The Master T-Operator for Inhomogeneous XXX Spin Chain and mKP Hierarchy

Following the approach of [Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., J. High Energy Phys. 2013 (2013), no. 9, 064, 65 pages, arXiv:1112.3310], we show how to construct the master T-operator for the quantum inhomogeneous GL(N) XXX spin chain with twisted boundary conditions. It satisfies the bilinear identity and Hirota equations for the classical mKP hierarchy. We also characterize the class of solutions to the mKP hierarchy that correspond to eigenvalues of the master T-operator and study dynamics of their zeros as functions of the spectral parameter. This implies a remarkable connection between the quantum spin chain and the classical Ruijsenaars-Schneider system of particles.


Introduction
The master T -operator was introduced in [2] (in a preliminary form, it was previously discussed in [14]). It is a generating function for commuting conserved quantities of quantum spin chains and associated integrable vertex models which unifies the transfer matrices on all levels of the nested Bethe ansatz and Baxter's Q-operators in one commuting family.
It was also proven in [2] that the master T -operator, as a function of infinitely many auxiliary parameters ("times"), one of which being the quantum spectral parameter, satisfies the same hierarchy of bilinear Hirota equations as the classical tau-function does. This means that any eigenvalue of the master T -operator is a tau-function of a classical integrable hierarchy. For finite spin chains with GL(N )-invariant R-matrices this tau-function is a polynomial in the quantum spectral parameter. The close connection of the spin chain spectral problem with integrable many-body systems of classical mechanics comes from the dynamics of zeros of the polynomial tau-functions. This is a further development of earlier studies [15,19,38,39,40] clarifying the role of the Hirota bilinear difference equation [11,24] in quantum integrable models.
In this paper we review the results of [2] and make the connection with classical many-body systems more precise. The presentation here is deliberately made as close as possible to that of [3], where a similar correspondence between the quantum Gaudin model and the classical This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html XXX chain mKP hierarchy master T -operator ←→ τ -function spectral parameter ←→ the t 0 -variable higher transfer matrices ←→ Plücker coordinates Moreover, from the explicit form of the R-matrix and the Yang-Baxter equation it follows that this tau-function is a polynomial in u = t 0 . Therefore, according to [20,12], the dynamics of its roots in t i with i ≥ 1 is given by equations of motion of the rational RS system of particles. It should be mentioned that the method for deriving the dynamics of roots is similar to that used in deriving the Bethe equations in Sklyanin's separation of variables method [35,36]. This is not particularly surprising because, according to [30] (see also [19,38]), the nested Bethe ansatz equations themselves can be understood as an integrable many-body system of RS type in discrete time.
The XXX-RS correspondence implies that the "inhomogeneities at sites" u i in the XXXchain should be identified with initial coordinates of the RS particles while eigenvalues of the spin chain Hamiltonians are their initial velocities. Eigenvalues of the Lax matrix for the rational RS model coincide with eigenvalues of the twist matrix (with certain multiplicities). Therefore, with fixed integrals of motion in the RS model determined by invariants of the twist matrix, there are a finite number of solutions for their values which correspond to different eigenstates of the spin chain. In other words, the eigenstates of the spin chain Hamiltonians are in one-to-one correspondence with (a finite number of) intersection points of two Lagrangian submanifolds in the phase space of the classical RS model. One of them is the hyperplane of fixed u i 's and another is the submanifold of constant levels of the RS Hamiltonians in involution. In short, the dictionary of the XXX-RS correspondence is as follows: XXX chain Ruijsenaars-Schneider inhomogeneities at the sites ←→ initial coordinates eigenvalues of Hamiltonians ←→ initial momenta twist parameters ←→ integrals of motion This "quantum-classical correspondence" was also discussed [7,8,29] in the context of supersymmetric gauge theories and branes.

