Quantum Trace Formulae for the Integrals of the Hyperbolic Ruijsenaars-Schneider model
aa r X i v : . [ h e p - t h ] J un HU-EP-19/02, ZMP-HH/19-2
Quantum Trace Formulae for the Integrals ofthe Hyperbolic Ruijsenaars-Schneider model
Gleb Arutyunov, a Rob Klabbers b and Enrico Olivucci a a II. Institut f¨ur Theoretische Physik, Universit¨at Hamburg, Luruper Chaussee 149, 22761 Hamburg, GermanyZentrum f¨ur Mathematische Physik, Universit¨at Hamburg, Bundesstrasse 55, 20146 Hamburg,Germany b Institut f¨ur Physik und IRIS Adlershof, Humboldt-Universit¨at zu Berlin, Zum Großen Windkanal 6, D-12489Berlin, Germany
Abstract:
We conjecture the quantum analogues of the classical trace formulae for the integrals ofmotion of the quantum hyperbolic Ruijsenaars-Schneider model. This is done by departing from theclassical construction where the corresponding model is obtained from the Heisenberg double by thePoisson reduction procedure. We also discuss some algebraic structures associated to the Lax matrixin the classical and quantum theory which arise upon introduction of the spectral parameter. ontents R -matrices and the L -operator 153.3 Spectral parameter and quantum L -operator 18 A.1 Lax matrix and its Poisson structure 23A.2 Dirac bracket 28
B Derivation of the spectral-dependent r -matrices 32 – 1 – Introduction
The Ruijsenaars-Schneider (RS) models [1, 2] continue to provide an outstanding theoretical laboratoryfor the study of various aspects of Liouville integrability, both at the classical and quantum level, see,for instance, [5–10]. Also, new interesting applications of these type of models were recently found inconformal field theories [11].In this work we study some aspects related to the quantum integrability of the RS model withthe hyperbolic potential. Recall that the definition of quantum integrability relies on the existence ofa quantisation map which maps a complete involutive family of classical integrals of motion into a setof commuting operators on a Hilbert space. In general, there are different ways to choose a functionalbasis for this involutive family which is mirrored by the ring structure of the corresponding commutingoperators. In particular, a classical integrable structure, most conveniently encoded into a Lax pair(
L, M ), produces a set of canonical integrals which are simply the eigenvalues of the Lax matrix. Theircommutativity relies on the existence of the classical r -matrix [12]. Provided this matrix exists onecan build up different classical involutive families represented, for instance, by elementary symmetricfunctions of the eigenvalues of L or, alternatively, by traces Tr L k for k ∈ Z . Concerning the particularclass of the RS hyperbolic models, the quantisation of a family of elementary symmetric functionsassociated to a properly chosen L is well known and given by the Macdonald operators [2, 13]. In thispaper we conjecture the quantum analogues of Tr L k built up in terms of the same L -operator that isused to generate Macdonald operators through the determinant type formulae [14, 15]. In fact thereappear two commuting families I ± k that are given by the quantum trace formulae I ± k = Tr (cid:0) C t L ¯ R t R t ± L . . . L ¯ R t R t ± L (cid:1) , as quantisation of the classical integrals Tr L k . In particular, R and ¯ R are two quantum dynamical R -matrices that depend rationally on the variables Q i = e q i , where q i , i = 1 , . . . , N are coordinates, andsatisfy a system of equations of Yang-Baxter type. Also, R is a parametric solution of the standardquantum Yang-Baxter equation. Departing from I ± k and introducing q = e − ¯ h , we then find thatthese integrals are related to the Macdonald operators S k through the q -deformed analogues of thedeterminant formulae that in the classical case relate the coefficients of characteristic polynomial of L with invariants constructed out of Tr L k . The commutativity of I ± k and their relation to Macdonaldoperators has been checked by explicit computation for sufficiently large values of N .We arrive to this expression for I ± k through the following chain of arguments. It is known that theCalogero-Moser-Sutherland models and their RS generalisations can be obtained at the classical levelthrough the hamiltonian or Poisson reduction applied to a system exhibiting free motion on one of thesuitably chosen initial finite- or infinite-dimensional phase spaces [16]-[23], [3–6]. For instance, the RSmodel with the rational potential is obtained by the hamiltonian reduction of the cotangent bundle T ∗ G = G ⋉ G , where G ia Lie group and G is its Lie algebra. In [19] the corresponding reduction wasdeveloped for the Lie group G = GL( N, C ) by employing a special parametrisation for the Lie algebra-valued element ℓ = T QT − ∈ G , where Q is a diagonal matrix and T is an element of the Frobeniusgroup F ⊂ G . An analogous parametrisation is used for the group element g = U P − T − ∈ G , where U is Frobenius and P is diagonal. If one writes Q i = q i and P i = exp p i , then ( p i , q i ) is a system ofcanonical variables with the Poisson bracket { p i , q j } = δ ij . In the new variables the Poisson structureof the cotangent bundle is then described in terms of the triangular dynamical matrix r satisfyingthe classical Yang-Baxter equation (CYBE) and of another matrix ¯ r . The cotangent bundle is easily For the definition of other quantities, see the main text. – 2 –uantised, in particular, the algebra of quantum T -generators is T T = T T R and its consistencyis guaranteed by the fact that the matrix R , being a quantisation of r , is triangular, R R = , andobeys the quantum Yang-Baxter equation. The quantum L -operator is then introduced as L = T − gT and it is an invariant under the action of F . In [19] the same formula for I k as given above was derivedby eliminating from the commuting operators Tr g k = Tr T L k T − an element T .To build up the hyperbolic RS model, one can start from the Heisenberg double associated to aLie group G . As a manifold, the Heisenberg double is G × G and it has a well-defined Poisson structurebeing a deformation of the one on T ∗ G [24]. However, an attempt to repeat the same steps of thereduction procedure meets an obstacle: since the action of G on the Heisenberg double is Poisson,rather than hamiltonian, the Poisson bracket of two Frobenius invariants, { L , L } , is not closed, i.e. it is not expressed via L ’s alone. Moreover, for the same reason, the Poisson bracket { p i , p j } does notvanish on the Heisenberg double. On the other hand, a part of the non-abelian moment map generatessecond class constraints and to find the Poisson structure on the reduced manifold one has to resort tothe Dirac bracket construction. In this paper we work out the Dirac brackets for Frobenius invariantsand show in detail how the cancellation of the non-invariant terms happens on the constraint surface.This leads to the canonical set of brackets for the degrees of freedom ( p i , q i ) on the reduced manifold,the physical phase space of the RS model. However, continuing along the same path as in the rationalcase [19] does not seem to yield { T, L } and { T, T } brackets. The variable T is not invariant withrespect to the stability subgroup of the moment map and computation of such brackets requires fixinga gauge, which makes the whole approach rather obscure. Moreover, the very simple and elegantbracket { L , L } emerging on the reduced phase space looks the same as in the rational case, with oneexception: now the r -matrix r entering this bracket is not skew-symmetric, i.e. r = r . We thenfind a quantisation of r : a simple quantum R -matrix R + satisfying R +12 R − = , where R − isanother solution of the quantum Yang-Baxter equation. In the absence of the triangular property for R +12 , assuming, for instance, the same algebra for T ’s as in the rational case - that is T T = T T R +12 - would be inconsistent. Thus, at this point we simply conjecture that the integrals of the hyperbolicmodel have absolutely the same form as in the rational case, with the exception that the rational R -matrices are replaced by their hyperbolic analogues, which we explicitly