Liouville reflection operator, affine Yangian and Bethe ansatz

In these notes we study integrable structure of conformal field theory by means of Liouville reflection operator/Maulik-Okounkov $R$-matrix. We discuss the relation between $RLL$ and current realization of the affine Yangian of $\mathfrak{gl}(1)$. We construct the family of commuting transfer matrices related to the Intermediate Long Wave hierarchy and derive Bethe ansatz equations for their spectra discovered by Nekrasov and Okounkov and independently by one of the authors. Our derivation mostly follows the one by Feigin, Jimbo, Miwa and Mukhin, but is adapted to the conformal case.


Introduction
There is a large class of 2D QFT's defined by Toda action where ϕ = (ϕ 1 , . . . , ϕ N ) is the N−component bosonic field and (α 1 , . . . , α N ) is the set of linearly independent vectors. The theory (1.1), properly coupled to a background metric, defines a conformal field theory. However, it is well known, that under some conditions on the set (α 1 , . . . , α N ) it also enjoys enlarged conformal symmetry usually referred as W −algebra [1]. There is a class of such distinguishable sets (α 1 , . . . , α N ) with semi-classical behavior α r = be r for all r = 1, . . . , N, where e r are finite in the limit b → 0. The vectors e r have to be simple roots of a semi-simple Lie algebra g of rank N. An interesting question arises if one perturbs the theory (1.1) by an additional exponential field S 0 → S 0 + λ e α N+1 ·ϕ d 2 x. (1.2) Typically this perturbation breaks down all the W −algebra symmetry down to Poincaré symmetry. However, there is a special class of perturbations, called the integrable ones, which survive an infinite symmetry of the original theory in a very non-trivial way [2]. Namely, one can argue that there are infinitely many mutually commuting local Integrals of Motion I λ s andĪ λ s which are perturbative in λ where (I s ,Ī s ) are defined in CFT. Thus any integrable perturbation (1.2) inherits a distinguishable set of local IM's I s in conformal field theory. The seminal program devoted to calculation of simultaneous spectra of I s has been initiated by Bazhanov, for sl(2)/KdV case. The culmination was the discovery [6] of Gaudin-like equations for the spectrum. In current notes we use an alternative approach, based on affine Yangian symmetry. We consider the case of sl(n) symmetry. Actually, it will be convenient for us to extend the theory by adding an auxiliary non-interacting bosonic field, leading to the action where the last term, corresponding to the affine root of sl(n), is known to lead to an integrable perturbation. With the last term dropped, the theory (1.3) defines the conformal field theory, whose symmetry algebra can be described by quantum Miura-Gelfand-Dikii transformation [7,8] Q∂ By symmetry arguments, it is clear that local Integrals of Motion I s should belong to the intersection of these two W −algebras. In particular, one can check that (for n large enough) n i<j (h i · ∂ϕ)(h j · ∂ϕ) dx,  where indeed satisfy this requirement. We note that in (1.6) we excluded trivial IM's build out of U(1) field J = 1 n k ∂ϕ k . (1.7) In general, one expects an existence of local Integrals of Motion for all s = 0(mod n). This point of view that IM's should belong to intersection of two W -algebras given by (1.4) and (1.5) automatically implies that the intertwining operator T 1 will be itself an Integral of Motion. The operator T 1 will be primarily important for us. We call it Knizhnik-Zamolodchikov operator (see section 4). Actually it is natural to define more operators, which will map between different W −algebras corresponding to different permutations of factors in (1.4). The Maulik-Okounkov R−matrix [9] corresponds to elementary transposition while the operator T 1 introduced in (1.8) corresponds to the long cycle permutation The operator R i,j acts in the tensor product of two Fock representations of Heisenberg algebra with the highest weight parameters u i and u j and its matrix depends on the difference u i − u j . Then it follows immediately from the definition (1.9) that R i,j (u i − u j ) satisfies the Yang-Baxter equation and hence the whole machinery of quantum inverse scattering method can be applied. In particular, one can construct a family of commuting transfer-matrices on n−sites T(u) = Tr ′ R 0,1 (u − u 1 )R 0,2 (u − u 2 ) . . . R 0,n−1 (u − u n−1 )R 0,n (u − u n ) Fu . (1.10) At u = u 1 one has R 0,1 = P 0,1 a permutation operator and hence T(u 1 ) = R 1,2 R 1,3 . . . R 1,n−1 R 1,n = T 1 , (1.11) which implies that T(u) commutes with local Integrals of Motion I s and can be taken as a generating function.
In (1.10) the notation Tr ′ corresponds to certain regularization of the trace, which goes through the introduction of the twist parameter q k is the level operator in auxiliary space F u . Remarkably, the introduction of the twist parameter does not spoil the integrability, that is the twist deformed transfer-matrices T q (u) = Tr q L (0) 0 R 0,1 (u − u 1 )R 0,2 (u − u 2 ) . . . R 0,n−1 (u − u n−1 )R 0,n (u − u n ) Fu , (1.12) still commute. On the level of local Integrals of Motion (1.6) this deformation corresponds to the nonlocal deformation I s → I s (q) called quantum ILW n (Intermediate Long Wave) integrable system [10]. In particular I 1 (q) = 1 2π 1 2 n k=1 (∂ϕ k ) 2 dx, (1. 13) where D is the non-locality operator whose Fourier image is We note that the limit q → 1 is a little subtle since the operator D is singular at q → 1 and hence some eigenvalues of I s (q) become infinite. However, one can show that on a subspace spanned by eigenfunctions with finite eigenvalues the modes of the J(z) field (1.7) are not exited. It implies in particular that The spectrum of ILW n integrable system is governed by finite type Bethe ansatz equations which have been conjectured by Nekrasov and Okounkov 1 and independently by one of the authors in [10] q j =i (1.14) such that the eigenvalues of I s (q) are symmetric polynomials in Bethe roots Equations (1.14)-(1.15) have been checked in [10] by explicit calculations on lower levels. A proof has been suggested in [11] in the trigonometric (q-deformed) case. We note that Bethe ansatz equations (1.15) are simplified drastically for q ±1 → 0, which is equivalent to D(k) → ±|k|. The limit of ILW n system at q ±1 → 0 is known as BO n integrable system (Benjamin-Ono). The basis of its eigenfunctions stands behind AGT correspondence [12]. Namely, it has been shown in [13,14] that the matrix elements of semi-degenerate W n -primary fields, dressed by suitably chosen U(1) vertex operators, sandwiched between the BO n eigenfunctions coincide with bi-fundamental contribution to the Nekrasov partition function [15] for corresponding quiver gauge theory.
The Maulik-Okounkov R−matrix defines in a standard way the Yang-Baxter algebra (RLL algebra). We note that R i,j intertwines two representations of Heisenberg algebra (1.9) ( gl(1) current algebra). Since the matrix elements of R i,j are rational functions of the highest weight/spectral parameter (see below), it is natural to call the corresponding Yang-Baxter algebra the Yangian of gl(1), or affine Yangian of gl(1). The algebra under the same name has been introduced by Tsymbaliuk in [16]. It has been given by explicit commutation relations (the so called current realization). We will show that both algebras do not literally coincide, but rather Tsymbaliuk's algebra Y gl(1) is obtained from the Yang-Baxter algebra YB gl(1) by factorization over infinite-dimensional center.
As we already mentioned, the Yangian Y gl(1) is the rational counterpart of the trigonometric algebra called Ding-Iohara-Miki algebra or quantum toroidal gl(1) algebra [16]. This algebra has been extensively studied by Feigin and collaborators in [11,[17][18][19]. Another but equivalent approach through the methods of geometric representation theory was developed by Okounkov and collaborators [20,21]. We borrow many ideas developed in [11,[17][18][19] and [20,21] for our study. In particular, Bethe anzatz equations as well as Bethe vectors can be found in [11] in the q−deformed case and in [21]. This paper is organized as follows. In section 2 we define the main actor of our study -Liouville reflection operator/Maulik-Okounkov R−matrix and discuss its general properties and various representations. In section 3 we study corresponding RLL algebra and discuss its relation to affine Yangian of gl(1). In section 4 we introduce quantum Integrals of Motion corresponding to ILW system and prove Bethe ansatz equations for the spectrum. In section 5 we give some conclusions an emphasize future possible directions of study. In appendices we present some explicit formulae and calculations used in the main text.

