Algebraic Bethe ansatz for o 2n+1 -invariant integrable models
aa r X i v : . [ m a t h - ph ] A ug Algebraic Bethe ansatz for o n +1 -invariant integrable models A. Liashyk a,b , S. Z. Pakuliak c,d a Skolkovo Institute of Science and Technology, Moscow, Russia b National Research University Higher School of Economics, Russia c Moscow Institute of Physics and Technology, Dolgoprudny, Moscow reg., Russia d Laboratory of Theoretical Physics, JINR, Dubna, Moscow reg., Russia
Abstract
A class of o n +1 -invariant quantum integrable models is investigated in theframework of algebraic Bethe ansatz method. A construction of the o n +1 -invariantBethe vector is proposed in terms of the Drinfeld currents for the double of Yangian D Y ( o n +1 ). Action of the monodromy matrix entries onto off-shell Bethe vectorsfor these models is calculated. Recursion relations for these vectors were obtained.The action formulas can be used to investigate structure of the scalar products ofBethe vectors in o n +1 -invariant models. Quantum integrable systems where the algebraic Bethe ansatz may be applied are de-scribed by the monodromy matrices that satisfy quadratic relations defined by some R -matrix. Main goal of the algebraic Bethe ansatz is to construct state vectors or Bethevectors for integrable models from matrix elements of the corresponding monodromies.Eigenvalue property of the Bethe vectors with respect to the transfer matrix uses thesequadratic commutation relations between monodromy matrix entries and leads to theBethe equations on the parameters of Bethe vectors.In this paper, we will consider quantum integrable systems whose monodromy ma-trices satisfy the commutation relations with the o n +1 -invariant R -matrix proposed in[1, 2] and solved in [3]. This R -matrix is associated with an infinite-dimensional algebra,that is double of Yangian D Y ( o n +1 ) [4, 5]. Generating series of generators of this algebra [email protected], [email protected] T -operator that will satisfy the same quadratic com-mutation relations as the monodromy matrix of o n +1 -invariant integrable system. Thecoincidence of these commutation relations allows to construct state vectors in terms ofgenerators of the algebra D Y ( o n +1 ) and use the properties of this algebra to study thespace of states in o n +1 -invariant integrable system.The advantages of this approach to the algebraic Bethe ansatz are that doubles ofYangians and quantum affine algebras, in addition to the R -matrix formulation, also havea ’new’ realization in terms of formal series or ’currents’ proposed by [5]. For quantumaffine algebra U q ( b gl n ) the equivalence of the two formulations was proved in [6], and forsimilar infinite-dimensional algebras associated with algebras of the series B , C and D analogous equivalences were recently proved in [7–10].The main tools in proving the equivalence of various realizations of quantum affinealgebras and doubles of Yangians are the so-called Gaussian coordinates of T -operators.These objects turned out to be very useful in research of the space of states in quan-tum integrable models associated with trigonometric and rational deformations of affinealgebra b gl n and their super-symmetric extensions [11–13]. Gaussian coordinates may berelated to the currents in the ’new’ realization of the Yangians and quantum affine al-gebras [5] using projections to the intersections of the different types Borel subalgebras.The Bethe vectors itself can be constructed as projections from the products of currents[14].Until recently the description of the space of states in the quantum integrable mod-els using the current realization of the corresponding infinite-dimensional algebras wasavailable only for the models associated with serie A algebras. Recent results publishedin [7–10] open the possibility to develop tools for the investigation of the quantum in-tegrable models associated with infinite-dimensional algebras for B , C and D series. Inparticular, the relations between Gaussian coordinates for the Yangian doubles of B , C and D series and projections of the corresponding currents were established in [15]. Notealso the papers [16–18], where the different methods of investigations of the orthogonaland symplectic quantum integrable models were developed.The present paper is a continuation of the research started in [19] where a descriptionof the off-shell Bethe vectors for the integrable models associated with the algebra o wasobtained. In many cases, to investigate the space of states of quantum integrable modelone does not need the explicit formulas for the Bethe vectors in terms of monodromyentries. It is sufficient to explore the explicit formulas of the actions of the monodromymatrix elements onto off-shell Bethe vectors. In [20] such action formulas were presentedin case of the supersymmetric gl ( m | n )-invariant integrable models. In this paper weobtain the explicit formulas for such an action for the Bethe vectors in the o n +1 -invariantquantum integrable models.The paper is composed as follows. Section 2 is devoted to definition of the class of o n +1 -invariant quantum integrable models based on R -matrix found by A.B. Zamolod-2hikov and Al.B. Zamolodchikov. Main results of our research is presented in the sec-tion 3 together with demonstration of the different reductions of the main action formulato already known cases. Last section 4 contains the proof of two technical statementsnecessary for the calculation of the action formulas. These proofs use the projectionmethod formulated for this case in [15]. o n +1 -invariant R -matrix and RT T algebra
Let N = 2 n + 1 for positive integer n ≥ e i,j are N × N matrices with the onlynonzero entry equal to 1 at the intersection of the i -th row and j -th column. We will useintegers to number matrix entries of the operators in End( C N ): − n ≤ i, j ≤ n . Let P (permutation operator) and Q be operators acting in C N ⊗ C N P = n X i,j = − n e i,j ⊗ e j,i , Q = n X i,j = − n e i,j ⊗ e − i, − j . We denote by R ( u, v ) o n +1 -invariant R -matrix [1] R ( u, v ) = I ⊗ I + c P u − v − c Q u − v + cκ n , (2.1)where I = P ni = − n e i,i is the identity operator in C N , c is a constant, u and v are arbitrarycomplex parameters called spectral parameters and κ = N/ − n − / . Due to the properties
