NUPHB14536S0550-3213(18)30373-010.1016/j.nuclphysb.2018.12.025Quantum Field Theory and Statistical SystemsGeneralized sℓ(2) Gaudin algebra and corresponding Knizhnik–Zamolodchikov equationI.Saloma⁎isalom@ipb.ac.rsN.Manojlovićbcnmanoj@ualg.ptN.Cirilo Antóniodnantonio@math.ist.utl.ptaInstitute of Physics, University of Belgrade, P.O. Box 57, 11080 Belgrade, SerbiaInstitute of PhysicsUniversity of BelgradeP.O. Box 57Belgrade11080SerbiabGrupo de Física Matemática da Universidade de Lisboa, Campo Grande, Edifício C6, PT-1749-016 Lisboa, PortugalGrupo de Física Matemática da Universidade de LisboaCampo GrandeEdifício C6LisboaPT-1749-016PortugalcDepartamento de Matemática, F.C.T., Universidade do Algarve, Campus de Gambelas, PT-8005-139 Faro, PortugalDepartamento de Matemática, F.C.T.Universidade do AlgarveCampus de GambelasFaroPT-8005-139PortugaldCenter for Functional Analysis, Linear Structures and Applications, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, PortugalCenter for Functional Analysis, Linear Structures and ApplicationsInstituto Superior TécnicoUniversidade de LisboaAv. Rovisco PaisLisboa1049-001Portugal⁎Corresponding author.Editor: Hubert SaleurAbstractThe Gaudin model has been revisited many times, yet some important issues remained open so far. With this paper we aim to properly address its certain aspects, while clarifying, or at least giving a solid ground to some other. Our main contribution is establishing the relation between the off-shell Bethe vectors with the solutions of the corresponding Knizhnik–Zamolodchikov equations for the non-periodic sℓ(2) Gaudin model, as well as deriving the norm of the eigenvectors of the Gaudin Hamiltonians. Additionally, we provide a closed form expression also for the scalar products of the off-shell Bethe vectors. Finally, we provide explicit closed form of the off-shell Bethe vectors, together with a proof of implementation of the algebraic Bethe ansatz in full generality.1IntroductionHistorically, Gaudin model was first proposed almost half a century ago [1–3], and has promptly gained attention primarily due to its long-range interactions feature [4,5]. It was shortly generalized to different underlining simple Lie algebras, as well as to trigonometric and elliptic types, cf. [6–9] and the references therein. The non-periodic boundary conditions were treated somewhat later [10–17], while in [18,19] we have derived the generating function of the sℓ(2) Gaudin Hamiltonians with boundary terms and obtained the spectrum of the generating function with the corresponding Bethe equations. The very latest developments are taking the field in various new directions, e.g. [20–23] which shows that the topic is still very attractive.However, in spite of the substantial interest for the topic, certain issues have not yet been, to our knowledge, fully addressed. First and foremost, we note that the relation of the Knizhnik–Zamolodchikov (KZ) equations [24] with the Gaudin sℓ(2) model [25,26] with non-periodic boundary was not yet established for arbitrary spins. Hikami comes close to this goal in his paper [10], but does not tackle the issue in full generality – namely, he constrains his analysis to a special case of equal spins at all nodes, moreover fixing