Maulik-Okounkov R-matrix as Liouville reflection operator
It is clear from the definition (1.9) that R i,j trivially commutes with the center of mass field ϕ i + ϕ j , that is where R is the Liouville reflection operator for the U(1) current algebra which is defined as We note that here the U(1) current has different normalization compared to one used before. In particular the commutation relations for modes of J(x) are differ by the factor of 2 [a m , a n ] = m 2 δ m,−n .
The relation (2.3) can be used for calculation of R. Consider highest weight representation of the U(1) algebra (2.2). It is generated by the negative mode operators a −k from the vacuum state |u : a 0 |u = u|u , a n |u = 0 for n > 0.
a k a n−k + (2a 0 ± inQ)a n , L and R vac (u) is an eigenvalue for the vacuum state. In the following we will usually take R vac (u) = 1. (2.5) Using (2.4) as a set of equations one can compute the matrix of R. For example at the level 1 one has Similarly, at the level 2 one obtains , .
Apart from explicit expressions on lower levels the reflection operator is not known in a closed form. However it shares several properties allowing to judge about its structure: Poles. It is clear that apart from the normalization factor the operator R is a meromorphic functions of the momentum u. In fact, it can be argued that it has only simple poles located at the Kac points u = u m,n = i mb 2 + n 2b , m, n > 0, (2.6) i.e. R(u) can be written in the form Indeed it is well known that for the values (2.6) the map from the Fock module F u to Verma module V ∆ given by the L + generators has a kernel. More precisely all the states of the form where |χ m,n def = (L + −1 ) mn + . . . |u is a special state called co-singular vector, vanish at u = u m,n . Explicitly, one has At the same time the reflected states do not vanish for u = u m,n which implies that R should exhibit a singularity at (2.8), namely a simple pole, which implies (2.7). We note that the formula (2.7) is a reminiscent of the Alyosha Zamolodchikov's recurrence formula for conformal block [22]. In particular, one can use (2.7) as a tool for calculation of the matrix of R(u).
Relation to Liouville S−matrix The Liouville reflection operator R is closely related to the Liouville S−matrix introduced in Zamolodchikov's paper [23]. Namely, they differ by the sign change operator πJ(x) = −J(x)π as R(u) = πŜ(u).
According to (2.4) the S-matrixŜ(u) acts between different Fock modules F uŜ Expression through Screening operators. Given the stress energy tensor T + = −J 2 + Q∂J with J = ∂ϕ, one finds that the exponential fields e 2b ±1 ϕ(z) satisfy Then suppose that u = −u m,n for m, n ≥ 0. In this case one can define a closed contour C such that the Large momenta expansion. We note that R 1,2 coincides with the KZ operator (1.11) for n = 2 and hence R commutes with the system of local Integrals of Motion of quantum KdV (mKdV) system It can be shown that R is an exponent of semi-local (non-polynomial) Integral of Motion 2 The formula (2.13) is rather symbolic and requires a regularization prescription to make sense. It can be defined as a large u expansion. Namely, if one splits J into constant and zero-mean parts J = u +J, then the expansion coefficients are zeta-valued regularized (similar regularization is used in the definition of local IM's (2.12)). So that (2.13) leads to large u expansion (2.14) We note that in (2.13) and (2.14) the normalization is different from the one used before, i.e. R vac (u) = 1.
Free-fermion point. One can show that R admits simple representation at the free-fermion point c = −2. Namely, if one uses boson-fermion correspondence to represent where (ψ(x), ψ + (x)) is the chiral part of Dirac fermion, then up to normalization factor one has an explicit formula (see appendix E) For c = −2 formula is more complicated and (2.15) will include multiple fermion terms.
Smirnov's fermion formula. There is also Smirnov's formula for Maulik-Okounkov R−matrix involving an infinite product of fermionic operators [25]. Unfortunately, we do not known any practical use of it for our purposes. Given the Liouville reflection operator R the Maulik-Okounkov matrix R i,j is given by substitution (2.1). We also found it convenient to take highest weight parameters to be purely imaginary. More precisely in (2.1) we replace

