Universal Bethe Ansatz and Scalar Products of Bethe Vectors

An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum affine algebra Uq(gl₃) is given. This result is obtained in the framework of the universal Bethe ansatz, using presentation of the universal Bethe vectors in terms o...

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Datum:	2010
Hauptverfasser:	Belliard, S., Pakuliak, S., Ragoucy, E.
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Veröffentlicht:	Інститут математики НАН України 2010
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spelling	irk-123456789-1465272019-02-10T01:25:23Z Universal Bethe Ansatz and Scalar Products of Bethe Vectors Belliard, S. Pakuliak, S. Ragoucy, E. An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum affine algebra Uq(gl₃) is given. This result is obtained in the framework of the universal Bethe ansatz, using presentation of the universal Bethe vectors in terms of the total currents of a ''new'' realization of the quantum affine algebra Uq(gl₃). 2010 Article Universal Bethe Ansatz and Scalar Products of Bethe Vectors / S. Belliard, S. Pakuliak, E. Ragoucy // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 81R50 DOI:10.3842/SIGMA.2010.094 http://dspace.nbuv.gov.ua/handle/123456789/146527 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum affine algebra Uq(gl₃) is given. This result is obtained in the framework of the universal Bethe ansatz, using presentation of the universal Bethe vectors in terms of the total currents of a ''new'' realization of the quantum affine algebra Uq(gl₃).
format	Article
author	Belliard, S. Pakuliak, S. Ragoucy, E.
spellingShingle	Belliard, S. Pakuliak, S. Ragoucy, E. Universal Bethe Ansatz and Scalar Products of Bethe Vectors Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Belliard, S. Pakuliak, S. Ragoucy, E.
author_sort	Belliard, S.
title	Universal Bethe Ansatz and Scalar Products of Bethe Vectors
title_short	Universal Bethe Ansatz and Scalar Products of Bethe Vectors
title_full	Universal Bethe Ansatz and Scalar Products of Bethe Vectors
title_fullStr	Universal Bethe Ansatz and Scalar Products of Bethe Vectors
title_full_unstemmed	Universal Bethe Ansatz and Scalar Products of Bethe Vectors
title_sort	universal bethe ansatz and scalar products of bethe vectors
publisher	Інститут математики НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/146527
citation_txt	Universal Bethe Ansatz and Scalar Products of Bethe Vectors / S. Belliard, S. Pakuliak, E. Ragoucy // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 28 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT belliards universalbetheansatzandscalarproductsofbethevectors AT pakuliaks universalbetheansatzandscalarproductsofbethevectors AT ragoucye universalbetheansatzandscalarproductsofbethevectors
first_indexed	2025-07-11T00:11:26Z
last_indexed	2025-07-11T00:11:26Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 094, 22 pages Universal Bethe Ansatz and Scalar Products of Bethe Vectors? Samuel BELLIARD †, Stanislav PAKULIAK ‡ and Eric RAGOUCY § † Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Italy E-mail: belliard@bo.infn.it ‡ Institute of Theoretical & Experimental Physics, 117259 Moscow, Russia Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow reg., Russia Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Moscow reg., Russia E-mail: pakuliak@theor.jinr.ru § Laboratoire de Physique Théorique LAPTH, CNRS and Université de Savoie, BP 110, 74941 Annecy-le-Vieux Cedex, France E-mail: eric.ragoucy@lapp.in2p3.fr Received October 25, 2010; Published online December 14, 2010 doi:10.3842/SIGMA.2010.094 Abstract. An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum affine algebra Uq(ĝl3) is given. This result is obtained in the framework of the universal Bethe ansatz, using presentation of the universal Bethe vectors in terms of the total currents of a “new” realization of the quantum affine algebra Uq(ĝl3). Key words: Bethe ansatz; quantum affine algebras 2010 Mathematics Subject Classification: 17B37; 81R50 1 Introduction The problem of computing correlation functions is one of the most challenging problem in the field of quantum integrable models, starting from the establishment of the Bethe ansatz method in [1]. For the models where algebraic Bethe ansatz [2, 3, 4, 5] is applicable, this problem can be reduced to the calculation of scalar products of off-shell Bethe vectors. These latters are Bethe vectors where the Bethe parameters are not constrained to obey the Bethe Ansatz equations anymore. For gl2-based integrable models, these scalar products were calculated in [6, 7, 8] and are given by the sums over partitions of two sets of the Bethe parameters. Lately, it was shown by N. Slavnov [9], that if one set of Bethe parameters satisfies Bethe equations (which guarantees that the Bethe vectors are eigenvectors of the transfer matrix), then the formula for scalar products can be written in a determinant form. This form is very useful to get an integral presentation for correlation functions [10, 11, 12, 13] in the thermodynamic limit. There is a wide class of quantum integrable models associated with the algebra glN (N > 2). An algebraic Bethe ansatz for these type models is called hierarchical (or nested) and was introduced by P. Kulish and N. Reshetikhin [14]. This method is based on a recursive procedure which reduces the eigenvalue problem for the transfer matrix for the model with glN symmetry to an analogous problem for the model with glN−1 symmetry. Assuming that the problem for ?This paper is a contribution to the Proceedings of the International Workshop “Recent Advances in Quantum Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2010.html mailto:belliard@bo.infn.it mailto:pakuliak@theor.jinr.ru mailto:eric.ragoucy@lapp.in2p3.fr http://dx.doi.org/10.3842/SIGMA.2010.094 http://www.emis.de/journals/SIGMA/RAQIS2010.html 2 S. Belliard, S. Pakuliak and E. Ragoucy N = 2 is solved, this method allows to find hierarchical Bethe equations. Explicit formulas for the hierarchical Bethe vectors in terms of the matrix elements of the glN monodromy matrix can be found in [15], but these complicated expressions are very difficult to handle. The solution of this problem, namely the formulas for the off-shell Bethe vectors in terms of monodromy matrix, was found in [16]. These vectors are called universal, because they have the same structure for the different models sharing the same hidden symmetry. This