find. That this conjectureyields integrals of motion can then be verified by tedious but direct computation and indeed holdstrue. Working out explicit expressions for these integrals for small numbers N of particles we findthe determinant formulae relating these integrals to the standard basis of Macdonald operators. Therest of the paper is devoted to the model whose formulation includes the spectral parameter. Neitherfor the rational nor for the hyperbolic case the spectral parameter is actually needed to demonstratetheir Liouville integrability, but its introduction leads to interesting algebraic structures and clarifiesthe origin of the shifted Yang-Baxter equation [22] and its scale-violating solutions.The paper is organised as follows. In the next section we show how to obtain the hyperbolicRS model by the Poisson reduction of the Heisenberg double. This includes the derivation of thePoisson algebra of the Lax matrix via the Dirac bracket construction. We also introduce the spectralparameter and build up the theory based on spectral parameter-dependent (baxterised) r -matrices.We also describe a freedom in the definition of r -matrices that does not change the Poisson algebraof L ’s. In section 3 we consider the corresponding quantum theory. Finding the hyperbolic quantum R -matrices R ± and ¯ R , we conjecture our main formula for the quantum integrals I ± k and explain howit is related to the basis of the Macdonald operators. The rest of the section is devoted to the quantum In the rational case there is only one family, R ± → R . In [22] this problem was avoided by looking at those entries of L only that commute with the second class constraints. – 3 –axterised R -matrices and the quantum L -operator algebra. We show that in spite of the fact thatthe constant R -matrices satisfy the usual system of quantum Yang-Baxter equations, their baxterisedcounterparts instead obey its modification that involves rescalings of the spectral parameter with thequantum deformation parameter q = e − ¯ h . Some technical details are relegated to two appendices. Allconsiderations are done in the context of holomorphic integrable systems. We start with recalling the construction of the classical Heisenberg double associated to the group G = GL( N, C ). Let the entries of matrices A, B ∈ G generate the coordinate ring of the algebra offunctions on the Heisenberg double. The Heisenberg double is a Poisson manifold with the followingPoisson brackets { A , A } = − r − A A − A A r + + A r − A + A r + A , { A , B } = − r − A B − A B r − + A r − B + B r + A , { B , A } = − r + B A − B A r + + B r − A + A r + B , { B , B } = − r − B B − B B r + + B r − B + B r + B . (2.1)Here and elsewhere in the paper we use the standard notation where the indices 1 and 2 denote thedifferent matrix spaces. The matrix quantities r ± are the following r -matrices r + = + 12 N X i =1 E ii ⊗ E ii + N X i
This equation can be elementary solved for W − and we get W − = N X i,j =1 β Q − i − ω Q − j c j Q j E ij , (2.16)– 5 –here we introduced c j = (e t T ) j . The condition W − ∈ F gives a set of equations to determine thecoefficients c j : N X j =1 V ij c j Q j = 1 , ∀ i . Here V is a Cauchy matrix with entries V ij = β Q − i − ω Q − j . We apply the inverse of VV − ij = 1 β ( Q − i − ω − Q − j ) N Q a =1 ( ω Q − i − Q − a ) N Q a = i ( Q − i − Q − a ) N Q a =1 ( ω − Q − j − Q − a ) N Q a = j ( Q − j − Q − a ) , to obtain the following formula for the coefficients c j c j = Q j N X j =1 V − ij = (1 − ω ) β N Q a = j ( Q − j − ω − Q − a ) N Q a = j ( Q − j − Q − a ) = N − ω − ω N N Y a = j Q j − ω Q a Q j − Q a , (2.17)where we substituted β from (2.11). Finally, inverting W − we find W itself W ij ( Q ) = Q i c i ( V − ) ij = N Q a = i ( Q − j − ω Q − a ) N Q a = j ( Q − j − Q − a ) . (2.18)It is obvious, that eq.(2.10) becomes equivalent to the following two constraints U = T W ( Q ) , e t T = c t , (2.19)where T, U ∈ F , and the quantities W ( Q ), c ( Q ) are given by (2.18) and (2.17), respectively. Anysolution of e t T = c t can be constructed as T = hT , where T is a particular solution of this equationand h is a Frobenius group element which satisfies the additional constraint e t h = e t . In fact, thesubgroup of ̥ ⊂ F ⊂ G determined by the conditions ̥ = { h ∈ G : h e = e , e t h = e t } , (2.20)constitutes the stability group of the moment map determined by the element n . Note that dim C F = N − N and dim C ̥ = ( N − . We do not include in ̥ the one dimensional dilatation subgroup C ∗ ≃ { h ∈ G : h = c , c = 0 } , because its actionon the phase space is not faithful. – 6 –ow we can define a family of G -invariant dynamical systems taking the combination L = W ( Q ) P − as their Lax matrix. Explicitly, L = n X i,j =1 (1 − ω ) Q i Q i − ω Q j N Y a = j ω Q j − Q a Q j − Q a P − j E ij . (2.21)After specifying the proper reality conditions, this L becomes nothing else but the Lax matrix of theRS family with the hyperbolic potential. Note that on the constrained surface the A, B -variables takethe following form A ( P, Q , h ) = hT QT − h − , B ( P, Q , h ) = hT LT − h − , h ∈ ̥ . The reduced phase space can be singled out by fixing the gauge to, for instance, h = 1. Its dimensionover C is 2 N − ( N − − dim C ̥ = 2 N . Now we turn to the analysis of the Poisson structure of the reduced phase space. We find from (2.1)the following formula { Q j , B } = B X kl T lj Q j T − jk E lk . (2.22)Next, we need to determine the bracket between Q j and P i . We have { Q j , P i } = δP i δA mn { Q j , A mn } + δP i δB mn { Q j , B mn } . Here the first bracket on the right-hand side vanishes because all Q j commute with A . To computethe second bracket, we consider the variation of B = U P − T − U − δB T P = U − δU − P − δP . Note that this formula does not include the variation δT . This is because T is solely determined by A , so so is its variation. The condition δU e = 0 allows one to find δP i δB mn = − X r P i U − im ( T P ) nr . We thus have { Q j , P i } = − X r P i U − im ( T P ) nr ( BT ) mj Q j T − jn = − Q i P i δ ij , (2.23)and similarly one can check the bracket { Q i , Q j } = 0. These formulae suggests to employ the expo-nential parametrisation for both P and Q , that is, to set P i = exp p i , Q i = exp q i , The systems whose hamiltonians are invariant under the action of G . The spectral invariants of A are central in the Poisson subalgebra of A , the latter is described by the Semenov-Tian-Shansky bracket [24] given by the first line in (2.1). – 7 –here ( p i , q i ) satisfy the canonical relations { p i , q j } = δ ij .An ̥ -invariant extension of the Lax matrix away from the reduced phase space is naturally givenby the following Frobenius invariant L = T − BT , (2.24)where T is an element of the Frobenius group entering the factorisation (2.12). The Poisson bracketof Q j with components of L is computed in a straightforward manner { Q j , L mn } = { Q j , ( T − BT ) mn } = X p ( T − B ) mp X kl T lj Q j T − jk ( E lk ) ps T sn = L mn Q n δ jn , which is perfectly compatible with the form (2.21) of the Lax matrix on the reduced space. In matrixform the previous formula reads as { Q , L } = Q L C , ¯ C = N X j =1 E jj ⊗ E jj . (2.25)As to the brackets between the entries of L , this time they cannot be represented in terms of L alone but also involve T . Ultimately, such a structure is a consequence of the fact that the actionof the Poisson-Lie group G on the phase space is Poisson rather than hamiltonian, so that thereis an obstruction for the Poisson bracket of two Frobenius invariants to also be such an invariant.In addition, computing the Dirac brackets of L one cannot neglect a non-trivial contribution fromthe second class constraints and, therefore, the analysis of the Poisson structure for L requires, asan intermediate step, to understand the nature of the constraints (2.10) imposed in the process ofreduction. The same argument holds for the Poisson brackets between any of the Frobenius invariants W = T − U and P , showing as a particular case that P i ’s have a non-vanishing Poisson algebra onthe Heisenberg double. We save the details of the corresponding analysis for appendix B and presenthere the final result for the Poisson bracket between the entries of the Lax matrix on the reducedphase space { L , L } = r L L − L L r + L ¯ r L − L ¯ r L . (2.26)Clearly, the bracket (2.26) has the same form as the corresponding bracket for the rational RS model[19] albeit with new dynamical r -matrices for which we got the following explicit expressions r = N X i = j (cid:16) Q j Q ij E ii − Q i Q ij E ij (cid:17) ⊗ ( E jj − E ji ) , ¯ r = N X i = j Q i Q ij ( E ii − E ij ) ⊗ E jj ,r = N X i = j Q i Q ij ( E ij ⊗ E ji − E ii ⊗ E jj ) , (2.27) At the level of quantisation, this fact prevents one from obtaining the quantum RS model starting from the algebraof the quantum Heisenberg double. Indeed, doing so one should later restore the canonical commutation relations of(