P Q = Q P = Q , P = I ⊗ I , Q = N Q this R -matrix obeys the Yang-Baxter equation R , ( u , u ) · R , ( u , u ) · R , ( u , u ) = R , ( u , u ) · R , ( u , u ) · R , ( u , u )in C N ⊗ C N ⊗ C N (here subscripts of R denote spaces C N in which R -matrix acts) andsatisfies the unitarity condition R ( u, v ) R ( v, u ) = (cid:18) − c ( u − v ) (cid:19) I ⊗ I . (2.2)Let T ( u ) be an operator valued N × N matrix satisfying the quadratic commutationrelations R ( u, v ) ( T ( u ) ⊗ I ) ( I ⊗ T ( v )) = ( I ⊗ T ( v )) ( T ( u ) ⊗ I ) R ( u, v ) . (2.3)3hese commutation relations can be written in terms of the matrix entries T i,j ( u ) T ( u ) = n X i,j = − n e i,j T i,j ( u ) , [ T i,j ( u ) , T k,l ( v )] = cu − v ( T k,j ( v ) T i,l ( u ) − T k,j ( u ) T i,l ( v )) ++ cu − v + cκ δ k, − i n X p = − n T p,j ( u ) T − p,l ( v ) − δ l, − j n X p = − n T k, − p ( v ) T i,p ( u ) ! . (2.4)As a direct consequence of the commutation relations (2.4) the matrix elements shouldsatisfy the relations [8] n X m = − n T − m, − i ( u − cκ ) T m,j ( u ) = n X m = − n T i,m ( u ) T − j, − m ( u − cκ ) = z ( u ) δ i,j , (2.5)where z ( u ) is a central elements in the RT T algebra (2.4). In what follows we set thiscentral element to 1. The quadratic relations (2.5) imposes nontrivial relations betweenentries T i,j ( u ). o n +1 -invariant integrable models We denote the
RT T algebra (2.3) as B algebra. The algebraically dependent elements ofthis algebra are operators T i,j [ ℓ ], ℓ ≥ − n ≤ i, j ≤ n gathered into generating series T i,j ( u ) = δ ij + X ℓ ≥ T i,j [ ℓ ]( u/c ) − ℓ − . (2.6)The zero modes operators T i,j [0] ≡ T i,j will play a special role below.The generating series (2.6) can be used to construct o n +1 -invariant integrable models.Let H be a Hilbert space of states of such a model, which can be considered as represen-tation space for the algebra B . In order to treat this model by the algebraic Bethe ansatzmethod, the physical space of the model H must include a special vector or referencestate | i such that T i,j ( u ) | i = 0 , − n ≤ j < i ≤ n,T i,i ( u ) | i = λ i ( u ) | i , − n ≤ i ≤ n, (2.7)where λ i ( u ) characterize the concrete model. They are free functional parameters modulothe relations (2.17) which follow from (2.5). As a result of the commutation relations(2.3) the transfer matrix T ( z ) = n X i = − n T i,i ( z ) (2.8)4enerates a commuting set of integrals of the model: [ T ( u ) , T ( v )] = 0.Let t iℓ ∈ C , i = 0 , , . . . , n − ℓ = 1 , . . . , r i be generic complex parameters, where thepositive integers r i ≥ i . These numbersare cardinalities of the sets ¯ t i : r i = t i and if some r i = 0 then the corresponding set ¯ t i is empty: ¯ t i = ∅ . We will gather these parameters into sets¯ t = { ¯ t , . . . , ¯ t n − } , ¯ t i = { t i , . . . , t ir i } , i = 0 , , . . . , n − . (2.9)For ℓ = 1 , . . . , r i , the notation ¯ t iℓ = { ¯ t i \ t iℓ } stands for the set ¯ t i with the parameter t iℓ omitted. Then, ¯ t iℓ has cardinality t iℓ = r i −
1. We will call the parameters t iℓ as the Bethe parameters .In what follows we will need the following rational functions g ( u, v ) = cu − v , f ( u, v ) = u − v + cu − v , h ( u, v ) = f ( u, v ) g ( u, v ) = u − v + cc . (2.10)To simplify presentation of our results we introduce the set of functions f s ( u, v ) = f ( u, v ) = u − v + c/ u − v , s = 0 ,f ( u, v ) , s = 1 , . . . , n − . (2.11)There is an equality f ( u, v ) = f ( u + c/ , v ) f ( u, v ) . We will use a shorthand notation for the products of functions of one or two variables.For example, λ k (¯ u ) = Y u i ∈ ¯ u λ k ( u i ) , f ( u, ¯ v ) = Y v j ∈ ¯ v f ( u, v j ) , f (¯ u, ¯ v ) = Y u i ∈ ¯ u Y v j ∈ ¯ v f ( u i , v j ) . If any set in these formulas is empty, then the corresponding product is equal to 1.The algebraic Bethe ansatz allows to describe the physical states of the model in termsof Bethe vectors. The vectors B (¯ t ) are called on-shell Bethe vectors if the set ¯ t of theBethe parameters satisfies the so-called Bethe equations α s ( t sℓ ) = f s ( t sℓ , ¯ t sℓ ) f s (¯ t sℓ , t sℓ ) f (¯ t s +1 , t sℓ ) f ( t sℓ , ¯ t s − ) , ℓ = 1 , . . . , r s , s = 0 , , . . . , n − , (2.12)where we assume ¯ t − = ¯ t n = ∅ and define functions α s ( z ) = λ s ( z ) λ s +1 ( z ) . (2.13)5n-shell Bethe vectors are eigenstates of the transfer matrix T ( z ) B (¯ t ) = τ ( z ; ¯ t ) B (¯ t ) . (2.14)To describe the eigenvalue τ ( z ; ¯ t ) it is convenient to introduce the notation z s = z − c (cid:18) s − (cid:19) , s = 0 , , . . . , n − , n . (2.15)Note that z n = z − cκ . The eigenvalue τ ( z ; ¯ t ) in (2.14) is equal to τ ( z ; ¯ t ) = λ ( z ) f (¯ t , z ) f ( z, ¯ t ) ++ n X s =1 (cid:16) λ s ( z ) f (¯ t s , z ) f ( z, ¯ t s − ) + λ − s ( z ) f (¯ t s − , z s − ) f ( z s , ¯ t s ) (cid:17) , (2.16)where λ − j ( z ) = 1 λ n ( z n ) n − Y s = j λ s +1 ( z s ) λ s ( z s ) = 1 λ j ( z j ) n Y s = j +1 λ s ( z s − ) λ s ( z s ) (2.17)for j = 0 , , . . . , n .The Bethe equations (2.12) can be obtained as vanishing of the residues of the eigen-value τ ( z ; ¯ t ) (2.16) at the values z = t sℓ or z s = t sℓ . If the Bethe parameters ¯ t are free,such vectors are called off-shell Bethe vectors. There exist different methods to describethe off-shell Bethe vectors B (¯ t ) ∈ H in terms of polynomials of the non-commuting mon-odromy matrix elements T i,j ( u ), i < j acting onto reference vector | i . But it appearsthat in order to calculate physically interesting quantities in such integrable models, onedoes not need to have fully explicit formulas for these vectors. In many cases it is sufficientto know explicit formulas for the action of the monodromy matrix elements T i,j ( z ) ontooff-shell Bethe vectors. One can prove that the set of these vectors is closed under thisaction, which means that the product T i,j ( z ) · B (¯ t ) maybe written as a linear combinationof the vectors of the same structure.The search of the explicit formulas for such an action in o n +1 -invariant integrablemodels is the main goal of this paper. In this section we present the main result of this paper. Besides collections of the Betheparameters ¯ t defined by (2.9) we introduce the collection of sets ¯ w ¯ w = { ¯ w , ¯ w , . . . , ¯ w n − } , ¯ w s = { t s , . . . , t sr s , z, z s } , (3.1)with z s defined by (2.15). Let ¯ w n = { z, z n } be an auxiliary set of the cardinality 2.6he action of the monodromy matrix element T i,j ( z ), − n ≤ i, j ≤ n onto off-shellBethe vector B (¯ t ) is given by the sum over partitions of the sets { ¯ w s I , ¯ w s II , ¯ w s III } ⊢ ¯ w s , s = 0 , , . . . , n such that cardinalities of the sets ¯ w s I and ¯ w s III are w s I = Θ( i + s ) + Θ( i − s − , w s III = Θ( s − j ) + Θ( − j − s − . (3.2)Here Θ( p ) is a Heaviside step function defined for integers p ∈ Z Θ( p ) = ( , p ≥ , , p < . The cardinalities of the sets ¯ w s I and ¯ w s III may be equal to 0, 1 or 2 depending on theinterrelations between integers i , j and s . Let us define σ i = 2Θ( i − − . (3.3)It is clear that σ i = − i ≤ σ i = 1 for i >
0. Note also, that according to (3.2) w n I = w n III = 1 for all values of the indices − n ≤ i, j ≤ n .Let us define the rational functions of two variables γ s ( u, v ) = f ( u, v ) = u − v + c/ u − v , s = 0 ,f ( u, v ) h ( u, v ) h ( v, u ) = c ( u − v )( v − u + c ) , s = 1 , . . . , n − . (3.4)The main result of this paper may be formulated as following Theorem 3.1.
