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

Dmitry Kolyaskin1,2, Alexey Litvinov2,3 and Arkady Zhukov1,2
1\. Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia
2\. Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia
3\. Center for Advanced Studies, Skolkovo Institute of Science and Technology,
143026 Moscow, Russia

###### Abstract

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
$\widehat{\mathfrak{gl}}(1|1)$.

##  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
$\widehat{\mathfrak{gl}}(1)$, or $Y(\widehat{\mathfrak{gl}}(1))$ 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 $Y(\widehat{\mathfrak{gl}}(1))$ and bosonic
$W_{1+\infty}$ algebra has been established. In particular, some of low lying
generators of $W_{1+\infty}$ algebra were expressed explicitly in terms of the
affine Yangian generators. Later in [7, 8] Gaberdiel et all studied
$\mathcal{N}=2$ superconformal version of $W_{1+\infty}$ algebra and defined,
what they called ”the supersymmetric affine Yangian algebra”. The analysis in
[7, 8] has been using the fact that $\mathcal{N}=2$ $W_{1+\infty}$ algebra
contains two commuting bosonic $W_{1+\infty}$ subalgebras. The corresponding
Yangian algebra were built in the so called gluing basis: having two
explicitly commuting $Y(\widehat{\mathfrak{gl}}(1))$ algebras and additional
fermionic generators, which transform as bi-fundamental representations (see
also [9, 10] for later developments).

In [11] (see also [12, 13]) the current realization of
$Y(\widehat{\mathfrak{gl}}(1|1))$ has been given111We note that [11] aimed to
construct various Yangian algebras from geometry and contains much more
examples. The algebra $Y(\widehat{\mathfrak{gl}}(1|1))$ corresponds to the
resolved conifold geometry and the so called pyramid partitions.. Moreover
later in [13] it has been explicitly stated that
$Y(\widehat{\mathfrak{gl}}(1|1))$ defined in [11] (together with Serre
relations, which appear later in [13]) should correspond to $\mathcal{N}=2$
$W_{1+\infty}$ algebra in the same way as $Y(\widehat{\mathfrak{gl}}(1))$
corresponds to bosonic $W_{1+\infty}$ algebra.

In this paper we relate $\mathcal{N}=2$ $W-$algebras and
$Y(\widehat{\mathfrak{gl}}(1|1))$. We follow another approach similar to [4].
Namely, we define the R-matrix, which acts in the tensor product of two
typical representations of $\mathfrak{gl}(1|1)$ current algebra, then define
the $RLL$ algebra and derive current relations of
$Y(\widehat{\mathfrak{gl}}(1|1))$ out of it. This paper is organized as
follows. In section 2 we review $\mathcal{N}=2$ $W_{n}-$algebras of type $A$
and their free field representations. In section 3 we introduce
$\widehat{\mathfrak{gl}}(1|1)$ algebra and its representations through
$bc\beta\gamma$-system. In section 4 we define the $R-$matrix for
$\widehat{\mathfrak{gl}}(1|1)$. In sections 5 and 6 we relate $RLL$ algebra
and current realization of $Y(\widehat{\mathfrak{gl}}(1|1))$. Sections 4, 5
and 6 constitute main results of our paper. In section 7 we provide concluding
remarks and directions for future studies.

##  2 $\mathcal{N}=2$ superconformal $W-$algebras

The $\mathcal{N}=2$ $W_{n+1}$-algebras, which play the key role in our study,
correspond to $SU(n+1)/U(n)$ Kazama-Suzuki models [14]. These algebras contain
one $\mathcal{N}=2$ multiplet of holomorphic currents for each integers spin
$s=1,\dots,n$. In particular, for $s=1$ one has the currents
$(J,G^{+},G^{-},T)$ of spins $(1,\frac{3}{2},\frac{3}{2},2)$, which form
$\mathcal{N}=2$ Virasoro algebra (whose commutation relations we present in
the form of OPE’s)

$\displaystyle
T(z)T(w)=\frac{c}{2(z-w)^{4}}+\frac{2T(w)}{(z-w)^{2}}+\frac{\partial
T(w)}{z-w}+\dots,$ (2.1) $\displaystyle
T(z)J(w)=\frac{J(w)}{(z-w)^{2}}+\frac{\partial J(w)}{z-w}+\dots,$
$\displaystyle T(z)G^{\pm}(w)=\frac{3G^{\pm}(w)}{2(z-w)^{2}}+\frac{\partial
G^{\pm}(w)}{z-w}+\dots,$ $\displaystyle J(z)J(w)=\frac{c}{3(z-w)^{2}}+\dots,$
$\displaystyle J(z)G^{\pm}(w)=\pm\frac{G^{\pm}(w)}{z-w}+\dots,$ $\displaystyle
G^{+}(z)G^{-}(w)=\frac{c}{3(z-w)^{3}}+\frac{J(w)}{(z-w)^{2}}+\frac{T(w)+\frac{1}{2}\partial
J(w)}{z-w}+\dots$

with other OPE’s regular. Commutation relations for higher spin currents
become much more involved and are, in principle, not sufficiently illuminating
and convenient to handle.