P, Q ) sub-algebra by imposing an analogue of the Dirac constraints at the quantum level. The quadratic and linear forms of the r -matrix structure for the RS model have been investigated in [25–29]. – 8 –here we introduced the notation Q ij = Q i − Q j . This structure can be obtained as well after thecomputation of the Dirac brackets of W and P on the reduced phase space { W , W } = [ r , W W ] (2.28) { W , P } = [¯ r , W ] P (2.29) { P , P } = 0 , (2.30)using the decomposition L = W P − . Remarkably the imposition of Dirac constraints makes thePoisson subalgebra { P i } abelian, allowing the interpretation of components p i = log P i as particlemomenta. Concerning the properties of the matrices (2.27) and the Lax matrix, we note the following:first, r is expressed via r and ¯ r as r = r + ¯ r − ¯ r . (2.31)Second, the matrix r is degenerate, det r = 0, and it obeys the characteristic equation r = − r .Moreover, in contrast to the rational case [19], r is not symmetric, rather it has the property r + r = C − ⊗ . (2.32)Third, it is a matter of straightforward calculation to verify that the Lax matrix (2.21) obeys thePoisson algebra relations (2.26), provided the bracket between the components of Q and P is given by(2.23),(2.30). Finally, as a consequence of the Jacobi identities, the matrices (2.27) satisfy a systemof equations of Yang-Baxter type. In particular, for r one has just the standard CYBE[ r , r ] + [ r , r ] + [ r , r ] = 0 . (2.33)In addition, there are two more equations involving r and ¯ r [¯ r , ¯ r ] + { ¯ r , p } − { ¯ r , p } = 0 , [ r , ¯ r ] + [ r , ¯ r ] + [¯ r , ¯ r ] + { r , p } = 0 . (2.34)The matrix r satisfies the classical analogue of the Gervais-Neveu-Felder equation [31, 32][ r , r ] + [ r , r ] + [ r , r ] + { r , p } − { r , p } + { r , p } = 0 . (2.35)It is elementary to verify that the quantities I k = Tr L k (2.36)are in involution with respect to (2.27). This property of I k is, of course, inherited from the sameproperty for Tr B k on the original phase space (2.1). We refer to (2.36) as the classical trace formula . Here we introduce a Lax matrix depending on a spectral parameter and discuss the associated algebraicstructures and an alternative way to exhibit commuting integrals.To start with, we point out one important identity satisfied by the Lax matrix (2.21). Accordingto the moment map equation (2.15), we have ω Q − W Q = W h + βω e ⊗ e t U i − , (2.37)– 9 –he inverse on the right-hand side of the last expression can be computed with the help of the well-known Sherman-Morrison formula and we get ω Q − W Q = W h − − ω N N e ⊗ e t U i = W − − ω N N e ⊗ c t W , (2.38)where we used the fact that W is a Frobenius matrix, so that W e = e. Here the vector c hascomponents (2.17) and satisfies the relation e t T = c t . Multiplying both sides of (2.19) with P − weobtain the following identity ω Q − L Q = L − − ω N N e ⊗ c t L , (2.39)for the Lax matrix (2.21).Evidently, we can consider L ′ = ω Q − L Q (2.40)as another Lax matrix since the evolution equation of the latter is of the Lax form˙ L ′ = [ M ′ , L ′ ] , M ′ = Q − M Q − Q − ˙ Q , (2.41)where M is defined by the hamiltonian flow of L . Note that one can add to M ′ any function of L ′ without changing the evolution equation for L ′ , which defines a class of equivalent M ′ ’s. Now, it turnsout that due to the special dependence of L on the momentum, M and M ′ fall in the same equivalenceclass. To demonstrate this point, it is enough to consider the simplest hamiltonian H = Tr L for whichthe matrix M is given by M = N X i = j Q j Q ij L ij ( E ii − E ij ) , (2.42)It follows from (2.25) that for the flow generated by this hamiltonian Q − ˙ Q = Q − { H, Q } = − N X i =1 L ii E ii . Therefore, M ′ = Q − M Q − Q − ˙ Q = N X i = j Q j Q ij L ij (cid:16) E ii − Q j Q i E ij (cid:17) + N X i =1 L ii E ii . Taking into account that Q j / ( Q ij Q i ) = 1 / Q ij − / Q i , we then find M ′ = N X i = j Q j Q ij L ij ( E ii − E ij ) + N X i = j Q − i L ij Q j E ij + N X i =1 L ii E ii = M + L ′ . Hence, M ′ is in the same equivalence class as M and, therefore, we can take the dynamical matrix M to be the same for both L and L ′ .The above observation motivates to introduce a Lax matrix depending on a spectral parameterjust as a linear combination of L and L ′ . Namely, we can define L ( λ ) = L − λ L ′ , (2.43)– 10 –here λ ∈ C is the spectral parameter. The matrix L ( λ ) has a pole at zero and the original matrix L is obtained from L ( λ ) in the limit λ → ∞ , in particular, H = lim λ →∞ Tr L ( λ ) = Tr L . (2.44)The evolution equation for L ( λ ) must, therefore, be of the form˙ L ( λ ) = { H, L ( λ ) } = [ M, L ( λ )] , (2.45)where M is the expression (2.42).The next task is to compute the Poisson brackets between the components of (2.43). We aim atfinding a structure similar to (2.26), namely, { L ( λ ) , L ( µ ) } = r ( λ, µ ) L ( λ ) L ( µ ) − L ( λ ) L ( µ ) r ( λ, µ )+ L ( λ )¯ r ( µ ) L ( µ ) − L ( µ )¯ r ( λ ) L ( λ ) , (2.46)where r ( λ, µ ), r ( λ, µ ) and ¯ r ( λ ) are some spectral-parameter-dependent r -matrices. We show how toderive these r -matrices in appendix B. Our considerations are essentially based on the identity (2.39).To state the corresponding result, we need the matrix σ = N X i = j ( E ii − E ij ) ⊗ E jj . (2.47)The minimal solution for the spectral-dependent r -matrices realising the Poisson algebra (2.46) isthen found to be r ( λ, µ ) = λr + µr λ − µ + σ λ − − σ µ − , ¯ r ( λ ) = ¯ r + σ λ − ,r ( λ, µ ) = r ( λ, µ ) + ¯ r ( µ ) − ¯ r ( λ ) = λr + µr λ − µ . (2.48)The matrices r and r are skew-symmetric in the sense that r ( λ, µ ) = − r ( µ, λ ) , r ( λ, µ ) = − r ( µ, λ ) . (2.49)Further, one can establish implications of the Jacobi identity satisfied by (2.46) for these r -matrices. Introducing the dilatation operator acting on the spectral parameter D λ = λ ∂∂λ , we find that the r -matrix r ( λ, µ ) does not satisfy the standard CYBE but rather the following modi-fication thereof[ r ( λ, µ ) , r ( λ, τ )] + [ r ( λ, µ ) , r ( µ, τ )] + [ r ( λ, τ ) , r ( µ, τ )] = (2.50)= − ( D λ + D µ ) r ( λ, µ ) + ( D λ + D τ ) r ( λ, τ ) − ( D τ + D µ ) r ( µ, τ ) . The explanation of its minimal character will be given later. – 11 –ollowing [23], we refer to (2.50) as the shifted classical Yang-Baxter equation . This equation can berewritten in the form of the standard Yang-Baxter equation[ˆ r ( λ, µ ) , ˆ r ( λ, τ )] + [ˆ r ( λ, µ ) , ˆ r ( µ, τ )] + [ˆ r ( λ, τ ) , ˆ r ( µ, τ )] = 0 . for the matrix differential operator ˆ r ( λ, µ ) = r ( λ, µ ) − D λ + D µ . (2.51)There are also two more equations involving the matrix ¯ r [ r ( λ, µ ) , ¯ r ( λ ) + ¯ r ( µ )] + [¯ r ( λ ) , ¯ r ( µ )] + P − { r ( λ, µ ) , P } = (2.52)= − ( D λ + D µ ) r ( λ, µ ) + ( D λ ¯ r ( λ ) − D µ ¯ r ( µ ))and [¯ r ( λ ) , ¯ r ( λ )] + P − { ¯ r ( λ ) , P } − P − { ¯ r ( λ ) , P } = − D λ (¯ r ( λ ) − ¯ r ( λ )) . (2.53)One can check that relations (2.50), (2.52) and (2.53) guarantee the fulfilment of the Jacobi identityfor the brackets (2.25) and (2.26). Note that r is scale-invariant: ( D λ + D µ ) r ( λ, µ ) = 0, implyingthat it depends on the ratio λ/µ . This property does not hold, however, for r and ¯ r .The solution we found for the spectral-dependent dynamical r -matrices is minimal in the sensethat there is a freedom to modify these r -matrices without changing the Poisson