Yang-Baxter algebra
The Maulik-Okounkov R-matrix defines the Yang-Baxter algebra in a standard way Here L i (u) is treated as an operator in some quantum space, a tensor product of n Fock spaces in our case, and as a matrix in auxiliary Fock space F u . The algebra (3.1) becomes an infinite set of quadratic relations between the matrix elements labeled by two partitions Since (3.1) is defined by the rational R-matrix which intertwines two representations of the Heisenberg algebra, which is the same as gl (1), it looks natural to associate it to the Yangian algebra of gl(1) introduced by Tsymbaliuk in [16]. In fact the algebras do not literally coincide. We will show that (3.1) is related to the Yangian of gl(1) by factorization over its center. This is similar to the well known fact that the Yangians of gl(n) and of sl(n) are differ by central elements [26]. We note that, compared to the non-affine case, the center of (3.1) is infinite dimensional. We will denote the Yang-Baxter algebra as YB gl(1) , reserving the notation Y gl(1) for Tsymbaliuk's algebra. In discussions below we will mainly follow the analysis of the relation between the RLL and current realizations performed in [27] for quantum groups. We introduce three basic currents of degree 0, 1 and −1 (see appendix B for more details) as well as an auxiliary current (as we will see (3.10a) it also belongs to the Cartan sub-algebra of YB gl(1) ) As follows from definition of the R-matrix these currents admit large u expansion As we will see below, it proves convenient to introduce higher currents labeled by 3D partitions. In particular, on level 2 one has three e λ (u) currents (3.5) and similarly (3.6) As we will see below these currents are algebraically depending on the basic ones (3.2).
It will be more convenient to use Nekrasov epsilon notations rather than Liouville notations. Formally, they are obtained by replacing central charge parameter together with the normalization of bosonic fields: Altogether, this leads to the following Miura transformation We also have to scale our basic current e(u) and f (u) as 3.1 Current realisation of the Yang-Baxter algebra YB gl(1) Using the definition (3.2) and (3.3) and explicit expression for the R-matrix on first three levels one finds (see appendix B for details) as well as ee, f f relations and Serre relations In the relations above we have used the following notations The higher currents e λ and f λ in (3.10e)-(3.10g) are related to (3.5) and (3.6) by change of notations (3.7)-(3.8) and by certain scaling factors.
We note that the terms shown by blue in (3.10b)-(3.10f) depend only on one parameter either u or v. We call such terms local. If one writes the commutation relations for the modes of the currents (3.4), local terms affect only few of them. Indeed if we apply does not contribute and we obtain The local term appears if we apply (3.11) with j = −1 Similarly, applying where σ k are elementary symmetric polynomials in ǫ j . However, taking either i or j negative allow to express the higher currents e λ in terms of commutators and similar expressions for f λ (u). Using the relations (3.13) one can express generators of the Yangian L ∅,λ (u) with |λ| = 2 as and similarly for L λ,∅ (u) with e k being replaced by f k . These equations as well as (3.12) suggest that generic generator L λ,µ (u) can be obtained as an adjoint action of e k and f k generators on h(u). Using the RLL relations (3.1) at level 3 one can find In general, we have found nice representation for the generating function In this section we will show that the algebra Y B gl(1) contains a huge center. Namely for any singular vector |s of W n algebra in the space of n bosons we assign a central element D s (3.21). First representative of this series is related to the operator ψ(u) as .
In representation of Y B gl(1) in the space of n bosons F u 1 ⊗ . . . F un the element D 1,1 (u) acts by the function: which we call the weight of representation.
In order to see it we note that the algebra (3.10) contains additional Hamiltonian ψ(u) which commutes with h(v). One can derive that Using the relation which immediately follows from (3.10b) at u = v + ǫ 3 , we may transform the operator ψ(u) to the more convenient form

Using another identity
where −1 |∅ u ⊗ |∅ u+ǫ 3 is a singular vector of a W -algebra which appears in the tensor product of two Fock spaces F u 1 ⊗ F u 2 at u 2 = u 1 + ǫ 3 . Indeed it can be checked that under the resonance condition u 2 = u 1 + ǫ 3 the vector |s 1,1 is annihilated by positive modes of the W −currents defined by Miura formula: Due to the property that singular vector is annihilated by all positive modes of W −currents it follows that the R-matrix acts trivially on the tensor product of the vacuum and the singular vector. In our particular case we have Relation (3.15) implies the commutativity of h(v) and ψ(u) and ensures that the Hamiltonian ψ(u) acts on the vacuum |∅ v by the highest weight We also found by explicit calculation that commutes with e(v) and f (v) and so belongs to the center of RLL algebra 3 . In order to understand this phenomenon, let us note that R-matrix between two vector spaces which are representations of W ∞ algebra is completely (up to a normalization constant) fixed by the eigenvalues of zero modes W 0 of W -currents on vacuum and intertwining identity We will consider two representations of W -algebra, one in the space of one boson, and other in the space of finite number of bosons n. We take two different representations of W n algebra -one is the standard Fock representation and the other is the highest weight representation arising from the singular vector |s u . Let us compute exchanging relation of higher Hamiltonian h s (u) = u s|L|s u and the current e(v). On general grounds, it has the form where F s (u − v) is some rational function. Let us concentrate on the first term of (3.17), because local terms are fixed by a demand that l.h.s of (3.17) doesn't have poles 4 . According to the RLL relation, the is the weight of representation arising from the singular vector |s In order to calculate the matrix element (3.18) let us act by the minus first mode of intertwining identity (3.16), specified to the case n 1 =1, The desired matrix element can be found by solving a linear system and excluding all W (k) −1 modes in the l.h.s of (3.19). However one can avoid this complicated calculation simply by substitution iǫ 3 Thus we find that the exchanging function in (3.17) depends only on the polynomial P s (u): For example, explicit calculation for singular vector on the first level |s 1,1 = (a More generally for a singular vector in W 2 algebra s m,n at level mn Let us note that the same polynomial corresponds to a vacuum vector in two Fock spaces and so belongs to the center of Y B( gl(1)). Indeed: However, as we have seen, all local terms came up with poles which should be canceled with residues of non local term. Since non-local terms do not have poles no local terms allowed. Thus, we proved Exchanging relation with f (u) is similar, and hence we prove that D m,n (u) is indeed belongs to the center of Y B( gl (1)).
In general, any singular vector of W n algebra in the space of n Fock modules gives rise to central element of Y B( gl(1)). As we explained exchanging relations of higher Hamiltonian h s with e(v), f (v) currents are encoded in a single polynomial (3.20). And the element: is central.