Instead, we will use free-field representation of $\mathcal{N}=2$
$W_{n+1}$-algebras, which is obtained by Hamiltonian reduction of
$\widehat{\mathfrak{sl}}(n+1|n)$ [15, 16, 17]. We work in $\mathcal{N}=1$
formalism and introduce $2n$ real holomorphic superfields
$\vec{\bm{\Phi}}(z,\theta)=(\bm{\Phi}_{1}(z,\theta),\dots,\bm{\Phi}_{2n}(z,\theta))$

$\begin{gathered}\bm{\Phi}_{k}(z,\theta)\overset{\text{def}}{=}\Phi_{k}(z)+\theta\chi_{k}(z),\\\
\Phi_{i}(z)\Phi_{j}(w)=-\frac{\delta_{ij}}{2}\log(z-w)+\dots,\quad\chi_{i}(z)\chi_{j}(w)=\frac{\delta_{ij}}{2}\frac{1}{z-w}+\dots{}\end{gathered}$
(2.2)

Let $\vec{\alpha}_{r}=(\alpha_{r,1},\dots,\alpha_{r,2n})$ with $r=1,\dots,2n$
be the simple fermionic roots of $\mathfrak{sl}(n+1|n)$ with the Gram matrix

$\big{(}\vec{\alpha}_{r}\cdot\vec{\alpha}_{s}\big{)}=\begin{pmatrix}0&-1&0&\cdots&\cdots\\\
-1&0&1&\cdots&\cdots\\\ 0&1&0&\cdots&\cdots\\\
\cdots&\cdots&\cdots&\cdots&\cdots\\\
\cdots&\cdots&\cdots&\cdots&\cdots\end{pmatrix},$ (2.3)

$\vec{\omega}_{r}$ being the dual basis
$(\vec{\omega}_{r}\cdot\vec{\alpha}_{s})=\delta_{rs}$ and

$\vec{h}_{r}\overset{\text{def}}{=}(-1)^{r}\left(\vec{\omega}_{r}-\vec{\omega}_{r-1}\right)\quad\text{for}\quad
r=1,\dots,2n+1,$ (2.4)

where $\vec{\omega}_{0}=\vec{\omega}_{2n+1}=0$. Then $\mathcal{N}=2$
$W_{n+1}$-algebra is defined by [15, 16, 17]

$\begin{gathered}:\left(b^{-1}D-D\big{(}\vec{h}_{1}\cdot\vec{\bm{\Phi}}(z,\theta)\big{)}\right)\left(b^{-1}D-D\big{(}\vec{h}_{2}\cdot\vec{\bm{\Phi}}(z,\theta)\big{)}\right)\dots\left(b^{-1}D-D\big{(}\vec{h}_{2n+1}\cdot\vec{\bm{\Phi}}(z,\theta)\big{)}\right):=\\\
=\left(b^{-1}D\right)^{2n+1}+\bm{W}_{\frac{1}{2}}(z,\theta)\left(b^{-1}D\right)^{2n}+\bm{W}_{1}(z,\theta)\left(b^{-1}D\right)^{2n-1}+\bm{W}_{\frac{3}{2}}(z,\theta)\left(b^{-1}D\right)^{2n-2}+\dots{}\end{gathered}$
(2.5)

where

$D=\frac{\partial}{\partial\theta}-\theta\partial.$ (2.6)

We have $\bm{W}_{\frac{1}{2}}(z,\theta)=0$,

$\bm{W}_{1}(z,\theta)=J(z)+i\theta(G^{+}(z)-G^{-}(z)),\quad
b\bm{W}_{\frac{3}{2}}(z,\theta)=iG^{-}(z)+\theta\left(T(z)+\frac{1}{2}\partial
J(z)\right)$ (2.7)

etc. The currents (2.7) satisfy commutation relations of $\mathcal{N}=2$
Virasoro algebra (2.1) with the central charge

$c=3n\left(1+\frac{n+1}{b^{2}}\right).$ (2.8)

Higher currents in (2.5) provide its extension.

In fact, it is more convenient to work in $\mathfrak{gl}(n+1|n)$ rather than
in $\mathfrak{sl}(n+1|n)$ terms. In order to do it, we extend our field as
$\vec{\bm{\Phi}}(z,\theta)=(\bm{\Phi}_{0}(z,\theta),\bm{\Phi}_{1}(z,\theta),\dots,\bm{\Phi}_{2n}(z,\theta))$.
In this case we can choose simple roots as

$\left(\vec{\alpha}_{2r+1}\cdot\vec{\bm{\Phi}}(z,\theta)\right)=i\bm{\Phi}_{2r}(z,\theta)-\bm{\Phi}_{2r+1}(z,\theta),\quad\left(\vec{\alpha}_{2r}\cdot\vec{\bm{\Phi}}(z,\theta)\right)=\bm{\Phi}_{2r-1}(z,\theta)-i\bm{\Phi}_{2r}(z,\theta).$
(2.9)