bracket (2.46). Firstof all, there is a trivial freedom of shifting r and r as r → r + f ( λ/µ ) ⊗ , r → r + f ( λ/µ ) ⊗ , (2.54)where f is an arbitrary function of the ratio of the spectral parameters. This redefinition affectsneither the bracket (2.26) nor equations (2.50), (2.52), (2.53).Second, one can redefine ¯ r and r as r ( λ, µ ) → r ( λ, µ ) − s ( λ ) ⊗ + ⊗ s ( µ )¯ r ( λ ) → ¯ r ( λ ) − s ( λ ) ⊗ , (2.55)where s ( λ ) is an arbitrary matrix function of the spectral parameter. Owing to the structure ofthe bracket (2.46) this redefinition of the r -matrices produces no effect on the latter, as r remainsunchanged, while the matrix s decouples from the right-hand side of the LL bracket (see (2.46)). Forgeneric s ( λ ), redefinition (2.55) affects , however, equations (2.50), (2.52), (2.53). In particular, thereexists a choice of s ( λ ) which turns the shifted Yang-Baxter equations for ¯ r and r into the conventionalones, where the derivative terms on the right hand side of (2.50), (2.52) and (2.53) are absent. Onecan take, for instance, s ( λ ) = 1 N N X i = j Q i Q ij ( E ii − E ij ) + 1 λ − N N X i = j ( E ii − E ij ) . (2.56)With the last choice the matrix ¯ r ( λ ) becomes¯ r ( λ ) = 1 λ − X i = j λ Q i − Q j Q ij ( E ii − E ij ) ⊗ (cid:16) E jj − N (cid:17) , An example of such a redefinition that does not affect the shifted Yang-Baxter equation corresponds to the choice s ( λ ) = f ( λ ) , where f is an arbitrary function of λ . – 12 –hile for r ( λ, µ ) one finds r ( λ, µ ) = λr m + µr m λ − µ + ρ λ − − ρ µ − , (2.57)where ρ = X i = j ( E ii − E ij ) ⊗ (cid:16) E jj − N (cid:17) and the modified r -matrix is r m = N X i = j (cid:16) Q j Q ij E ii − Q i Q ij E ij (cid:17) ⊗ ( E jj − E ji ) − N X i = j Q i Q ij ( E ii − E ij ) ⊗ + 1 N X i = j Q i Q ij ⊗ ( E ii − E ij ) . (2.58)The modified r -matrix still solves the CYBE and obeys the same relation (2.32).There is no symmetry operating on r -matrices that would allow one to remove the scale-non-invariant terms from these matrices. Clearly, the r -matrices satisfying the shifted version of the Yang-Baxter equations have a simpler structure than their cousins subjected to the standard Yang-Baxterequations. This fact plays an important role when it comes to quantisation of the corresponding modeland the associated algebraic structures. We also point out that the r -matrices we found here throughconsiderations in appendix B also follow from the elliptic r -matrices of [23] upon their hyperbolicdegeneration, albeit modulo the shift symmetries (2.54) and (2.55).From (2.46) one then finds { Tr L ( λ ) , L ( µ ) } = [Tr L ( λ )( r ( λ, µ ) + ¯ r ( µ )) , L ( µ )] , which, upon taking the limit λ → ∞ , yields the Lax equation (2.45) with M given by (2.42). Theconserved quantities are, therefore, I k ( λ ) = Tr L ( λ ) k , k ∈ Z . The determinant det( L ( λ ) − ζ ), whichgenerates I k ( λ ) in the power series expansion over the parameter ζ , defines the classical spectral curve det( L ( λ ) − ζ ) = 0 , ζ, λ ∈ C . (2.59) At the classical level we obtained the hyperbolic RS model by means of the Poisson reduction of theHeisenberg double. It is therefore natural to start with the quantum analogue of the Heisenberg double.The Poisson algebra (2.1) can be straightforwardly quantised in the standard spirit of deformationtheory. We thus introduce an associative unital algebra generated by the entries of matrices
A, B modulo the relations [33] R − − A R + A = A R − − A R + , R − − B R + A = A R − − B R − , R − A R + B = B R − − A R + , R − − B R + B = B R − − B R + , (3.1)– 13 –nd they can be regarded as the quantisation of the Poisson relations (2.1). The quantum R -matriceshere are defined as follows: first, we consider the following well-known solution of the quantum Yang-Baxter equation R = n X i = j E ii ⊗ E jj + e ¯ h/ n X i =1 E ii ⊗ E ii + ( e ¯ h/ − e − ¯ h/ ) n X i>j E ij ⊗ E ji . (3.2)Using this R one can construct two more solutions R ± of the quantum Yang-Baxter equation, namely, R +12 = R , R − = R − . (3.3)These solutions are, therefore, related as R +21 R − = , (3.4)and they also satisfy R + − R − = ( e ¯ h/ − e − ¯ h/ ) C , (3.5)where C is the split Casimir. In the limit ¯ h → R ± expand as R ± = 1 + ¯ h r ± + o (¯ h ) , (3.6)where r ± are the classical r -matrices (2.2). Further, we point out that ˆ R ± = C R ± satisfy the Heckecondition b R ± ∓ ( e ¯ h/ − e − ¯ h/ ) ˆ R ± − = ( ˆ R ± − e ± ¯ h/ )( ˆ R ± + e ∓ ¯ h/ ) = 0 . (3.7)The first, or alternatively, the last line in (3.1) is a set of defining relations for the correspondingsubalgebra that describes a quantisation of the Semenov-Tian-Shansky bracket, the latter has a set ofCasimir functions generated by C k = Tr A k . In the quantum case an analogue Tr A k can be defined bymeans of the quantum trace formula C k = Tr q A k = Tr( DA k ) , q = e − ¯ h , where D is a diagonal matrix D = diag( q , q , . . . , q n ). The elements C k are central in the subalgebragenerated by A . Indeed, by successively using the permutation relations for A , one gets A R + A k R − = R − A k R − − A . We then multiply both sides of this relation by D and take the trace in the first matrix space A Tr ( D R + A k R − ) = Tr ( D R − A k R − − ) A . It remains to notice that Tr ( D R + A k R − ) = Tr ( D R − A k R − − ) = Tr q A k · , so that A Tr q A k = Tr q A k A , (3.8) i.e. Tr q A k is central in the subalgebra generated by A . Analogously, the I k = Tr q B k are central inthe algebra generated by B and, in particular, the I k form a commutative family.In principle, we can start with (3.1) and try to develop a proper parametrisation of the ( A, B )generators suitable for reduction. It is an interesting path that should lead to understanding howto implement the Dirac constraints at the quantum level. We will find, however, a short cut to thealgebra of the quantum L -operator. – 14 – .2 Quantum R -matrices and the L -operator An alternative route to the quantum R -matrices and to the corresponding L -operator algebra is basedon the observation that in the classical theory, the Poisson brackets between the entries of the Laxmatrix have the same structure (2.26) for both rational and hyperbolic cases. As a consequence, theequations satisfied by the classical rational and hyperbolic r -matrices are also the same. This shouldalso be applied to the equations obeyed by the corresponding quantum R -matrices. We thus assumethat the matrices R and ¯ R for the hyperbolic RS model satisfy the system of equations R R R = R R R (3.9)and R ¯ R ¯ R = ¯ R ¯ R P R P − , (3.10)¯ R P ¯ R P − = ¯ R P ¯ R P − . (3.11)and have the standard semi-classical limit where they match the classical r -matrices (2.27). Here andin the following ( Q i , P i ) satisfy the quantum algebra Q i Q j = Q j Q i P i P j = P j P i [ P i , Q j ] = ( e ¯ h − Q j P j δ ij , (3.12)being the standard quantisation of the Poisson algebra on the reduced phase space (2.23),(2.30). Infact, it is not difficult to guess a proper solution for these R -matrices based on the analogy with therational case. For R we can take R = exp ¯ hr , (3.13)where r is given on the first line of (2.27). In the following we adopt the notation R + = R . Since theclassical r -matrix satisfies the property r = − r , the exponential in (3.13) can be easily evaluated andwe find R + = + (1 − q ) N X i = j (cid:16) Q j Q ij E ii − Q i Q ij E ij (cid:17) ⊗ ( E jj − E ji ) . (3.14)A direct check shows that (3.14) is a solution of (3.9).In comparison to the rational model, a new feature is that there exists yet another solution R − ofthe Yang-Baxter equation, namely, R − = − (1 − q − ) N X i = j ( E ii − E ij ) ⊗ (cid:16) Q i Q ij E jj − Q j Q ij E ji (cid:17) . (3.15)These solutions are related as R +21 R − = , (3.16) i.e. precisely in the same way as their non-dynamical counterparts, cf. (3.4). Furthermore, thematrices R ± satisfy equation R + − q R − = (1 − q ) C . (3.17)– 15 –hey are also of Hecke type and the matrices ˆ R ± = CR ± have the following property( ˆ R ± − )( ˆ R ± + q ± ) = 0 . (3.18)Concerning the generalisation of equation (3.10) to the hyperbolic case, we can imagine twodifferent versions - one involving R + and another