The quantum spin chain
Consider generalized quantum integrable spin chains with GL(N )-invariant R-matrix Here u is the spectral parameter and I is the unity matrix. By e ab we denote the basis in the space of N × N matrices such that e ab has only one non-zero element (equal to 1) at the place ab: (e ab ) cd = δ ac δ bd . Note that P = ab e ab ⊗ e ba is the permutation matrix in the space C N ⊗ C N with the defining property P (v ⊗ w) = w ⊗ v for any vectors v, w ∈ C N , so the R-matrix can be written as R(u) = I ⊗ I + 1 u P . The GL(N )-invariance of this R-matrix means that g ⊗ g R(u) = R(u)g ⊗ g for any g ∈ GL(N ).
A more general GL(N )-invariant R-matrix is e ab ⊗ π λ (e ba ), (2.1) which acts in the tensor product of the vector representation space C N and an arbitrary finitedimensional irreducible representation π λ of the algebra U (gl(N )) with highest weight λ. We identify λ with the Young diagram λ = (λ 1 , λ 2 , . . . , λ ) with = (λ) non-zero rows, where λ i ∈ Z + , λ 1 ≥ λ 2 ≥ · · · ≥ λ > 0. By e ab we denote the generators of the algebra U (gl(N )) with the commutation relations e ab e a b − e a b e ab = δ a b e ab − δ ab e a b . In this notation we have e ab = π (1) (e ab ), where π (1) is the N -dimensional vector representation corresponding to the 1-box diagram λ = (1). For working in multiple tensor product spaces like (C N ) ⊗n = C N ⊗ · · · ⊗ C N n the following notation is convenient. For any g ∈ End (C N ) we write g (i) = I ⊗(i−1) ⊗g ⊗I ⊗(n−i) ∈ End (C N ) ⊗n .
In particular, the generators of GL(N ) can be realized as e (i) ab := I ⊗(i−1) ⊗e ab ⊗I ⊗(n−i) . They commute for any i = j because they act in different spaces. In this notation, P ij = P ji = a,b e (i) ab e (j) ba (i = j) is the operator acting by permutation of the i-th and j-th tensor factors in the space (C N ) ⊗n . We have: P ij g (j) = g (i) P ij , and P ij g (k) = g (k) P ij for k = i, j.
Fix a matrix g ∈ GL(N ) called the twist matrix which is assumed to be diagonalizable. A family of commuting operators acting in the space V = (C N ) ⊗n (quantum transfer matrices or T -operators) can be constructed as where u i are arbitrary complex parameters which are assumed to be all distinct. The trace is taken in the auxiliary space V λ where the representation π λ is realized. By R j0 λ (u) we denote the R-matrix (2.1) acting in the tensor product of the j-th local space C N of the chain and the space V λ labeled by 0. More precisely, let us denote (2.1) symbolically as Here the operator b i acts in the auxiliary space V λ . It follows from the Yang-Baxter equation that the T -operators with the same g, u i commute for all u, λ and can be simultaneously diagonalized. The normalization used above is such that T ∅ (u) = I ⊗n . Another useful normalization is In this normalization all T λ (u) and all their eigenvalues are polynomials in u of degree n.
For n = 0 the transfer matrix (2.2) is the character of g in the representation π λ : T λ (u) = tr π λ g := χ λ (g), It is given by the Schur polynomial s λ (y) of the variables y = {y 1 , y 2 , . . .}, where y k = 1 k tr g k : (the Jacobi-Trudi formula). Here the complete symmetric polynomials h k (y) = s (k) (y) are defined by It is convenient to set h k = 0 for k < 0. Let w 1 , . . . , w N be the eigenvalues of g ∈ GL(N ) realized as an element of End (C N ). Then y k = 1 k (w k 1 + · · · + w k N ) and [23]). This formula implies that χ ∅ (g) = s ∅ (y) = 1. A more explicit construction of the quantum transfer matrices T λ (u) was suggested in [16]. It uses the special derivative operator on the group GL(N ) called there the co-derivative operator. In fact it is a sort of "matrix logarithmic derivative". The precise definition is as follows. Let g be an element of the group GL(N ) and f be any function of g with values in End (L), where L is the space of any GL(N )-representation. The (left) co-derivative is defined as The right hand side belongs to End (C N ⊗ L). In particular, the result of the action of D on a scalar function is a linear operator in C N , acting by D twice we get an operator in C N ⊗ C N and so on. For example: When the number of tensor factors is more than two another notation is more convenient. Let V i ∼ = C N be copies of C N and V = V 1 ⊗ · · · ⊗ V n as before. Then, applying the matrix derivatives to a scalar function f several times, we can embed the result into End(V) according to the formulas , and so on. The lower indices of D show in which tensor factors the resulting operator acts non-trivially. In this notation, the examples given above read: . For many other formulas of this type see [2,Appendix D]. According to [16] the transfer matrix (2.2) can be represented as a chain of operators of the form 1 + D i u−u i acting on the character: For simplicity we assume that the twist matrix g is diagonal: g = diag (w 1 , w 2 , . . . , w N ). By analogy with the Gaudin model, one may introduce (non-local) spin chain "Hamiltonians" as residues of the T (1) (u) at u = u i : Explicitly, they have the form: (Here and below we write For example, for n = 3 we have: , Similar non-local operators were discussed in [10].
It is easy to check that the operators be the decomposition of the Hilbert space of the spin chain V into the direct sum of eigenspaces of the operators M a with eigenvalues m a ∈ Z ≥0 , a = 1, . . . , N . Then eigenstates of the H i 's belong to the spaces V({m a }). Since a e aa = I is the unit matrix, a M a = nI ⊗n , and hence N a=1 m a = n.
Note also that 3 The master T -operator and the mKP hierarchy