In this case the map (2.5) takes very simple form

$\left(b^{-1}D-iD\bm{\Phi}_{0}(z,\theta)\right)\left(b^{-1}D-D\bm{\Phi}_{1}(z,\theta)\right)\dots\left(b^{-1}D-iD\bm{\Phi}_{2n}(z,\theta)\right)$
(2.10)

We denote the corresponding algebra as $W_{\mathfrak{gl}(n+1|n)}$. Similarly
$W_{\mathfrak{gl}(n|n)}$ corresponds to (2.10) with the factor
$\left(b^{-1}D-iD\bm{\Phi}_{0}(z,\theta)\right)$ dropped.

It is important to note that both algebras $W_{\mathfrak{gl}(n+1|n)}$ and
$W_{\mathfrak{gl}(n|n)}$ can be defined as commutants of certain set of
screening operators [17]. For $W_{\mathfrak{gl}(n|n)}$ it will be convenient
to use complex fields ($k=1,\dots,n$)

$\displaystyle\bm{X}_{k}(z,\theta)\overset{\text{def}}{=}\bm{\Phi}_{2k-1}(z,\theta)+i\bm{\Phi}_{2k}(z,\theta)=X_{k}(z)+\theta\psi_{k}(z),$
(2.11)
$\displaystyle\bm{X}^{*}_{k}(z,\theta)\overset{\text{def}}{=}\bm{\Phi}_{2k-1}(z,\theta)-i\bm{\Phi}_{2k}(z,\theta)=X_{k}^{*}(z)+\theta\psi_{k}^{*}(z),$

where $(X_{k}(z),X_{k}^{*}(z),\psi(z),\psi^{*}(z))$ have the following OPE’s

$X_{i}(z)X_{j}^{*}(w)=-\delta_{ij}\log(z+w)+\dots,\quad\psi_{i}(z)\psi_{j}^{*}(w)=\frac{\delta_{ij}}{z-w}+\dots$
(2.12)

In these terms the screening operators for $W_{\mathfrak{gl}(n|n)}$ algebra
$(S_{1},S_{1,2},S_{2},S_{2,3},\dots,S_{n-1,n},S_{n})$ have the form [17]

$\displaystyle S_{k}=b^{-1}\oint
e^{b\big{(}\vec{\alpha}_{2k-1}\cdot\vec{\bm{\Phi}}(z,\theta)\big{)}}dzd\theta=b^{-1}\oint
e^{b\bm{X}_{k}^{*}(z,\theta)}dzd\theta=\oint\psi^{*}_{k}(z)e^{bX^{*}_{k}(z)}dz,$
(2.13) $\displaystyle S_{k,k+1}=2b^{-1}\oint
e^{b\big{(}\vec{\alpha}_{2k}\cdot\vec{\bm{\Phi}}(z,\theta)\big{)}}dzd\theta=2b^{-1}\oint
e^{\frac{b}{2}\left(\bm{X}_{k}(z,\theta)-\bm{X}_{k+1}(z,\theta)-\bm{X}^{*}_{k}(z,\theta)-\bm{X}^{*}_{k+1}(z,\theta)\right)}dzd\theta=$
$\displaystyle=\oint
e^{\frac{b}{2}\left(X_{k}(z)-X_{k+1}(z)-X^{*}_{k}(z)-X^{*}_{k+1}(z)\right)}\left(\psi_{k}(z)-\psi_{k+1}(z)-\psi^{*}_{k}(z)-\psi^{*}_{k+1}(z)\right)dz.$

We note that all screening fields (2.13) are fermionic ones. There are also
bosonic screening fields [17].

##  3 $\mathfrak{gl}(1|1)$ current algebra and $bc\beta\gamma$-system

The commutation relations of $\widehat{\mathfrak{gl}}(1|1)$ algebra [18] can
be written in terms of OPE of two bosonic currents $J^{E}(z)$, $J^{N}(z)$ and
two fermionic currents $J^{\pm}(z)$:

$\displaystyle J^{N}(z)J^{E}(w)=\frac{\kappa}{(z-w)^{2}}+\dots$ (3.1)
$\displaystyle
J^{+}(z)J^{-}(w)=\frac{\kappa}{(z-w)^{2}}+\frac{J^{E}(w)}{(z-w)^{2}}+\dots$
$\displaystyle J^{\pm}(z)J^{N}(w)=\mp\frac{J^{\pm}(w)}{z-w}+\dots$