R − , that is, R ± ¯ R ¯ R = ¯ R ¯ R P R ± P − , (3.19)It appears that there exists a unique matrix ¯ R which satisfies both these equations. It is given by¯ R = − N X i = j q Q i − Q i q Q i − Q j ( E ii − E ij ) ⊗ E jj . (3.20)and its inverse is ¯ R − = − (1 − q ) N X i = j Q i Q ij ( E ii − E ij ) ⊗ E jj . (3.21)The matrix (3.20) also obeys (3.11),¯ R P ¯ R P − = ¯ R P ¯ R P − . (3.22)Introducing R = ¯ R − R ¯ R , (3.23)we find R + = + (1 − q ) N X i = j Q i Q ij ( E ij ⊗ E ji − E ii ⊗ E jj ) ,R − = − (1 − q − ) N X i = j Q j Q ij ( E ij ⊗ E ji − E ii ⊗ E jj ) . (3.24)These matrices satisfy the Gervais-Neveu-Felder equation R ± P − R ± P R ± = P − R ± P R ± P − R ± P . (3.25)and are related to each other as R +21 R − = . (3.26)They also have another important property, usually referred to as the zero weight condition [32],[ P P , R ± ] = 0 . (3.27)Finally, the quantum L -operator is literally the same as its classical counterpart (2.21), of coursewith the natural replacement of p i by the corresponding derivative L = N X i,j =1 Q i − ω Q i Q i − ω Q j b j ⊤ j E ij , b j = N Y a = j ω Q j − Q a Q j − Q a , (3.28)– 16 –here ω = e − γ and ⊤ j is the operator ⊤ j = e − ¯ h ∂∂qj . On smooth functions f ( Q , . . . , Q N ) it acts as( ⊤ j f )( Q , . . . , Q N ) = f ( Q , . . . , q Q j , . . . Q N ) . It is a straightforward exercise to check that this L -operator satisfies the algebraic relations R +12 L ¯ R − L = L ¯ R − L R +12 ,R − L ¯ R − L = L ¯ R − L R − . (3.29)with the R -matrices given by (3.14), (3.15), (3.20) and (3.24). The consistency of these relationsfollow from (3.16) and (3.26). One can alternatively derive equations (3.29) by direct quantisation of(2.28)-(2.29), where the classical matrix is chosen to be r or, equivalently, − r W W R ± = R ± W W , (3.30) W ¯ R P = ¯ R P W , (3.31)whose consistency follows from the same R -matrices relations. The algebraic relation (3.30) is alsoknown as the quantum Frobenius group condition [19].Concerning commuting integrals, the Heisenberg double has a natural commutative family I k =Tr q B k . It is not clear, however, how these integrals can be expressed via L , because we are lackingan analogue of the quantum factorisation formula B = T LT − , where T and L would be subjected towell-defined algebraic relations. Instead, what we could do is to conjecture the same formula as wasobtained for quantum integrals in the rational case [19], where now the R -matrices are those of thehyperbolic model. Interestingly, the existence of two R -matrices, R ± , should give rise to two familiesof commuting integrals I ± k . Borrowing the corresponding expression from the rational case [19], weconjecture the following quantum trace formulae I ± k = Tr (cid:0) C t L ¯ R t R t ± L . . . L ¯ R t R t ± L (cid:1) , (3.32)as quantisation of the classical integrals (2.36). In (3.32) the number k on the right-hand side gives anumber of L ’s and t stands for the transposition in the second matrix space. In particular, C t = N X i,j =1 E ij ⊗ E ij is a one-dimensional projector and from (3.14), (3.15) and (3.20) we get¯ R t R t + 12 = + (1 − q ) X i,j Q i Q i − q Q j E ij ⊗ ( E ij − E jj ) , (3.33)¯ R t R t − = + (1 − q ) X i,j (cid:20) Q j Q i − q Q j E ij ⊗ ( E ij − E jj ) + 1 q ( E ii − E ij ) ⊗ E jj (cid:21) . (3.34)Commutativity of I ± k is then verified by tedious but direct computation which we do not reproducehere, rather our goal is to present a formula which relates I ± k with the commuting family given byMacdonald operators. In fact, ⊤ j = P − j , we use ⊤ j to signify that we talk about a particular representation for L . – 17 –e denote by { S k } a commutative family of Macdonald operators , where S k = ω k ( k − X J ⊂{ ,...,n }| J | = k Y i ∈ Jj J ω Q i − Q j Q i − Q j Y i ∈ J ⊤ i . (3.35)The Macdonald operators have the following generating function: det( L − ζ ) : = N X k =0 ( − ζ ) N − k S k , S = 1 , (3.36)where ζ is a formal parameter, L is the Lax operator (3.28). Under the sign : : of normal ordering theoperators p j and q j are considered as commuting and upon algebraic evaluation of the determinantall ⊤ j are brought to the right. In the classical theory the normal ordering is omitted and thecorresponding generating function yields classical integrals of motion that are nothing else but thespectral invariants of the Lax matrix.We found an explicit formula that relates the families { I ± k } and S k . To present it, we need thenotion of a q -number [ k ] q associated to an integer k [ k ] q = k − X n =0 q n = 1 − q k − q , (3.37)so that [ k ] = k , which corresponds to the limit ¯ h →
0. Then S k is expressed via I + m or I − m as S k = 1[ k ! ] q ± (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I ± [ k − q ± · · · I ± I ± [ k − q ± · · · · · · · ... I ± k − I ± k − · · · · [1] q ± I ± k I ± k − · · · · I ± (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.38)These formulae can be inverted to express each integral I ± k as the determinant of a k × k matrixdepending on S j , namely, I ± k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S · · · q ± S S · · · ... ... · · · · · · k ] q ± S k S k − S k − · · · S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.39) L -operator The quantum L -operator depending on the spectral parameter is naturally introduced as a normalordered version of its classical counterpart L ( λ ) = (1 − ω ) λ N X i,j =1 λ Q i − ωe − ¯ h/ Q j Q i − ω Q j b j ⊤ j E ij = L − ω e ¯ h/ λ Q − L Q , (3.40)where b j are the same as in (3.28). This L -operator satisfies the following quadratic relation R ( λ, µ ) L ( µ ) ¯ R − ( λ ) L ( λ ) = L ( λ ) ¯ R − ( µ ) L ( µ ) R ( λ, µ ) , (3.41)– 18 –here R ( λ, µ ) = ¯ R − ( λ ) R ( λ, µ ) ¯ R ( µ ) . (3.42)In (3.41) the quantum R -matrices are R ( λ, µ ) = λe ¯ h/ R + − µe − ¯ h/ R − λ − µ − e ¯ h/ − e − ¯ h/ e ¯ h/ λ − X + e ¯ h/ − e − ¯ h/ e − ¯ h/ µ − X . ¯ R ( λ ) = ¯ R − e ¯ h − e ¯ h/ λ − X . (3.43)Here R + and R − are the solutions (3.14) and (3.15) of the quantum Yang-Baxter equation, ¯ R is (3.20)and we have introduced the matrix X ≡ X , X = N X i,j =1 E ij ⊗ E jj . (3.44)This matrix satisfies a number of simple relations with ¯ R and R ± , which are¯ RX = X ¯ R (3.45)and R − X = X R − , R − X − X R − = (1 − q − )( X − X ) ,R + X = X R + , R + X − X R + = − (1 − q )( X − X ) . (3.46)We also present the formula for the inverse of ¯ R ( λ )¯ R ( λ ) − = ¯ R − + e ¯ h − e ¯ h/ λ − e ¯ h X . (3.47)With the help of this formula and (3.43) one can show that (3.42) boils down to R ( λ, µ ) = λe ¯ h/ R + − µe − ¯ h/ R − λ − µ , (3.48)where R ± are the same as given by (3.24). We note also the relation R ( λ, µ ) R ( µ, λ ) = R ( λ, µ ) R ( µ, λ ) = ( e ¯ h/ λ − e − ¯ h/ µ )( e − ¯ h/ λ − e ¯ h/ µ )( λ − µ ) . (3.49)Finally, in addition to (3.42) there is one more relation between R ( λ, µ ) and R ( λ, µ ), namely, R ( λ, µ ) = P − ¯ R ( µ ) P R ( λ, µ ) P − ¯ R − ( λ ) P . (3.50)An interesting observation is that the combination R YB ( λ, µ ) = λe ¯ h/ R + − µe − ¯ h/ R − λ − µ solves the usual quantum Yang-Baxter equation with the spectral parameter. However, the full R -matrix in (3.43) differs from R YB by the terms that violate scale invariance. As a result, this matrixobeys the shifted version of the quantum Yang-Baxter equation, namely, R ( λ, µ ) R ( q λ, q τ ) R ( µ, τ ) = R ( q µ, q τ ) R ( λ, τ ) R ( q λ, q µ ) . (3.51)– 19 –n addition, there are two more equations – the one involving both R and ¯ R , and the other involving¯ R only, R ( λ, µ ) ¯ R ( q λ ) ¯ R ( µ ) = ¯ R ( q µ ) ¯ R ( λ ) P R ( q λ, q µ ) P − , (3.52)¯ R ( λ ) P ¯ R ( q λ ) P − = ¯ R ( λ ) P ¯ R ( q λ ) P − . (3.53)It is immediately recognisable that equations (3.51), (3.52) and (3.53) are a quantum analogue (quanti-sation) of the classical equations (2.50), (2.52) and (2.53), respectively. In the semi-classical expansion R ( λ, µ ) = + ¯ hr ( λ, µ ) + o (¯ h ) , ¯ R ( λ ) = + ¯ hr ( λ ) + o (¯ h ) (3.54)the matrices (3.43) yield r ( λ, µ ) = λr + µr λ − µ + σ λ − − σ µ − (cid:16) λ + µλ − µ − λ − µ − (cid:17) ⊗ , ¯ r ( λ ) = ¯ r + σ λ − − ⊗ λ − , which is different from the canonical classical r -matrices (2.48) by allowed symmetry