R-matrix formulation of affine Yangian of $\hat{\mathfrak{gl}}(1|1)$

We study $\mathcal{N}=2$ superconformal field theory and define the R-matrix which acts as an intertwining operator between different realizations of $\mathcal{N}=2$ $W-$algebras of type $A$. Using this R-matrix we define $RLL$ algebra and relate it to current realization of affine Yangian of $\hat{\mathfrak{gl}}(1|1)$.


Introduction
It has been known for a while that integrability in CFT can be studied using the affine Yangian symmetry. The basic example -the affine Yangian of gl(1), or Y ( gl(1)) for shortness, has been introduced in R-matrix formulation by Maulik and Okounkov [1]. The R-matrix of [1] is closely related to the Liouville reflection operator [2], which intertwines two Miura representations for Virasoro algebra. The algebra under the same name has been introduced by Tsymbaliuk in [3] by explicit commutation relations (the so called current realization). In [4] it has been shown that both algebras differ by an infinite-dimensional center.
In [5,6] the relation between Y ( gl(1)) and bosonic W 1+∞ algebra has been established. In particular, some of low lying generators of W 1+∞ algebra were expressed explicitly in terms of the affine Yangian generators. Later in [7,8] Gaberdiel et all studied N = 2 superconformal version of W 1+∞ algebra and defined, what they called "the supersymmetric affine Yangian algebra". The analysis in [7,8] has been using the fact that N = 2 W 1+∞ algebra contains two commuting bosonic W 1+∞ subalgebras. The corresponding Yangian algebra were built in the so called gluing basis: having two explicitly commuting Y ( gl(1)) algebras and additional fermionic generators, which transform as bi-fundamental representations (see also [9,10] for later developments).
In this paper we relate N = 2 W −algebras and Y ( gl(1|1)). We follow another approach similar to [4]. Namely, we define the R-matrix, which acts in the tensor product of two typical representations of gl(1|1) current algebra, then define the RLL algebra and derive current relations of Y ( gl(1|1)) out of it. This paper is organized as follows. In section 2 we review N = 2 W n −algebras of type A and their free field representations. In section 3 we introduce gl(1|1) algebra and its representations through bcβγ-system. In section 4 we define the R−matrix for gl(1|1). In sections 5 and 6 we relate RLL algebra and current realization of Y ( gl(1|1)). Sections 4, 5 and 6 constitute main results of our paper. In section 7 we provide concluding remarks and directions for future studies.
3 gl(1|1) current algebra and bcβγ-system The commutation relations of gl(1|1) algebra [18] can be written in terms of OPE of two bosonic currents J E (z), J N (z) and two fermionic currents J ± (z): The parameter κ called the level can be set κ = 1 by the simple rescaling J E (z) → κJ E (z) and J + (z) → κJ + (z). It implies that gl(1|1) algebra admits free field representation [19] via the bcβγ-system. Namely, bcβγ-system is the system of complex fermions α(z), α * (z) and complex bosons a(z), a * (z) with OPE's (other OPE's are trivial) Then one can realize the currents J E (z), J N (z) and J ± (z) as bilinears (Wick normal ordering is assumed) For our purposes it will be convenient to further realize (α(z), α * (z), a(z), a * (z)) by the complex superfield X(z, θ) = X(z) + θψ(z), X * (z, θ) = X * (z) + θψ * (z), as the Wick ordered fields (3.4) Here b is an arbitrary parameter, which can be re-scaled out by the transformation X → bX, X * → b −1 X * , ho we will keep it as it is. It is important to note that the currents (3.4) can be equivalently defined by the following Wakimoto type screening operator 2 We note also that (3.4) admit the following simple superfield expression with κ = −2 − b −2 . In gl(1|1) case (ϕ 1 (z) − ϕ 2 (z)) is replaced by X * (z) and β(z) by ψ * (z).
In terms of the fields (X, X * , ψ, ψ * ) the representation of gl(1|1) takes the form In these notes we consider the so called typical (or generic) representations of gl(1|1). One distinguishes between NS and R modules. We consider NS module for simplicity. We define the mode expansion of the currents on the cylinder z = e −ix : x ∼ x + 2π (3.9) with commutation relations (other commutation relations are trivial) Then the Fock module F u , where u = (u, u * ), is generated from the vacuum state |u by the action of the creation operators where λ and λ * are partitions and s and s * are strict partitions (fermionic partitions) 14) The character of F u has the form