Zero twist integrable system
The Yang-Baxter algebra YB( gl (1)) contains commutative subalgebra spanned by modes of the current h(u). If one consider a representation of the YB(ĝl(1)) algebra on n sites, this integrable system is known to coincide with matrix generalization of quantum Benjamin-Ono integrable hierarchy. It attracted some attention because it is directly related to AGT representation for conformal blocks [12]. Much is known about this integrable system. In particular, its spectra and eigenfunctions can be written rather explicitly. Suppose, one has an eigenvector of h(u) Then one can try to create new states by repetitive application of e(v). Using (3.10b), one finds that and hence in general e(v)|Λ is not an eigenvector of h(u). However if e(v)|Λ develops a singularity at some value v = x, typically a pole, then the second term in the r.h.s. of (3.22) is negligible and we have a new eigenvector Similar argument applies to the operator ψ(u) and to any higher Hamiltonian h s (u) from the previous section. Using (3.23)-(3.24), one can generate any eigenvector from the vacuum state by successive application of e(u). We note that the operators e(u) do not commute. However the structure of commutation relations (3.10e) implies the following property provided that x and y are simple poles and that y = x + ǫ α . The properties (3.23)-(3.24) and (3.25) are used to show that the eigenstates are in correspondence with tuples of Young diagrams or more generally with 3D partitions. In order to demonstrate how it works, we take our quantum space to be the tensor product of n Fock modules generated from the vacuum state Our normalization of h(u), which is inherited from our normalization of the R−matrix (2.5), implies that h(u)|∅ = |∅ . Then it follows from the definition of ψ(u) (3.3) that Moreover the vacuum state is annihilated by f (u) while the new states are generated by the modes of e(u). In principle, one can rewrite a generic state in F x 1 ⊗ · · · ⊗ F xn as an integral for some function ρ λ (u) (see [28,29] for explicit formulae on lowest levels). The eigenfunctions of h(u) provide another basis | λ in F x 1 ⊗ · · · ⊗ F xn which has very simple form in terms of e(u) generators We will specify the proportionality coefficient in (3.27) later. In fact it depends on the order in which we perform the integrations. The contours in (3.27) go counterclockwise around simple poles located at the contents of Young diagrams in λ. By definition a content of a cell with coordinates The order of the contours C i in (3.27) should follow the order of any standard Young tableaux associated to λ (k) . Different choices of the ordering would lead to the same state which might differ by a factor, later we will provide a formula for eigenvector | λ which is independent of the ordering (see (3.33)). The state defined by (3.27) is an eigenstate of h(u) and ψ(u) with eigenvalues We note that (3.28) follows immediately from (3.10b), (3.14) and (3.26) provided that the surrounded singularities of the integrand in (3.27) are all simple poles. This statement can be proven by induction in level N: • The base of induction. Let us consider generic states at level one: e(u)|∅ . In order to find it's poles we use (3.10d) which implies that poles of e(u)|∅ are located exactly at u = x k and hence are the eigenstates of h(u).
• Let us assume that up to level N the operators e(u) and f (u) act as follows where the amplitudes E( λ, λ + ) and F ( λ, λ − ) are given by In (3.29) the sets addable( λ) and removable( λ) corresponds to the sets of all boxes which can be either added or removed from λ.
• We have to show that e(u)| λ with | λ| = N has poles at addable points. Consider u poles of the following vector There are two sources of poles in the r.h.s of (3.32): the eigenvalue of ψ(u) and the e(u)f (v)| λ term. It is easy to show that both terms have poles only at addable and removable points. Formula (3.30) provides exact cancellation of poles at removable points, which implies the statement.
Finally, we provide the normalized formula (3.27) for the eigenvector | λ which agrees with formulas (3.29)

ILW Integrals of Motion and Bethe ansatz
Consider the monodromy matrix on n sites T q (u) defined by (1.12). One can easily see that T q (u) admits the following large u expansion where Λ(u, q) is a normalization factor and I 1 and I 2 are the first ILW n Integrals of Motion (1.13). As explained in Introduction among other Integrals of Motion there is a particular one called KZ integral 5 Using the fact that R 0,1 (0) = P 0,1 a permutation operator, one finds As announced in Introduction the simultaneous spectrum of T q (u) is governed by Bethe ansatz equations (1.15). In this section we will prove these equations.