construction requires a very complicated procedure of calculation of the trace of projected tensor powers of the monodromy matrix. It was performed in [17], but only on the level of the evaluation representation of Uq(ĝlN ) monodromy matrix. There is a new, alternative approach to the construction of universal Bethe vectors for glN symmetry models using current realizations of the quantum affine algebras [18] and using a Ding–Frenkel isomorphism between current and L-operators realizations of the quantum affine algebra Uq(ĝlN ) [19]. This approach allows to obtain explicit formulas for the universal Bethe vectors in terms of the current generators of the quantum affine algebra Uq(ĝlN ) for arbitrary highest weight representations. It was proved in [20] that the two methods of construction of the universal Bethe vectors coincide on the level of the evaluation representations. Furthermore, it was shown in [21] that the eigenvalue property of the hierarchical universal Bethe vectors can be reformulated as a problem of ordering of the current generators in the product of the universal transfer matrix and the universal Bethe vectors. It was proved that the eigenvalue property appears only if parameters of the universal Bethe vectors satisfy the universal Bethe equations of the analytical Bethe ansatz [22]. The universal Bethe vectors in terms of current generators have an integral presentation, as an integral transform with some kernel of the product of the currents. In the Uq(ŝl2) case, this integral representation produces immediately an integral formula for the scalar product of off- shell Bethe vectors [23] which is equivalent to the Izergin–Korepin formula. In this article, we present an integral presentation of the universal off-shell Bethe vectors based on the quantum affine algebra Uq(ĝl3). These integral formulas lead to integral formulas for scalar products with some kernel. The corresponding formula (5.6) is the main result of our paper. The problem left to be done is to transform the integral form we have obtained to a determinant form which can be very useful for the application to quantum integrable models associated to glN symmetry algebra. 2 Universal Bethe vectors in terms of L-operator 2.1 Uq(ĝl3) in L-operator formalism Let Eij ∈ End(C3) be a matrix with the only nonzero entry equals to 1 at the intersection of the i-th row and j-th column. Let R(u, v) ∈ End(C3 ⊗ C3)⊗ C[[v/u]], R(u, v) = ∑ 1≤i≤3 Eii ⊗ Eii + u− v qu− q−1v ∑ 1≤i<j≤3 (Eii ⊗ Ejj + Ejj ⊗ Eii) + q − q−1 qu− q−1v ∑ 1≤i<j≤3 (uEij ⊗ Eji + vEji ⊗ Eij) be a trigonometric R-matrix associated with the vector representation of gl3. Let q be a complex parameter different from zero or a root of unity. The algebra Uq(ĝl3) (with zero central charge and the gradation operator dropped out) is an associative algebra with unit, generated by the modes L±i,j [±k], k ≥ 0, 1 ≤ i, j ≤ 3 of the Universal Bethe Ansatz and Scalar Products of Bethe Vectors 3 L-operators L±(z) = ∞∑ k=0 3∑ i,j=1 Eij ⊗ L±i,j [±k]z∓k, subject to the relations R(u, v) · (L±(u)⊗ 1) · (1⊗ L±(v)) = (1⊗ L±(v)) · (L±(u)⊗ 1) · R(u, v), R(u, v) · (L+(u)⊗ 1) · (1⊗ L−(v)) = (1⊗ L−(v)) · (L+(u)⊗ 1) · R(u, v), (2.1) L+ j,i[0] = L−i,j [0] = 0, 1 ≤ i < j ≤ 3, (2.2) L+ k,k[0]L−k,k[0] = 1, 1 ≤ k ≤ 3. (2.3) Actually, we will not impose the condition (2.3) since the universal Bethe vectors will be con- structed only from one L-operator, say L+(z). Subalgebras formed by the modes L±[n] of the L-operators L±(z) are the standard Borel subalgebras Uq(b±) ⊂ Uq(ĝlN ). These Borel subalgebras are Hopf subalgebras of Uq(ĝlN ). Their coalgebraic structure is given by the formulae ∆ ( L±i,j(u) ) = N∑ k=1 L±k,j(u)⊗ L±i,k(u). 2.2 Universal off-shell Bethe vectors We will follow the construction of the off-shell Bethe vectors due to [16]. Let L(z) = ∞∑ k=0 3∑ i,j=1 Eij⊗ Li,j [k]z −k be the L-operator1 of the Borel subalgebra Uq(b+) of Uq(ĝl3) satisfying the Yang– Baxter commutation relation with a R-matrix R(u, v). We use the notation L(k)(z) ∈ ( C3 )⊗M ⊗ Uq(b+) for L-operator acting nontrivially on k-th tensor factor in the product ( C3 )⊗M for 1 ≤ k ≤M . Consider a series in M variables T(u1, . . . , uM ) = L(1)(u1) · · ·L(M)(uM ) · R(M,...,1)(uM , . . . , u1), (2.4) with coefficients in ( End(C3) )⊗M ⊗ Uq(b+), where R(M,...,1)(uM , . . . , u1) = ←−∏ M≥j>1 ←−∏ j>i≥1 R(ji)(uj , ui). (2.5) In the ordered product of R-matrices (2.5), the R(ji) factor is on the left of the R(ml) factor if j > m, or j = m and i > l. Consider the set of variables {t̄, s̄} = {t1, . . . , ta; s1, . . . , sb} . Following [16], let B(t̄, s̄) = ∏ 1≤j≤b ∏ 1≤i≤a qsj − q−1ti sj − ti × (tr(C3)⊗(a+b) ⊗ id) ( T(t1, . . . , ta; s1, . . . , sb)E⊗a 21 ⊗ E⊗b 32 ⊗ 1 ) . (2.6) The element T(t̄, s̄) in (2.6) is given by (2.4) with obvious identification. The coefficients of B(t̄, s̄) are elements of the Borel subalgebra Uq(b+). 1We omit superscript + in this L-operator, since will consider only positive standard Borel subalgebra Uq(b +) here and below. 4 S. Belliard, S. Pakuliak and E. Ragoucy We call vector v a right weight singular vector if it is annihilated by any positive mode Li,j [n], i > j, n ≥ 0 of the matrix elements of the L+(z) operator and is an eigenvector of the diagonal matrix entries L+ i,i(z): L+ i,j(z)v = 0, i > j, L+ i,i(z)v = λi(z)v, i = 1, . . . , 3. (2.7) For any right Uq(ĝl3)-module V with a right singular vector v, denote BV (t̄, s̄) = B(t̄, s̄)v. (2.8) The vector valued function BV (t̄, s̄) was called in [16, 17] universal off-shell Bethe vector. We call vector v′ a left weight singular vector if it is annihilated by any positive mode Li,j [n], i < j, n ≥ 0 of the matrix elements of the L-operator L+(z) and is an eigenvector of the diagonal matrix entries L+ i,i(z): 0 = v′L+ i,j(z), i < j, v′L+ i,i(z) = µi(z)v′, i = 1, . . . , 3. (2.9) For any left Uq(ĝl3)-module V ′ with a left singular vector v′, denote CV ′(τ̄ , σ̄) = v′C(τ̄ , σ̄), (2.10) where {τ̄ , σ̄} = {τ1, . . . , τa;σ1, . . . , σb} , and C(τ̄ , σ̄) = ∏ 1≤j≤b ∏ 1≤i≤a qσj − q−1τi σj − τi × (tr(C3)⊗(a+b) ⊗ id) ( T(τ1, . . . , τa;σ1, . . . , σb)E⊗a 12 ⊗ E⊗b 23 ⊗ 1 ) . Our goal is to calculate the scalar product 〈CV ′(τ̄ , σ̄),BV (t̄, s̄)〉. (2.11) There is a direct way to solve this problem, using the exchange relations of the L-operators matrix elements and the definitions of the singular weight vectors. However, this approach is a highly complicated combinatorial problem. Instead, we will use another presentation of the universal Bethe vectors given recently in the paper [20, 24], using current realization of the quantum affine algebra Uq(ĝl3) and method of projections introduced in [25] and developed in [26]. 