shifts. Thus,(3.43) should be regarded as a quantisation of the classical r -matrices satisfying the shifted Yang-Baxter equation. In this respect it is interesting to point out that the corresponding quantisation ofthe r -matrices solving the usual CYBE remains unknown.Finally, the algebra (3.41) should be completed by the following additional relations encoding thecommutation properties of L with Q L Q = Q L Ω , Q − L = L Q − Ω , (3.55)where Ω = − (1 − q ) ¯ C .Now we derive a couple of important consequences of the algebraic relation (3.41). Namely, we es-tablish the quantum Lax representation, similar to the rational case, and also prove the commutativityof the operators Tr L ( λ ) for different values of the spectral parameter.Following considerations of the dynamics in the classical theory, we take H = lim λ →∞ Tr L ( λ ) as thehamiltonian. From (3.41) we getTr h R ( µ, λ ) L ( λ ) ¯ R − ( µ ) i L ( µ ) = L ( µ ) Tr h ¯ R − ( λ ) L ( λ ) R ( µ, λ ) i , (3.56)where (3.48) was used. A straightforward computation reveals that the traces on the left and theright-hand side of the last expression are equal and that, for instance, e ¯ h/ Tr h ¯ R − ( λ ) L ( λ ) R ( µ, λ ) i = Tr L ( λ ) − M ( λ, µ ) , (3.57)where M ( λ, µ ) = ( e ¯ h − λλ − µ µ − e − ¯ h/ λ − e ¯ h/ L ( λ )+ e ¯ h − λ − e ¯ h/ N X i = j λe − ¯ h Q j − e − ¯ h/ Q i Q i − e − ¯ h Q j L ij ( λ )( E ii − E ij ) . (3.58)– 20 –hus, equation (3.56) turns intoTr L ( λ ) L ( µ ) − L ( µ )Tr L ( λ ) = [ M ( λ, µ ) , L ( µ )] . (3.59)From (3.58) we, therefore, derive the quantum-mechanical operator MM = lim λ →∞ M ( λ, µ ) = ( e ¯ h − N X i = j e − ¯ h Q j Q i − e − ¯ h Q j L ij ( E ii − E ij )= ( e ¯ h − N X i = j L ij Q j Q ij ( E ii − E ij ) , (3.60)where in the last expression we commuted the entries of L ij to the left so that it formally coincideswith its classical counterpart (2.42). In the limit λ → ∞ , (3.59) becomes the quantum Lax equation.Note that in the derivation of this equation we did not use any concrete form of L ; we only use thatit factorises as L = W P − , where W is a function of coordinates only.Taking the trace of (3.59), one getsTr L ( λ )Tr L ( µ ) − Tr L ( µ )Tr L ( λ ) = Tr[ M ( λ, µ ) , L ( µ )] . (3.61)A priori the trace of the commutator on the right-hand side might not be equal to zero, becauseit involves matrices with operator-valued entries. An involved calculation that uses representation(3.40) shows that it nevertheless vanishes , identically for λ and µ . Fortunately, there is a simpleand transparent way to show the commutativity of traces of the Lax operator, which directly relies onthe algebraic relations (3.56), thus bypassing the construction of the quantum Lax pair. Indeed, letus multiply both sides of (3.41) with P − ¯ R ( λ ) P R − ( λ, µ ) and take the trace with respect to bothspaces. We get Tr h P − ¯ R ( λ ) P L ( µ ) ¯ R − ( λ ) L ( λ ) i =Tr h P − ¯ R ( λ ) P R − ( λ, µ ) L ( λ ) ¯ R − ( µ ) L ( µ ) R ( λ, µ ) i . From (3.50) we have P − ¯ R ( λ ) P R − ( λ, µ ) = R − ( λ, µ ) P − ¯ R ( µ ) P , so that the right-hand side of the above equation can be transformed asTr h P − ¯ R ( λ ) P L ( µ ) ¯ R − ( λ ) L ( λ ) i = (3.62)Tr h R − ( λ, µ ) P − ¯ R ( µ ) P L ( λ ) ¯ R − ( µ ) L ( µ ) R ( λ, µ ) i . Further progress is based on the fact that the matrices ¯ R ( λ ) and ¯ R − ( λ ) are diagonal in the secondspace. We represent it in factorised form¯ R ( λ ) = N X j =1 G j ( λ ) ⊗ E jj , (3.63) For this result to hold, the presence in (3.58) of the first term proportional to L ( λ ) is of crucial importance. – 21 –ee (3.43), (3.20) and (3.44). Therefore, P − ¯ R ( λ ) P = N X j =1 P − j G j ( λ ) P j ⊗ E jj . (3.64)Although this expression involves the shift operator, it commutes with any function of coordinates q j ,because when pushed through (3.64), this function will undergo the shifts of q j in opposite directionswhich compensate each other. Similarly,¯ R − ( λ ) = N X j =1 G j ( λ ) − ⊗ E jj = N X j =1 ( ⊗ E jj )( G j ( λ ) − ⊗ ) . Consider first the left-hand side of (3.62)Tr h N X j =1 N X k =1 ( P − j G j ( λ ) P j ⊗ E jj L ( µ ) E kk )( G k ( λ ) − ⊗ ) L ( λ ) i . Using the cyclic property of the trace in the second space, this expression is equivalent toTr h N X j =1 N X k =1 ( P − j G j ( λ ) P j ⊗ L ( µ ) E jj E kk )( G k ( λ ) − ⊗ ) L ( λ ) i . Taking into account that L = W P − and the commutativity of P − j G j ( λ ) P j with any function ofcoordinates, we arrive atTr h N X j =1 ( ⊗ W ( µ ))( P − j G j ( λ ) P j ⊗ P − j E jj ) ¯ R − ( λ ) L ( λ ) i = Tr L ( µ )Tr L ( λ ) . Now we look at the right-hand side of (3.62): using the cyclic property of the trace, the matrix R ( λ, µ )can be moved to the left where it cancels with its inverse. This manipulation is allowed because L ( λ )and L ( µ ) produce together a factor P − P − with which R ( λ, µ ) commutes due to the zero weightcondition (3.27). Also, the individual entries of R ( λ, µ ) are freely moved through P − ¯ R ( µ ) P ,because of the diagonal structure of the latter matrix in the first matrix space, analogous to thesimilar property of (3.64). Then, to eliminate ¯ R ( µ ), one employs the same procedure as was usedfor the left-hand side of (3.62) and the final result is Tr L ( λ )Tr L ( µ ). This proves the commutativityof traces of the Lax matrix for different values of the spectral parameter.We finally remark that writing the analogue of (3.36) with spectral parameter dependent Laxoperator [14, 15] : det( L ( λ ) − ζ ) : = N X k =0 ( − ζ ) N − k S k ( λ ) , (3.65)the quantities S k ( λ ) are commuting integrals and they are related to Macdonald operators (3.35) bya simple coupling- and spectral parameter-dependent rescaling S k ( λ ) = λ − k ( λ − ω k e − ¯ h/ )( λ − e − ¯ h/ ) k − S k . (3.66)– 22 – Conclusions
We have discussed the hyperbolic RS model in the context of Poisson reduction of the Heisenbergdouble [22]: we derive its Poisson structure and show that only on the reduced phase space does thePoisson algebra of the Lax matrix close and take a form very similar to the Lax matrix of the rationalRS model [19]. We find a quantisation of the L -operator algebra governed by new R matrices R ± , alongwith a quantisation of the classical integrals in the form of quantum trace formulae I k (see (3.32)).We show how these quantum integrals are related to the well-known Macdonald operators throughdeterminant formulae. Along the way we present a second Lax matrix that we can use to introduce aspectral parameter in the model. At the classical level this yields r -matrices that satisfy the shiftedYang-Baxter equation due to scale-violating terms. We show that this L -operator algebra admits aquantisation as well, with new R matrices satisfying the shifted quantum Yang-Baxter equation.A particularly interesting observation is that one cannot obtain the quantum L -operator algebrafrom the quantum Heisenberg double in the same way as was done for the quantum cotangent bundle.It would be interesting to pursue the question whether and how one can impose the Dirac constraintsafter quantisation in order to reconstruct the quantum L -operator algebra. A first step in that directioncould be finding an analytic proof that the Dirac bracket for L on the reduced phase space is closed forgeneral N . Another interesting question is to find the relation between our quantum trace formulaeand the commuting traces obtained by the fusion procedure [34, 35] for the equations (3.29). Inaddition, it would be interesting to extend our results to the RS models with spin, in particular, tothose discussed in [8], as well as to find an analogue of the formulae (3.32) for the model with ellipticpotential or for other series of Lie algebras. Constructing the quantum spin versions of these modelscould further aid the understanding of the RS type models that appear in the study of conformalblocks as in [11]. Acknowledgements
We would like to thank Sylvain Lacroix for interesting discussions. The work of G. A. and E. O.is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Ger-many’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306. The work of E.O. is alsosupported by the DFG under the Research Training Group 1670.