Off-shell Bethe vector
The basic ingredient of algebraic Bethe ansatz is the construction of the so-called off-shell Bethe vector. For the case of Y ( gl(1)) algebra they have been introduced in [21] by the methods of geometric representation theory, here we use an equivalent but more direct approach. We take the tensor product of n + N Fock spaces, with n "quantum" and N "auxiliary" spaces Consider the special state in the auxiliary space where the contour C k encircles the point x k in counter-clockwise direction. The vector |χ x is an eigenvector of zero twist integrable system We note also convenient formula which is an immediate consequence of (3.29) and (3.30). Now we define the off-shell Bethe vector as [21] |B The off-shell Bethe vector |B(x) can be represented by the following picture PSfrag replacements Consider the matrix element between |B(x) u and generic state which is non-zero only if Following [30], we call ω λ (x|u) the weight function. It can be simplified by noting that the matrix element of Lax operator L λ,∅ (u) can be expressed through h(u) and f (z) via contour integral 6 where each contour C k goes clockwise around ∞ and u − ǫ 3 . Using (4.7) the weight function (4.6) can be rewritten as Then the matrix element in (4.8) can be explicitly computed using (3.10c) and (4.4). One obtains where (z 1 , . . . , z N ) = (z 1 , . . . z λ (n) ) and the function has been obtained from F λ ( z|u) as a result of application of (3.10c). We note that as explained in appendix D the local terms do not appear in (4.10) if one extends the integration contour to include all new singularities (4.11). It implies that the integral shrinks to the points x and one obtains Let us note finally that this last equation implies the well known co-product property of weight function: Where the sum is over partitions of set of indices I = 1, 2, ...N into two sets (I 1 , I 2 ) of lengths (N 1 , N 2 ), we also denote all x i variables from set I a x (a) .

Diagonalization of KZ Integral
The action of the KZ Integral of Motion on off-shell Bethe vector |B(x) u is very simple and can be explained by the following picture Projecting this equation on arbitrary state, one obtains u ∅|a If we require that |B(x) u is an eigenstate for T 1 we have to demand which should hold for any set of partitions λ. The eigenvalue T 1 (u) can be found from (4.14) by taking For generic λ the eigenstate equation (4.14) implies the integral identity 1 ) . . .
which holds provided that x obeys Bethe ansatz equations go around simple poles located at the points x 1 , . . . , x N (see (4.10)), the local terms in (3.10c) and (3.10f) do not contribute and we will have where z denotes the set of all z One can easily show that under Bethe ansatz equations (4.15) each additional factor D(z (1) k |z) in (4.16) equals to 1, which implies the statement.

reduces (4.19) to (4.20) after relabeling of integration variables. The statement (4.18) follows.
Of course these considerations are correct modulo choice of integration contour. Integrals of the form (4.17) have been discussed in details in the literature [31,32]. Following these approaches, we treat the integral (4.17) as a sum over residues, the poles contributing to the integral are in one to one correspondence with a collection of n 3D partitions, with fixed floor shape λ: Here we treat the 3D partition as 2D Young diagram filled with integer numbers n (k) i,j such that:

Diagonalization of I 2 Integral
The diagonalization problem of KZ integral given above does not work for n = 1. Specially for this case and also for academic purposes we consider diagonalization problem for I 2 (q) IM (1.13). We have to remember that we have changed normalization in (3.7)-(3.8). It is also convenient to subtract the vacuum eigenvalue and ǫ 3 2 I 1 (q) from I 2 (q). Altogether, one has where the last two terms are added for a convenience, in order to have more simple formula for the eigenvalues (4.27).
Since our total Fock space splits into quantum and auxiliary parts (4.2), it will be convenient to add an upper index toĨ 2 (q), either x, u or (x, u) referring to auxiliary, quantum or total spaces respectively. The key observation is that the Integral of MotionĨ (x,u) 2 (q) is almost the sum of terms acting separately on spaces F u and F x plus a cross term We will show that on-shell Bethe vector, is an eigenvector ofĨ u 2 (q) where x k obeys Bethe ansatz equations (4.15). We start with off-shell Bethe vector (4.5) and insert Integral of Motion for a system with zero twist q = 0 acting on x space We have the following chain of arguments 1. Since |χ x is an eigenvector of zero twist integrable system (4.3), it is also an eigenvector forĨ (0) which acts on the whole (x, u) spaceĨ because the last two terms vanish on |∅ u . It implies 3. From the definition of R(x, u) (see (4.5)) we have and hence x ∅|R(x, u)Ĩ x,u 4. One has a remarkable property which holds provided that x satisfy (4.15). 5. One can replace J x k → J x k + J u k in the r.h.s. of (4.25) and use the property to obtain on Bethe ansatz equations (4.15).
In the above reasoning (4.25) requires explanation. In Appendix D we have shown that Using this formula, one finds explicitly where C i n are the binomial coefficients. Consider a matrix element of the l.h.s. of (4.25) with generic state u ∅|a It can be rewritten as 1 ) . . .
which is equivalent to (4.25). In the first line in (4.29) we have used definition of L λ,∅ (u), in the second line (4.28), (4.7) and definition (4.9). While in the third line we have used argument similar to the one in (4.16) that is dragging all f (ξ j )'s to the right, abandoning local terms in commutation relations (3.10c) and (3.10f) and using the fact that all the factors (here z denotes the set of all ξ j and z are equal to q on Bethe ansatz equations (4.15).

Okounkov-Pandharipande equation
We saw that there are two related problems: diagonalization of KZ integral and solution of KZ difference equation. Both problems can be solved in terms of Bethe vector. Similarly to KZ case, both counterparts exists for local Integrals of Motion. Let us consider the following equation [33]: We will show now that this equation is solved by the same function (4.17). In order to do that let us notice that while acting (4.17) differential operator q d dq is equal to multiplication on x k . The later can be expressed as an action of Integral of Motion in auxiliary space: Then we may repeat all the steps from previous section and found that under the integral over Bethe roots: action of I x 2 (0) on auxiliary space is equal to the action of I v 2 (q) on the quantum space. The only problematic point is number 4, let us explain it more details.
Let us consider the matrix element of the wave function: On the other hand, if we insert J x k from the left we will have: As we explain in Appendix D (see (D.14)), the function g k (x i ) is transnational invariant under the matrix element. And under the integral (4.17) we can freely perform simultaneous shift of all x i for i > N − k.
After this shift factor D(x i |y j ) will be canceled, and we arrive to the desired identity u ∅|a where the " ∼ " means equivalence under the integral (4.17). Other steps are completely similar to the ones in section 4.4. And we recover the equation (4.30).