these spins to the value 12. He also does not provide the expression for the norms of the eigenvectors of the Gaudin Hamiltonians, which can be obtained from the KZ approach. One of our goals here is to improve on both of these points: we successfully establish the relation between solutions of the corresponding KZ equations with the off-shell Bethe vectors in the case of arbitrary spins and derive the norm formula.Superior to the formula for norm of the on-shell Bethe vectors is a formula for scalar product of arbitrary off-shell Bethe vectors. Following an approach laid in [27], we derive such an expression pertinent to the non-periodic sℓ(2) case for arbitrary spins, in a closed form. The expression involves a sum of certain matrix determinants and its significance stems from the fact that it represents the first step towards the correlation functions.En route to our treatment of the KZ equations, we present a closed form expression for the off-shell Bethe vectors and prove the implementation of the algebraic Bethe ansatz in full generality (for arbitrary reflection matrices and to arbitrary number of excitations). Such a development was a result of a suitable change of generalized Gaudin algebra basis (as compared to the one used in [19]), combined with observation of certain algebraic relations that we came across. The resulting simplifications have also facilitated calculations related to KZ equations.The paper is structured as follows. In the next section, we introduce some standard notions while nevertheless relying heavily on the notation and conclusions of our previous paper [19], to which we direct the reader as a preliminary. The third section is devoted to the task of deriving the general off-shell form of the Bethe vectors and to proving its validity. As a key step to this end we, within the same section, first present a new basis of the generalized Gaudin algebra [28,29], and point to its advantages. In the fourth section we finally turn to KZ equations, establishing their relation to the previously derived Bethe vectors and obtaining the norm formula. In the same section we also present the novel formula for the scalar product of off-shell Bethe vectors. Finally, we summarize our results in the last section.2PreliminariesThe generating function of the sℓ(2) Gaudin Hamiltonians with boundary terms was derived in [19]. Besides, the suitable Lax operator, accompanied by the corresponding linear bracket and an appropriate non-unitary r-matrices, as well as the transfer matrix, were also obtained. In this section we will briefly review only the most relevant of these results, while for the details of the notations and derivation we refer to the [19].We study the sℓ(2) Gaudin model with N sites, characterised by the local space Vm=C2sm+1 and inhomogeneous parameter αm, implying non-periodic boundary conditions. The relevant classical r-matrix was given e.g. in [6], r(λ)=−Pλ, where P is the permutation matrix in C2⊗C2.In the case of periodic boundary conditions, this structure is essentially sufficient (after proceeding in the standard manner) to obtain the complete solution of the system [6], together with the corresponding correlation functions [30]. However, the non-periodic case which is the