The parameter $\kappa$ called the level can be set $\kappa=1$ by the simple
rescaling $J^{E}(z)\rightarrow\kappa J^{E}(z)$ and $J^{+}(z)\rightarrow\kappa
J^{+}(z)$. It implies that $\widehat{\mathfrak{gl}}(1|1)$ algebra admits free
field representation [19] via the $bc\beta\gamma$-system. Namely,
$bc\beta\gamma$-system is the system of complex fermions $\alpha(z)$,
$\alpha^{*}(z)$ and complex bosons $\mathrm{a}(z)$, $\mathrm{a}^{*}(z)$ with
OPE’s (other OPE’s are trivial)

$\alpha(z)\alpha^{*}(w)=\frac{1}{z-w}+\dots,\qquad\mathrm{a}(z)\mathrm{a}^{*}(w)=\frac{1}{z-w}+\dots$
(3.2)

Then one can realize the currents $J^{E}(z)$, $J^{N}(z)$ and $J^{\pm}(z)$ as
bilinears (Wick normal ordering is assumed)

$\bm{E}\overset{\text{def}}{=}\begin{pmatrix}\frac{J^{E}}{2}+J^{N}&J^{+}\\\
J^{-}&\frac{J^{E}}{2}-J^{N}\end{pmatrix}=\begin{pmatrix}\alpha^{*}\alpha&\alpha^{*}\mathrm{a}\\\
\mathrm{a}^{*}\alpha&\mathrm{a}^{*}\mathrm{a}\end{pmatrix}$ (3.3)

For our purposes it will be convenient to further realize
$(\alpha(z),\alpha^{*}(z),\mathrm{a}(z),\mathrm{a}^{*}(z))$ by the complex
superfield $\bm{X}(z,\theta)=X(z)+\theta\psi(z)$,
$\bm{X}^{*}(z,\theta)=X^{*}(z)+\theta\psi^{*}(z),$ as the Wick ordered fields

$\displaystyle\alpha=\psi e^{-\frac{X}{b}},\quad$
$\displaystyle\mathrm{a}=e^{-\frac{X}{b}},$ (3.4)
$\displaystyle\alpha^{*}=\psi^{*}e^{\frac{X}{b}},\quad$
$\displaystyle\mathrm{a}^{*}=\left(\psi\psi^{*}-b\partial
X^{*}\right)e^{\frac{X}{b}}.$

Here $b$ is an arbitrary parameter, which can be re-scaled out by the
transformation $X\rightarrow bX$, $X^{*}\rightarrow b^{-1}X^{*}$, 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 operator222We
remind that $\widehat{\mathfrak{gl}}(2)_{\kappa}$ algebra admits free field
representation in terms of two bosonic fields $\varphi_{1}(z)$,
$\varphi_{2}(z)$ and $\beta\gamma$-system realized by Wakimoto screening
operator [20] $\mathcal{S}=\oint
e^{b(\varphi_{1}(z)-\varphi_{2}(z))}\beta(z)dz,$ (3.5) with
$\kappa=-2-b^{-2}$. In $\widehat{\mathfrak{gl}}(1|1)$ case
$(\varphi_{1}(z)-\varphi_{2}(z))$ is replaced by $X^{*}(z)$ and $\beta(z)$ by
$\psi^{*}(z)$.

$\mathcal{S}=b^{-1}\oint
e^{b\bm{X}^{*}(z,\theta)}dzd\theta=\oint\psi^{*}(z)e^{bX^{*}(z)}dz.$ (3.6)

We note also that (3.4) admit the following simple superfield expression

$\mathrm{a}(z)-\frac{\theta}{b}\alpha(z)=e^{-\frac{\bm{X}(z,\theta)}{b}},\quad\alpha^{*}(z)+\frac{\theta}{b}\mathrm{a}^{*}(z)=D\bm{X}^{*}(z,\theta)e^{\frac{\bm{X}(z,\theta)}{b}}.$
(3.7)

In terms of the fields $(X,X^{*},\psi,\psi^{*})$ the representation of
$\widehat{\mathfrak{gl}}(1|1)$ takes the form

$\bm{E}=\begin{pmatrix}b^{-1}\partial X-\psi\psi^{*}&\psi^{*}\\\
\partial\psi-b\psi\partial X^{*}&\psi\psi^{*}-b^{-1}\partial X-b\partial
X^{*}\end{pmatrix}$ (3.8)

In these notes we consider the so called typical (or generic) representations
of $\widehat{\mathfrak{gl}}(1|1)$. One distinguishes between NS and R modules.
We consider NS module for simplicity. We define the mode expansion of the
currents on the cylinder $z=e^{-ix}$: $x\sim x+2\pi$

$\begin{gathered}\partial
X(x)=\sum_{k\in\mathbb{Z}}X_{k}e^{-ikx},\quad\partial
X^{*}(x)=\sum_{k\in\mathbb{Z}}X^{*}_{k}e^{-ikx},\\\
\psi(x)=i^{\frac{1}{2}}\sum_{r\in\mathbb{Z}+\frac{1}{2}}\psi_{r}e^{-irx},\quad\psi^{*}(x)=i^{\frac{1}{2}}\sum_{r\in\mathbb{Z}+\frac{1}{2}}\psi^{*}_{r}e^{-irx}.\end{gathered}$
(3.9)

with commutation relations (other commutation relations are trivial)

