 52 and ˆQ 1 are integrals of motion. A proof is outlined in Appendix B. As h varies from 0 to, the conserved charge ˆQ 1 interpolates between ˆQ 1 free and ˆT 0 fixed : 2 ˆQ 1 h 0 ˆQ 1 free 2 In 9], we exhibited a different set of charges ˆQ 1 old, ˆQ 1 old which interpolate instead between ˆQ 1 free 11

13 h i h2 m e2iϕ ˆT0 fixed. 53 GoingtoFouriermodes,wehave 3 ˆQ 1 = ˆQ 1 free 2i h2 m e2iϕ T 0, 54 where ˆQ 1 free is given by Eq. 49. We also find higher-derivative IM which in terms of Fourier modes are given by n1 ˆQ n = ˆQ nfree 4 h2 2 m ei3ϕφ k Q n2k Q 2kn k=1 n3 2i h2 m e2iϕ 2 n1 2 T2 k1 T n2k1 T 2k1n k=1 n3 4 h4 m 2 e4iϕ n1 n ˆQ 1 fixed 4 k1 k ˆQ n2k fixed, n odd 3, k=1 ] ˆT n = ˆT nfree 2 h2 e iφϕ e iφϕ 4 h4 ˆT m m 2 n2fixed 4i h2 e 2iϕ Q n1 m Q n3 e 2iϕ Q n1 ] Q n3, n even 2, 55 where Q ± n and T n are given by Eqs. 37, 38, Q ± nfixed and T nfixedare given by Eq. 45, and Q ± nfree and T nfreeare given by Eq. 50. See Appendix B for details. Note that these IM also interpolate between free and fixed IM as h varies from 0 to. and ˆQ 1 fixed : ˆQ 1 old h 0 h ˆQ 1 free e 4iϕ ˆQ 1 fixed. The old charges have the drawback of being nonlocal in time; indeed, they are constructed in terms of the quantities Cψ andc ψ, where Cψ isdefinedineq Note that according to our prescription 43, 44 for going to Fourier modes, the boundary term in Eq. 52 gives no contribution. 12

14 etc. These IM obey the following commutation relations: ˆQ1, ˆQ 1] = 2ˆT 2 4i h2 e 2iϕ ˆQ1 e 2iϕ ˆQ 1, m ˆQ3, ˆQ ] 1 = 4i h2 m e2iϕ ˆQ3 2 h2 h 2 m eiφ3ϕ ˆT2 4i m e2iϕ 2 h4 m 2 eiφϕ ˆQ 1, ˆQ3, ˆQ 1] = 2ˆT 4 4i h2 m e2iϕ ˆQ3 2 h2 m eiφϕ ˆT2 4i h2 m e2iϕ ˆQ 1 8i h4 m 2eiφϕˆQ1, ˆT2, ˆQ 1 ] ˆT2, ˆQ 3 ] = 8ˆQ 3 8 = 4ˆQ h2 m eiφϕ 1 h2 m 1 2 h2 m eiφϕ Boundary S matrix revisited ˆQ 1 8 h2 m eiφ3ϕ ˆQ 1, ] e iφϕ e iφϕ 2 h4 ˆQ 3 ˆQ 1 8 m 2 h 2 m eiφ3ϕ 2 h4 m 2 e2iφϕ ˆQ 1, 56 The IM ˆQ 1 and ˆQ 1 as well as the higher IM correctly determine the boundary S matrix. Indeed, the commutation relations of ˆQ1 and ˆQ 1 with Aθ are given by see Eqs. 54, 49,37, 38 ˆQ1,Aθ ] = Zθ Aθ, ˆQ 1,Aθ ] = Z θ Aθ, 57 where the 2 2 matrices Zθ andz θ are given by c bθ c Zθ =2, Z aθ c aθ θ=2 bθ c, 58 and aθ =e θ, bθ=e 2iφϕθ, c = ie iφϕ i h2 m e2iϕ. 59 We next observe that ˆQ 1 Aθ 0 B = Rθ ˆQ 1 Aθ 0 B = Rθ Zθ Aθ 0 B 60 = Zθ Aθ 0 B = Zθ Rθ Aθ 0 B