subject of our present consideration is substantially more involved. In this case, of relevance is the classical reflection equation [31–33]:(2.1)r12(λ−μ)K1(λ)K2(μ)+K1(λ)r21(λ+μ)K2(μ)==K2(μ)r12(λ+μ)K1(λ)+K2(μ)K1(λ)r21(λ−μ). In [19] we have derived the general form of the K-matrix solution, and have shown that it can be, without any loss of generality, brought into the upper triangular form:(2.2)K(λ)=(ξ−λνλψ0ξ+λν), where neither of the parameters ξ,ψ,ν depends on the spectral parameter λ.In the course of our analysis in [19] we arrived to the generalized sℓ(2) Gaudin algebra [28,29] with generators e˜(λ),h˜(λ) and f˜(λ). To facilitate later comparison with the new basis, we give the three nontrivial relations:(2.3)[h˜(λ),e˜(μ)]=2λ2−μ2(e˜(μ)−e˜(λ)),(2.4)[h˜(λ),f˜(μ)]=−2λ2−μ2(f˜(μ)−f˜(λ))−2ψν(λ2−μ2)ξ(μ2h˜(μ)−λ2h˜(λ))−ψ2(λ2−μ2)ξ2(μ2e˜(μ)−λ2e˜(λ)),(2.5)[e˜(λ),f˜(μ)]=2ψν(λ2−μ2)ξ(μ2e˜(μ)−λ2e˜(λ))−4λ2−μ2((ξ2−μ2ν2)h˜(μ)−(ξ2−λ2ν2)h˜(λ)), as well as the form of generating function of the Gaudin Hamiltonians in [19]:(2.6)τ(λ)=2λ2(h˜2(λ)+2ν2ξ2−λ2ν2h˜(λ)−h˜′(λ)λ)−2λ2ξ2−λ2ν2(f˜(λ)+ψλ2νξh˜(λ)+ψ2λ24ξ2e˜(λ)−ψνξ)e˜(λ).In [19] we tried to implement the algebraic Bethe ansatz based on these generators. Although the approached looked promising and resulted in the conjecture for the spectra of the generating function τ(λ) and the corresponding Gaudin Hamiltonians, the expression for the Bethe vector φM(μ1,μ2,…,μM), for an arbitrary positive integer M, was missing. It turned out, as we show in the following section, that the full implementation of the algebraic Bethe ansatz in this case requires to define a new set of generators which will enable explicit expressions for the Bethe vectors as well as the algebraic proof of the off shell action of the generating function τ(λ) and its spectrum.3New generators and the eigenvectorsIn the algebraic Bethe ansatz it is essential to find the commutation relations between the generating function and a product of the creation operators in a closed form. To this end, with the aim to simplify the relations (2.4) and (2.5) as well as the expression (2.6), we introduce new generators e(λ),h(λ) and f(λ) as the following linear combinations of the previous ones:(3.1)e(λ)=e˜(λ),h(λ)=h˜(λ)+ψ2ξνe˜(λ),f(λ)=f˜(λ)+ψξνh˜(λ)+ψ24ν2e˜(λ). It is straightforward to check that in the new basis we still have(3.2)[e(λ),e(μ)]=[h(λ),h(μ)]=[f(λ),f(μ)]=0, while the key simplification occurs in the three nontrivial relations which are now given by(3.3)[h(λ),e(μ)]=2λ2−μ2(e(μ)−e(λ)),(3.4)[h(λ),f(μ)]=−2λ2−μ2(f(μ)−f(λ)),(3.5)[e(λ),f(μ)]=−4λ2−μ2((ξ2−μ2ν2)h(μ)−(ξ2−λ2ν2)h(λ)).By using these generators the expression for the generating function of the Gaudin Hamiltonians with boundary terms (2.6) also simplifies. We invert the relations (3.1) and obtain the expression for the generating function in terms of the new generators(3.6)τ(λ)=2λ2(h2(λ)+2ν2ξ2−λ2ν2h(λ)−h′(λ)λ)−2λ2ξ2−λ2ν2f(λ)e(λ). Evidently we have achieved our first objective, as the relations (3.3)–(3.5) and the expression (3.6) are much simple than before. Below we will demonstrate how these new results facilitate the study of the Bethe vectors.As in [19], we define the vacuum Ω+ which is annihilated by e(λ), while being an eigenstate for