The action of the monodromy matrix element T i,j ( z ) , − n ≤ i, j ≤ n ontooff-shell Bethe vector B (¯ t ) is given by the explicit formula T i,j ( z ) · B (¯ t ) = − σ i σ − j λ n ( z ) g ( z , ¯ t ) κ h ( z, ¯ t ) X part B ( ¯ w II ) n − Y s =0 α s ( ¯ w s III ) × n − Y s =0 γ s ( ¯ w s I , ¯ w s II ) γ s ( ¯ w s II , ¯ w s III ) γ s ( ¯ w s I , ¯ w s III ) h ( ¯ w s +1 II , ¯ w s I ) h ( ¯ w s +1 III , ¯ w s I ) h ( ¯ w s +1 III , ¯ w s II ) g ( ¯ w s +1 I , ¯ w s II ) g ( ¯ w s +1 I , ¯ w s III ) g ( ¯ w s +1 II , ¯ w s III ) . (3.5) Here we use notation ¯ w II = { ¯ w II , ¯ w II , . . . , ¯ w n − II } . The sets ¯ w s for s = 0 , , . . . , n are dividedinto subsets { ¯ w s I , ¯ w s II , ¯ w s III } ⊢ ¯ w s and summation in (3.5) goes over these partitions withcardinalities described by (3.2) . Recall that functions α s ( z ) are defined by (2.13).7 emark 3.1. It follows from the action formula that effectively the sum over partitionsof the set ¯ w n reduces only to one term when ¯ w n I = { z n } and ¯ w n III = { z } . Indeed, letus assume that ¯ w n I = { z } since w n I = 1. Then the product g ( z, ¯ w n − II ) − g ( z, ¯ w n − III ) − in denominator of the second line of (3.5) yields z ∈ ¯ w n − I otherwise we got zero term.Considering other factors g ( ¯ w s +1 I , ¯ w s II ) − g ( ¯ w s +1 I , ¯ w s III ) − for s = 0 , n − z should be in the set ¯ w s I for all s . Analogously, letus assume that ¯ w n III = { z n } . The product h ( z n , ¯ w n − I ) h ( z n , ¯ w n − II ) yields z n − ∈ ¯ w n − III sinceotherwise we got zero contribution because h ( z n , z n − ) = 0. Continuing we prove thatassumption ¯ w n III = { z n } yields that z s ∈ ¯ w s III for all s . But (3.5) has a factor f ( ¯ w I , ¯ w III ) ∼ f ( z, z ) ≡
0. This proves that our initial assumptions that ¯ w n I = { z } and ¯ w n III = { z n } leadto the zero contribution into sum over partitions. Remark 3.2. ( Reduction to o n − -invariant Bethe vectors .) By the same argumentsas in the previous remark we may observe that the action formula respect the hierar-chical embedding of the o n − -invariant RT T algebra into o n +1 -invariant RT T algebradescribed in lemma (3.6) of the paper [8]. It means, that if ¯ t n − = ∅ then we can provethat ¯ w n − I = { z n − } and ¯ w n − III = { z } and that the action formula (3.5) for the valuesof the indices − n + 1 ≤ i, j ≤ n − o n − -invariantintegrable model. To prove the statement of the theorem 3.1 we will use so called zero-modes method . Todescribe it we modify slightly the definition of the algebra B given in the section 2.1. Let K = diag( χ − n , . . . , χ − , , χ , . . . , χ n )be a diagonal matrix with non-zero entries χ i ∈ C . Sometimes we will use in the formulasbelow the notation χ = 1. The diagonal matrix K satisfies the relation R ( u, v ) · K ⊗ K = K ⊗ K · R ( u, v ) (3.6)if χ − i = χ − i , i = 1 , . . . , n. (3.7)So it has only n independent parameters χ i for i = 1 , . . . , n . Due to (3.6), the monodromymatrix T K ( u ) = K · T ( u ) satisfies the same commutation relations (2.4). The onlydifference will be that instead of the expansion (2.6) we will have T K i,j ( u ) = χ i δ ij + cu T i,j + O ( u − ) . (3.8) Recall that we denoted zero modes T i,j [0] as T i,j for − n ≤ i, j ≤ n . h T i,j ( u ) , T k,l i = δ i,l χ i T k,j ( u ) − δ k,j χ j T i,l ( u ) − δ i, − k χ l T − l,j ( u ) + δ − l,j χ k T i, − k ( u ) . (3.9)Proof of the theorem 3.1 is based on the commutation relations (3.9) and two propo-sitions. Proposition 3.1.
The action of the zero mode operators T j,i onto off-shell Bethe vectors B (¯ t ) for ≤ i < j ≤ n is given by the equality T j,i · B (¯ t ) = X part B (¯ t II ) n − Y s =1 g (¯ t s II , ¯ t s − I ) g (¯ t s I , ¯ t s − II ) g (¯ t s I , ¯ t s − I ) 1 h (¯ t s II , ¯ t s I ) h (¯ t s I , ¯ t s II ) × χ j j − Y s = i α s (¯ t s I ) f (¯ t s I , ¯ t s − ) f s (¯ t s II , ¯ t s I ) h (¯ t s I , ¯ t s − I ) − χ i j − Y s = i f (¯ t s +1 , ¯ t s I ) f s (¯ t s I , ¯ t s II ) h (¯ t s +1 I , ¯ t s I ) ! , (3.10) where the functions α s ( t ) are defined by (2.12) and the sum goes over partitions { ¯ t s I , ¯ t s II } ⊢ ¯ t s with cardinalities t s I = 1 for s = i, . . . , j − and t s I = 0 for other s . Remark 3.3. If χ j = 1 and the Bethe parameters ¯ t satisfy the Bethe equations (2.12)the on-shell Bethe vectors become highest weight vectors for the algebra o n +1 . Literally,it means that in this case T j,i · B (¯ t ) = 0.Note that the product in the first line of (3.10) is going effectively from s = i to s = j . Proposition 3.2.
The action of monodromy matrix element T − n,n ( z ) onto off-shell Bethevector B (¯ t ) (4.9) is regular and given by the relation T − n,n ( z ) · B (¯ t ) = − κ g ( z , ¯ t ) h ( z, ¯ t n − ) h ( z, ¯ t ) g ( z n , ¯ t n − ) λ n ( z ) B ( ¯ w ) , (3.11) and the collection of sets ¯ w is defined by (3.1) . Proofs of the propositions 3.1 and 3.2 will be given in section 4 using identificationof
RT T algebra B with the standard Borel subalgebra of the Yangian double D Y ( o n +1 )[4]. This infinite-dimensional algebra was investigated in [7, 15] and we will use certainprojections onto intersections of the different types Borel subalgebras studied in [14] forthe quantum affine algebras.As a direct consequence of (3.9) the zero-modes operators obey the relations h T i,j , T k,l i = δ i,l χ i T k,j − δ k,j χ j T i,l − δ i, − k χ l T − l,j + δ − l,j χ k T i, − k . (3.12) In what follows we remove the superscript in the notation of the monodromy matrix elements T K i,j ( u )always assuming expansion (3.8).
9s well as for the generating series T i,j ( u ) the quadratic relation (2.5) impose sev-eral relations to the zero-modes operators. Substituting expansion (3.8) into (2.5) andequating terms at u and u − we obtain (3.7) and ( − n ≤ i, j ≤ n ) χ − j T i,j + χ i T − j, − i = 0 . (3.13)It follows from (3.7) and (3.13) that T − j, − i = − χ − i χ − j T i,j , − n ≤ i, j ≤ n (3.14)and this equality for j = − i implies that T i, − i = − T i, − i = 0 , − n ≤ i ≤ n . Equality (3.14) and commutation relations (3.12) allow to express all zero-modes throughthe algebraically independent set of generators T i,i , T i − ,i , T i,i − for i = 1 , . . . , n .Let us sketch the proof of the theorem 3.1 using zero-modes method. We start from(3.11) and commutation relation[ T − n,n ( z ) , T n,i ] = − χ n T − n,i ( z ) − χ i T − i,n ( z ) (3.15)for 0 ≤ i ≤ n − T − i,n ( z ).This can be achieved by applying the equality (3.15) to the off-shell Bethe vector B (¯ t )and equating the coefficients at χ i in left and right hand side of this equality. This canbe done because χ i are independent parameters.Next we start from the action of the entry T ,n ( z ) and applying the commutationrelations [ T ,n ( z ) , T i, ] = T i,n ( z ) − δ i,n χ n T , ( z )to B (¯ t ) for 1 ≤ i ≤ n we obtain the action of the rest entries T i,n ( z ) from the last columnof monodromy matrix T ( z ).At the next step we explore already calculated actions of the entries T i,n ( z ), − n ≤ i ≤ n and the commutation relation[ T i,n ( z ) , T n,j ] = − χ n T i,j ( z ) − δ i, − n χ j T − j,n ( z ) + δ i,j χ i T n,n ( z )to obtain the action formulas of the entries T i,j ( z ) for all i and 0 ≤ j ≤ n − T i, ( z ) for all i and use the commutationrelation [ T i, ( z ) , T j, ] = χ j T i, − j ( z ) − δ i, − j T , ( z ) + δ i, χ i T j, ( z )to obtain action formulas for remaining entries T i, − j ( z ), − n ≤ i ≤ n and 1 ≤ j ≤ n . Performing these calculations we can always equate the terms at the independentparameters χ j , 0 ≤ j ≤ n . (cid:3) .2 Recurrence relations The action formulas (3.5) allows to obtain the recurrence relations for the off-shell Bethevectors in o n +1 -invariant integrable models. For gl ( m | n )-invariant supersymmetric mod-els such recurrence relations were obtained in [21]. Proposition 3.3.