15 In the first line, we first use the definition 17 of the boundary S matrix, and we then use the commutation relations together with the fact ˆQ 1 0 B = 0; and in the second line we reverse the order of operations. We conclude that An analogous analysis with ˆQ 1 yields Zθ Rθ = Rθ Zθ. 62 Z θ Rθ = Rθ Z θ. 63 Solving these two relations for Rθ leads directly to the boundary S matrix given by Eqs. 19, 33, 34, up to the unitarization factor. 4 Away from the free Fermion point β 2 4π We would like to explicitly construct generalizations of the integrals of motion ˆQ 1 and ˆQ 1 for the sine-gordon model with boundary for β 2 4π. As a preliminary step in this direction, we make the following observation. Define the quantities ˆQ and ˆQ by ] ˆQ = 2 ᾱ 1 Q α 1k Q 2i eiξ 1 q T, k q q 1 ˆQ = 2 k ᾱ 1 Q α 1k k Q 2i eiξ k q T ] 1, 64 q q 1 where Q ±, Q±,andT have commutation relations with Aθ given by Eq. 13. A generalization of the argument given in Eqs using ˆQ and ˆQ instead of ˆQ1 and ˆQ 1, respectively, leads to the boundary S matrix Rθ given by Eq. 19. For q 1, one can verify with the help of Eqs. 33 and 34 that the charges ˆQ and ˆQ reduce to the expression 54 for ˆQ 1 in terms of Fourier modes and its Hermitian conjugate, respectively. It remains to construct ˆQ and ˆQ explicitly in terms of the local sine-gordon field Φx, t, to demonstrate their conservation, and to identify their algebra. 5 Discussion For the case q = 1, we have seen that the boundary quantum group is a one-parameter h family of infinite-dimensional subalgebras of twisted sl2. For fixed h and free 14

16 h = 0 boundary conditions, the subalgebras can be described very explicitly. See Eqs. 46 and 51, respectively. However, for general values of h, the subalgebra is evidently more complicated see Eq. 56, and we have not yet succeeded in giving a complete characterization of its structure. We remark that, to our knowledge, the general subject of subalgebras of affine Lie algebras remains largely unexplored. We emphasize that for the general case q 1, the most pressing problems are to explicitly construct the fractional-spin IM in terms of the local sine-gordon field, and to identify the algebra of these IM. The boundary sine-gordon field theory 16 can be regarded as a free scalar conformal field theory with both bulk and boundary integrable perturbations 12], 7]. Within this framework, it should be easier to treat the special case of only a boundary perturbation i.e., conformal bulk. It would also be interesting to identify analogues of fractional-spin IM for integrable quantum spin chains, which can be solved by the Bethe Ansatz. A related question is whether a systematic construction of the fractional-spin IM can be found, in analogy with the so-called quantum inverse scattering QISM construction of local IM. Finally, we remark that it should be possible to construct fractional-spin IM for other integrable field theories with boundaries. 6 Acknowledgments The work presented here was initiated by A. B. Zamolodchikov, and represents an account of a joint ongoing investigation with him. We are grateful to him for his invaluable advice, kind hospitality, and for giving us the permission to present some of our joint results here. We also thank O. Alvarez, D. Bernard, F. Essler, A. LeClair, P. Pearce, V. Rittenberg, and L. Vinet for discussions; and we acknowledge the hospitality at Bonn University, Rutgers University, and the Benasque Center for Physics, where part of this work was performed. This work was supported in part by DFG Ri 317/13-1, and by the National Science Foundation under Grant PHY Appendix A: Higher-derivative IM for fixed and free boundary conditions We explain here our strategy for constructing higher-derivative integrals of motion. We begin with the case of fixed boundary conditions 40. The first step is to observe using the 15

17 field equations and boundary conditions that the following charges are conserved: q 2n1 fixed = 2i t 2nfixed = m 2n1 0 2 m 2n 0 dx ψn dx ψ n ψ n n1 ψ e 2iφϕ ψ n ψ n1 ], ψ n ] n ψ, 65 where n is a nonnegative integer, and n denotes n th -order time derivative. For n =0,these IM coincide with ˆQ 1 fixed and ˆT 0 fixed, respectively. See Eq. 42. The second step is to express these IM in terms of Fourier modes. Using the prescription 43, 44, we find that q 2n1 fixed = 2 dθ cosh 2n θ e θ e 2iφϕθ] A θ A θ, t 2nfixed = dθ cosh 2n θ A θ A θ A θ A θ ]. 66 The third step is to express these IM in terms of the basis 37, 38 by writing cosh θ in terms of exponentials e ±θ : q 1 fixed = 2 Q 1 e2iφϕ Q 1], q 3 fixed = 1 Q 2 3 Q 1 e 2iφϕ Q 1 Q 3] 1 2 q 1 fixed,. t 0 fixed = 2T 0, t 2 fixed = 1 2 T 2 T t 0 fixed,. 67 Finally, it is important to note that the IM q 2n1 fixed and t 2nfixed for n > 0 are not irreducible ; i.e., they are sums of terms which are separately conserved. The irreducible conserved quantities evidently are ˆQ ± nfixed = Q ± n Q± 2n e±2iφϕ Q ± n2 Q± n, n odd 1, ˆT nfixed = T n T n, n even 0, 68 which is the result stated in Eq