R-matrix
In order to define the R-matrix for gl(1|1), we take two copies of gl(1|1) algebra defined by two screening fields and define the additional screening field The commutant of this additional screening is actually larger and can be associated with the W (gl(2|2)) algebra introduced in section 2. We note that the screening operatorS 1,2 given by (4.2) differs from S 1,2 given by (2.13) by the replacement in the prefactor However two algebras defined by (S 1 , S 1,2 , S 2 ) and by (S 1 ,S 1,2 , S 2 ) are actually isomorphic and are related by marginal deformations. It can be seen as follows. Consider the Lax operator where X = Φ 1 + iΦ 2 and perform the formal scale transformation which does not change commutation relations. Then one has where (we use D 2 = −∂) and Having defined the Lax operator L (0) , we define the R−matrix as an intertwining operator We note that (4.10) automatically implies the Yang-Baxter equation The W gl(2|2) algebra defined by L 2 is obtained from the one defined by L 1 L 2 in the limit λ → 0. It consists of eight currents and where all the densities are Wick ordered. We note that (J 12 , G 12 , G * 12 , T 12 ) form N = 2 Virasoro algebra with the central charge c = 6. (4.15) At the same time the currents (Ψ * 12 , U 12 , J 12 , G 12 ) correspond to the diagonal current E 1 + E 2 . Working in a particular representation, the relations (4.10) can be used to define the matrix of R 12 . We consider the tensor product of two NS representations (3.12): Since the normalization of R 12 is not fixed from (4.10), it can be chosen at will. We fix it by R 12 |∅ = |∅ . Then using (4.10) one can find the action of R 12 on any state in F u ⊗F v . In particular, using two equations we find that Similarly, using we find that and Proceeding in this way, one can compute the matrix of R 12 at any given order.
In general, it is clear that the matrix elements of R 12 are some rational functions of three parameters Indeed, from the relation [R 12 , U 12 ] = 0 it follows that R 12 does not depend on u + v. Generic expression for R 12 is unlikely can be obtained in a closed form. However, some particular cases and expansions might be useful. In particular, following [4,21], one can develop the expansion of R 12 (u, v) at large u − v.

RLL algebra
In this section we study RLL algebra provided by the R−matrix defined in section 4 where as usual L 1 (u) and L 2 (v) are treated as matrices acting in gl(1|1) modules F u and F v correspondingly (see (3.12)) those entries are operators acting in some unspecified space (called quantum space). Then (5.1) is equivalent to the infinite set of quadratic relations between the matrix elements where |k and |l are arbitrary states from F u . However, for various reasons it is more convenient to use the so called current realization of the same algebra. The isomorphism between the R−matrix and current realizations is well understood in the case of classical algebras (see e.g. [22]). For affine algebras it has been studied in [4] for gl(1) and in [21] for gl(2) cases.
We will proceed exactly as in [4,21]. We note however, that there is a conceptual difference compared to gl(1) and gl(2) cases. Namely, the R-matrix constructed in section 4 is rather unusual since unlike most other known R-matrices it cannot be written as a function of the difference of two spectral parameters. The well known example of non-difference R-matrix has been found by Shastry [23] and depends on two spectral parameters individually rather than on their difference. Our R-matrix depends on three combinations of four spectral parameters. Nevertheless, we will show that the relations (5.1) imply that some of the currents are functions of only one spectral parameter, while the others are expressed in terms of them.
This phenomenon can be demonstrated at lower levels. Let us define the following currents and The terms shown in blue are called local terms, since they depend only on u, but not on v.
Consider for example the relation (5.5), set v − v * = u − u * and multiply it by h −1 (u) from the left. Then the first term in the r.h.s. of (5.5) vanishes and one obtains e(v + v * , u − u * ) = e(u + u * , u − u * ) =⇒ e = e(u − u * ). (5.9) Similarly one finds f = f (u + u * ), e * = e * (u + u * ), and f * = f * (u − u * ). (5.10) The relations similar to (5.9)-(5.10) persist at higher levels as well. Let us consider the basic currents at level 1 a(u) and their linear combinations Using commutation relations (A.1) and (A.2) presented in the appendix, one finds Also the currents at level −1 (5.15) satisfy similar relations with h(u), which imply In addition to (5.13) and (5.16) there are composite currents, which also depend on one parameter instead of two. Namely, two level 1 currents (5.17) and two level −1 currents The derivation of (5.17) and (5.18) is straightforward, yet tedious task and is based on commutation relation with h(u).