For n ≥ one has following recurrence relations for the off-shell Bethevector B (¯ t ) B (¯ t , . . . , ¯ t n − , { ¯ t n − , z } ) = 1 h ( z, ¯ t n − ) λ n ( z ) × n − X i = − n X part σ i +1 T i,n ( z ) · B (¯ t II ) h (¯ t n − II , z ) g ( z, ¯ t n − II ) n − Y s =0 γ s (¯ t s I , ¯ t s II ) n − Y s =1 h (¯ t s II , ¯ t s − I ) g (¯ t s I , ¯ t s − II ) , (3.16) where sum in (3.16) goes over partitions { ¯ t s I , ¯ t s II } ⊢ ¯ t s with different cardinalities of ¯ t s I fordifferent terms depending on i for s ≤ n − as t s I = ( , ≤ s ≤ i − , , i ≤ s ≤ n − for ≤ i ≤ n − , t s I = ( , ≤ s ≤ − i − , , − i ≤ s ≤ n − for − n + 1 ≤ i ≤ − and for s = n − t n − I = ( , − n + 1 ≤ i ≤ n − , , i = − n . (3.18)Proof of this proposition can be performed in a similar way as in [21]. We have to usethe action formulas (3.5) in the right hand side of (3.16) and prove that the right handside identically coincide with the left hand side in this equality. The recurrence relationsin case of n = 1 were proved in [19]. (cid:3) The statement of the proposition 3.3 is a recursion procedure which allows to reducethe set of the Bethe paremetrs ¯ t n − until this set becomes empty. This yields the presenta-tion of the off-shell o n +1 -invariant Bethe vector B (¯ t , . . . , ¯ t n − ) as a linear combinationsof the products of the matrix entries T i,n ( t n − ℓ ), − n ≤ i ≤ n − o n − -invariant Bethe vector B (¯ t , . . . , ¯ t n − ) (see remark 3.2). Then we re-peat this procedure to express o n − -invariant Bethe vector B (¯ t , . . . , ¯ t n − ) as a linearcombinations of the products of the matrix entries T i,n − ( t n − ℓ ), − n + 1 ≤ i ≤ n − o n − -invariant Bethe vector B (¯ t , . . . , ¯ t n − ) and so on. Finally, one may obtain a11resentation of the off-shell Bethe vector as polynomial of the matrix entries T i,j ( t j − ℓ )with rational coefficients acting on the reference vector | i with − j ≤ i ≤ j − ≤ j ≤ n .Note that at each step this recurrence procedure is in accordance with embedding ofthe smaller algebra B n − into the bigger algebra B n described in [8] (see remark 3.2).For example, first nontrivial Bethe vectors in o -invariant model B ( t ; t ) = T , ( t ) | i λ ( t ) + 1 g ( t , t ) T , ( t ) T , ( t ) | i λ ( t ) λ ( t ) , B ( t ; { t , t } ) = h ( t , t ) h (¯ t , ¯ t ) T , ( t ) T , ( t ) | i λ ( t ) λ ( t ) + 1 g ( t , t ) h (¯ t , ¯ t ) T , ( t ) T , ( t ) | i λ ( t ) λ ( t )+ 1 g (¯ t , t ) h (¯ t , ¯ t ) T , ( t ) T , ( t ) T , ( t ) | i λ ( t ) λ ( t ) λ ( t ) , B ( { t , t } ; t ) = − T − , ( t ) | i λ ( t ) + f ( t , t ) g ( t , t ) T , ( t ) T , ( t ) | i λ ( t ) λ ( t ) + f ( t , t ) g ( t , t ) T , ( t ) T , ( t ) | i λ ( t ) λ ( t ) − g ( t , ¯ t ) h ( t , t ) T , ( t ) T − , ( t ) | i λ ( t ) λ ( t ) + 1 g ( t , ¯ t ) f ( t + c/ , t ) T , ( t ) T , ( t ) T , ( t ) | i λ ( t ) λ ( t ) λ ( t ) . First nontrivial off-shell Bethe vector in o -invariant model is B ( t ; t ; t ) = T , ( t ) | i λ ( t ) + 1 g ( t , t ) T , ( t ) T , ( t ) | i λ ( t ) λ ( t ) + 1 g ( t , t ) T , ( t ) T , ( t ) | i λ ( t ) λ ( t )+ 1 g ( t , t ) g ( t , t ) T , ( t ) T , ( t ) T , ( t ) | i λ ( t ) λ ( t ) λ ( t ) . To obtain these formulas one has to apply recurrence relations (3.16) and similar relationsfor the o -invariant Bethe vectors presented in [19].Besides recurrence relations given by the proposition 3.3 corresponding to the lastcolumn of monodromy matrix one can obtain from the action formulas the recurrencerelation with respect to the first row of the monodormy matrix. We formulate this relationas the following Proposition 3.4.
For n ≥ one has an alternative recurrence relations for the off-shellBethe vector B (¯ t ) B (¯ t , . . . , ¯ t n − , { ¯ t n − , z n − } ) = 1 h (¯ t n − , z n − ) λ − n +1 ( z ) × n X j = − n +1 X part ( − δ j,n h ( z n − , ¯ t n − II ) σ j T − n,j ( z ) · B (¯ t II ) g ( z n − , ¯ t n − II ) n − Y s =0 α s (¯ t s III ) γ s (¯ t s II , ¯ t s III ) n − Y s =1 h (¯ t s III , ¯ t s − II ) g (¯ t s II , ¯ t s − III ) , (3.19)12 here sum in (3.19) goes over partitions { ¯ t s II , ¯ t s III } ⊢ ¯ t s with cardinalities ( s < n − ) t s III = ( , ≤ s ≤ − j − , , − j ≤ s ≤ n − for − n + 1 ≤ j ≤ , t s III = ( , ≤ s ≤ j − , , j ≤ s ≤ n − for ≤ j ≤ n − and for s = n − t n − III = ( , − n + 1 ≤ j ≤ n − , , j = n . In (3.16) and (3.19) the sign factor σ i is defined by (3.3). B (¯ t ) Let us demonstrate in this section how the action formulas (3.5) reproduce the eigenvalue(2.16). We formulate it as following
Proposition 3.5.