h(λ):(3.7)h(λ)Ω+=ρ(λ)Ω+,withρ(λ)=1λ∑m=1N(smλ−αm+smλ+αm)=∑m=1N2smλ2−αm2.The next relevant remark is that the vector Ω+ is an eigenvector of the generating function τ(λ). To show this we use (3.6) and the action (3.7):(3.8)τ(λ)Ω+=χ0(λ)Ω+=2λ2(ρ2(λ)+2ν2ρ(λ)ξ2−λ2ν2−ρ′(λ)λ)Ω+.Our main aim in this section it to prove that the generator f(λ) (3.1) defines the Bethe vectors naturally, that is, to show that the Bethe vector in the general case is given by the following symmetric function of its arguments:(3.9)φM(μ1,μ2,…,μM)=f(μ1)⋯f(μM)Ω+. We stress that this was not possible in the old basis (of tilde operators), and thus the general form of the Bethe vector lacked in [19].The action of the generating function of the Gaudin Hamiltonians τ(λ) on φM(μ1,μ2,…,μM) is given by(3.10)τ(λ)φM(μ1,μ2,…,μM)=[τ(λ),f(μ1)⋯f(μM)]Ω++χ0(λ)φM(μ1,μ2,…,μM).The key part of the proof will be to determine the commutator in the first term of the righthand side. Due to the simplicity of the new commutation relations (3.3)–(3.5) we will show that it is now possible to evaluate this commutator in an algebraically closed form. As the first step we will calculate the commutator between the generating function (3.6) and a single generator f(λ). A straightforward calculation yields(3.11)[τ(λ),f(μ)]=−8λ2λ2−μ2f(μ)(h(λ)+ν2ξ2−λ2ν2)+8λ2λ2−μ2ξ2−μ2ν2ξ2−λ2ν2f(λ)(h(μ)+ν2ξ2−μ2ν2).For the general case, we assert that the following holds:(3.12)[τ(λ),f(μ1)⋯f(μM)]=f(μ1)⋯f(μM)∑i=1M−8λ2λ2−μi2(h(λ)+ν2ξ2−λ2ν2−∑j≠iM1λ2−μj2)+8λ2λ2−μ12ξ2−μ12ν2ξ2−λ2ν2f(λ)f(μ2)⋯f(μM)(h(μ1)+ν2ξ2−μ12ν2−∑j≠1M2μ12−μj2)⋮+8λ2λ2−μM2ξ2−μM2ν2ξ2−λ2ν2f(μ1)⋯f(μM−1)f(λ)×(h(μM)+ν2ξ2−μM2ν2−∑j=1M−12μM2−μj2).Our proof of this statement is based on the induction method: we assume that, for some integer M≥1, the above formula (i.e. the induction hypothesis) is satisfied and proceed to show that this assumption implicates the same relation for the product of M+1 operators. To this end we write(3.13)[τ(λ),f(μ1)⋯f(μM)f(μM+1)]=[τ(λ),f(μ1)⋯f(μM)]f(μM+1)+f(μ1)⋯f(μM)[τ(λ),f(μM+1)]. To evaluate the first term on the right-hand-side of (3.13) we use the induction assumption (3.12), while in the second term we apply (3.11) and obtain(3.14)[τ(λ),f(μ1)⋯f(μM+1)]=f(μ1)⋯f(μM)∑i=1M−8λ2λ2−μi2(h(λ)+ν2ξ2−λ2ν2−∑j≠iM1λ2−μj2)f(μM+1)+8λ2λ2−μ12ξ2−μ12ν2ξ2−λ2ν2f(λ)f(μ2)⋯f(μM)×(h(μ1)+ν2ξ2−μ12ν2−∑j≠1M2μ12−μj2)f(μM+1)⋮+8λ2λ2−μM2ξ2−μM2ν2ξ2−λ2ν2f(μ1)⋯f(μM−1)f(λ)×(h(μM)+ν2ξ2−μM2ν2−∑j≠MM2μM2−μj2)f(μM+1)+f(μ1)⋯f(μM)(−8λ2λ2−μM+12f(μM+1)(h(λ)+ν2ξ2−λ2ν2)+8λ2λ2−μM+12ξ2−μM+12ν2ξ2−λ2ν2f(λ)(h(μM+1)+ν2ξ2−μM+12ν2)). Then, using (3.4), we rearrange the terms having f(μM+1) on the right(3.15)[τ(λ),f(μ1)⋯f(μM+1)]=f(μ1)⋯f(μM+1)∑i=1M−8λ2λ2−μi2(h(λ)+ν2ξ2−λ2ν2−∑j≠iM1λ2−μj2)+f(μ1)⋯f(μM)∑i=1M−8λ2λ2−μi2(−2λ2−μN+12(f(μM+1)−f(λ)))+f(μ1)⋯f(μM+1)−8λ2λ2−μM+12(h(λ)+ν2ξ2−λ2ν2)+8λ2λ2−μ12ξ2−μ12ν2ξ2−λ2ν2f(λ)f(μ2)⋯f(μM+1)(h(μ1)+ν2ξ2−μ12ν2−∑j≠1M2μ12−μj2)+8λ2λ2−μ12ξ2−μ12ν2ξ2−λ2ν2f(λ)f(μ2)⋯f(μM)(−2μ12−μM+12(f(μM+1)−f(μ1)))⋮+8λ2λ2−μM2ξ2−μM2ν2ξ2−λ2ν2f(μ1)⋯f(μM−1)f(λ)f(μM+1)×(h(μM)+ν2ξ2−μM2ν2−∑j=1M−12μM2−μj2)+8λ2λ2−μM2ξ2−μM2ν2ξ2−λ2ν2f(μ1)⋯f(μM−1)f(λ)(−2μM2−μM+12(f(μM+1)−f(μM)))+8λ2λ2−μM+12ξ2−μM+12ν2ξ2−λ2ν2f(μ1)⋯f(μM)f(λ)(h(μM+1)+ν2ξ2−μM+12ν2). The next step is to add similar terms