Difference equations and norms of Bethe eigenvectors
In this section we will treat to be purely imaginary. As we already discussed in section As a consequence of KZ and OP equations this scalar product obey.
and can be computed at the point q = 0: We note that the constant C(N, ) is independent on the spectral parameters u i and the twist q. Let us compute the integral (4.17) for general q , but in the limit → 0. In this limit (4.17) can be taken by the saddle point method. Using we find that the integration kernel in (4.17) turns to the exponent of the Yang-Yang function: Computing the integral (4.33) by saddle point, we found: where H(Y) is a Hessian: Comparing to the (4.32), we immediately recover the Slavnov's determinant formula for the norms of on-shell Bethe vectors [34] The formula (4.37) can be rewritten in a different way:

Concluding remarks
This paper represents our efforts to understand the affine Yangian of gl(1) and its role in integrability of conformal field theory. Many aspects have not been touched. Below we present some open problems and preliminary results that will be left for future work.
Other representations of the Yangian. As we have seen, the commutation relations of YB( gl(1)) (3.10) are symmetric with respect to permutations of ǫ k . It implies that the algebra YB( gl(1)) admits three types of Fock modules F (k) u with k = 1, 2, 3. Taking a representation of generic type F (k 1 ) u 1 ⊗ F (k 2 ) u 2 ⊗ · · · ⊗ F (kn) un , will lead to ILW type integrable system corresponding to more general W algebras introduced in [35,36]. The corresponding Miura transformation is explicitly known [37,38]. All the results obtained in our paper can be generalized with a mild modification to this case. We collect some details in appendix E.
Massive deformation of ILW n integrable system. The twist deformation of CFT integrable system (1.13) leads to certain τ −deformation of Toda action (1.3). Namely, for our choice of twist deformation (1.12), one exponent in (1.3) gets replaced by its non-local counterpart The corresponding classical field theory called non-local gl(n) Toda field theory is known to be integrable in a Lax sense [39,40]. Its quantization has not been studied in the literature so far. The simplest model of this kind is a free boson perturbed by a single exponent This model has an interesting feature, in a finite volume of circumference L = πnτ : (x ∼ x + L), relabeling the fields: φ(x + πkτ ) def = φ k (x) we found that non local theory (5.1) in a volume L = πnτ is mapped to a local affine A n Toda in volume τ . So it will be interesting to study the S matrix and the spectrum of theory (5.1) in finite volume.
Relation to Sklyanin's results. Bethe anzatz equations similar to the ones studied in this paper were recently obtained by Sklyanin in a slightly different context of quantization of the first Hamiltonian structure of KP equation [41]. We believe that our results may be relevant in this context.
Integrable systems corresponding to different root systems. It is interesting to discover analytic continuation of general W(g) algebra, associated to lie algebra g, and their relation to Yangian structures. Some results in the q−deformed case has been obtained in a recent paper [42]. Namely, there were proposed a new algebra K which is an analytic continuation of W(B n , D n , D n ). As W 1+∞ algebra is related to Affine Yangian of gl(1), it turns out that K algebra is related to Sklyanin boundary algebra [43] associated to Affine Yangian. However authors were unable to find Bethe eigenvectors and Bethe equations in this case, which is an interesting open question.
ODE/IM correspondence. The spectrum of untwisted integrable systems (i.e. at q = 1) can be studied by means of ODE/IM correspondence (see eg [44] for review). We do not known any transparent relation between these two approaches. In particular the transfer matrices are quite different and we were unable to relate them. We note also that algebraic equations for the spectrum are rather different in two approaches. In Yangian approach one has BA equations (4.15), while on ODE/IM side the spectrum is given by Gaudin-like equations (see for example [6,45,46]). It looks similar to the known duality between trigonometric Gaudin and rational XXX models [47], however it has not been clarified yet (see discussions in [19]).
Yangian Double. The algebra called Yangian Double has been introduced in [48] following Drinfeld's quantum double construction [49]. The Yangian Double seems to be more appropriate for construction of Bethe vectors by the so called "method of projections" developed in [50,51] (see [52,53] for latest results). We introduce Yangian Double of Y B( gl(1)) in Appendix C and discuss some its properties. Unfortunately, we were unable to repeat the procedure executed in [52,53] and define the off-shell Bethe vector as a projection of a state build of "total" currents. This is an interesting open problem.
where J(x) = k =0 a k e −ikx in the form [9] Solving (A.1) one can find first few densities g k (x) explicitly: 3) and more disgusting expression for g 8 For all densities g k (x) in (A.3) we used zeta-function regularization. For example where :: stands for the Wick ordering. Explicit formula (A.2) is useful for us, because it provides a relation between Yangian currents (3.2) and W n (z) currents. For example for the first few modes: Of course these formulas are only true in a bosonic representation, however it is easy to analytically continue them to the arbitrary number of bosons: Where L ±1 is a special W (2) current, such that: Let us also note that there is easily established pattern in the densities (A.3). Namely, the first terms in (A.3) can be written as Using this observation one can formally do the resumation in (A.2).
which admits the derivative expansion The expansion (A.5) suggests the following general form where U 2k+2 (J x , J xx , . . . ) is a homogeneous and even with respect to the transformation J → −J density of degree 2k + 2. It would be interesting to find the densities U 2k+2 (J x , J xx , . . . ) exactly. One can also compute the R(u) operator in the "free fermion" point c = −2. Namely, take Q = − i √ 2 in (A.2) and represent the current J(x) by the complex fermion ψ(x) as Then one can check that (see also appendix E)