The action of the transfer matrix (2.8) onto off-shell Bethe vectors B (¯ t ) which follows from (3.5) is T ( z ) · B (¯ t ) = τ ( z ; ¯ t ) B (¯ t ) + · · · , (3.20) where eigenvalue τ ( z ; ¯ t ) is given by the equality (2.16) and · · · stands for the termswhich are vanishing if Bethe equations (2.12) are satisfied. It is clear that first term in the right hand side of (3.20) will correspond to the socalled ’wanted’ terms in the right hand side of the diagonal elements actions T ℓ,ℓ ( z ) whichcorresponds to the partitions such that ¯ w II = ¯ t . It is convenient to consider separately theaction of T − ℓ, − ℓ ( z ) and T ℓ,ℓ ( z ) for ℓ = 0 , , . . . , n − , n . According to (3.2) the cardinalitiesof the sets ¯ w s I and ¯ w s III for the actions of T − ℓ, − ℓ ( z ) are s = 0 , . . . , ℓ − , w s I = 0 and w s III = 2 ,s = ℓ, . . . , n − , w s I = 1 and w s III = 1 , and for the action of T ℓ,ℓ ( z ) s = 0 , . . . , ℓ − , w s I = 2 and w s III = 0 ,s = ℓ, . . . , n − , w s I = 1 and w s III = 1 .
13n the first case of the action T − ℓ, − ℓ ( z ) the sets ¯ w s I and ¯ w s III which will correspond tothe wanted terms are for s = 0 , . . . , ℓ − , ¯ w s I = ∅ , ¯ w s III = { z, z s } ,s = ℓ, . . . , n − , ¯ w s I = { z } , ¯ w s III = { z s } or ¯ w s I = { z s } , ¯ w s III = { z } . (3.21)In the second case of the action T ℓ,ℓ ( z ) the sets ¯ w s I and ¯ w s III for the wanted terms are s = 0 , . . . , ℓ − , ¯ w s I = { z, z s } , ¯ w s III = ∅ ,s = ℓ, . . . , n − , ¯ w s I = { z } , ¯ w s III = { z s } or ¯ w s I = { z s } , ¯ w s III = { z } . (3.22)The cases ℓ = 0 in (3.21) and in (3.22) both correspond to the action of T , ( z ) and in thiscase all sets ¯ w s I and ¯ w s III have cardinalities 1. On the other hand the cases ℓ = 1 , . . . , n in(3.21) and in (3.22) correspond to the action of T − ℓ, − ℓ ( z ) and T ℓ,ℓ ( z ) respectively. For theaction of T − ℓ, − ℓ ( z ) the term in (3.5) which produces the wanted term corresponds to thepartition ¯ w s III = { z, z s } for s = 0 , . . . , ℓ − w s III = { z } , ¯ w s I = { z s } for s = ℓ, . . . , n − T ℓ,ℓ ( z ) we have ¯ w s I = { z, z s } for s = 0 , . . . , ℓ − w s III = { z } , ¯ w s I = { z s } for s = ℓ, . . . , n − T − ℓ, − ℓ ( z ) to the eigenvalue (2.16). Forthe factors depending on the functional parameters λ j ( z ) we have λ n ( z ) n − Y s =0 α s ( z ) ℓ − Y s =0 α s ( z s ) = λ n ( z ) n − Y s =0 λ s ( z ) λ s +1 ( z ) ℓ − Y s =0 λ s ( z s ) λ s +1 ( z s ) == λ ( z ) λ ( z ) λ ℓ ( z ℓ − ) ℓ − Y s =1 λ s ( z s ) λ s ( z s − ) = 1 λ ℓ ( z ℓ ) n Y s = ℓ +1 λ s ( z s − ) λ s ( z s ) = λ − ℓ ( z ) , where we have used the relations (2.17) for the functional parameters. One may fur-ther check that wanted terms of (3.5) for the action T − ℓ, − ℓ ( z ) reduces to the func-tion λ − ℓ ( z ) f (¯ t ℓ − , z ℓ − ) f ( z ℓ , ¯ t ℓ ) restoring part of the eigenvalue (2.16). Analogously, onemay check that the wanted terms of (3.5) for the action of T ℓ,ℓ ( z ) produces the term λ ℓ ( z ) f (¯ t ℓ , z ) f ( z, ¯ t ℓ − ) of the eigenvalue (2.16).Unfortunately, the unwanted terms marked by dots in (3.20) cannot be presentedin the nice form. One can investigate all these terms case by case and verify that allthese terms will be proportional to the differences of the left and right hands sides ofthe equalities (2.12) and will disappear if the Bethe equations for the parameters ¯ t aresatisfied. (cid:3) gl n -invariant Bethe vectors Let us consider formula (3.5) for i = 1, j = n and ¯ t = ∅ . For these values of indices σ = 1, σ − n = − w s III = 0 for s = 0 , . . . , n − w I = 2 and w s I = 1 for s = 1 , . . . , n −
1. Since ¯ w s III = ∅ for all s = 0 , . . . , n − t = ∅ , the set ¯ w II = ∅ is empty and the action formula (3.5) simplifies to (recall thatthere is no summation over partition of the set ¯ w n = { z, z n } and ¯ w n I = { z n } , ¯ w n III = { z } ,see remark 3.1) T ,n ( z ) · B ( ∅ , ¯ t , . . . , ¯ t n − ) = − λ n ( z ) κ h ( z, ¯ t n − ) g ( z n , ¯ t n − ) × X part B ( ∅ , ¯ w II , . . . , ¯ w n − II ) 1 h ( ¯ w n − I , z n − ) n − Y s =1 h ( ¯ w s II , ¯ w s − I ) g ( ¯ w s I , ¯ w s II ) h ( ¯ w s II , ¯ w s I ) g ( ¯ w s I , ¯ w s − II ) . (3.23)The set ¯ w I is equal to { z, z = z + c/ } . Then because of the factor h ( ¯ w II , ¯ w I ) the sumover partitions of the set ¯ w reduces to the single partition ¯ w II = { ¯ t , z } and ¯ w I = { z = z − c/ } . Next the factor h ( ¯ w II , ¯ w I ) g ( ¯ w I , ¯ w II ) = h ( ¯ w II , z − c/ g ( ¯ w I , z ) g ( ¯ w I , ¯ t )reduces summation over partitions of the set ¯ w to a single partition ¯ w II = { ¯ t , z } and¯ w I = { z = z − c/ } . Continuing we find that sum over partitions of all sets ¯ w s , s = 1 , , . . . , n − w s II = { ¯ t s , z } and ¯ w s I = { z s } s = 1 , , . . . , n − . For these sets and ¯ w I = { z, z = z + c/ } , ¯ w II = ∅ the product in (3.23) is equal to1 h ( ¯ w n − I , z n − ) n − Y s =1 h ( ¯ w s II , ¯ w s − I ) g ( ¯ w s I , ¯ w s II ) h ( ¯ w s II , ¯ w s I ) g ( ¯ w s I , ¯ w s − II ) = − κ h (¯ t , z ) g ( z n , ¯ t n − ) . Summarizing we can write the action (3.23) in the form T ,n ( z ) · B ( ∅ , ¯ t , . . . , ¯ t n − ) = λ n ( z ) h (¯ t , z ) h ( z, ¯ t n − ) B ( ∅ , ¯ η , . . . , ¯ η n − ) , (3.24)where ¯ η s = { ¯ t s , z } , s = 1 , . . . , n −