3 Current realization of Uq(ĝl3) 3.1 Gauss decompositions of L-operators The relation between the L-operator realization of Uq(ĝl3) and its current realization [18] is known since the work [19]. To build an isomorphism between these two realizations, one has to consider the Gauss decomposition of the L-operators and identifies linear combinations of some Gauss coordinates with the total currents of Uq(ĝl3) corresponding to the simple roots of gl3. Recently, it was shown in [24] that there are two different but isomorphic current realization of Uq(ĝl3). They correspond to different embeddings of smaller algebras into bigger ones and to different type of Gauss decompositions of the fundamental L-operators. These two different Universal Bethe Ansatz and Scalar Products of Bethe Vectors 5 current realizations have different commutation relations, different current comultiplications and different associated projections onto intersections of the current and Borel subalgebras of Uq(ĝl3). Our way to calculate the scalar product of Bethe vectors (2.11) is to use an alternative form to expressions (2.8) and (2.10) for the universal Bethe vectors. It is written in terms of projections of products of currents onto intersections of the current and Borel subalgebras of Uq(ĝl3). In this case, the universal Bethe vectors can be written as some integral and the calculation of the scalar product is reduced to the calculation of an integral of some rational function. For the L-operators fixed by the relations (2.1) and (2.2), we consider the following decom- positions into Gauss coordinates F±j,i(t), E±i,j(t), j > i and k±i (t): L±i,j(t) = F±j,i(t)k + i (t) + ∑ 1≤m<i F±j,m(t)k±m(t)E±m,i(t), 1 ≤ i < j ≤ 3, (3.1) L±i,i(t) = k±i (t) + ∑ 1≤m<i F±i,m(t)k±m(t)E±m,i(t), i = 1, 2, 3, (3.2) L±i,j(t) = k±j (t)E±j,i(t) + ∑ 1≤m<j F±j,m(t)k±m(t)E±m,i(t), 3 ≥ i > j ≥ 1. (3.3) Using the arguments of [19], we may obtain for the linear combinations of the Gauss coordi- nates (i = 1, 2) Fi(t) = F+ i+1,i(t)− F−i+1,i(t), Ei(t) = E+ i,i+1(t)− E−i,i+1(t), and for the Cartan currents k±i (t), the commutation relations of the quantum affine algebra Uq(ĝl3) with zero central charge and the gradation operator dropped out. In terms of the total currents Fi(t), Ei(t) and of the Cartan currents k±i (t), these commutation relations read (qz − q−1w)Ei(z)Ei(w) = Ei(w)Ei(z)(q−1z − qw), (3.4) (q−1z − qw)Ei(z)Ei+1(w) = Ei+1(w)Ei(z)(z − w), (3.5) k±i (z)Ei(w) ( k±i (z) )−1 = z − w q−1z − qw Ei(w), (3.6) k±i+1(z)Ei(w) ( k±i+1(z) )−1 = z − w qz − q−1w Ei(w), (3.7) k±i (z)Ej(w) ( k±i (z) )−1 = Ej(w), if i 6= j, j + 1, (3.8) (q−1z − qw)Fi(z)Fi(w) = Fi(w)Fi(z)(qz − q−1w), (3.9) (z − w)Fi(z)Fi+1(w) = Fi+1(w)Fi(z)(q−1z − qw), (3.10) k±i (z)Fi(w) ( k±i (z) )−1 = q−1z − qw z − w Fi(w), (3.11) k±i+1(z)Fi(w) ( k±i+1(z) )−1 = qz − q−1w z − w Fi(w), (3.12) k±i (z)Fj(w) ( k±i (z) )−1 = Fj(w), if i 6= j, j + 1, (3.13) [Ei(z), Fj(w)] = δi,jδ(z/w)(q − q−1) ( k−i+1(z)/k − i (z)− k+ i+1(w)/k+ i (w) ) , (3.14) with Serre relations Symz1,z2 ( Ei(z1)Ei(z2)Ei±1(w)− (q + q−1)Ei(z1)Ei±1(w)Ei(z2) + Ei±1(w)Ei(z1)Ei(z2) ) = 0, 6 S. Belliard, S. Pakuliak and E. Ragoucy Symz1,z2 ( Fi(z1)Fi(z2)Fi±1(w)− (q + q−1)Fi(z1)Fi±1(w)Fi(z2) + Fi±1(w)Fi(z1)Fi(z2) ) = 0. Formulae (3.4)–(3.14) should be considered as formal series identities describing the infinite set of relations between modes of the currents. The symbol δ(z) entering these relations is a formal series ∑ n∈Z zn. 3.2 Borel subalgebras and projections on their intersections We consider two types of Borel subalgebras in the algebra Uq(ĝl3). Borel subalgebras Uq(b±) ⊂ Uq(ĝl3) are generated by the modes of the L-operators L(±)(z) respectively. For the generators in these subalgebras, we can use instead modes of the Gauss coordinates (3.1)–(3.3), E±i,i+1(t), F±i+1,i(t), k ± j (t). Other types of Borel subalgebras are related to the current realizations of Uq(ĝlN ) given in the previous subsection. We consider first the current Borel subalgebras generated by the modes of the currents Ei(t), Fi(t), k±j (t). The Borel subalgebra UF ⊂ Uq(ĝl3) is generated by modes of the currents Fi[n], k+ j [m], i = 1, 2, j = 1, 2, 3, n ∈ Z and m ≥ 0. The Borel subalgebra UE ⊂ Uq(ĝl3) is generated by modes of the currents Ei[n], k−j [−m], i = 1, 2, j = 1, 2, 3, n ∈ Z and m ≥ 0. We will consider also the subalgebra U ′F ⊂ UF , generated by the elements Fi[n], k+ j [m], i = 1, 2, j = 1, 2, 3, n ∈ Z and m > 0, and the subalgebra U ′E ⊂ UE generated by the elements Ei[n], k−j [−m], i = 1, 2, j = 1, 2, 3, n ∈ Z and m > 0. In the following, we will be interested in the intersections U−f = U ′F ∩ Uq(b−), U+ F = UF ∩ Uq(b+), U−E = UE ∩ Uq(b−), U+ e = U ′E ∩ Uq(b+), and will describe properties of the projections on these intersections. We call UF and UE the current Borel subalgebras. In [18] the current Hopf structure for the algebra Uq(ĝl3) has been defined as: ∆(D) (Ei(z)) = 1⊗ Ei(z) + Ei(z)⊗ k−i+1(z) ( k−i (z) )−1 , ∆(D) (Fi(z)) = Fi(z)⊗ 1 + k+ i+1(z) ( k+ i (z) )−1 ⊗ Fi(z), (3.15) ∆(D) ( k±i (z) ) = k±i (z)⊗ k±i (z). With respect to the current Hopf structure, the current Borel subalgebras are Hopf subalgebras of Uq(ĝl3). One may construct the whole algebra Uq(ĝl3) from one of its current Borel subalgebras using the current Hopf structure and the Hopf pairing 〈Ei(z), Fj(w)〉 = (q − q−1)δijδ(z/w), 〈ψ−i (z), k+ i+1(w)〉 = 〈k−i (z), ψ+ i (w)〉−1 = q − q−1z/w 1− z/w , (3.16) 〈ψ−i (z), k+ i (w)〉 = 〈k−i+1(z), ψ + i (w)〉−1 = q−1 − qz/w 1− z/w , where ψ±i (t) = k±i+1(t) ( k±i (t) )−1 , i = 1, 2. (3.17) Universal Bethe Ansatz and Scalar Products of Bethe Vectors 7 Formulas (3.16) can be obtained from the commutation relations (3.4)–(3.14) using the commu- tator rules (5.4) in the quantum double. One can check [26, 27] that the intersections U−f and U+ F , respectively U+ e and U−E , are subalgebras and coideals with respect to the Drinfeld coproduct (3.15): ∆(D)(U+ F ) ⊂ U+ F ⊗ Uq(ĝl3), ∆(D)(U−f ) ⊂ Uq(ĝl3)⊗ U−f , ∆(D)(U+ e ) ⊂ U+ e ⊗ Uq(ĝl3), ∆(D)(U−E ) ⊂ Uq(ĝl3)⊗ U−E , and that the multiplication m in Uq(ĝl3) induces an isomorphism of vector spaces m : U−f ⊗ U + F → UF , m : U+ e ⊗ U−E → UE . According to the general theory presented in [26], we introduce the projection operators P+ f : UF ⊂ Uq(ĝl3)→ U+ F , P−f : UF ⊂ Uq(ĝl3)→ U−f , P+ e : UE ⊂ Uq(ĝl3)→ U+ e , P−e : UE ⊂ Uq(ĝl3)→ U−E . They are respectively defined by the prescriptions P+ f (f− f+) = ε(f−) f+, P−f (f− f+) = f− ε(f+), ∀ f− ∈ U−f , ∀ f+ ∈ U+ F , (3.18) P+ e (e+ e−) = e+ ε(e−), P−e (e− e+) = ε(e+) e−, ∀ e− ∈ U−E , ∀ e+ ∈ U+ e , (3.19) where ε : Uq(ĝl3)→ C is a counit map. It was proved in [28] that the projections P±f and P±e are adjoint with respect to the Hopf pairing (3.16) 〈e, P±f (f)〉 = 〈P∓e (e), f〉. Denote by UF an extension of the algebra UF formed by infinite sums of monomials that are ordered products ai1 [n1] · · · aik [nk] with n1 ≤ · · · ≤ nk, where ail [nl] is either Fil [nl] or k+ il [nl]. Denote by UE an extension of the algebra UE formed by infinite sums of monomials that are ordered products ai1 [n1] · · · aik [nk] with n1 ≥ · · · ≥ nk, where ail [nl] is either Eil [nl] or k−il [nl]. It was proved in [26] that (1) the action of the projections (3.18) can be extended to the algebra UF ; (2) for any f ∈ UF with ∆(D)(f) = ∑ i f ′ i ⊗ f ′′i we have f = ∑ i P − f (f ′i) · P + f (f ′′i ); (3) the action of the projections (3.19) can be extended to the algebra UE ; (4) for any e ∈ UE with ∆(D)(e) = ∑ i e ′ i ⊗ e′′i we have e = ∑ i P + e (e′′i ) · P−e (e′i). 3.3 Definition of the composed currents We introduce the composed currents2 E1,3(w) and F3,1(y) which are defined by the formulas E1,3(w) = ∮ C0 dz z E2(z)E1(w)− ∮ C∞ dz z E1(w)E2(z) q − q−1w/z 1− w/z , F3,1(y) = ∮ C0 dv v F1(y)F2(v)− ∮ C∞ dv v F2(v)F1(y) q − q−1y/v 1− y/v , (3.20) 2Different definitions exist for composed currents, we choose the one, such that the corresponding projections of them coincide with the Gauss coordinates (3.1). 8 S. Belliard, S. Pakuliak and E. Ragoucy where the contour integrals ∮ C0,∞ dz z g(z) are considered as integrals around zero and infinity points respectively. The composed currents E1,3(w) and F3,1(y) belong to the completed alge- bras UE and UF , respectively. Let us remind that, according to these completions, we have to understand the product of currents E2(z)E1(w) as an analytical ‘function’ without singularities in the domain \|z\| � \|w\|. Analogously, the product F1(y)F2(v) is an analytical ‘function’ in the domain \|y\| � \|v\|. For practical calculation, the contour integrals in definitions (3.20) can be understood as the formal integrals of a Laurent series g(z) = ∑ k∈Z g[k]z−k picking up its zero mode coefficient g[0]. Deforming contours in the defining formulas for the composed currents we may rewrite them differently E1,3(w) = − res z=w E2(z)E1(w) dz z , F3,1(y) = − res v=y F1(y)F2(v) dv v , (3.21) or E2(z)E1(w) = q − q−1w/z 1− w/z E1(w)E2(z) + δ(w/z)E1,3(w), F1(y)F2(v) = q − q−1y/v 1− y/v F2(v)F1(y) + δ(y/v)F3,1(y). (3.22) Formulas (3.21) are convenient for the presentation of the composed currents as products of simple root currents E1,3(w) = (q−1 − q)E1(w)E2(w), F3,1(y) = (q−1 − q)F2(y)F1(y). (3.23) Formulas (3.22) are convenient to calculate the commutation relation between total and half- currents. This will be done lately. First, we calculate the formal integrals in the formulas (3.20) to obtain E1,3(w) = E2[0]E1(w)− q−1E1(w)E2[0]− ( q − q−1 ) ∞∑ k=0 E1(w)E2[−k]wk, F3,1(y) = F1(y)F2[0]− qF2[0]F1(y)− ( q − q−1 ) ∞∑ k=1 F2[−k]F1(y)yk. Here we used the series expansion q − q−1w/z 1− w/z = q + ( q − q−1 ) ∞∑ k=1 (w/z)k = q−1 + ( q − q−1 ) ∞∑ k=0 (w/z)k. Introducing now the half-currents E±2 (w) = ± ∑ k>0 k≤0 E2[k]w−k, F±2 (w) = ± ∑ k≥0 k<0 F2[k]w−k. and using the decomposition of the algebra Uq(ĝl3) into its standard positive and negative Borel subalgebras and the definition of the screening operators E2(E1(w)) = E2[0]E1(w)− q−1E1(w)E2[0], F2(F1(y)) = F1(y)F2[0]− qF2[0]F1(y), Universal Bethe Ansatz and Scalar Products of Bethe Vectors 9 we may write E2(E1(w)) = E1,3(w) + (q−1 − q)E1(w)E−2 (w), F2(F1(y)) = F3,1(y) + (q−1 − q)F−2 (y)F1(y), or E2(E1(w)) = (q−1 − q)E1(w)E+ 2 (w), F2(F1(y)) = (q−1 − q)F+ 2 (y)F1(y). (3.24) To obtain the latter relation, we used formulas (3.23) and relation between total and half-currents E2(w) = E+ 2 (w)− E−2 (w) and F2(w) = F+ 2 (w)− F−2 (w). 4 Universal Bethe vectors and projections The goal of this section is to obtain the representations for the left and right universal Bethe vectors in terms of the integral transform of the products of the total currents. This will generalize the results obtained for Uq(ŝl3) in the paper [28]. The calculation of the scalar product after that will be reduced to the calculation of the exchange relations between products of total currents. 4.1 Universal Bethe vectors through currents It was shown in the papers [20, 24] that the universal right Bethe vectors BV (t̄, s̄) = P+ f (F1(t1) · · ·F1(ta)F2(s1) · · ·F2(sb)) a∏ i=1 k1(ti) b∏ i=1 k2(si)v can be identified with some projection of products of total currents. Using the same method, we may prove that the left Bethe vectors CV ′(τ̄ , σ̄) = v′ a∏ i=1 k1(τi) b∏ i=1 k2(σi)P+ e (E2(σb) · · ·E2(σ1)E1(τa) · · ·E1(τ1)) (4.1) can be also identified with projection of products of total currents3. So the problem of calculating the scalar product 〈CV ′(τ̄ , σ̄),BV (t̄, s̄)〉 is reduced to the exchange relations between projections. Fortunately, to perform this exchange, we have to calculate only modulo the ideals in the algebra Uq(ĝl3) which are annihilated by the left/right singular vectors. One could calculate these projections to present them in the form of a sum of products of projections of simple and composed root currents (see formulas (4.4) and (4.5) below). However, this calculation has the same level of difficulty as the exchange relations of Bethe vectors in terms of L-operators. The idea of the present paper is to rewrite projection formulas (4.4) and (4.5) in terms of integrals of total simple root currents, and then to compute the exchange of products of total currents. In this way, we will obtain an integral representation for the scalar product of the off-shell Bethe vectors. This calculation is much more easy, since the commutation relations of the simple roots total currents are rather simple. 3In (4.1) all operators are acting to the left onto left singular vector v′. 