Appendix B. Affine Yangian commutation relations
Here we consider commutation relations of the Yangian algebra (3.1) in components. Similar analysis has been performed in [38]. We use the following notations and admit the convention that the operators acts in "quantum" space from up to down. It is also convenient to define according to (3.2) We will introduce currents e λ (u) and f λ (u) associated to 3D partitions. There are 3 currents on level 2 (3.5), 6 currents on level 3 (see (B.12)) etc. Similar expressions one has for f λ (u). All these and other generators of the Yang-Baxter (3.1) admit large u expansion which is inherited from the large u expansion of the R−matrix (A.2). In particular, while the higher currents are expected to behave at u → ∞ as The relations of the Yang-Baxter algebra (3.1) appear from the tensor product of two Fock spaces F u and F v . We will use the following notations for the bra and ket highest weight states in The action of the zero-mode a 0 on the vacuum state |u is he and hf relations: where the contours C ∞ go concentrically around ∞ in such a way that all singularities of the function F (z) are kept inside of these contours. Similarly to (B.4) one also obtains the relation which enables one to rewrite ee and f f relations: On level 2 we have three independent equations In the r.h.s of any of these relation one has a linear combination of five terms One can always find special linear combination of three equations (B.5) which kills contributions of last two terms. Explicitly, one has where We note that using (B.4) one can express 7 and hence the relation (B.6) can be rewritten as where the higher currents e λ (u) are given by (3.5).
Other two relations from (B.5) are equivalent to commutation relations between h(u) end e λ (v) (similar to (B.4)) and similar for e (v) . 7 We also have more general relation

Serre relations
Formulas at level 3 becomes tough, however they are straightforward. For example for e(u)e (v) we will have: We will specify "locals" terms later (see (B.11)), here we just want to point out that l.h.s by definition doesn't have any poles and so does the r.h.s. This condition will imply some additional relations, most of them will be non local, and we are not gonna discuss them. However we note that the multiplier doesn't have pole at u = v + Q. Surprisingly, there is a "local" term with pole at this point. Setting the residue to zero, one finds a relation We will have similar relation for e(u)e (v). And also one trivial relation which follows from the fact that e(u)e(u) 2 = e(u) 2 e(u) = e(u) 3 . As a result we will have 6 independent currents at level 3, which is equal to the number of 3d young diagrams with 3 boxes. In practice we used this three relations in order to exclude composite currents (e (u)e(u), e (u)e(u), e (u)e(u)). This three relations may look mysterious, however, after explicit calculation we found that this three relations are equivalent to Serre relations, the later could be written down in terms of currents as [e(x σ 1 ), e (x σ 2 ) + e (x σ 2 ) + e (x σ 2 )] = 0 (B.10) Namely, using the quadratic relations, we may reorder any polynomial in e(u i ) in a way that it will contain only monomials with ordered arguments: e(u i 1 )e(u i 2 )...e(u in ) , i 1 < i 2 ... < i n , this could be done explicitly with the formulas (B.11). After doing this procedure with Serre relations we found that they proportional to linear combination of three currents, and so equal to zero in Yangian algebra. Finally, after imposing three relations (B.9) we will have explicitly: with the higher currents given by In principle we may go further, and calculate quadratic relations at next levels, however as we already have shown, the algebra is generated by the h(u), e(u) and f (u) currents, so in principle we don't need to use higher currents. The only problem is to prove that quadratic and Serre relations are the only ones which currents e(u) obeys.

Relations in ǫ notations
We see that there is an S 3 symmetry associated to permutation of the triple (b, b −1 , −Q). In fact it is more convenient to go to epsilon notations: It is also convenient to change a normalization of the highest weight/spectral parameters, together with the whole normalization of basic fields: Then the relation (B.7) takes apparently symmetric form In definition of matrix elements L λ,µ (u) we define the state | , as well as any state |λ to be normalized as λ|λ = 1 in any notation. Note that here we used a Maulik-Okounkov R matrix, which breaks the symmetry between ǫ 1 , ǫ 2 , ǫ 3 , so that we have a selected ǫ 3 . In fact, there exist additional R (1/2) f matrices with either ǫ 1 or ǫ 2 selected (see appendix E). and The price for this relations is that product of two currents e(u)e(v) has poles at points v = u + ǫ α and similar for f (u)f (v). Yangian Double may be useful for understanding the structure of the relations (B.11) between higher currents. Let us, for example, derive the relation between e(u) and the higher current e (v). First of all we have Now let us find an exchange relation between e(ξ) and e (v). Using (C.4), one finds Exchange relation with the first term of (C.5) is simple, because local terms doesn't contribute to the residue e(ξ)res u=v−b e + (u)e(v) = res u=v−b e + (u)e(v) e(ξ)ḡ Combining this together, we find the relation which reproduces non-local part of the relation between the half currents (B.11). We note that (B.11) contains more information. In particular they contain cubic Serre relation (3.10g).