$[X_{m},X_{n}^{*}]=m\delta_{m,-n},\quad\\{\psi_{r},\psi_{s}^{*}\\}=\delta_{r,-s}.$
(3.10)

Then the Fock module $\mathcal{F}_{\bm{u}}$, where $\bm{u}=(u,u^{*})$, is
generated from the vacuum state $|\bm{u}\rangle$

$\begin{gathered}X_{n}|\bm{u}\rangle=X^{*}_{n}|\bm{u}\rangle=\psi_{r}|\bm{u}\rangle=\psi^{*}_{r}|\bm{u}\rangle=0\quad\text{for}\quad
n>0,\,r>0,\\\ X_{0}|\bm{u}\rangle=-iu|\bm{u}\rangle,\quad
X_{0}^{*}|\bm{u}\rangle=-iu^{*}|\bm{u}\rangle,\end{gathered}$ (3.11)

by the action of the creation operators

$\mathcal{F}_{\bm{u}}=\textrm{Span}\left(\psi_{-\bm{s}^{*}}^{*}\psi_{-\bm{s}}X^{*}_{-\bm{\lambda}^{*}}X_{-\bm{\lambda}}|\bm{u}\rangle\right),\quad\text{where}\quad
X_{-\bm{\lambda}}=X_{-\lambda_{1}}X_{-\lambda_{2}}\dots\,\text{etc}.,$ (3.12)

where $\bm{\lambda}$ and $\bm{\lambda}^{*}$ are partitions

$\bm{\lambda}=\\{\lambda_{1}\geq\lambda_{2}\geq\dots\\},\qquad\bm{\lambda}^{*}=\\{\lambda_{1}^{*}\geq\lambda_{2}^{*}\geq\dots\\},$
(3.13)

and $\bm{s}$ and $\bm{s}^{*}$ are strict partitions (fermionic partitions)

$\bm{s}=\\{s_{1}>s_{2}>\dots\\},\qquad\bm{s}^{*}=\\{s_{1}^{*}>s_{2}^{*}>\dots\\},$
(3.14)

The character of $\mathcal{F}_{\bm{u}}$ has the form

$\chi(q,t)\overset{\text{def}}{=}\textrm{Tr}(q^{L_{0}}t^{c_{0}})\Bigl{|}_{\mathcal{F}_{\bm{u}}}=\prod_{k=1}^{\infty}\frac{\left(1+tq^{k-\frac{1}{2}}\right)\left(1+t^{-1}q^{k-\frac{1}{2}}\right)}{(1-q^{k})^{2}},$
(3.15)

where

$L_{0}=\sum_{k>0}\left(X_{-k}X_{k}^{*}+X_{-k}^{*}X_{k}\right)+\sum_{s>0}s\left(\psi_{-s}\psi^{*}_{s}+\psi_{-s}^{*}\psi_{s}\right),\quad
c_{0}=\sum_{s>0}\left(\psi_{-s}\psi^{*}_{s}-\psi_{-s}^{*}\psi_{s}\right).$
(3.16)

##  4 R-matrix

In order to define the R-matrix for $\widehat{\mathfrak{gl}}(1|1)$, we take
two copies of $\widehat{\mathfrak{gl}}(1|1)$ algebra defined by two screening
fields

$\mathcal{S}_{1}=\int\psi_{1}^{*}(z)e^{bX_{1}^{*}(z)}dz\quad\text{and}\quad\mathcal{S}_{2}=\int\psi_{2}^{*}(z)e^{bX_{2}^{*}(z)}dz$
(4.1)

and define the additional screening field

$\tilde{\mathcal{S}}_{1,2}\overset{\text{def}}{=}\int(\psi_{1}(z)-\psi_{2}(z))e^{\frac{b}{2}\left(X_{1}(z)-X_{2}(z)-X_{1}^{*}(z)-X_{2}^{*}(z)\right)}dz,$
(4.2)

which commutes with the diagonal $\widehat{\mathfrak{gl}}(1|1)$ current
$\bm{E}_{1}+\bm{E}_{2}$.

The commutant of this additional screening is actually larger and can be
associated with the $W(\mathfrak{gl}(2|2))$ algebra introduced in section 2.
We note that the screening operator $\tilde{\mathcal{S}}_{1,2}$ given by (4.2)
differs from $S_{1,2}$ given by (2.13) by the replacement in the prefactor

$\psi_{1}(z)-\psi_{2}(z)\rightarrow\psi_{1}(z)-\psi_{2}(z)-\psi_{1}^{*}(z)-\psi_{2}^{*}(z).$
(4.3)

However two algebras defined by
$(\mathcal{S}_{1},\mathcal{S}_{1,2},\mathcal{S}_{2})$ and by
$(\mathcal{S}_{1},\tilde{\mathcal{S}}_{1,2},\mathcal{S}_{2})$ are actually
isomorphic and are related by marginal deformations. It can be seen as
follows. Consider the Lax operator