10 S. Belliard, S. Pakuliak and E. Ragoucy 4.2 Calculation of the universal off-shell Bethe vectors Before presenting the formulas for the universal off-shell Bethe vectors in terms of the current generators, we have to introduce the following notations. Consider the permutation group Sn and its action on the formal series of n variables defined, for the elementary transpositions σi,i+1, as follows π(σi,i+1)G(t1, . . . , ti, ti+1, . . . , tn) = q−1 − q ti/ti+1 q − q−1 ti/ti+1 G(t1, . . . , ti+1, ti, . . . , tn). The q-depending factor in this formula is chosen in such a way that each product Fa(t1) · · · Fa(tn) is invariant under this action. Summing the action over all the group of permutations, we obtain the operator Sym t̄ = ∑ σ∈Sn π(σ) acting as follows Sym t̄G(t̄) = ∑ σ∈Sn ∏ `<`′ σ(`)>σ(`′) q−1 − q tσ(`′)/tσ(`) q − q−1 tσ(`′)/tσ(`) G(σt). The product is taken over all pairs (`, `′), such that conditions ` < `′ and σ(`) > σ(`′) are satisfied simultaneously. According to the results of the papers [26, 27], the calculation of the universal off-shell Bethe vectors is reduced to the calculation of the projections P+ f (F1(t1) · · ·F1(ta)F2(s1) · · ·F2(sb)) (4.2) for the right Bethe vectors and4 P+ e (E2(σb) · · ·E2(σ1)E1(τa) · · ·E1(τ1)) (4.3) for the left Bethe vectors. The calculation was detailed in [28]. Here, we present the result of calculations and give several comments on how it was performed. Proposition 1. The projections (4.2) and (4.3) are given by the series P+ f (F1(t1) · · ·F1(ta)F2(s1) · · ·F2(sb)) = min{a,b}∑ k=0 1 k!(a− k)!(b− k)! Symt̄,s̄ ( P+ f (F1(t1) · · ·F1(ta−k)F3,1(ta−k+1) · · ·F3,1(ta)) × P+ f (F2(sk+1) · · ·F2(sb))Z(ta, . . . , ta−k+1; sk, . . . , s1) ) (4.4) and 5 P+ e (E2(σb) · · ·E2(σ1)E1(τa) · · ·E1(τ1)) = min{a,b}∑ m=0 1 m!(a−m)!(b−m)! Symτ̄ ,σ̄ ( P+ e (E2(σb) · · ·E2(σm+1)) (4.5) × P+ e (E1,3(τa) · · ·E1,3(τa−m+1)E1(τa−m) · · ·E1(τ1))Y (τa, . . . , τa−m+1;σm, . . . , σ1) ) , 4For further convenience, we will denote the spectral parameters of the right Bethe vectors by latin symbols, and those of the left vectors by greek ones. 5The ordering of the variables in the rational series in (4.4) differs from the ordering in the corresponding series in the paper [28]. This is because the rational series (4.6) are defined differently with respect to the paper [28]. Universal Bethe Ansatz and Scalar Products of Bethe Vectors 11 where Y (t1, . . . , tn;x1, . . . , xn) = n∏ i=1 1 1− xi/ti i−1∏ j=1 q−1 − qxi/tj 1− xi/tj = n∏ i=1 1 1− xi/ti n∏ j=i+1 q−1 − qxj/ti 1− xj/ti , (4.6) Z(t1, . . . , tn;x1, . . . , xn) = Y (t1, . . . , tn;x1, . . . , xn) n∏ i=1 xi ti . Note that the kernels (4.6) are defined in such a way, that they have only k simple poles at the point t1, . . . , tk with respect to the variable xk, k = 1, . . . , n. These kernels appear in the integral presentation of the projections of the products of the same simple root currents (see (4.14) below). The proof of the formulas (4.4) and (4.5) is similar to the proof presented in the paper [28]. We will not repeat this calculations here, but for completeness, we collect all necessary formulas. As a first step, we present the products of currents F2(s1) · · ·F2(sb) and E2(σb) · · ·E2(σ1) in a normal ordered form using properties of the projections given at the end of the Subsection 3.2: F2(s1) · · ·F2(sb) = b∑ k=0 1 k!(b− k)! Syms̄ ( P−f (F2(s1) · · ·F2(sk)) · P+ f (F2(sk+1) · · ·F2(sb)) ) , E2(σb) · · ·E2(σ1) = b∑ m=0 1 m!(b−m)! Symσ̄ ( P+ e (E2(σb) · · ·E2(σm+1)) · P−e (E2(σm) · · ·E2(σ1)) ) . To evaluate the projections in formulas (4.4) and (4.5), we commute the negative projections P−f (F2(s1) · · ·F2(sk)) to the left through the product of the total currents F1(t1) · · · F1(ta) in case of (4.4) and commute the negative projections P−e (E2(sm) · · ·E2(s1)) to the right through the product of the total currents E1(τa) · · ·E1(τ1) in (4.5). To perform this commutation we use P−f (F2(s1) · · ·F2(sk)) = (−1)k F−2 (s1; s2, . . . , sk) · · ·F−2 (sk−1; sk)F−2 (sk), P−e (E2(σm) · · ·E2(σ1)) = (−1)m E−2 (σm)E−2 (σm−1;σm) · · ·E−2 (σ1;σ2, . . . , σm), (4.7) and F3,1(t)F−2 (s1; s2, . . . , sk) = q−1s1 − qt s1 − t F−2 (s1; s2, . . . , sk, t)F3,1(t), E−2 (σ1;σ2, . . . , σm)E1,3(τ) = q−1σ1 − qτ σ1 − τ E1,3(τ)E−2 (σ1;σ2, . . . , σm, τ). The expressions F−2 (s1; s2, . . . , sk) = F−2 (s1)− k∑ `=2 s1 s` φs` (s1; s2, . . . , sk)F−2 (s`), E−2 (σ1;σ2, . . . , σm) = E−2 (σ1)− m∑ `=2 φσ` (σ1;σ2, . . . , σm)E−2 (σ`) are linear combinations of the half-currents, while φs` (s1; s2, . . . , sk) = k∏ j=2, j 6=` s1 − sj s` − sj k∏ j=2 q−1s` − qsj q−1s1 − qsj 12 S. Belliard, S. Pakuliak and E. Ragoucy are rational functions satisfying the normalization conditions φsj (si; s2, . . . , sk) = δij , i, j = 2, . . . , k. One also needs the commutation relations between negative half-currents and the total currents F1(t)F−2 (s) = qs− q−1t s− t ( F−2 (s)− (q − q−1)s qs− q−1t F−2 (t) ) F1(t) + s t− s F3,1(t), E−2 (σ)E1(τ) = qσ − q−1τ σ − τ E1(τ) ( E−2 (σ)− (q − q−1)τ qσ − q−1τ E−2 (τ) ) + τ σ − τ E1,3(τ), and the identity∏ i<j q−1ti − qtj ti − tj Symt̄ (Y (tn, . . . , t1;ω s̄)) = ∏ i<j q−1si − qsj si − sj Syms̄ ( Y (ω′t̄; sn, . . . , s1) ) valid for arbitrary permutations ω and ω′ of the sets s̄ and t̄, respectively. 4.3 Integral presentation of the projections (4.4) and (4.5) The projections (4.4) and (4.5) are given as a product of projection of currents. As already mentioned, this form is not convenient to obtain scalar products. We give a new representation in term of a multiple integral over the product of simple root currents: Proposition 2. P+ e (E2(σb) · · ·E2(σ1)E1(τa) · · ·E1(τ1)) = ∮ dν1 ν1 · · · ∮ dνb νb ∮ dµ1 µ1 · · · ∮ dµa µa E(τ̄ , σ̄; µ̄, ν̄)E1(µ1) · · ·E1(µa)E2(νb) · · ·E2(ν1), P+ f (F1(t1) · · ·F1(ta)F2(s1) · · ·F2(sb)) (4.8) = ∮ dy1 y1 · · · ∮ dyb yb ∮ dx1 x1 · · · ∮ dxa xa F(t̄, s̄; x̄, ȳ)F2(y1) · · ·F2(yb)F1(xa) · · ·F1(x1), where the kernels E(τ̄ , σ̄; µ̄, ν̄) and F(t̄, s̄; x̄, ȳ) are given by the series E(τ̄ , σ̄; µ̄, ν̄) = Symτ̄ ,σ̄ min{a,b}∑ m=0 (q−1 − q)m m!(a−m)!(b−m)! ∏ m<i<j≤b σi − σj q−1σi − qσj × ∏ 1≤i<j≤a−m a−m<i<j≤a τi − τj q−1τi − qτj Y (τa, . . . , τa−m+1;σm, . . . , σ1)Z(τa, . . . , τ1;µa, . . . , µ1) × Z(µa, . . . , µa−m+1, σm+1, . . . , σb; ν1, . . . , νb) a−m∏ j=1 b∏ i=m+1 q−1 − qνi/µj 1− νi/µj  (4.9) and F(t̄, s̄; x̄, ȳ) = Symt̄,s̄ min{a,b}∑ k=0 (q−1 − q)k k!(a− k)!