appropriately(3.16)[τ(λ),f(μ1)⋯f(μM+1)]=f(μ1)⋯f(μM+1)∑i=1M−8λ2λ2−μi2(h(λ)+ν2ξ2−λ2ν2−∑j≠iM+11λ2−μj2)+f(μ1)⋯f(μM+1)−8λ2λ2−μM+12(h(λ)+ν2ξ2−λ2ν2−∑j=1M1λ2−μj2)+8λ2λ2−μ12ξ2−μ12ν2ξ2−λ2ν2f(λ)f(μ2)⋯f(μM+1)(h(μ1)+ν2ξ2−μ12ν2−∑j≠1M+12μ12−μj2)⋮+8λ2λ2−μM2ξ2−μM2ν2ξ2−λ2ν2f(μ1)⋯f(μM−1)f(λ)f(μM+1)×(h(μM)+ν2ξ2−μM2ν2−∑j≠MM+12μM2−μj2)+8λ2λ2−μM+12ξ2−μM+12ν2ξ2−λ2ν2f(μ1)⋯f(μM)f(λ)(h(μM+1)+ν2ξ2−μM+12ν2)+f(μ1)⋯f(μM)f(λ)∑i=1M(−8λ2λ2−μi22λ2−μM+12+8λ2λ2−μi22μi2−μM+12ξ2−μi2ν2ξ2−λ2ν2). Using the following identity(3.17)−λ2λ2−μi21λ2−μM+12+λ2λ2−μi21μi2−μM+12ξ2−μi2ν2ξ2−λ2ν2=λ2λ2−μM+121μi2−μM+12ξ2−μM+12ν2ξ2−λ2ν2, for i=1,…,N, we can bring together all the terms in the last two lines of (3.16) and obtain the final expression(3.18)[τ(λ),f(μ1)⋯f(μM+1)]=f(μ1)⋯f(μM+1)∑i=1M+1−8λ2λ2−μi2(h(λ)+ν2ξ2−λ2ν2−∑j≠iM+11λ2−μj2)+8λ2λ2−μ12ξ2−μ12ν2ξ2−λ2ν2f(λ)f(μ2)⋯f(μM+1)(h(μ1)+ν2ξ2−μ12ν2−∑j≠1M+12μ12−μj2)⋮+8λ2λ2−μN2ξ2−μM2ν2ξ2−λ2ν2f(μ1)⋯f(μM−1)f(λ)f(μM+1)×(h(μM)+ν2ξ2−μM2ν2−∑j≠MM+12μM2−μj2)+8λ2λ2−μM+12ξ2−μM+12ν2ξ2−λ2ν2f(μ1)⋯f(μM)f(λ)×(h(μM+1)+ν2ξ2−μM+12ν2−∑j=1M2μM+12−μj2). Since we have already explicitly showed that the induction hypothesis is valid for M=1 (the (3.11) is a special case of (3.12)), this completes our proof of (3.12) by induction.Now, using the result (3.12), we finally find the off shell action (3.10) of the generating function τ(λ) on φM(μ1,μ2,…,μM) to be:(3.19)τ(λ)φM(μ1,μ2,…,μM)=χM(λ,μ1,μ2,…,μM)φM(μ1,μ2,…,μM)+8λ2λ2−μ12ξ2−μ12ν2ξ2−λ2ν2(ρ(μ1)+ν2ξ2−μ12ν2−∑j≠1M2μ12−μj2)φM(λ,μ2,…,μM)⋮(3.20)+8λ2λ2−μM2ξ2−μM2ν2ξ2−λ2ν2(ρ(μM)+ν2ξ2−μM2ν2−∑j=1M−12μM2−μj2)φM(μ1,…,μM−1,λ), and the eigenvalue is(3.21)χM(λ,μ1,μ2,…,μM)=χ0(λ)−∑i=1M8λ2λ2−μi2(h(λ)+ν2ξ2−λ2ν2−∑j≠iM1λ2−μj2). The above off shell action of the generating function also contains the M unwanted terms which vanish when the following Bethe equations are imposed on the parameters μ1,…,μM,(3.22)ρ(μi)+ν2ξ2−μi2ν2−∑j≠iM2μi2−μj2=0, where i=1,2,…,M.Hence we have showed that the symmetric function φM(μ1,μ2,…,μM) defined in (3.9) is the Bethe vector of the generating function τ(λ) corresponding to the eigenvalue χM(λ,μ1,μ2,…,μM), stated above (3.21). With this proof we close the topic of the implementation of the algebraic Bethe ansatz for this model.4Solutions to the Knizhnik–Zamolodchikov equationsFinding the off-shell action on Bethe vectors in the previous section was, in this approach, a necessary prerequisite for solving of the corresponding Knizhnik–Zamolodchikov equations [25,27]. In this context the local realization of Gaudin algebra basis operators is also relevant:(4.1)e(λ)=−2∑m=1Nξ−αmνλ2−αm2Sm+,(4.2)h(λ)=2∑m=1N1λ2−α2m(Sm3−ψ2νSm+),(4.3)f(λ)=2∑m=1Nξ+αmνλ2−αm2(Sm−+ψνS3m−ψ24ν2Sm+), where Sm3, Sm±, are the usual spin generators at the local node m (see [19]). In this local realization the vacuum vector Ω+ has the form(4.4)Ω+=ω1⊗⋯⊗ωN∈H, where vector ωm belongs to local node Hilbert space Vm=C2s+1 and:(4.5)Sm3ωm=smωmandS+mωm=0.The Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function τ(λ) at poles λ=±αm [19] and in order to make the paper self contained, we state these result also here:(4.6)Resλ=αmτ(λ)=4HmandResλ=−αmτ(λ)=(−4)H˜m, yielding:(4.7)Hm=∑n≠mNS→m⋅S→nαm−αn+∑n=1N(Km(αm)S→mKm−1(αm))⋅S→n+S→n⋅(Km(αm)S→mKm−1(αm))2(αm+αn), and(4.8)H˜m=∑n≠mNS→m⋅S→nαm−αn+∑n=1N(Km(−αm)S→mKm−1(−αm))⋅S→n+S→n⋅(Km(−αm)S→mKm−1(−αm))2(αm+αn).It follows from the above relations and (3.21) that the eigenvalues of the Gaudin Hamiltonians (4.7) and (4.8) can be derived as the residues of χM(λ,μ1,…,μM), obtained in the previous section, at the poles λ=±αm [19]. It turns out that the respective eigenvalues of the Hamiltonians (4.7) and (4.8) coincide:(4.9)Em,M=14Resλ=αmχM(λ,μ1,…,μM)=E˜m,M=sm(sm+1)2αm+αmsm(ν2ξ2−αm2ν2+∑n≠mN2snαm2−αn2)−2αmsm∑i=1M1αm2−μi2. When all the spin sm are set to one half, these energies, as well as the Bethe equations, coincide with the expressions obtained in [14] (up to normalisation; for the connection of the corresponding notations, cf. [19]).The key observation in what follows will be that by taking the residue of both sides of the equation (3.19) at λ=αn, using (4.6), (4.7) and (4.9), and dividing both sides of the equation by the factor of four one obtains(4.10)HnφM(μ1,μ2,…,μM)=En,MφM(μ1,μ2,…,μM)+∑j=1M2αn2αn2−μj2ξ2−μj2ν2ξ2−αn2ν2××(ρ(μj)+ν2ξ2−μj2ν2−∑k≠jM2μj2−μk2)ξ+αnναn×(Sn−+ψνSn3−ψ24ν2Sn+)φM−1(μ1,…,μjˆ,…,μM), here the notation μjˆ means that the argument μj is not present.The solutions to the Knizhnik–Zamolodchikov equations we seek in the form of contour integrals over the variables μ1,μ2,…,μM [25,27]:(4.11)ψ(α1,α2,…,αN)=∮⋯∮ϕ(μ→|α→)φM(μ→|α→)dμ1⋯dμM, where the integrating factor ϕ(μ→|α→) is a scalar function(4.12)ϕ(μ→|α→)=exp(S(μ→|α→)κ) obtained by exponentiating a function S(μ→|α→) [34]. As in [10], from now on, the K-matrix parameters take fixed values ψ=ξ=0 and ν=1. For these values it is straightforward to check that i) the Gaudin Hamiltonians are Hermitian; and ii) Hamiltonians (4.7) and (4.8) coincide.We find that the proper form of S(μ→|α→) in this case is:(4.13)S(μ→|α→)=∑n=1Nsn(sn−1)2ln(αn)+∑n<mNαnαmln(αn2−αm2)+∑j=1Mln(μj)+∑j<kMln(μj2−μk2)−∑j=1M∑n=1Nsnln(αn2−μj2). In order to show this, it is important to notice that the function ϕ(μ→|α→) as defined above also satisfies the following equations(4.14)κ∂αnϕ=En,Mϕ,(4.15)κ∂μjϕ=βM(μj)ϕ, where(4.16)βM(μj):=−μj(ρ(μj)−1μj2−∑k≠jM2μj2−μk2). Introducing the notation(4.17)φ˜M−1(j,n):=Sn−φM−1(μ1,…,μjˆ,…,μM) the equation (4.10) can be expressed in the following form(4.18)HnφM(μ1,μ2,…,μM)=En,MφM(μ1,μ2,…,μM)+∑j=1M(−2)μjαn2−μj2βM(μj)φ˜M−1(j,n).Using the definition of φM (3.9) and the local realisation of the generator f(μ) (4.3) it follows that(4.19)∂αnφM=(−2)∑j=1M∂μj(μjφ˜M−1(j,n)μj2−αn2).Then it is straightforward to show that(4.20)κ∂αn(ϕφM)=Hn(ϕφM)+κ∑j=1M∂μj((−2)μjμj2−αn2ϕφ˜M−1(j,n)). A closed contour integration of ϕφM with respect to the variables μ1,μ2,…,μM will cancel the contribution from the terms under the sum in (4.20) and therefore ψ(α1,α2,…,αN) given by (4.11) satisfies the Knizhnik–Zamolodchikov equations(4.21)κ∂αnψ(α1,α2,…,αN)=Hnψ(α1,α2,…,αN).Moreover, the interplay between the Gaudin model and the Knizhnik–Zamolodchikov equations, once the Bethe equations are imposed(4.22)∂S∂μj=βM(μj)=−μj(∑m=1N2smμj2−αm2−1μj2−∑k≠jM2μj2−μk2)=0, enabled us to determine the on-shell norm of the Bethe vectors(4.23)‖φM(μ1,μ2,…,μM)‖2=2Mdet(∂2S∂μj∂μk).It turns out to be possible to derive also a stronger formula than the one above for the norms [27]. Indeed, we calculate the following expression for the off-shell scalar product of arbitrary two Bethe vectors:(4.24)Ω+⁎e(λ1)e(λ2)⋯e(λM)f(μM)⋯f(μ2)f(μ1)Ω+=4M∑σ∈SMdetMσ, where SM is the symmetric group of degree M and the M×M matrix Mσ is given