1. This action coincides with the action given bylemma 4.2 of the paper [20]. Indeed, if we renormalize the Bethe vectors of this paper˜ B (¯ t , . . . , ¯ t n − ) = Q n − s =1 h (¯ t s , ¯ t s ) Q n − s =2 h (¯ t s , ¯ t s − ) B ( ∅ , ¯ t , . . . , ¯ t n − ) (3.25)we obtain from (3.24) T ,n ( z ) · ˜ B (¯ t , . . . , ¯ t n − ) = λ n ( z ) ˜ B ( { ¯ t , z } , . . . , { ¯ t n − , z } ) , (3.26)15hich literally coincide with the action of monodromy matrix entry T ,n ( z ) onto off-shellBethe vectors ˜ B (¯ t ) in gl n -invariant integrable models.Let us note that formulas (4.14) at ¯ t = ∅ for i = 1 , . . . , n − T i +1 ,i · ˜ B (¯ t , . . . , ¯ t n − ) = r i X ℓ =1 ˜ B (¯ t , . . . , ¯ t i − , ¯ t iℓ ; ¯ t i +1 , . . . , ¯ t n − ) × (cid:18) χ i +1 λ i ( t iℓ ) λ i +1 ( t iℓ ) f (¯ t iℓ , t iℓ ) f (¯ t i +1 , t iℓ ) − χ i f ( t iℓ , ¯ t iℓ ) f ( t iℓ , ¯ t i − ) (cid:19) (3.27)for renormalized vectors ˜ B (¯ t ) (3.25).Since the action formulas (3.26) and (3.27) coincide with the statements of the lemma4.2 of the paper [20] they can be used to restore the action of entries T i,j ( z ) onto Bethevectors ˜ B (¯ t , . . . , ¯ t n − ) in the framework of zero-modes method. So far we have demon-strated that the action formulas (3.5) of T i,j ( z ) for 1 ≤ i, j ≤ n onto off-shell Bethevectors B ( ∅ , ¯ t , . . . , ¯ t n − ) with empty set ¯ t = ∅ leads to the action formulas in gl n -invariant models which was calculated in [20].Let us also check that recurrence relations given by the proposition 3.3 yield thecorresponding relations found in [21]. Formula (3.16) in case of ¯ t = ∅ becomes B ( ∅ , ¯ t , . . . , ¯ t n − , { ¯ t n − , z } ) = n − X i =1 X part T i,n ( z ) · B ( ∅ , ¯ t , . . . , ¯ t i − , ¯ t i II , . . . , ¯ t n − II , ¯ t n − ) λ n ( z ) h ( z, ¯ t n − ) h (¯ t n − , z ) g ( z, ¯ t n − II ) × n − Y s = i g (¯ t s I , ¯ t s II ) h (¯ t s II , ¯ t s I ) h (¯ t n − , ¯ t n − I ) g (¯ t i I , ¯ t i − ) n − Y s = i +1 h (¯ t s II , ¯ t s − I ) g (¯ t s I , ¯ t s − II ) . The sum over i in (3.16) reduces to the interval 1 ≤ i ≤ n − t s I = ∅ for 1 ≤ s ≤ i − t n − I = ∅ according to(3.18). This recurrence relation for the renormalized Bethe vector (3.25)˜ B (¯ t , . . . , ¯ t n − , { ¯ t n − , z } ) = n − X i =1 T i,n ( z ) λ n ( z ) X part ˜ B (¯ t , . . . , ¯ t i − , ¯ t i II , . . . , ¯ t n − II , ¯ t n − ) × g ( z, ¯ t n − I ) f ( z, ¯ t n − ) Q n − s = i f (¯ t s I , ¯ t s II ) Q n − s = i +1 g (¯ t s I , ¯ t s − I ) Q n − s = i f (¯ t s I , ¯ t s − )coincides identically with equation (4.4) of the paper [21] in case m = 0. o -invariant models In [22] we have calculated the action formulas for o -integrable model. Let us verify thatgeneral formula (3.5) reduces to these action formulas at n = 1. Cardinalities of the sets16 w I and ¯ w III according to (3.2) will be in this case ( − ≤ i, j ≤ w I = Θ( i ) + Θ( i −
1) = i + 1 , w III = Θ( − j ) + Θ( − j −
1) = 1 − j . Equality (3.5) reduces to T i,j ( z ) · B (¯ t ) = ( − δ i, − + δ j, − λ ( z )2 X part B ( ¯ w II ) α ( ¯ w III ) f ( ¯ w I , ¯ w II ) f ( ¯ w II , ¯ w III ) f ( ¯ w I , ¯ w III ) h ( z, ¯ w III ) h ( ¯ w I , z + c/ w = { ¯ t , z, z + c/ } . Here we used the fact that for − ≤ i ≤ σ i ( − i = σ − i and σ i = ( − δ i, +1 . D Y ( o n +1 ) As we already mentioned above in order to prove the propositions 3.1 and 3.2 we identifythe
RT T algebra B with the Borel subalgebra in the Yangian double D Y ( o n +1 ). Thisinfinite dimensional algebra is generated by two T -operators T ± ( u ) which satisfy thecommutation relations R ( u, v ) ( T µ ( u ) ⊗ I ) ( I ⊗ T ν ( v )) = ( I ⊗ T ν ( v )) ( T µ ( u ) ⊗ I ) R ( u, v ) (4.1)with the same R -matrix (2.1) for µ, ν = ± . We set the corresponding central elementsgiven by the equality (2.5) for T -operators T ± ( u ) equal to 1. We assume following seriesexpansion of the generating series T ± i,j ( u ) T ± i,j ( u ) = χ i δ ij ± X ℓ ≥ ℓ< T i,j [ ℓ ]( u/c ) − ℓ − . (4.2)This expansion allows to identify monodromy matrix T ( u ) of o n +1 -invariant model with T -operator T + ( u ).Algebra D Y ( o n +1 ) has also so called ’current’ realization [5] described in details in[7, 15]. The link between RT T and ’current’ realizations is established through
Gausscoordinates which can be defined as follows T ± i,j ( u ) = X max( i,j ) ≤ s ≤ n F ± s,i ( u ) k ± s ( u )E ± j,s ( u ) , (4.3)where we assume that F ± i,j ( u ) = E ± j,i ( u ) = 0 for i < j and F ± i,i ( u ) = E ± i,i ( u ) = 1 for i = − n, . . . , n . It was shown in [7, 15] that the set of the Gauss coordinatesF ± i +1 ,i ( u ) , E ± i,i +1 ( u ) , ≤ i ≤ n − , k ± j ( u ) , ≤ j ≤ n D Y ( o n +1 ). Commutation relations of this algebra contains also currents k ± ( u ).The modes of these currents k [ ℓ ], ℓ ∈ Z can be expressed through modes k s [ ℓ ], ℓ ∈ Z , s = 1 , . . . , n of the algebraically independent currents by the relations k ± ( u + c/ k ± ( u ) = n Y s =1 k ± s ( u − c ( s − / k ± s ( u − c ( s − / .RT T commutation relations (4.1) for the Yangian double D Y ( o n +1 ) can be presentedin terms of the formal generating series or currents F i ( u ) = F + i +1 ,i ( u ) − F − i +1 ,i ( u ) = X ℓ ∈ Z F i [ ℓ ] u − ℓ − ,E i ( u ) = E + i,i +1 ( u ) − E − i,i +1 ( u ) = X ℓ ∈ Z E i [ ℓ ] u − ℓ − (4.4)for 0 ≤ i ≤ n −
1. We will write explicitly only those commutation relations in termsof the currents which will be relevant for the calculations (see, for example, (4.6), (4.13)and (4.1) below). One can find the whole set of the relations between currents in [7, 15].