The master T -operator
The master T -operator for the spin chain can be defined as . .} is an infinite set of "time parameters". These operators commute for all values of the parameters: implies that the expansion of T (u, t) in the Schur functions is The sum is taken over all Young diagrams including the empty one. Therefore, the T -operators T λ (u) can be restored from the master T -operator according to the formula In particular, .
Given z ∈ C, we will use the standard notation t ± [z −1 ] for the following special shift of the time variables: As we shall see below, T (u, t ± [z −1 ]) regarded as functions of z with fixed t plays an important role. Here we only note that equation (3.3) implies that T (u, 0 ± [z −1 ]) is the generating series for T -operators corresponding to the one-row and one-column diagrams respectively:

The bilinear identity and Hirota equations
The following statement was proved in [2]: Theorem 3.1. The master T -operator (3.1) satisfies the bilinear identity for the mKP hierarchy [6,13]: for all t, t and u, u , where the integration contour C [0,∞] encircles the cut [0, ∞] between 0 and ∞ (including the points 0 and ∞) and does not enclose any singularities coming from the T -factors.
This means that each eigenvalue of the master T -operator is a tau-function of the mKP hierarchy. Equation (3.2) is the expansion of the tau-function in Schur polynomials [9,31,33]. The functional relations for quantum transfer matrices [4,5,21,22] can be then interpreted as Plücker relations for coefficients of the expansion.
Setting u = u and t k = t k − 1 in (3.5) and taking the residues we arrive at the 3-term Hirota equation , we obtain another 3-term Hirota equation Due to (3.9) (see below), it can be formally regarded as a particular case of (3.6) in the limit z 3 → 0.

The Baker-Akhiezer functions
According to the general scheme, the Baker-Akhiezer (BA) function and its adjoint corresponding to the tau-function (3.1) are given by the formulas [6,13] For brevity, we will refer to both ψ and ψ * as BA functions. In terms of the BA functions, the bilinear identity (3.5) can be written as Using the definition (3.1), we have: T u, t − z −1 = z −N u − u n + D n · · · u − u 1 + D 1 det(zI − g)e tr ξ(t,g) , T u, t + z −1 = z N u − u n + D n · · · u − u 1 + D 1 e tr ξ(t,g) det(zI − g) .
Note that because (det g) −1 D det g = D + 1, we have For the BA functions we can thus write: . (3.11) From these formulas we see that z −u e −ξ(t,z) ψ u (t; z) is a polynomial in z −1 of degree N while z u e ξ(t,z) ψ * u (t; z) is a rational function of z with poles at the points z = w i (eigenvalues of the matrix g) of at least first order because of det(zI − g) in the denominator. Moreover, since each co-derivative raises the order of the poles, these poles may be actually of a higher order, up to n + 1. Also, as is seen from the second formula, this function has a zero of order N at z = 0. (We assume that w a = 0.) Regarded as functions of u, both z −u ψ u and z u ψ * u are rational functions of u with n zeros and n poles which are simple in general position. From (3.1) and (3.10), (3.11) it follows that The BA functions satisfy the following differential-difference equations: where We also note the formulas for the stationary BA functions ψ u (z) := ψ u (0; z), ψ * u (z) := ψ * u (0; z) which directly follow from (3.10), (3.11): .
Below we will also need the relation (Here res ∞ (. . .) ≡ 1 2πi ∞ (. . .) and 1 2πi ∞ z −1 dz = 1.) This relation can be derived from the bilinear identity (3.5) in the following way. Applying ∂ t m and putting u = u + 1, t k = t k afterwards, we get: The first term is regular at z = 0 and thus contributes to the integral by the residue at ∞ while the second term has residues at the points 0, ∞ and the both contribute to the integral. Using (3.9), we can find the residues: Dividing both sides by T (u, t)T (u + 1, t), we obtain (3.17).