A Derivation of the Poisson structure
A.1 Lax matrix and its Poisson structure
Consider the following matrix function on the Heisenberg double L = T − BT , (A.1)where T is the Frobenius solution of the factorisation problem (2.12). On the reduced space L turnsinto the Lax matrix of the hyperbolic RS model. For this reason we continue to call (A.1) the Laxmatrix and below we compute the Poisson brackets between the entries of L considered as functionson the Heisenberg double. This will constitute the first step towards evaluation of the correspondingDirac bracket.The standard manipulations give { L , L } = T L L − L T L − L T L + L L T (A.2)+ T − T − { B B } T T + B L − L B − B L + L B , – 23 –here we defined the following quantities T = T − T − { T , T } , B = T − T − { T , B } T . By using (2.1), we get T − T − { B , B } T T = − ˇ r − L L − L L ˇ r + + L ˇ r − L + L ˇ r + L . Here we introduced the dressed r -matricesˇ r ± = T − T − r ± T T , (A.3)which have proved themselves to be a useful tool for the present calculation. The dressed r -matriceshave essentially the same properties as their undressed counterparts, most importantly,ˇ r + − ˇ r − = C , (A.4)because C is an invariant element. Thus, for (A.2) we get { L , L } = ( T − ˇ r − ) L L + L L ( T − ˇ r + ) + L (ˇ r − − T ) L + L (ˇ r + − T ) L + B L − L B − B L + L B . (A.5)Now we proceed with evaluation of T . Taking onto account that T satisfies (2.14), in componentswe have T ij,kl = T − ip T − kq δT pj δA mn δT ql δA rs { A mn , A rs } (A.6)= X a = j X b = l Q ja Q lb ( δ ia T nj T − am + δ ij T na T − jm )( δ kb T sl T − br + δ kl T sb T − lr ) { A mn , A rs } = X a = j X b = l Q ja Q lb ( δ ia δ kb ζ aj,bl + δ ij δ kb ζ ja,bl + δ ia δ kl ζ aj,lb + δ ij δ kl ζ ja,lb ) . Here Q ij = Q i − Q j and we introduced the concise notation ζ = T − T − { A , A } T T . Using (2.1) and the fact that A = T Q T − , we find that ζ = − ˇ r − Q Q − Q Q ˇ r + + Q ˇ r − Q + Q ˇ r + Q . With the help of (A.4) we find in components ζ ij,kl = − Q ij (ˇ r − ij,kl Q kl + C ij,kl Q k ) , where C ij,kl = δ jk δ il are the entries of C . Substitution of this tensor into (A.6) yields the followingexpression T ij,kl = X a = j X b = l (cid:16) − δ ia δ kb ˇ r − aj,bl + δ ij δ kb ˇ r − ja,bl + δ ia δ kl ˇ r − aj,lb − δ ij δ kl ˇ r − ja,lb (cid:17) + X a = j X b = l Q lb (cid:16) δ ia δ kb C aj,bl Q b − δ ij δ kb C ja,bl Q b + δ ia δ kl C aj,lb Q l − δ ij δ kl C ja,lb Q l (cid:17) . – 24 –n the first line the summation can be extended to all values of a and b , because the expression whichis summed vanishes for a = j and independently for b = l . For the same reason, we have extended thesummation over a in the second line, where we also substitute the explicit value for C ij,kl = δ jk δ il . Inthis way we find T ij,kl = X ab (cid:16) − δ ia δ kb ˇ r − aj,bl + δ ij δ kb ˇ r − ja,bl + δ ia δ kl ˇ r − aj,lb − δ ij δ kl ˇ r − ja,lb (cid:17) + X a X b = l Q lb (cid:16) δ ia δ kb δ al δ jb Q b − δ ij δ jl δ kb δ ab Q b + δ ia δ kl δ ab δ jl Q l − δ ij δ kl δ al δ jb Q l (cid:17) . This further yields the following expression T ij,kl = − ˇ r − ij,kl + δ ij X a ˇ r − ia,kl + δ kl X a ˇ r − ij,ka − δ ij δ kl X ab ˇ r − ia,kb + X b = l Q lb (cid:16) δ kb δ il δ jb Q b − δ ij δ jl δ kb Q b + δ kl δ ib δ jl Q l − δ ij δ kl δ jb Q l (cid:17) . Here the second line can be written in the concise form as the matrix element r Q ij,kl of the followingmatrix r Q = X a = b Q b Q ab ( E aa − E ab ) ⊗ ( E bb − E ba ) (A.7)Therefore, T ij,kl = r Q ij,kl − ˇ r − ij,kl + δ ij X a ˇ r − ia,kl + δ kl X a ˇ r − ij,ka − δ ij δ kl X ab ˇ r − ia,kb . Hence, T = r Q − ˇ r − + a + b − c . (A.8)where we introduced three r -matrices, a , b and c with entries a ij,kl = δ ij X a ˇ r − ia,kl , b ij,kl = δ kl X a ˇ r − ij,ka , c ij,kl = δ ij δ kl X ab ˇ r − ia,kb . (A.9)Needless to say, the bracket thus obtained is compatible with the Frobenius condition (2.14), whichmeans that X a T ia,kl = 0 , X a T ij,ka = 0 , for any values of the free indices.Now we turn our attention to B , which in components reads as B ij,kl = X a = j Q ja ( δ ia η aj,kl + δ ij η ja,kl ) , where we introduced the notation η = T − T − { A , B } T T . – 25 –ith the help of (2.1) we get η = − ˇ r − Q L − Q L ˇ r − + Q ˇ r − L + L ˇ r + Q , and by using (A.4) obtain for components the following expression η aj,kl = Q ja ( L ks ˇ r − aj,sl − ˇ r − aj,ks L sl ) + L ks C aj,sl Q j . With this expression at hand, we get B ij,kl = X a = j (cid:16) δ ia ( L ks ˇ r − aj,sl − ˇ r − aj,ks L sl ) − δ ij ( L ks ˇ r − ja,sl − ˇ r − ja,ks L sl ) (cid:17) + L ks X a = j Q ja (cid:16) δ ia δ al δ js Q j + δ ij δ jl δ as Q a (cid:17) . Here the summation in the first line can be extended to include the term with a = j because the lattervanishes. The second line can be conveniently written as a matrix element of some r -matrix. Namely, B ij,kl = L ks (cid:16) ˇ r − ij,sl − δ ij X a ˇ r − ja,sl (cid:17) − (cid:16) ˇ r − ij,ks − δ ij X a ˇ r − ja,ks (cid:17) L sl + L ks X a = b Q b Q ab ( E aa − E ab ) ij ⊗ ( E ba ) sl . In matrix form B = L (ˇ r − − a ) − (ˇ r − − a ) L + L d , (A.10)where a is the same matrix as in (A.8) and we introduced d = X a = b Q b Q ab ( E aa − E ab ) ⊗ E ba . (A.11)We also need B = L (ˇ r − − a ) − (ˇ r − − a ) L + L d , Since ˇ r − = − ˇ r +12 , we have B = − L (ˇ r +12 + a ) + (ˇ r +12 + a ) L + L d . (A.12)Now everything is ready to obtain the bracket (A.5). Substituting in (A.5) expressions (A.8), (A.10)and (A.12), we conclude that (A.5) has the structure { L , L } = k +12 L L + L L k − + L s − L + L s +12 L , (A.13)where the coefficients are k +12 = r Q + C + ( a + b − c ) ,k − = r Q + d − d + ( a + b − c ) ,s +12 = − r Q − d − ( a + b − c ) ,s − = − r Q − C + d − ( a + b − c ) . (A.14)– 26 –irst, we note that these coefficients satisfy k + + k − + s + + s − = 0 , (A.15)which guarantees that spectral invariants of L are in involution on the Heisenberg double. Second, in(A.14) the apparent dependence on the variable T occurs in the single combination a + b − c .To make further progress, consider a = C a C , as C acts as the permutation. We have, written in components,( a ) ij,kl = C im,kn ( a ) mr,ns C rj,sl = δ mk δ in (cid:16) δ mr X a ˇ r − ma,ns (cid:17) δ js δ rl = δ kl X a ˇ r − ka,ij = − δ kl X a ˇ r + ij,ka . Therefore, ( a + b ) ij,kl = − δ kl X a ˇ r + ij,ka + δ kl X a ˇ r − ij,ka = − δ kl X a C ij,ka = − X a δ kl δ jk δ ia = − X ab ( E ab ) ij ⊗ ( E bb ) kl . The dependence on T disappears and we find a simple answer a + b = − X ab E ab ⊗ E bb . (A.16)The only T -dependence is in the coefficient c . This coefficient cannot be simplified