(b− k)! ∏ k<i<j≤b si − sj q−1si − qsj × ∏ 1≤i<j≤a−k a−k<i<j≤a ti − tj q−1ti − qtj Z(ta, . . . , ta−k+1; sk, . . . , s1)Y (ta, . . . , t1;xa, . . . , x1) × Y (xa, . . . , xa−k+1, sk+1, . . . , sb; y1, . . . , yb) a−k∏ j=1 b∏ i=k+1 q−1 − qyi/xj 1− yi/xj  . (4.10) Universal Bethe Ansatz and Scalar Products of Bethe Vectors 13 The proof of these formulas is given in the next subsection. Let us explain the meaning of the integral formulas for the projections (4.8). There is a preferable order of integration in these formulas. First, we have to calculate the integrals over variables νi and yi, i = 1, . . . , b, respectively, and then calculate the integrals over µj and xj , j = 1, . . . , a. Example 1. Let us illustrate how it works in the simplest example a = b = 1 and for projection P+ f (F1(t)F2(s)). We have P+ f (F1(t)F2(s)) = ∮ dy y ∮ dx x F(t, s;x, y)F2(y)F1(x), where F(t, s;x, y) = Y (t;x)Y (s; y) q−1 − qy/x 1− y/x + (q−1 − q)Z(t, s)Y (t;x)Y (x; y). Integration over y with the first term of the kernel yields to∮ dy y 1 1− y/s q−1 − qy/x 1− y/x F2(y)F1(x) = q−1x− qs x− s F+ 2 (s;x)F1(x) = F1(x)F+ 2 (s), due to the commutation relations (4.19). Integration over y with the second term of the kernel produces (q−1 − q)F+ 2 (x)F1(x) = F2 (F1(x)) , according to the formulas (3.24). Finally, integration over x in both terms produces the result for the projection in this simplest case. The general case can be treated analogously. Of course, one can first integrate over x and then over y. However in this case, the calculation of the integrals for the projection becomes more involved and requires more complicated commutation relations between half-currents. 4.4 Proof of the integral presentation of the projections (4.4) and (4.5) Integral representation for the projections of the same type of currents P+ f (Fi(s1) · · ·Fi(sb)) and P+ e (Ei(σb) · · ·Ei(σ1)) (i = 1, 2) were obtained in [28]. They can be obtained from the calculation of these projections P+ f (Fi(s1) · · ·Fi(sb)) = F+ i (s1)F+ i (s2; s1) · · ·F+ i (sb; sb−1, . . . , s1), P+ e (Ei(σb) · · ·Ei(σ1)) = E+ i (σb;σb−1, . . . , σ1) · · ·E+ i (σ2;σ1)E+ i (σ1), (4.11) where F+ i (sk; sk−1, . . . , s1) and E+ i (σk;σk−1, . . . , σ1) are linear combinations of the half-currents F+ i (sk; sk−1, . . . , s1) = F+ i (sk)− k−1∑ `=1 sk s` ϕs` (sk; sk−1, . . . , s1)F+ i (s`), E+ i (σk;σk−1, . . . , σ1) = E+ i (σk)− k−1∑ `=1 ϕσ` (σk;σk−1, . . . , σ1)F+ i (σ`), with coefficients being rational functions ϕs` (sk; sk−1, . . . , s1) = k−1∏ j=1, j 6=` sk − sj s` − sj k−1∏ j=1 qs` − q−1sj qsk − q−1sj . There is a very simple analytical proof of the formulas (4.11) given in [28]. 14 S. Belliard, S. Pakuliak and E. Ragoucy Example 2. Let us illustrate this method on one example: the first relation in (4.11) with b = 2. Indeed, from the commutation relation of the total currents Fi(s1) and Fi(s2), and due to the integral presentation of negative half-currents F−i (s2) = − ∫ dy y s2/y 1− s2/y Fi(y), (4.12) we know that P+ f (Fi(s1)Fi(s2)) = F+ i (s1)F+ i (s2) + s2 s1 X(s1) q−1s1 − qs2 , (4.13) where X(s1) is an unknown algebraic element which depends only on the spectral parameter s1. This element can be uniquely defined from the relation (4.13) setting s1 = s2 and using the fact that F 2 i (s) = 0. The general case can be treated analogously (see details in [28]). Formulas (4.7) can be proved in the same way. Using now the integral form of the half-currents F+ i (s) = ∫ dy y 1 1− y/s Fi(y), E+ i (s) = ∫ dy y y/s 1− y/s Ei(y), one can easily obtain integral formulas for (4.11): P+ f (Fi(s1) · · ·Fi(sb)) = ∏ 1≤i<j≤b si − sj q−1si − qsj × ∫ dy1 y1 · · · dyb yb Fi(y1) · · ·Fi(yb)Y (s1, . . . , sb; y1, . . . , yb), P+ e (Ei(σb) · · ·Ei(σ1)) = ∏ 1≤i<j≤b σi − σj q−1σi − qσj × ∫ dν1 ν1 · · · dνb νb Ei(νb) · · ·Ei(ν1)Z(σ1, . . . , σb; ν1, . . . , νb). (4.14) According to the structure of the kernels (4.6), the integrands in (4.14) have only simple poles with respect to the integration variables y1 and ν1 in the points s1 and σ1 respectively, while with respect to the variables yb and νb they have simple poles in the points sj and σj , j = 1, . . . , b. Due to q-symmetric prefactors in the integrals (4.14), the integrals themselves are symmetric with respect to the spectral parameters sj and σj , j = 1, . . . , b, respectively. The integral form for the projections of the strings P+ f (F1(t1) · · ·F1(ta−k)F3,1(ta−k+1) · · ·F3,1(ta)) and P+ e (E1,3(τa) · · ·E1,3(τa−m+1)E1(τa−m) · · ·E1(τ1)) is a more delicate question. To present them as integrals, we use arguments of [28] and formu- las (3.24). The point is that the analytical properties of the reverse strings P+ f (F3,1(ta) · · ·F3,1(ta−k+1)F1(ta−k) · · ·F1(t1)) and P+ e (E1(τ1) · · ·E1(τa−m)E1,3(τa−m+1) · · ·E1,3(τa)) Universal Bethe Ansatz and Scalar Products of Bethe Vectors 15 are the same as the analytical properties of the product of the simple root currents F1(ta)· · ·F1(t1) and E1(τ1)· · ·E1(τa). Therefore the calculation of projection of the reverse string can be done along the same steps as for the product of simple root currents. In order to relate the projection of the string and projection of the reverse string, we need the commutation relations F1(t1)F3,1(t2) = qt1 − q−1t2 t1 − t2 F3,1(t2)F1(t1), E1,3(τ2)E1(τ1) = qτ1 − q−1τ2 τ1 − τ2 E1(τ1)E1,3(τ2) and the fact (proved in [28]) that under projections we can freely exchange currents without taking into account the δ-function terms. As result, we get P+ f (F1(t1) · · ·F1(ta−k)F3,1(ta−k+1) · · ·F3,1(ta)) = ∏ 1≤i≤a−k a−k<j≤a qti − q−1tj ti − tj ∏ 1≤i<j≤a−k a−k<i<j≤a qti − q−1tj q−1ti − qtj × ←−∏ a≥`>a−k P+ f (F3,1(t`; t`+1, . . . , ta)) ←−∏ a−k≥`≥1 F+ 1 (t`; t`+1, . . . , ta) = ∏ 1≤i<j≤a−k a−k<i<j≤a ti − tj q−1ti − qtj ∫ dx1 x1 · · · dxa xa Y (ta, . . . , t1;xa, . . . , x1) × F2(F1(xa)) · · ·F2(F1(xa−k+1))F1(xa−k) · · ·F1(x1). (4.15) Analogously P+ e (E1,3(τa) · · ·E1,3(τa−m+1)E1(τa−m) · · ·E1(τ1)) = ∏ 1≤i≤a−k a−k<j≤a qτi − q−1τj τi − τj ∏ 1≤i<j≤a−k a−k<i<j≤a qτi − q−1τj q−1τi − qτj × −→∏ 1≤`≤a−m E+ 1 (t`; t`+1, . . . , ta) −→∏ a−k<`≤a P+ e (E1,3(t`; t`+1, . . . , ta)) = ∏ 1≤i<j≤a−m a−m<i<j≤a τi − τj q−1τi − qτj ∫ dµ1 µ1 · · · dµa µa Z(τa, . . . , τ1;µa, . . . , µ1) × E1(µ1) · · ·E1(µa−m)E2(E1(µa−m+1)) · · ·E2(E1(µa)). (4.16) Here we used the notations ←−∏ a≥`≥1 A` = AaAa−1 · · ·A2A1, −→∏ 1≤`≤a B` = B1B2 · · ·Ba−1Ba for products of non-commutative terms and the identities P+ f (F3,1(t)) = P+ f (F2(F1(t))) = F2 ( P+ f (F1(t)) ) = F2 ( F+ 1 (t) ) , P+ e (E1,3(τ)) = P+ e (E2(E1(τ))) = E2 ( P+ e (E1(τ)) ) = E2 ( E+ 1 (τ) ) , on commutativity of the screening operators and the projections proved in [28]. The last step before getting integral formulas for universal Bethe vectors is to present products of screening operators acting on total currents, F2(F1(xk)) · · ·F2(F1(x1)), E2(E1(µ1)) · · ·E2(E1(µm)), 16 S. Belliard, S. Pakuliak and E. Ragoucy as an integral using formulas (3.24). The presentation follows from the following chain of equa- lities