Zeros of the master T -operator as Ruijsenaars-Schneider particles
The eigenvalues of the master T -operator are polynomials in the spectral parameter u: T (u, t) = e t 1 tr g+t 2 tr g 2 +··· n k=1 (u − u k (t 1 , t 2 , . . .)).
The roots of each eigenvalue have their own dynamics in the times t k . These dynamics are known to be given by the rational RS model [32] (see [12,20], which extend the methods developed by Krichever [18] and Shiota [34] for dymamics of poles of solutions to the KP hierarchy). The inhomogeneity parameters of the spin chain play the role of coordinates of the RS particles at . Therefore, Comparing with (2.3), we conclude that the initial velocities are equal (up to sign) to the eigenvalues of the spin chain Hamiltonians: This unexpected connection between the quantum spin chain and the classical RS model was mentioned in [2]. A similar relation between quantum Hamiltonians in Gaudin model and velocities of particles in the classical Calogero-Moser model was found in [26,27] within a different framework, see also [25,28] for further developments.

Lax pair for the RS model from dynamics of poles
Following Krichever's method [18], let us derive equations of motion for the t 1 -dynamics of the u i 's. Essentially, the derivation below is not specific to the master T -operator case but only depends on the polynomiality of the tau-function. The specific part is the particular normalization of the BA functions.
It is convenient to denote t 1 = t and put all other times to zero since they are irrelevant for this derivation. Correspondingly, we will write T (u, t) instead of T (u, t) and ∂ t u k =u k , etc. From (3.15) we see that The method of [18] is to perform the pole expansion of the linear problem (3.13) for the BA function ψ. From (3.10) we have the pole expansion of the BA function where c 0 (z) = det(I − z −1 g) (see (3.12)). Substituting this into (3.13), we obtain The l.h.s. is a rational function of u with first order poles at u = u i and u = u i − 1 (possible poles of the second order cancel automatically) vanishing at infinity. Therefore, to solve the linear problem it is enough to cancel all the poles. Representing the l.h.s. as a sum of simple pole terms and equating the coefficients in front of each pole to zero, we get the following system of equations for i = 1, . . . , n: These equations can be rewritten in the matrix form: where c = (c 1 , c 2 , . . . , c n ) t , 1 = (1, 1, . . . , 1) t are n-component vectors and the n × n matrices U = U (t), Y = Y (t), T = T (t) are given by The compatibility condition of the system (4 where M :=U Q +QT −U −1 TU Q . A straightforward calculation shows that M = W Q, where W is the diagonal matrix W = diag(W 1 , . . . , W n ) with elements .
Since Q is a non-degenerate matrix, the matrix equation [T, Y ] −Ẏ = 0 is equivalent toÜ + W = 0. At the same time one can easily check that and so the compatibility condition for the linear system (4.2) isÜ + W = 0 which yields the equations of motion for the RS model with n particles , i = 1, . . . , n. and the matrices Y , T form the Lax pair for the model. The matrix Y is the Lax matrix for the RS model. As is seen from (4.7), the time evolution preserves its spectrum, i.e., the coefficients J k of the characteristic polynomial det(zI − Y (t)) = n k=0 J k z n−k are integrals of motion. In a similar way, substituting the adjoint BA function into the adjoint linear problem (3.14), we get where c * t = (c * 1 , c * 2 , . . . , c * n ) and 1 t = (1, 1, . . . , 1). Note that 1 t T = 0. Using (4.2), (4.8), we find the solutions for the vectors c, c * : For the functions ψ, ψ * themselves we then have: Let us mention some properties of the matrices U , Y to be used in the calculations below. As is well known (and easy to check), the matrix [U, Y ] − Y has rank 1. More precisely, the matrices U , Y satisfy the commutation relation (here 1 ⊗ 1 t is the n×n matrix of rank 1 with all entries equal to 1).
Lemma 4.1. For any k ≥ 0 the following equality holds: Indeed, we have: ] but the last trace is 0 as trace of a commutator.