18 We follow a similar procedure for constructing higher-derivative IM for the case of free boundary conditions 47. For this case, we have instead of 65 the following reducible conserved charges: q 2n1 free = 2i 0 dx { m 2n1 ψn n1 ψ me iφϕ ψ n t 2n2 free = 2 m 2n2 0 dx { ψn iφϕ e ψn e 2iφϕ ψ n ]} n ψ, ψ n ψ n1 ψ n1 ψ n ψ n ψ n1 m 2 ψ n ψ n1 ψ n1 e iφϕ ψn n1 ψ ψ n ψ n1 ]}, 69 where n is a nonnegative integer, and the prime denotes differentiation with respect to x. Going to Fourier modes and expressing the result in terms of the basis 37, 38, we eventually arrive at the irreducible IM given in Eq Appendix B: IM for general boundary conditions In this Appendix we outline a proof that, for general boundary conditions, the quantity ˆQ 1 given by Eq. 52 is an integral of motion. We also indicate how to construct the higher-derivative IM. We consider first the expression 52 for ˆQ 1 except without the boundary term: P 0 dx { ψ ψ e 2iφϕ ψ ψ me iφϕ h 2 e 2iϕ ψ ψ ψ ψ }. 70 Differentiating with respect to time, we obtain using the field equations 21 a sum of boundary terms: P = 0 dx 1 ψ ψ e 2iφϕ ψ ψ me iφϕ h 2 e 2iϕ ψ ψ ψ ψ ] = { ψ ψ e 2iφϕ ψ ψ h 2 e 2iϕ ψ ψ ψ ψ m ψ ψ e 2iφϕ ψ ψ e iφϕ ψ ψ ψ ψ ] } x=0, 71 where the prime denotes differentiation with respect to x. Our objective is to express the result as a time derivative of a local boundary term. We proceed by eliminating ψ ± in favor 17

19 of ψ ± using the following identities 9] which can be derived from the boundary conditions 31, 32: ψ e iφϕ ψ e iφ C ψ ] x=0 = 0, ψ e iφϕ ψ e iφ C ψ ] x=0 = 0, 72 where the quantity Cψ is defined by Cψ = h 2 e iϕ ψ e iϕ ψ. 73 We find in this way that P = 0 e iϕ C ψ ψ x=0 = 0 e iφϕ ψ x=0 ψ. 74 Therefore, the quantity ˆQ 1 P e iφϕ ψ ψ x=0 75 is an integral of motion. In a similar manner, we find the following reducible higher-derivative IM: q 2n1 = 2i 0 { dx m 2n1 ψn h 2 2iϕ e ψn1 ψn1 h 2 2iϕ e ψn e 2iφϕ ψ n h 2 ψ n1 n1 ψ h 2 ψ n me iφϕ ψ n t 2n = 2 0 dx m 2n ψ n ψn h 2 ψ n1 e iφϕ ψn1 { ψn h 2 e h 2 ψ n1 n ψ h 2 e 2iϕ ψ n1 h 2 ψn ]} ψn1 h 2 2iϕ e ψn1, 2iϕ ψn1 h 2 2iϕ e ψn1 ψn n ψ h 2 ψ n1 m 2 ψ n ψ n1 ψ n m 2 mh 2 ψn1 ψ n1 ψ n1 ψ n1 cosφϕ ψn1 ψ n1 ψ n1 ψ n1 d dt e iφϕ ψn1 ψ n1 ψ n n1 ψ ψ n1 ] ψ n1 ψ n ψ n1 ]}, 76 where n 1, and n denotes n th -order time derivative. For h = 0, these charges reduce to those given in Eq. 69. Following the procedure described in Appendix A, we obtain the irreducible IM given in Eq

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