Current relations
We define the parameters and the currents It is also convenient to define the currents labeled by pyramid partitions (see (5.12), (5.15), (5.17) and One can derive the following commutation relations (see appendix A for details) The relations (6.7)-(6.12) should be understood in terms of modes of basic currents expanded at u = ∞ e k (u) = e We note that local terms in (6.7)-(6.12) depend only on one spectral parameter and do not contribute to commutation relations for modes with (i, j) > 1. The relations with either i ≤ 1 or j ≤ 1 can be used to express higher currents in terms of basic ones. In particular, one finds from (6.9) 1 , e 2 (u) + e 1 , e 2 (u) , 1 , e 2 (u) + e 1 , e 2 (u) , 2 , e 1 (u) , 2 , e 1 (u) . (6.14) Similar relations hold for higher currents (6.6) as well. We note that (6.7)-(6.12) coincide exactly with commutation relations for Y ( gl(1|1)) listed in [11]. It is important to note that RLL-algebra contains more currents than Y ( gl(1|1)). In particular, it has the current h(u), which is not in Y ( gl(1|1)). Moreover this current depends on two spectral parameters rather than one. The precise relation between two algebras, RLL and Y ( gl(1|1)), should be described by studying the center of the first along the lines of [4]. We postpone this study to future publication.
Also the algebra Y ( gl(1|1)) contains Serre relations. Their derivation from RLL algebra is straightforward task, which we expect to return in future publication (see [4] for similar derivation for Y ( gl(1))).

Conclusion
In this paper we derived current relations of Y ( gl(1|1)) (6.7)-(6.12) starting from RLL realization. First, we defined the R-matrix, which serves as an intertwining operator for certain N = 2 superconformal algebra. The remarkable property of this R−matrix is that it depends on 4 spectral parameters instead of 2. However, we have shown that the relations of RLL algebra imply that some of the currents depend only on one spectral parameter. In particular, we defined the currents e i (u), f i (u) and ψ i (u) which satisfy the relations (6.7)-(6.12).
There are several open questions, which we plan to address in future publications: • The relations (6.7)-(6.12) should be supplemented by Serre relations to define the correct algebra. The derivation of these relations starting from RLL algebra is rather tedious. It is expected that Serre relations should be of quartic order of total charge 0 (see e.g. [13]). This requires studying of the R−matrix acting in 39-dimensional space (as can be easily seen from the character formula (3.15)).
• Using current realization of Y ( gl(1|1)) it is straightforward to derive Bethe ansatz equations following the lines of [4,21]. These Bethe ansatz equations will correspond in particular to quantum version of N = 2 KdV hierarchy and its W n generalizations. We plan to provide more details elsewhere.
ee commutation relation. We introduce the following notations Then equation can be rewritten in the explicit form where we considered the action of R-matrix on |∅ . It is important to note the presence of the "minus" sign in this equation due to taking into account the fermions.
Consider also the equation which takes the form (compare to (4.21)) Then multiplying (A.5) by h −1 (u)h −1 (v) on the left and using (A.7), we get the anticommutation relation ee * commutation relation. We consider the equation In principle, one can add to this formula an arbitrary linear combination of RLL-relations at level 1 with charge 0: but we leave only those terms which, after multiplying by h −1 (u)h −1 (v) from the left, turn out to be local: The coefficients α, β, γ are found from the condition of cancellation of the terms in the right side of the RLL-relation (A.11). Their explicit expressions are .
(A. 13) Here and below, we use the following notation Then, taking into account the action of the R-matrix on the states of level 1 with charge 0, we get Multiplying this expression by h −1 (u)h −1 (v) on the left and using commutation relations h −1 (v)L ∅ψ (u) and h −1 (u)L ∅ψ * (v), one obtains , (A. 16) We see that, this commutation relation corresponds to (6.9). Then from the definition of the R-matrix one finds Let us also consider two auxiliary equations which can be rewritten explicitly as Multiplying (A.18) by h −1 (u)h −1 (v) from the right and using these relations, one finds from which one can obtain where (compare to (5.20)) So, this formula corresponds to (6.8). Similarly, one can get the commutation relation for e(u) and f * (v): The remaining relations from (6.7)-(6.12) can be derived in a similar way.