by(4.25)Mjjσ=−λj2ρ(λj)−μσ(j)2ρ(μσ(j))λj2−μσ(j)2−∑k≠jλk2+μσ(k)2(λj2−λk2)(μσ(j)2−μσ(k)2),(4.26)Mjkσ=−λk2+μσ(k)2(λj2−λσ(k)2)(μσ(j)2−μσ(k)2),forj,k=1,2,…,M. This formula (that can be proved by commuting e(λ) operators to the right and using mathematical induction) has obvious potential applications as the first step towards the general correlation functions. It should be noted that in [13] a related problem was analysed in the trigonometric case and under certain restrictions: local spins were all fixed to the value 12 and it was required that N=2M (in the notation of that paper). Our formula is more compact and valid for arbitrary spins and arbitrary number of excitations.5ConclusionIn this paper we addressed a number of open problems related to Gaudin model with non periodic boundary conditions.First, we obtained a new basis of the generalized sℓ(2) Gaudin algebra, in which the commutation relations and the generating function are manifestly simpler. This step allowed us to calculate Bethe vectors and off-shell action of the generating function upon them in a closed form, for arbitrary number of excitations. The obtained expressions we have proved by mathematical induction.Once having the general expressions for the Bethe vectors and for the corresponding eigenvalues, we could proceed to relate KZ equations with the Bethe vectors. Taking residues of the off-shell action at poles ±αm, we obtained both Gaudin Hamiltonians and their eigenvalues. By finding the appropriate form of the function S in (4.13), we managed to establish and prove relations (4.14) and (4.15) which led to solution to KZ equations. Proceeding in the same framework, we also obtained the expression for norms of Bethe vectors on shell. Moreover, we went a step further and provided a closed form formula for the scalar product of arbitrary two Bethe vectors.AcknowledgementsWe acknowledge partial financial support by the Foundation for Science and Technology (FCT), Portugal, project PTDC/MAT-GEO/3319/2014. I.S. was supported in part by the Ministry of Education, Science and Technological Development, Serbia, under grant number ON 171031.References[1]M.GaudinDiagonalisation d'une classe d'hamiltoniens de spinJ. Phys.37197610871098[2]M.GaudinLa fonction d'onde de Bethe1983MassonParis[3]M.GaudinThe Bethe Wavefunction2014Cambridge University Press[4]K.HikamiP.P.KulishM.WadatiIntegrable spin systems with long-range interactionChaos Solitons Fractals251992543550[5]K.HikamiP.P.KulishM.WadatiConstruction of integrable spin systems with long-range interactionJ. Phys. Soc. Jpn.619199230713076[6]E.K.SklyaninSeparation of variables in the Gaudin modelZap. Nauč. Semin. POMI1641987151169translation inJ. Sov. Math.472198924732488[7]B.JurčoClassical Yang–Baxter equations and quantum integrable systemsJ. Math. Phys.30198912891293[8]B.JurčoClassical Yang–Baxter equations and quantum integrable systems (Gaudin models)Quantum GroupsClausthal, 1989Lecture Notes in Phys.vol. 3701990219227[9]M.A.Semenov-Tian-ShanskyQuantum and classical integrable systemsIntegrability of Nonlinear SystemsLecture Notes in Physicsvol. 