RT T realization of the Yangian double and its ’current’ realization correspond tothe different choices of Borel subalgebras. Following the ideas of the paper [14] one candefine projections onto intersections of these different type Borel subalgebras. As it wasshown in several papers (see, for example, [13] for the super-symmetric Yangian double D Y ( gl ( m | n )) and references therein) these projections being applied to the products ofthe currents F i ( u ) (4.4) can be identified with off-shell Bethe vectors in the corresponding g -invariant integrable models.In [15] one can find a formal definition of the projections onto intersections of thedifferent type Borel subalgebras in the Yangian doubles. Very often one can use followingapproach to calculate these projections of the product of the currents F i ( u ). One shouldreplace each current by the difference of the Gauss coordinates according to the Ding-Frenkel formulas (4.4), expand all brackets and use commutation relations between Gausscoordinates which follow from (4.1) to order all monomials in such a way that all ’positive’Gauss coordinates F + j,i ( u ), i < j are on the right of all ’negative’ Gauss coordinates F − j,i ( u )in each monomial. Although we start from the Gauss coordinates F ± j,i ( u ) for j = i + 1the higher Gauss coordinates for j > i + 1 will appear during the process of the ordering.Then application of the projection P + f amounts to delete all ordered monomials composedfrom the Gauss coordinates which have at least one ’negative’ Gauss coordinate F − j,i ( u ) onthe left. Analogously, application of the projection P − f amounts to delete all monomialswhich have at least one ’positive’ Gauss coordinate F + j,i ( u ) on the right. Analogously, onecan define the applications of the projections P ± e to the products of the currents E i ( u ).Let ¯ t be a set of the generic parameters described in (2.9). For any scalar function x ( t, t ′ ) of two variables and a set ¯ t s = { t s , . . . , t sr s } , r s > δ x (¯ t s ) = Y ℓ>ℓ ′ x ( t sℓ , t sℓ ′ ) . (4.5)For any s = 0 , . . . , n − t s of the Bethe parameters we define the normalizedordered products of the currents F s (¯ t s ) = δ f s (¯ t s ) F s ( t sr s ) · · · F s ( t s ) for s = 0 , , . . . , n − , where f s are defined by (2.11). By definition we set F s ( ∅ ) = 1 for all s . It is clear fromthe commutation relation f s ( t, t ′ ) F s ( t ) F s ( t ′ ) = f s ( t ′ , t ) F s ( t ′ ) F s ( t ) , ≤ s ≤ n − t s .Let ¯ t be a set of generic parameters described by (2.9). Let us define the orderedproducts of the currents F (¯ t ) = F n − (¯ t n − ) F n − (¯ t n − ) · · · F (¯ t ) F (¯ t ) (4.7)and F (¯ t ) = n − Y s =1 g (¯ t s , ¯ t s − ) 1 h (¯ t s , ¯ t s ) F (¯ t ) . (4.8) Definition 4.1.
The o n +1 -invariant off-shell Bethe vector B (¯ t ) is defined by the actionof the projection P + f ( F (¯ t )) onto reference vector | i (2.7) B (¯ t ) = P + f ( F (¯ t )) | i . (4.9)Products of the currents (4.7) and (4.8) can be written in the ordered form [15] F (¯ t ) = X part f (¯ t I , ¯ t II ) n − Y s =1 f (¯ t s I , ¯ t s II ) f (¯ t s II , ¯ t s − I ) P − f (cid:0) F (¯ t I ) (cid:1) · P + f (cid:0) F (¯ t II ) (cid:1) (4.10)and F (¯ t ) = X part f (¯ t I , ¯ t II ) n − Y s =1 g (¯ t s I , ¯ t s II ) h (¯ t s II , ¯ t s − I ) h (¯ t s II , ¯ t s I ) g (¯ t s I , ¯ t s − II ) P − f ( F (¯ t I )) · P + f ( F (¯ t II ))= X part n − Y s =0 γ s (¯ t s I , ¯ t s II ) n − Y s =1 h (¯ t s II , ¯ t s − I ) g (¯ t s I , ¯ t s − II ) P − f ( F (¯ t I )) · P + f ( F (¯ t II )) , (4.11)19here functions γ s ( u, v ) are defined by (3.4) and summation goes over all possible par-titions of the sets ¯ t s onto nonintersecting subsets ¯ t s I and ¯ t s II such that { ¯ t s I , ¯ t s II } ⊢ ¯ t s and t s I + t s II = t s . Cardinalities of the sets ¯ t s I and ¯ t s II can be equal to zero. Notations ¯ t I and ¯ t II means the collections of the subsets ¯ t s I and ¯ t s II ¯ t I = { ¯ t I , ¯ t I , . . . , ¯ t n − I } and ¯ t II = { ¯ t II , ¯ t II , . . . , ¯ t n − II } . We assume that F ( ∅ ) ≡ F ( ∅ ) ≡ P ± f (1) = 1. Equations (4.2) and (4.3) yield the following series expansion k ± j ( u ) = χ j ± X ℓ ≥ ℓ< k j [ ℓ ]( u/c ) − ℓ − . Because of this expansion the zero-modes of T + j +1 ,j ( u ) for 0 ≤ j ≤ n − T + j +1 ,j = χ j +1 E j [0] , where E j [0] are the zero-modes of the currents E j ( z ) for j = 0 , . . . , n − o n +1 .To calculate the action of the operator E j [0] onto off-shell Bethe vector we use aspecial presentation for the ordered product of the currents F (¯ t ) (4.8) which follows from(4.11) P + f ( F (¯ t )) = F (¯ t ) + r X ℓ =1 f ( t ℓ , ¯ t ℓ ) f (¯ t , t ℓ ) g (¯ t , t ℓ ) F − , ( t ℓ ) F (¯ t ℓ ; ¯ t ; . . . ; ¯ t n − )++ n − X s =1 r s X ℓ =1 f ( t sℓ , ¯ t sℓ ) f (¯ t s +1 , t sℓ ) g (¯ t s +1 , t sℓ ) g ( t sℓ , ¯ t s − ) h (¯ t s , t sℓ ) h ( t sℓ , ¯ t s ) F − s +1 ,s ( t sℓ ) F (¯ t ; . . . ; ¯ t sℓ ; . . . ; ¯ t n − ) + · · · , (4.12)where · · · stands for terms which are annihilated by the projection P + f after the adjointaction of the zero mode E j [0].Using the commutation relations k ± ( u ) F ( v ) k ± ( u ) − = f ( u, v ) f ( v, u + c/ F ( v ) ,k ± i ( u ) F i ( v ) k ± i ( u ) − = f ( v, u ) F i ( v ) , ≤ i ≤ n − ,k ± i +1 ( u ) F i ( v ) k ± i +1 ( u ) − = f ( u, v ) F i ( v ) , ≤ i ≤ n − ,k ± i ( u ) F j ( v ) k ± i ( u ) − = F j ( v ) , j = i, i + 1 , ≤ j ≤ n − , ≤ i ≤ n (4.13)20nd formulas E j [0] F l ( v ) = F l ( v ) E j [0] + δ jl (cid:0) k + j ( v ) k + j +1 ( v ) − − k − j ( v ) k − j +1 ( v ) − (cid:1) ,E j [0]F − l +1 ,l ( v ) = F − l +1 ,l ( v ) E j [0] + δ jl (cid:0) k − j ( v ) k − j +1 ( v ) − − χ j χ − j +1 (cid:1) , where j, l = 0 , . . . , n − i = 0 , . . . , n − T i +1 ,i · B (¯ t ) = r i X ℓ =1 B (¯ t ; . . . ; ¯ t i − ; ¯ t iℓ ; ¯ t i +1 ; . . . ; ¯ t n − ) × (cid:18) χ i +1 λ i ( t iℓ ) λ i +1 ( t iℓ ) − χ i f i ( t iℓ , ¯ t iℓ ) f (¯ t i +1 , t iℓ ) f i (¯ t iℓ , t iℓ ) f ( t iℓ , ¯ t i − ) (cid:19) f i (¯ t iℓ , t iℓ ) g (¯ t i +1 , t iℓ ) h ( t iℓ , ¯ t i − ) h ( t iℓ , ¯ t i ) h (¯ t i , t iℓ ) . (4.14)In (4.12) and (4.14) we set ¯ t − = ¯ t n = ∅ and the products of the rational functionsdepending on these sets are equal to 1.If χ j = 1 and the Bethe parameters ¯ t satisfy the Bethe equations (2.12) the on-shell Bethe vectors become highest weight vectors for the algebra o n +1 . Performingcalculations of the zero modes actions (4.14) we omit the terms which are annihilatedby the action of the projection P + f . In particular, the terms containing the ratio of k − j ( v ) /k − j +1 ( v ) do not contribute to these actions.To prove proposition 3.1 it is sufficient to use the action (4.14) of the o n +1 simpleroots monodromy matrix elements zero modes and the commutation relations (3.12). (cid:3) According to Gauss decomposition (4.3) the monodromy matrix element T + − n,n ( z ) is T + − n,n ( z ) = F + n, − n ( z ) k + n ( z ) . (4.15)We will calculate the action of this element T + − n,n ( z ) onto ordered product of the currents P + f ( F (¯ t )). We need following Lemma 4.1.