or cancelled, sowe leave it in the present form. Substituting in (A.14) the matrices (A.7), (A.11) and (A.16) and,performing necessary simplifications, we obtain our final result for the coefficients of the bracket (A.13) k +12 = X a = b (cid:16) Q b Q ab E aa − Q a Q ab E ab (cid:17) ⊗ ( E bb − E ba ) − c ,k − = X a = b Q a Q ab E aa ⊗ E bb − X a = b Q a Q ab E ab ⊗ E ba − ⊗ − c ,s +12 = − X a = b Q a Q ab ( E aa − E ab ) ⊗ E bb + ⊗ + c ,s − = − X a = b Q b Q ab E aa ⊗ ( E bb − E ba ) + c . (A.17)In fact, the identity matrix ⊗ appearing in k − and s + can be omitted as it cancels out in theexpression (A.13). As was already mentioned, the only T -dependence left over is in the term c ,namely, ( c ) ij,kl = δ ij δ kl X ab ˇ r − ia,kb = δ ij δ kl T − im T − kn X ab r − ma,nb . (A.18)It is this term which violates the invariance of the bracket (A.13) under transformations from theFrobenius group. – 27 –o complete our discussion, we consider( c ) ij,kl = δ ij δ kl X ab ˇ r − ka,ib = − δ ij δ kl X ab ˇ r + ia,kb . This gives( c + c ) ij,kl = − δ ij δ kl X ab C ia,kb = − δ ij δ kl X ab δ ib δ ka = − δ ij δ kl = − ( ⊗ ) ij,kl , or in other words, c + c = − ⊗ . (A.19)Equation (A.19) leads to the following relations between the coefficiencients k +12 + k +21 = C − ⊗ ) , k − + k − = − C , s − = − s +21 . (A.20)Notice that the fact that the right-hand side of the first two expressions is an invariant tensor. Relations(A.20) guarantee that the bracket (A.13) is skew-symmetric.Following similar steps, we can derive the Poisson brackets involving other Frobenius invariantson the Heisenberg double, namely W ij and P i coordinates. Introducing the notations r hg ± = h − g − r ± h g , ( c hg ) ijkl = δ ij δ kl X α,β ( r hg − ) iαkβ , which, for Frobenius elements g, h satisfies c hg + c gh = − ⊗ , we can write { W , W } = [ r , W W ] + W c UT W + W c T U W − W W c UU − c T T W W { W , P } = P [¯ r , W ] + P W ( c UT − c UU ) + P ( c T U − c T T ) W { P , P } = P P ( c UT + c T U − c T T − c UU ) , (A.21)where matrices r and ¯ r are defined in (2.27). The c hg -like terms in the brackets (A.21) are notFrobenius invariants, despite the arguments of the brackets are so, as it happens for (A.13). Theseterms disappear after imposing Dirac constraints in the reduced phase space, as it will explicitly shownfor the LL -bracket in A.2. A.2 Dirac bracket
Here we outline the construction of the Dirac bracket between the entries of the Lax matrix (A.1).We argue that the contribution to the Dirac bracket coming from the second class constraints hasthe same matrix structure as (A.13) and that this contribution precisely cancels all the terms c in(A.17), so that the resulting coefficients describing the Dirac bracket on the constraint surface aregiven by expressions (2.27) in the main text.We start with the Poisson algebra of the non-abelian moment map { M , M } = − r + M M − M M r − + M r − M + M r + M . (A.22)– 28 –his is the Semenov-Tian-Shansky type bracket; it has N Casimir functions Tr( M k ) with k = 1 , . . . , N .On the constraint surface S the moment map is fixed to the following value M = ω + β e ⊗ e t , (A.23)see (2.10). Substituting this expression into the right-hand side of (A.22) yields the following answer M ij,kl ≡ { M ij , M kl } (cid:12)(cid:12)(cid:12) S = β h ( ω − N − β ( i − )) δ il − β δ jl + β Θ( l − j ) (A.24) − ( ω − N − β ( j − )) δ jk + β δ ik − β Θ( k − i ) i , where Θ is the Heaviside step function Θ( j ) = (cid:26) , j ≥ , , j < . (A.25)For any X ∈ Mat( N, C ) introduce the following quantities t (0) ( X ) ij = X ij − N X a X aj − N X a X ia + 1 N X ab X ab , i, j = 2 , . . . , N,t (1) ( X ) j = 1 N X ab X ab − N X a X aj , j = 2 , . . . , N ,t (2) ( X ) j = 1 N X ab X ab − N X a X ja , j = 2 , . . . , N ,t (3) ( X ) = 1 N X ab X ab . (A.26)From these quantities we construct the projectors π ( i ) that have the following action on X π (0) ( X ) = N X i,j =2 ( E − E i − E j + E ij ) t (0) ( X ) ij , π (1) ( X ) = N X j =2 a j t (1) ( X ) j , π (2) ( X ) = N X j =2 b j t (2) ( X ) j , π (3) ( X ) = N X i,j =1 E ij t (3) ( X ) . (A.27)where a j = N X i =1 ( E i − E ij ) , j = 2 , . . . , N ,b j = N X i =1 ( E i − E ji ) , j = 2 , . . . , N . In particular, π (0) projects on the Lie algebra of ̥ and π (3) – on the one-dimensional dilatationsubalgebra C ∗ . The completeness condition is X = X k =0 π ( k ) ( X ) . – 29 –rom (A.24) it is readily seen that { t (3) ( M ) , M kl } = 1 N X ab { M ab , M kl } = 0 . Analogously, we find { t (0) ( M ) ij , M kl } = { M ij − N P a M aj − N P a M ia , M kl } = 0 , i, j = 2 , . . . , N . Thus, projections π (0) ( M ) and π (3) ( M ) constitute ( N − + 1 = N − N + 2 constraints of thefirst class. Projections π (1) and π (2) yield a non-degenerate matrix of Poisson brackets and, therefore,represent 2( N −
1) constraints of the second class. This matrix should be inverted and used to definethe corresponding Dirac bracket. Even simpler, the matrix (A.24) has rank 2( N −
1) and we can useany non-degenerate submatrix of this rank to define the corresponding Dirac bracket.Now we derive the Poisson relations between the moment map M and the Lax matrix given by(A.1). First, we compute { M ij , T kl } = δT kl δA rs { M ij , A rs } = (A.28)= − (( r + M − M r − ) T ) ij,kl + T kl X a ( T − ( r + M − M r − )) ij,la . Deriving this formula, we have used (2.7), as well as the fact that T ∈ F . Next, we obtain { M ij , L kl } = L kl X sp ( T − ls − T − ks )( r + M − M r − ) ij,sp . (A.29)It is clear that the diagonal entries from this expression of L commute with all the constraints: { M ij , L kk } = 0, even without restricting to the constrained surface.On the constrained surface where M is given by (A.23), we have { M ij , L kl } (cid:12)(cid:12)(cid:12) S = ω − N L kl ( T − lj − T − kj ) + βL kl X sp ( T − ls − T − ks )Ω ijs . Here Ω ijs ≡ X p [ r + , (e ⊗ e t ) ] ij,sp = − δ is − ( j − ) δ js + Θ( s − i ) . (A.30)From the explicit expression (A.30) and the fact that T is an element of the Frobenius group, wefurther deduce that { t (0) ( M ) ij , L kl } (cid:12)(cid:12)(cid:12) S = 0 , { t (3) ( M ) ij , L kl } (cid:12)(cid:12)(cid:12) S = 0 . In other words, L commutes on the constraint surface with all constraints of the first class, indepen-dently on the value of T .With the help of (A.30) we obtain { M ij , L kl } (cid:12)(cid:12)(cid:12) S = L kl h ( ω − N − β ( j − ))( T − lj − T − kj )+ β ( T − li − T − ki ) + β X s>i ( T − ls − T − ks ) i . – 30 –aking into account that N X s>i ( T − ls − T − ks ) + N X si ( T − ls − T − ks ) − N X s
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