4951997314377[10]K.HikamiGaudin magnet with boundary and generalized Knizhnik–Zamolodchikov equationJ. Phys. A, Math. Gen.28199549975007[11]W.L.YangR.SasakiY.Z.ZhangZn elliptic Gaudin model with open boundariesJ. High Energy Phys.092004046[12]W.L.YangR.SasakiY.Z.ZhangAn−1 Gaudin model with open boundariesNucl. Phys. B7292005594610[13]K.HaoW.-L.YangH.FanS.Y.LiuK.WuZ.Y.YangY.Z.ZhangDeterminant representations for scalar products of the XXZ Gaudin model with general boundary termsNucl. Phys. B8622012835849[14]K.HaoJ.CaoT.YangW.-L.YangExact solution of the XXX Gaudin model with the generic open boundariesarXiv:1408.3012[15]N.ManojlovićI.SalomAlgebraic Bethe ansatz for the XXZ Heisenberg spin chain with triangular boundaries and the corresponding Gaudin modelNucl. Phys. B923201773106arXiv:1705.02235[16]N.ManojlovićI.SalomAlgebraic Bethe ansatz for the trigonometric sℓ(2) Gaudin model with triangular boundaryarXiv:1709.06419[17]N.CrampéAlgebraic Bethe ansatz for the XXZ Gaudin models with generic boundarySIGMA132017094[18]N.Cirilo AntónioN.ManojlovićI.SalomAlgebraic Bethe ansatz for the XXX chain with triangular boundaries and Gaudin modelNucl. Phys. B889201487108arXiv:1405.7398[19]N.Cirilo AntónioN.ManojlovićE.RagoucyI.SalomAlgebraic Bethe ansatz for the sℓ(2) Gaudin model with boundaryNucl. Phys. B8932015305331arXiv:1412.1396[20]B.VicedoC.YoungCyclotomic Gaudin models: construction and Bethe ansatzCommun. Math. Phys.343320169711024[21]V.CaudrelierN.CrampéClassical N-reflection equation and Gaudin modelsarXiv:1803.09931[22]E.A.YuzbashyanIntegrable time-dependent Hamiltonians, solvable Landau–Zener models and Gaudin magnetsAnn. Phys.3922018323339[23]N. Manojlović, N. Cirilo António, I. Salom, Quasi-classical limit of the open Jordanian XXX spin chain, in: Proceedings of the 9th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics, 18–23 September 2017, Belgrade, Serbia.[24]V.G.KnizhnikA.B.ZamolodchikovCurrent algebras and Wess–Zumino model in two dimensionsNucl. Phys. B247198483103[25]H.M.BabujianR.FlumeOff-shell Bethe ansatz equations for Gaudin magnets and solutions of Knizhnik–Zamolodchikov equationsMod. Phys. Lett. A922199420292039[26]B.FeginE.FrenkelN.ReshetikhinGaudin model, Bethe ansatz and critical levelCommun. Math. Phys.16619942762[27]P.P.KulishN.ManojlovićCreation operators and Bethe vectors of the osp(1|2) Gaudin modelJ. Math. Phys.4210200147574778[28]T.SkrypnykNon-skew-symmetric classical r-matrix, algebraic Bethe ansatz, and Bardeen–Cooper–Schrieffer-type integrable systemsJ. Math. Phys.502009033540[29]T.Skrypnyk“Z2-graded” Gaudin models and analytical Bethe ansatzNucl. Phys. B87032013495529[30]E.K.SklyaninGenerating function of correlators in the sℓ(2) Gaudin modelLett. Math. Phys.4731999275292[31]E.K.SklyaninBoundary conditions for integrable equationsFunkc. Anal. Prilozh.2119878687(in Russian); translation inFunct. Anal. Appl.2121987164166[32]E.K.SklyaninBoundary conditions for integrable systemsProceedings of the VIIIth International Congress on Mathematical PhysicsMarseille, 19861987World Sci. PublishingSingapore402408[33]E.K.SklyaninBoundary conditions for integrable quantum systemsJ. Phys. A, Math. Gen.21198823752389[34]N.ReshetikhinA.VarchenkoQuasiclassical asymptotics of solutions to the KZ equationsGeometry, Topology & Physics for Raul Bott, Conference Proceedings Lecture Notes Geometry Topology VI1995Int. PressCambridge, MA293322