F2(F1(xk)) · · ·F2(F1(x1)) = (q−1 − q)kF+ 2 (xk)F1(xk) · · ·F+ 2 (x2)F1(x2)F+ 2 (x1)F1(x1) = (q−1− q)k ∏ 1≤i<j≤k qxi−q−1xj xi − xj F+ 2 (xk)F+ 2 (xk−1;xk) · · ·F+ 2 (x1;x2, . . . , xk)F1(xk) · · ·F1(x1) = (q−1 − q)k ∫ dz1 z1 · · · dzz zk Y (xk, . . . , x1; zk, . . . , z1)F2(zk) · · ·F2(z1)F1(xk) · · ·F1(x1) (4.17) and E2(E1(µ1)) · · ·E2(E1(µm)) = (q−1 − q)kE1(µ1)E+ 2 (µ1) · · ·E1(µm)E+ 2 (µm) = (q−1 − q)m ∏ 1≤i<j≤m qµi − q−1µj µi − µj E1(µ1) · · ·E1(µm)E+ 2 (µ1;µ2, . . . , µm) · · ·E+ 2 (µm) (4.18) = (q−1 − q)m ∫ dρ1 ρ1 · · · dρz ρm Z(µm, . . . , µ1; ρm, . . . , ρ1)E1(µ1) · · ·E1(µm)E2(ρ1) · · ·E2(ρm), where we have used the commutation relation F1(xj)F+ 2 (xi;xi−1, . . . , xj−1) = qxi − q−1xj xi − xj F+ 2 (xi;xi−1, . . . , xj)F1(xj), E+ 2 (µi;µi+1, . . . , µj−1)E1(µj) = qµi − q−1µj µi − µj E1(µj)E+ 2 (µi;µi+1, . . . , µj). (4.19) Note that these commutation formulas are crucial for the integral formulas given below in (4.20) and (4.21). One can see that the right hand sides of these formulas are not ordered, while the left hand sides are. Example 3. Let us check the first equality in (4.19), in the simplest case. To calculate this exchange relation, we start from the definition of the composed currents F3,1(x) as given in (3.22) and apply to this relation the integral transformation∫ dy y 1 1− y/x1 . To calculate this integral, we decompose the kernel of the integrand as q − q−1x2/y 1− x2/y · 1 1− y/x1 = q − q−1x2/x1 1− x2/x1 · 1 1− y/x1 + (q − q−1) 1− x2/x1 · x2/y 1− x2/y . This leads to F1(x2)F+ 2 (x1) = q − q−1x2/x1 1− x2/x1 F+ 2 (x1)F1(x2) − (q − q−1) 1− x2/x1 F−2 (x2)F1(x2) + 1 1− x2/x1 F3,1(x2) = q − q−1x2/x1 1− x2/x1 ( F+ 2 (x1)− (q − q−1)x1 q − q−1x2/x1 F+ 2 (x2) ) F1(x2) = q − q−1x2/x1 1− x2/x1 F+ 2 (x1;x2)F1(x2), where we have used the definition of the negative half-current (4.12), the expression of the total composed current (3.23) and the Ding–Frenkel relation F2(x2) = F+ 2 (x2)− F−2 (x2). Universal Bethe Ansatz and Scalar Products of Bethe Vectors 17 After substituting formulas (4.17) and (4.18) into integral formulas for the projections of the string (4.15) and (4.16), we obtain, from the resolution of the hierarchical relations for the universal Bethe vectors (4.4) and (4.5), the following intermediate results P+ e (E2(σb) · · ·E2(σ1)E1(τa) · · ·E1(τ1)) = min{a,b}∑ m=0 (q−1 − q)m m!(a−m)!(b−m)! × Symτ̄ ,σ̄  ∏ 1≤i<j≤a−m a−m<i<j≤a τi − τj q−1τi − qτj Y (τa, . . . , τa−m+1;σm, . . . , σ1) × ∮ dν1 ν1 · · · ∮ dνm νm ∮ dµ1 µ1 · · · ∮ dµa µa × Z(τa, . . . , τ1;µa, . . . , µ1)Z(µa, . . . , µa−m+1; ν1, . . . , νm) × P+ e (E2(σb) · · ·E2(σm+1)) E1(µ1) · · ·E1(µa)E2(νm) · · ·E2(ν1)  (4.20) and P+ f (F1(t1) · · ·F1(ta)F2(s1) · · ·F2(sb)) = min{a,b}∑ k=0 (q−1 − q)k k!(a− k)!(b− k)! × Symt̄,s̄  ∏ 1≤i<j≤a−k a−k<i<j≤a ti − tj q−1ti − qtj Z(ta, . . . , ta−k+1; sk, . . . , s1) × ∮ dy1 y1 · · · ∮ dyk yk ∮ dx1 x1 · · · ∮ dxa xa × Y (ta, . . . , t1;xa, . . . , x1)Y (xa, . . . , xa−k+1; y1, . . . , yk) × F2(y1) · · ·F2(yk)F1(xa) · · ·F1(x1)P+ f (F2(sk+1) · · ·F2(sb))  . (4.21) The last step is to move to the left, in (4.20), the product of the total currents E1(µ1) · · · E1(µa) through the projection P+ e (E2(σb) · · ·E2(σm+1)) using the factorization formulas (4.11) and the commutation relations (4.19). Analogously, in (4.21), one has to move to the right the product of the total currents F1(xa) · · ·F1(x1) through the projection P+ f (F2(sk+1) · · ·F2(sb)), using again the factorization formulas (4.11) and the commutation relations (4.19). As result, we obtain the integral formulas (4.8) for the projections of the product of currents for the algebra Uq(ĝl3). 5 Scalar products of universal Bethe vectors 5.1 Commutation of products of total currents Formulas (4.8) show that in order to calculate the scalar product of the universal Bethe vectors, one has to commute the products of the total currents E(µ̄, ν̄) = E1(µ1) · · ·E1(µa) E2(νb) · · ·E2(ν1) and F(x̄, ȳ) = F2(y1) · · ·F2(yb) F1(xa) · · ·F1(x1). 18 S. Belliard, S. Pakuliak and E. Ragoucy According to the decomposition of the quantum affine algebra Uq(ĝl3) used in this paper, the modes of the total currents Fi[n], Ei[n + 1], k+ j [n], n ≥ 0 and a q-commutator E1,3[1] = E2[0]E1[1] − q−1E1[1]E2[0], belong to the Borel subalgebra Uq(b+) ∈ Uq(ĝl3). We define the following ideals in this Borel subalgebra. Definition 1. We note J , the left ideal of Uq(b+) generated by all elements of the form Uq(b+) · Ei[n], n > 0 and Uq(b+) · E1,3[1]. Equalities in Uq(b+) modulo element from the ideal J are denoted by the symbol ‘∼J ’. Definition 2. Let I be the right ideal of Uq(b+) generated by all elements of the form Fi[n] · Uq(b+) such that n ≥ 0. We denote equalities modulo elements from the ideal I by the sym- bol ‘∼I ’. We also define the following ideal in Uq(ĝl3): Definition 3. We denote by K the two-sided Uq(ĝl3) ideal generated by the elements which have at least one arbitrary mode k−j [n], n ≤ 0, of the negative Cartan current k−j (t). Equalities in Uq(ĝl3) modulo element of the ideal K are denoted by the symbol ‘∼K’. Equalities in Uq(ĝl3) modulo the right ideal I, the left ideal J and the two-sided ideal K will be denoted by the symbol ‘≈’. A right weight singular vector defined by the relations (2.7) is annihilated by the right action of any positive mode Ei[n], n > 0, the element E1,3[1] and is a right-eigenvector for k+ j (t), E+ i (τ) · v = 0, P+ e (E1,3(τ)) · v = 0, k+ j (τ) · v = Λj(τ)v, where Λj(τ) are some meromorphic functions, decomposed as a power series in τ−1. A left weight singular vector v′ defined by the relation (2.9) is annihilated by the left action of any nonnegative modes Fi[n], n ≥ 0 and is a left-eigenvector for k+ j (t), v′ · F+ i (t) = 0, v′ · k+ j (t) = Λ′j(t)v ′, where Λ′j(t) are also meromorphic functions. These facts follow from the relation between projections of the currents and the Gauss coordinates of the L-operator (3.1)–(3.3). We observe that the vectors P+ f (F1(t1) · · ·F1(ta)F2(s1) · · ·F2(sb)) · v (5.1) and v′ · P+ e (E2(σb) · · ·E2(σ1)E1(τa) · · ·E1(τ1)) (5.2) belong to the modules over the quantum affine algebra Uq(ĝl3) from the categories of the highest weight and lowest weight representations respectively. This is in accordance with the definition of the completions UE and UF and the corresponding projections given above. We assume the existence of a nondegenerate pairing 〈v′, v〉 and by the scalar product of the left and right universal Bethe vectors, we will understand the coefficient S(τ̄ , σ̄; t̄, s̄) in