$\mathcal{L}=\left(b^{-1}D-D\bm{\Phi}_{1}\right)\left(b^{-1}D-iD\bm{\Phi}_{2}\right)=\left(b^{-1}D\right)^{2}-D\bm{X}^{*}b^{-1}D+\frac{1}{2}D\bm{X}^{*}D\bm{X}+\frac{1}{2b}D^{2}\left(\bm{X}^{*}-\bm{X}\right),$
(4.4)

where $\bm{X}=\bm{\Phi}_{1}+i\bm{\Phi}_{2}$ and perform the formal scale
transformation

$\psi\rightarrow\lambda^{-\frac{1}{2}}\psi,\quad\psi^{*}\rightarrow\lambda^{\frac{1}{2}}\psi^{*},\quad\theta\rightarrow\lambda^{\frac{1}{2}}\theta,\quad\frac{\partial}{\partial\theta}\rightarrow\lambda^{-\frac{1}{2}}\frac{\partial}{\partial\theta},$
(4.5)

which does not change commutation relations. Then one has

$\mathcal{L}=\mathcal{L}^{(0)}+\lambda\mathcal{L}^{(1)},$ (4.6)

where (we use $D^{2}=-\partial$)

$\mathcal{L}^{(0)}=-b^{-2}\partial+\frac{1}{2}\left(b^{-1}\partial
X-\psi\psi^{*}-b^{-1}\partial
X^{*}\right)+\frac{1}{2b}\left(\partial\psi-b\psi\partial
X^{*}\right)\theta-b^{-1}\psi^{*}\frac{\partial}{\partial\theta}+b^{-1}\partial
X^{*}\theta\frac{\partial}{\partial\theta},$ (4.7)

and

$\mathcal{L}^{(1)}=\left(b^{-1}\psi^{*}\partial+\frac{1}{2}\psi^{*}\partial
X-\frac{1}{2b}\partial\psi^{*}\right)\theta.$ (4.8)

It is clear that if $\mathcal{L}_{1}\cdot\mathcal{L}_{2}$ ”commutes” with the
screening charges $(\mathcal{S}_{1},\mathcal{S}_{1,2},\mathcal{S}_{2})$ then
$\mathcal{L}_{1}^{(0)}\cdot\mathcal{L}_{2}^{(0)}$ ”commutes” with
$(\mathcal{S}_{1},\tilde{\mathcal{S}}_{1,2},\mathcal{S}_{2})$333The current
$A(z)$ is said to ”commute” with the screening field
$\mathcal{S}=\int\mathcal{V}(\xi)d\xi$ if
$\oint_{\mathcal{C}_{z}}\mathcal{V}(\xi)A(z)d\xi=0.$ (4.9) ..

Having defined the Lax operator $\mathcal{L}^{(0)}$, we define the $R-$matrix
as an intertwining operator

$\mathcal{R}_{ij}\mathcal{L}^{(0)}_{i}\mathcal{L}^{(0)}_{j}=\mathcal{L}^{(0)}_{j}\mathcal{L}^{(0)}_{i}\mathcal{R}_{ij},$
(4.10)

where
$\mathcal{L}^{(0)}_{j}\overset{\text{def}}{=}\mathcal{L}^{(0)}(X_{i},X_{i}^{*},\psi_{i},\psi_{i}^{*})$.
We note that (4.10) automatically implies the Yang-Baxter equation

$\mathcal{R}_{12}\mathcal{R}_{13}\mathcal{R}_{23}=\mathcal{R}_{23}\mathcal{R}_{13}\mathcal{R}_{12}.$
(4.11)

The $W_{\mathfrak{gl}(2|2)}$ algebra defined by
$\mathcal{L}_{1}^{(0)}\mathcal{L}_{2}^{(0)}$ is obtained from the one defined
by $\mathcal{L}_{1}\mathcal{L}_{2}$ in the limit $\lambda\rightarrow 0$. It
consists of eight currents