In the subalgebra B and for − n ≤ i < j ≤ n we have the equality P + f (cid:0) T + − n,n ( v )F − j,i ( u ) (cid:1) = 0 . (4.16)Let us consider the equality (4.1) for the values µ = − and ν = + and multiply itfrom the right by R ( v, u ). Using unitarity condition (2.2) we obtain (cid:18) − c ( u − v ) (cid:19) (cid:0) I ⊗ T + ( v ) (cid:1) (cid:0) T − ( u ) ⊗ I (cid:1) = R ( u, v ) (cid:0) T − ( u ) ⊗ I (cid:1) (cid:0) I ⊗ T + ( v ) (cid:1) R ( v, u ) .
21f we consider the element ( i, j ; − n, n ) with − n ≤ i < j ≤ n in this matrix equation weobtain (cid:18) − c ( v − u ) (cid:19) T + − n,n ( v ) T − i,j ( u ) = T − i,j ( u ) T + − n,n ( v ) + cu − v T −− n,j ( u ) T + i,n ( v )++ cv − u (cid:18) T − i,n ( u ) T + − n,j ( v ) + cu − v T −− n,n ( u ) T + i,j ( v ) (cid:19) . (4.17)Multiplying both sides of the equality (4.17) on the right by k − j ( u ) − and ordering allterms according to the circular ordering in the Yangian double D Y ( o n +1 ) described in[15] we obtain the statement of the lemma. All terms in RHS of (4.17) start with someGauss coordinates F − j,i ( t ), so they annihilated by the action of projection P + f . (cid:3) The total currents F i ( u ) (4.4) are defined for the values i = 0 , . . . , n −
1. It was shownin [15] using results of the paper [22] that the same differences of the Gauss coordinatesdefines the currents for the values i = − n, . . . , − F i ( u ) = − F − i − ( u + c ( i + 3 / . It was proved further in [15] that the Gauss coordinates of T -operators T ± ( u ) are relatedto the currents F i ( u ), for i = − n, . . . , n − + j,i ( v ) = P + f ( F i ( v ) · F i +1 ( v ) · · · F j − ( v ) · F j − ( v )) , (4.18)˜F − j,i ( v ) = P − f ( F i ( v ) · F i +1 ( v ) · · · F j − ( v ) · F j − ( v )) , (4.19)where − n ≤ i < j ≤ n and˜F ± j,i ( u ) = j − i − X ℓ =0 ( − ) ℓ +1 X j>i ℓ > ··· >i >i F ± i ,i ( u )F ± i ,i ( u ) · · · F ± i ℓ ,i ℓ − ( u )F ± j,i ℓ ( u ) . Let us introduce the set { z , z , . . . , z n − } of the generic complex parameters suchthat z s = z s ′ if s = s ′ for all s, s ′ = 0 , , . . . , n − z i = z i − c (cid:18) i − (cid:19) . (4.20)Then using (4.18) we may express monodromy element entry (4.15) through the currentsas follows T + − n,n ( z ) = ( − n P + f (cid:0) F n − (˜ z n − ) · · · F (˜ z ) · F ( z ) · · · F n − ( z n − ) (cid:1) k + n ( z n − ) (cid:12)(cid:12) z i = z , where parameters ˜ z i are defined by (4.20) for i = 0 , . . . , n − P − f ( F (¯ t I )) in this equality can be expressed in terms of the ’negative’ Gausscoordinates F − j,i ( u ) due to (4.19) we can calculate the action of matrix entries T + − n,n ( z )onto ordered product of the currents P + f ( F (¯ t )) as follows T + − n,n ( z ) · P + f (cid:0) F (¯ t ) (cid:1) = P + f (cid:0) T + − n,n ( z ) · F (¯ t ) (cid:1) . (4.21)Let us define an element T − n,n ( z , . . . , z n − ) T − n,n ( z , . . . , z n − ) = ( − n F n − (˜ z n − ) · · · F (˜ z ) · F ( z ) · · · F n − ( z n − ) k + n ( z n − ) . Then the action (4.21) can be written as T + − n,n ( z ) · P + f (cid:0) F (¯ t ) (cid:1) = P + f (cid:0) T − n,n ( z , . . . , z n − ) · F (¯ t ) (cid:1)(cid:12)(cid:12) z i = z . Using the commutation relations between currents (4.6), (4.13) and( t − t ′ − c ) F i ( t ) F i +1 ( t ′ ) = ( t − t ′ ) F i +1 ( t ′ ) F i ( t ) , ≤ i ≤ n − T − n,n ( z , . . . , z n − ) · F (¯ t ) as follows T − n,n ( z , . . . , z n − ) · F (¯ t ) = ( − n F ( ¯ w ) Q n − s =1 f ( z s , z s − ) f (¯ t s , z s − ) f ( z s − , ¯ t s − ) k + n ( z n − )+ · · · , (4.22)where · · · stands for the terms which have higher zeros when z ℓ = z ℓ ′ = z and do notcontribute into (4.23). The collection of sets ¯ w in (4.22) is¯ w = { ¯ w , . . . , ¯ w n − } and ¯ w i = { ¯ t i , z i , ˜ z i } . To obtain (4.22) we used a trivial identity f ( u − c, v ) f ( v, u ) = 1and the fact that we may replace any z ℓ by any z ℓ ′ since at the end we will set z ℓ = z ℓ ′ = z .It is obvious that we cannot set z ℓ = z in (4.22) since f ( z, z ) − = 0. But the actionof T − n,n ( z , . . . , z n − ) onto renormalized ordered product of currents F (¯ t ) (4.8) is non-singular and yields the action of T + − n,n ( z ) onto off-shell Bethe vector B (¯ t ) (4.9) T + − n,n ( z ) · B (¯ t ) = T + − n,n ( z ) · P + f ( F (¯ t )) | i = P + f (cid:0) T + − n,n ( z ) · F (¯ t ) (cid:1) | i == P + f (cid:0) T − n,n ( z , . . . , z n − ) · F (¯ t ) (cid:1)(cid:12)(cid:12) z i = z | i = − κ g ( z , ¯ t ) h ( z, ¯ t n − ) h ( z, ¯ t ) g ( z n , ¯ t n − ) λ n ( z ) B ( ¯ w ) , (4.23)where z , z n are defined in (2.15). This proves the proposition 3.2. (cid:3) Conclusion
This paper is a continuation of our work [19] to investigate the o n +1 -invariant quantumintegrable models which are defined by the (2 n + 1) × (2 n + 1) monodromy matricessatisfying the commutation relations (2.3) with o n +1 -invariant R -matrix (2.1). To de-scribe the space of states in these models we are using the method introduced in [14] anddeveloped in [13, 20] for the supersymmetric integrable models associated with Yangiandouble DY ( gl ( m | n )). In these papers the action of the upper-triangular monodromymatrix elements were used to describe the recurrent relations for the corresponding off-shell Bethe vectors and the action of the lower-triangular matrix elements were exploitedto find recurrence relations for the higher coefficients in the summation formula for theBethe vectors scalar products.Analogous program in case of o n +1 -invariant integrable models is far from comple-tion. The arguments of the paper [20] cannot be directly repeated since we yet do nothave clear picture on the structure of the summation formulas for the scalar productsand properties of the higher coefficients. We hope to investigate this picture in our forth-coming publications using the identification of the higher coefficients with the kernels inthe integral presentation of the Bethe vectors through the currents [23]. Acknowledgments
This work was carried out in Skolkovo Institute of Science and Technology under finan-cial support of Russian Science Foundation within grant 19-11-00275. Authors thankE. Ragoucy and N. A. Slavnov for fruitful discussions.
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