front of the pairing 〈v′, v〉 in the right hand side of equality 〈v′ · P+ e (E2(σb) · · ·E2(σ1)E1(τa) · · ·E1(τ1)) , P+ f (F1(t1) · · ·F1(ta)F2(s1) · · ·F2(sb)) · v〉 = S(τ1, . . . , τa, σ1, . . . , σb; t1, . . . , ta, s1, . . . , sb)〈v′, v〉. (5.3) Universal Bethe Ansatz and Scalar Products of Bethe Vectors 19 It is clear that the scalar product (2.11) differs from (5.3) by the product a∏ k=1 Λ1(tk)Λ′1(τk) b∏ m=1 Λ2(sm)Λ′2(σm). The problem of calculation of the scalar product of the universal Bethe vectors (5.3) is equiva- lent to the commutation of the projections entering the definitions of the vectors (5.1) and (5.2) modulo the left ideal J and the right ideal I. To calculate this commutation, we use the integral presentation of the projections (4.8), commute the total currents and then calculate the integrals. Since both projections belong to the positive Borel subalgebra Uq(b+), we can neglect the terms which contain the negative Cartan currents k−i (t) and perform the commutation of the total currents modulo the two-sided ideal K. Actually, in commuting the total currents, we will be interested only in terms which are products of combinations of the Uq(ĝl3) positive Cartan currents (3.17). All other terms will be annihilated by the weight singular vectors. Let us recall that elements E(µ̄, ν̄) and F(x̄, ȳ) are elements of the completed algebras UE and UF , which are dual subalgebras in Uq(ĝl3) considered as a quantum double. There is a nondegenerate Hopf pairing between these subalgebras, given by the formulas (3.16). For any elements a ∈ A and b ∈ B from two dual Hopf subalgebras A and B of the quantum double algebra D(A) = A⊕ B, there is a relation [26] 〈a(2), b(2)〉 b(1) · a(1) = a(2) · b(2)〈a(1), b(1)〉, (5.4) where ∆A(a) = a(1) ⊗ a(2) and ∆B(b) = b(1) ⊗ b(2). Let us apply formula (5.4) for a = E(µ̄, ν̄) = E and b = F(x̄, ȳ) = F . Using the current coproduct (3.15), we conclude that ∆(D)E = 1⊗ E + E ′ ⊗ E ′′, ∆(D)F = K+ ⊗F + F ′ ⊗F ′′, (5.5) where the element E ′ satisfies ε(E ′) = 0 and the element E ′′ contains at least one negative Cartan current k−i (τ). The element K+ in (5.5) takes the form K+ = a∏ i=1 ψ+ 1 (xi) b∏ j=1 ψ+ 2 (yj). The left hand side of the relation (5.4) have the form 〈E ,F〉 · K+ mod J̃ and the right hand side of the same relation is E · F mod K. The ideal J̃ , similar to the ideal J , is the left ideal in Uq(ĝl3) generated by the elements Uq(ĝl3) · Ei[n], i = 1, 2 and n ∈ Z. One can check that after integration in (4.8) the terms of the ideal J̃ which have non-positive modes of the currents E1(µk) and E2(νm) on the right will disappear and can be neglected. Alternatively, we can argue that these terms are irrelevant using cyclic ordering of the current or Cartan–Weyl generators, as it was done in the papers [26, 21]. As result, a general equality (5.4) for the given elements a = E(µ̄, ν̄) and b = F(x̄, ȳ) reads E(µ̄, ν̄) · F(x̄, ȳ) = 〈E(µ̄, ν̄),F(x̄, ȳ)〉 a∏ i=1 ψ+ 1 (xi) b∏ j=1 ψ+ 2 (yj) mod (K,J) modulo ideals K and J . This relation shows that instead of calculating the exchange relations for the product of the currents E(µ̄, ν̄) and F(x̄, ȳ) it is enough to calculate the pairing between them. 20 S. Belliard, S. Pakuliak and E. Ragoucy 5.2 Pairing and integral formula for scalar products To calculate the pairing, we will use the basic properties of pairing between dual Hopf subalgebras 〈a1a2, b〉 = 〈a1 ⊗ a2,∆B(b)〉, 〈a, b1b2〉 = 〈∆A(a), b2 ⊗ b1〉, where A = UE and B = UF . From these properties, we obtain 〈E(µ̄, ν̄),F(x̄, ȳ)〉 = b∏ i=1 a∏ j=1 q−1xj − qyi xj − yi a∏ i<j q−1xi − qxj qxi − q−1xj b∏ i<j q−1yi − qyj qyi − q−1yj × (q − q−1)a+b Sym x̄ ( a∏ i=1 δ(µi/xi) ) Symȳ ( b∏ i=1 δ(νi/yi) ) . Using the definition of the scalar product of the universal Bethe vectors (5.3) and integral presentations of the projections (4.8), we conclude Proposition 3. S(τ1, . . . , τa, σ1, . . . , σb; t1, . . . , ta, s1, . . . , sb) = (q − q−1)a+b ∮ dx1 x1 · · · ∮ dxa xa ∮ dy1 y1 · · · ∮ dyb yb b∏ i=1 a∏ j=1 q−1xj − qyi xj − yi × a∏ i<j q−1xi − qxj qxi − q−1xj b∏ i<j q−1yi − qyj qyi − q−1yj E(τ̄ , σ̄; x̄, ȳ)Symx̄,ȳ ( F(t̄, s̄; x̄, ȳ) ) × a∏ i=1 ψ+ 1 (xi) b∏ j=1 ψ+ 2 (yj), (5.6) where the rational series E(τ̄ , σ̄; x̄, ȳ) and F(t̄, s̄; x̄, ȳ) are given in (4.9) and (4.10). 6 Conclusions The kernels entering the formulas (4.8) can be q-symmetrized over integration variables due to the q-symmetric properties of the product of the total currents. In the gl2 case, this leads to the determinant representation of the kernel due to the identity n∏ i<j q−1ti − qtj ti − tj Symt̄ (Y (t̄, x̄)) = n∏ i<j q−1xi − qxj xi − xj Symx̄ (Y (t̄, x̄)) = ∏ i ti ∏ i,j(q −1ti − qxj)∏ i<j(ti − tj)(xj − xi) det ∣∣∣∣ 1 (ti − xj)(q−1ti − qxj) ∣∣∣∣ i,j=1,...,n , where the determinant on the right hand side is called an Izergin determinant. 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Phys. 145 (2005), 1373–1399, math.QA/0610433. http://dx.doi.org/10.1007/s00023-009-0416-x http://dx.doi.org/10.1007/s00023-009-0416-x http://arxiv.org/abs/0810.3135 http://dx.doi.org/10.1007/s00023-006-0280-x http://arxiv.org/abs/math-ph/0512037 http://dx.doi.org/10.1070/RM2007v062n04ABEH004430 http://dx.doi.org/10.1090/S1061-0022-2010-01110-5 http://arxiv.org/abs/0711.2821 http://dx.doi.org/10.1007/BF02773478 http://arxiv.org/abs/q-alg/9608005 http://dx.doi.org/10.1007/s00220-007-0351-y http://arxiv.org/abs/math.QA/0610398 http://dx.doi.org/10.1016/j.geomphys.2007.02.005 http://arxiv.org/abs/math.QA/0610517 http://dx.doi.org/10.1007/s11232-005-0167-x http://dx.doi.org/10.1007/s11232-005-0167-x http://arxiv.org/abs/math.QA/0610433 1 Introduction 2 Universal Bethe vectors in terms of L-operator 2.1 Uq(gl"0362gl3) in L-operator formalism 2.2 Universal off-shell Bethe vectors 3 Current realization of Uq(gl"0362gl3) 3.1 Gauss decompositions of L-operators 3.2 Borel subalgebras and projections on their intersections 3.3 Definition of the composed currents 4 Universal Bethe vectors and projections 4.1 Universal Bethe vectors through currents 4.2 Calculation of the universal off-shell Bethe vectors 4.3 Integral presentation of the projections (4.4) and (4.5) 4.4 Proof of the integral presentation of the projections (4.4) and (4.5) 5 Scalar products of universal Bethe vectors 5.1 Commutation of products of total currents 5.2 Pairing and integral formula for scalar products 6 Conclusions References

Universal Bethe Ansatz and Scalar Products of Bethe Vectors

Institution

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