$\begin{gathered}\Psi_{12}^{*}\overset{\text{def}}{=}\psi_{1}^{*}+\psi_{2}^{*},\quad
U_{12}\overset{\text{def}}{=}\partial X_{1}^{*}+\partial X_{2}^{*},\quad
J_{12}\overset{\text{def}}{=}\psi_{1}\psi_{1}^{*}+\psi_{2}\psi_{2}^{*}-\frac{1}{b}\left(\partial
X_{1}+\partial X_{2}\right),\\\
G_{12}\overset{\text{def}}{=}i\left(\psi_{1}\partial
X_{1}^{*}+\psi_{2}\partial
X_{2}^{*}-\frac{1}{b}\left(\partial\psi_{1}+\partial\psi_{2}\right)\right),\\\
G_{12}^{*}\overset{\text{def}}{=}i\left(\psi_{1}^{*}(\partial X_{1}-\partial
X_{2}^{*})+\psi_{2}^{*}(\partial X_{2}+\partial
X_{1}^{*})-\frac{1}{b}\left(\partial\psi_{1}^{*}-\partial\psi_{2}^{*}\right)\right),\\\
T_{12}\overset{\text{def}}{=}-\partial X_{1}\partial X_{1}^{*}-\partial
X_{2}\partial
X_{2}^{*}+\dfrac{1}{b}(\partial^{2}X_{1}^{*}-\partial^{2}X_{2}^{*})+\partial\psi_{1}\psi_{1}^{*}+\partial\psi_{2}\psi_{2}^{*}-\dfrac{1}{2}\partial
J_{12},\end{gathered}$ (4.12)
$\Theta_{12}\overset{\text{def}}{=}-\dfrac{1}{8b^{2}}((\partial X_{1}-\partial
X^{*}_{1})^{2}+(\partial X_{2}-\partial
X^{*}_{2})^{2})-\dfrac{1}{2b^{3}}(\partial^{2}X_{2}-\partial^{2}X_{2}^{*})+\dfrac{1}{4b}(\partial
X_{1}-\partial X_{1}^{*})\psi_{1}\psi_{1}^{*}+\\\ +\dfrac{1}{4b}(\partial
X_{2}-\partial X_{2}^{*})\psi_{2}\psi_{2}^{*}+\dfrac{1}{2b}\partial
X_{2}^{*}\psi_{1}^{*}\psi_{2}+\dfrac{1}{8}(\psi_{1}^{*}\partial\psi_{1}+\psi_{1}\partial\psi_{1}^{*}+\psi_{2}^{*}\partial\psi_{2}+\psi_{2}\partial\psi_{2}^{*})+\\\
+\dfrac{1}{2b^{2}}(\psi_{2}\partial\psi_{2}^{*}++\partial\psi_{2}\psi_{2}^{*}-\psi_{1}^{*}\partial\psi_{2}),$
(4.13)

and

$H_{12}\overset{\text{def}}{=}\dfrac{1}{4}\psi_{1}\psi_{1}^{*}\psi_{2}\partial
X_{2}^{*}+\dfrac{1}{4}\psi_{2}\psi_{2}^{*}\psi_{1}\partial X_{1}^{*}-\\\
-\frac{1}{4b}\left(\big{(}\partial X_{1}+\partial X_{1}^{*}\big{)}\partial
X_{2}^{*}\psi_{2}+\big{(}\partial X_{2}-\partial X_{2}^{*}\big{)}\partial
X_{1}^{*}\psi_{1}+\psi_{1}\psi_{1}^{*}\partial\psi_{2}+\psi_{2}\psi_{2}^{*}\partial\psi_{1}\right)+\\\
+\frac{1}{4b^{2}}\left(\big{(}\partial X_{1}+\partial X_{1}^{*}+2\partial
X_{2}^{*}\big{)}\partial\psi_{2}+\big{(}\partial X_{2}-\partial
X_{2}^{*}\big{)}\partial\psi_{1}+2\psi_{2}\partial^{2}X_{2}^{*}\right)-\frac{1}{2b^{3}}\partial^{2}\psi_{2},$
(4.14)

where all the densities are _Wick ordered_. We note that
$\left(J_{12},G_{12},G_{12}^{*},T_{12}\right)$ form $\mathcal{N}=2$ Virasoro
algebra with the central charge

$c=6.$ (4.15)

At the same time the currents $(\Psi^{*}_{12},U_{12},J_{12},G_{12})$
correspond to the diagonal current $\bm{E}_{1}+\bm{E}_{2}$.

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): $\mathcal{F}_{\bm{u}}\otimes\mathcal{F}_{\bm{v}}$ with
the vacuum state

$|\varnothing\rangle\overset{\text{def}}{=}|\bm{u}\rangle\otimes|\bm{v}\rangle.$
(4.16)

Since the normalization of $R_{12}$ is not fixed from (4.10), it can be chosen
at will. We fix it by
$\mathcal{R}_{12}|\varnothing\rangle=|\varnothing\rangle$. Then using (4.10)
one can find the action of $\mathcal{R}_{12}$ on any state in
$\mathcal{F}_{\bm{u}}\otimes\mathcal{F}_{\bm{v}}$. In particular, using two
equations