Appendix D. Special vector |χ and shuffle functions
In the later we will need a more detailed description of a subalgebra n + generated by currents f (z). Easy to understand that the subspace of the form L µ,∅ 1 (v 1 ) . . . L µ,∅n (v n ) may be identified with the subspace :n + (v) = h(v 1 ) . . . h(v n )n + . A particular result of this section is an explicit realization of this mapping.
First of all let us note that both spaces are graded by the number of f (ξ i ) currents in the monomial, let us note each graded component of corresponding algebras by n + N , n + N (v). It is a natural idea to identify elements of n + and n + (v) by their matrix elements in some representation: In order to unambiguously characterize the elements of n + N , n + N (v) we need a big enough set of representations and vectors |χ . Our choice is the following: let us pick an N Fock spaces: F x 1 ⊗ . . . F x N , and consider simplest vector of grade N : Then, our mapping (D.1) maps an element of n + N , n + N (v) to a rational function of N variables f (x 1 , . . . , x N ) obeying the so called "wheel" condition [11]: For n + N and additional condition: For n + N (v). This functions is a rational limits of Sh 0 and Sh 1 functions from [11]. The multiplication in algebra, implies the multiplication of Shuffle functions For n + , And It is clear from the RLL relation that R(u) matrix commute with W (1) current: Taking the matrix element over the auxiliary space ∅| . . . |µ for positive n we will get: where µ + n| def = µ|a n . It is also clear, that J n for n > 0 belongs to the sub-algebra n + . Indeed, explicit calculation of the large u limit of R(u) matrix (A.2) shows that: Then we get: where C i n are the binomial coefficients. Note that the function g(ξ) defined ambiguously, indeed algebra Y ( gl(1)) enjoys Serre relations (B.10) Indeed such an element lies in the kernel of the Shuffle map (D.4) In particular, commutativity of J n may be thought as a consequence of Serre relation, for example choosing i = j = k = 0 We should consider functions g n (ξ) modulo equivalence: g (1) n (ξ) ∼ g (2) n (ξ) + Ker(S 0 ) It is easy to understand that modulo this equivalence function g n (ξ) is invariant under the simultaneous shift of all variables ξ → ξ + we will use this fact in section 4.5.
As we announced, operators L(u) µ,∅ belongs to the subspace n + (u) |µ| : L λ,∅ (u) = 1 (2πi) |λ| · · · F λ (z) h(u)f (z |λ| ) . . . f (z 1 )dz 1 . . . dz |λ| , g (D. 15) where contours go clockwise around ∞ and all poles of F λ (z). Let us prove this statement, and find recurrence relations for the rational function F λ (z). Now in order to recover formula (D.15) we have to use relation (D.7) together with the formula (D.9). In order to reproduce (D.15) we have to reorder h and f current, in order to move h to the left, this can be done with the simple fact h(u)f (ξ)ξ n dξ 2πi (D. 16) Here in the first equality we used equation (3.10c), while in the second we used a simple fact that l.h.s of (3.10c) doesn't have pole at u = v + ǫ 3 , and so r.h.s does (h(u)f (u + ǫ 3 ) = f (u)h(u)), thus we may deform integration contour. Equation (D.7) together with (D.16) implies integral formula (D.15) together with recurrence representation for F λ (z|u) Appendix E. Other representations of YB gl (1) In this paper we were concentrated on an examples of "spin chain" with n sites and periodic boundary conditions, this setup corresponds to an affine A n Toda field theory. At the each site of our "spin chain" we should place a representation of RLL algebra. The generating function of IM's is equal to T (u) = Tr F 0 q n a (0) −n a (0) n R 0,1 (u − u 1 ) . . . R 0,n (u − u n ) One possibility is to choose R 0,k (u − u k ) to be the Maulik-Okounkov R−matrix. However we have already seen that RLL algebra in current realization is symmetric under permutation of three parameters ǫ α , in terms of usual parameters b, Q, b −1 this means a symmetry between b and Q = b + 1 b where b = ǫ 1 ǫ 2 . In order to see two additional representations of RLL algebra let us realize representation of W algebra in the space of two bosons as commutant of Screening charge, according to [35,36] there are three choices of screening currents. Our notation is that there exist three different types of representation of Y B( gl (1): u and F (3) u . We assign the screening charge S k to a tensor product of two Fock spaces of the same type F (k) u ⊗ F (k) v , and we assign "fermionic" screening charge S i,f to the tensor product of different Fock spaces F (j) with (i, j, k) = cycl (1,2,3). Fixing one Fock space to be of type 3, we will have three options for the other one S f,1 = e bφ 0 (z)+iβφ 1 (z) dz, S f,2 = e b −1 φ 0 (z)+iβφ 1 (z) dz, S ± 3 = e b ±1 (φ 0 (z)−φ 1 (z)) dz, While the third screening charge S 3 leads to the MO R−matrix (∂φ 0 (x)−∂φ 1 (x)) 2 − 1 3u 2 (∂φ 0 (x)−∂φ 1 (x)) 3 +o( 1 u 2 )] dx 2π , the first and the second screenings have dimension 1 2 and the corresponding W algebra admits free fermion representation. For example for the first screening, let us introduce two fermionic currents ψ(x) = e −ibux e bφ 0 (x)+iβφ 1 (x) (E.1) where iu is the zero mode of φ 0 (x). It is easy to check that they obeys free fermionic OPE's ψ(x)ψ † (y) = 1 sin(x − y) + reg, ψ(x)ψ(y) = reg, ψ † (x)ψ † (y) = reg.
Correspondingly, W (2) (z) current which commutes with S 1 is simply Intertwining relation implies One can find that the R Indeed under the adjoint action of R matrix fermions transform as: Such that (E.3) holds. Although formula for R matrix looks pretty simple, it's structure is quite complicated because one should remember that ψ(z) operator is nontrivial in terms of individual bosons (E.1) and (E.2). In order to find local integrals of motion we have to expand R matrix in powers of 1 u . Let us introduce a shorthand notation Φ(x) = ibφ 0 (x) + iβφ 1 (x) It is easy to find that : ψ † (x)ψ(x) := ∂Φ(x) (E.5) . . .
Using (E.4), (E.5) and (E.6), it is easy to find first non trivial integral of motion in the space of one boson Where F q is our quantum space, and F aux is an auxiliary space. As usual the generating function of Integrals of Motion is T (u) = Tr aux q n ana −n R aux,q = tr aux q n ana −n n 1 j=1 R 2,f (u−v j ) Expanding at large spectral parameter, easy to find first non trivial integral of motion: Where B is a n 1 × n 2 × n 3 block matrix:  Alternatively switching to epsilon notation: And basic fields normalized as follows: Where σ 3 = ǫ 1 ǫ 2 ǫ 3 .
Bethe ansatz for the models considered in this section could be derived along the same lines, the only difference is in the action of ψ(u) generators on vacuum. For a Fock space representation F = k ⊗F (α k ) u k we have: So that we will have the same Bethe equations as in (4.15), but with different source function