Eigenvalues of the Lax matrix
Here we prove that the eigenvalues of the Lax matrix Y are the same as the eigenvalues of the twist matrix g with appropriate multiplicities.
Theorem 4.2. The Lax matrix Y has eigenvalues w a with multiplicities m a ≥ 0 such that m 1 + · · · + m N = n.
Indeed, let us compare the large |u| expansions of (3.16) and (4.10). From (3.16) we have: The expansion of (4.10) at t = 0 gives (using Lemma 4.1): where we set Y 0 := Y (0). Therefore, we conclude that where M a is the operator (2.4). Hence we see that the M a is the "operator multiplicity" of the eigenvalue w a . In the sector V({m a }) the multiplicity becomes equal to m a .
Less formal arguments are as follows. The singularities of the vectors c(z, t), c * (z, t) as functions of z are the same as the singularities of the functions ψ, ψ * in the finite part of the complex plane. From (3.10) we see that c(z, t) has a pole of order N at z = 0 and no other poles. At the same time the first equation in (4.9) states that there are possible poles at eigenvalues of the matrix Y (t) (which do not depend on time). Therefore, they must be canceled by zeros of c 0 (z) = z −N det(zI − g) which are at z = w a and are assumed to be simple. If all eigenvalues of Y are distinct, such a cancellation is only possible if n ≤ N . However, the most interesting setting for the quantum spin chains is quite opposite: n > N or even n N (large chain length at a fixed rank of the symmetry algebra). We conclude that in this case the Lax matrix has to have multiple eigenvalues. At first glance, a multiple eigenvalue w a with multiplicity m a ≥ 2 might lead to an unwanted pole of ψ at z = w a coming from the higher order pole of the matrix (zI − Y ) −1 which now can not be cancelled by the simple zero of det(zI − g). In fact higher order poles do not appear in the vector (zI − Y ) −1 1 becauseU 1 is a special vector for the matrix Y which can be decomposed into N Jordan blocks of sizes m a ×m a . However, they do appear in the co-vector 1 t (zI − Y ) −1 and the function ψ * has multiple poles at z = w a (with multiplicities m a + 1).

Equations of motion in Hamiltonian form
The momenta v i canonically conjugate to the coordinates of the RS particles u i can be introduced by the formulȧ Then the Hamiltonian form of the RS equations of motion (4.6) is with the Hamiltonian This result was generalized to the difference KP hierarchy (which is essentially equivalent to the mKP hierarchy) in [12]: The H m 's are higher integrals of motion (Hamiltonians) for the RS model. They are known to be in involution [32]. This agrees with the commutativity of the mKP flows. For completeness, we give a derivation of (4.12) which is a version of the arguments from [12,34]. The main technical tool is equation (3.17) which states that Matching coefficients in front of the poles, we get Inserting here (4.9), we continue the chain of equalities: The next steps are to use the commutation relation (4.11) and notice that E ii Y =u i ∂Y ∂u i : The trace in the last term is which is equal to zero because the matrix U is diagonal. We are left with This proves the first equality in (4.12). Note that another form of the equation The proof of the second equality in (4.12) is more involved. Here we will closely follow [12]. First, using the Lax equationẎ = [T , Y ] and cyclicity of the trace, we take the t 1 -derivative of (4.13) to get: (recall thatT = T + Y , see (4.5)). With the help of this formula we can find ∂ tm v i : HereT is the matrixT (see (4.4)) with zeros on the main diagonal,T ij =T ij − δ ijTii . One can show that where C (i) is the matrix (here and below u ij ≡ u i − u j ). From this it immediately follows that ∂ tm v i = ∂ u i tr(Y m ), which is the second equality in (4.12). The most direct way to prove (4.14) is to calculate matrix elements of both sides. For example, matrix elements of the matrix A (i) are as follows: i.e., given the initial coordinates u i and the action variables H j = tr Y j 0 one has to find possible values of the initial velocitiesu i = −H i . This is equivalent to n algebraic equations for n quantities H 1 , . . . , H n .
In other words, the eigenstates of the quantum Hamiltonians correspond to the intersection points of two Lagrangian manifolds in the phase space of the RS model. One of them is the Lagrangian hyperplane defined by fixing the u i 's and the other one is the Lagrangian submanifold obtained by fixing values of the involutive integrals of motion H i 's, with the latter being determined by eigenvalues of the spin chain twist matrix. This purely classical prescription appears to be equivalent to the Bethe ansatz solution and solves the spectral problem for the quantum spin chain. Example 4.3. Consider the vector v a ∈ C N with components (v a ) b = δ ab . Since P ij (v a ) ⊗n = (v a ) ⊗n , the vector (v a ) ⊗n is an eigenstate for the Hamiltonians H i with the eigenvalues w a n j=1,j =i It is also an eigenvector for the operators M b with eigenvalues m b = nδ ab . The matrix (4.18) in this case is the n×n Jordan block with the only eigenvector 1 with eigenvalue w a and tr Y j 0 = nw j a .