$\mathcal{R}_{12}\left(\Psi_{12}^{*}\right)_{-\frac{1}{2}}|\varnothing\rangle=\left(\Psi_{21}^{*}\right)_{-\frac{1}{2}}|\varnothing\rangle,\qquad\mathcal{R}_{12}\left(G_{12}^{*}\right)_{-\frac{1}{2}}|\varnothing\rangle=\left(G_{21}^{*}\right)_{-\frac{1}{2}}|\varnothing\rangle,$
(4.17)

we find that

$\mathcal{R}_{12}\big{(}\psi_{1}^{*}\big{)}_{-\frac{1}{2}}|\varnothing\rangle=\frac{(u-v)-(u^{*}-v^{*})}{(u-v)-(u^{*}+v^{*})+b^{-1}}\big{(}\psi_{1}^{*}\big{)}_{-\frac{1}{2}}|\varnothing\rangle-\frac{2u^{*}-b^{-1}}{(u-v)-(u^{*}+v^{*})+b^{-1}}\big{(}\psi_{2}^{*}\big{)}_{-\frac{1}{2}}|\varnothing\rangle,$
(4.18)

and

$\mathcal{R}_{12}\big{(}\psi_{2}^{*}\big{)}_{-\frac{1}{2}}|\varnothing\rangle=-\frac{2v^{*}-b^{-1}}{(u-v)-(u^{*}+v^{*})+b^{-1}}\big{(}\psi_{1}^{*}\big{)}_{-\frac{1}{2}}|\varnothing\rangle+\frac{(u-v)+(u^{*}-v^{*})}{(u-v)-(u^{*}+v^{*})+b^{-1}}\big{(}\psi_{2}^{*}\big{)}_{-\frac{1}{2}}|\varnothing\rangle.$
(4.19)

Similarly, using

$\mathcal{R}_{12}\left(G_{12}\right)_{-\frac{1}{2}}|\varnothing\rangle=\left(G_{21}\right)_{-\frac{1}{2}}|\varnothing\rangle,\qquad\mathcal{R}_{12}\left(H_{12}\right)_{-\frac{1}{2}}|\varnothing\rangle=\left(H_{21}\right)_{-\frac{1}{2}}|\varnothing\rangle,$
(4.20)

we find that

$\mathcal{R}_{12}\big{(}\psi_{1}\big{)}_{-\frac{1}{2}}|\varnothing\rangle=\frac{(u-v)+(u^{*}-v^{*})}{(u-v)+(u^{*}+v^{*})+b^{-1}}\big{(}\psi_{1}\big{)}_{-\frac{1}{2}}|\varnothing\rangle+\frac{2v^{*}+b^{-1}}{(u-v)+(u^{*}+v^{*})+b^{-1}}\big{(}\psi_{2}\big{)}_{-\frac{1}{2}}|\varnothing\rangle,$
(4.21)

and

$\mathcal{R}_{12}\big{(}\psi_{2}\big{)}_{-\frac{1}{2}}|\varnothing\rangle=\frac{2u^{*}+b^{-1}}{(u-v)+(u^{*}+v^{*})+b^{-1}}\big{(}\psi_{1}\big{)}_{-\frac{1}{2}}|\varnothing\rangle+\frac{(u-v)-(u^{*}-v^{*})}{(u-v)+(u^{*}+v^{*})+b^{-1}}\big{(}\psi_{2}\big{)}_{-\frac{1}{2}}|\varnothing\rangle.$
(4.22)

Proceeding in this way, one can compute the matrix of $\mathcal{R}_{12}$ at
any given order.

In general, it is clear that the matrix elements of $\mathcal{R}_{12}$ are
some rational functions of three parameters

$u-v,\quad u^{*}\quad\text{and}\quad v^{*}.$ (4.23)

Indeed, from the relation $[\mathcal{R}_{12},U_{12}]=0$ it follows that
$\mathcal{R}_{12}$ does not depend on $u+v$. Generic expression for
$\mathcal{R}_{12}$ 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 $\mathcal{R}_{12}(\bm{u},\bm{v})$ at
large $u-v$.

##  5 $RLL$ algebra

In this section we study $RLL$ algebra provided by the $R-$matrix defined in
section 4

$\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})=\mathcal{L}_{2}(\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{R}_{12}(\bm{u},\bm{v}),$
(5.1)

where as usual $\mathcal{L}_{1}(\bm{u})$ and $\mathcal{L}_{2}(\bm{v})$ are
treated as matrices acting in $\widehat{gl}(1|1)$ modules
$\mathcal{F}_{\bm{u}}$ and $\mathcal{F}_{\bm{v}}$ 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

$\langle l|\mathcal{L}(u)|k\rangle,$ (5.2)

where $|k\rangle$ and $|l\rangle$ are arbitrary states from
$\mathcal{F}_{\bm{u}}$.

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 $\widehat{\mathfrak{gl}}(1)$ and in [21] for
$\widehat{\mathfrak{gl}}(2)$ cases.

We will proceed exactly as in [4, 21]. We note however, that there is a
conceptual difference compared to $\widehat{\mathfrak{gl}}(1)$ and
$\widehat{\mathfrak{gl}}(2)$ 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

$\begin{gathered}h(\bm{u})\overset{\text{def}}{=}\langle\varnothing|\mathcal{L}(\bm{u})|\varnothing\rangle,\quad
e(\bm{u})\overset{\text{def}}{=}h^{-1}(\bm{u})\langle\varnothing|\mathcal{L}(\bm{u})\psi_{-\frac{1}{2}}|\varnothing\rangle,\quad
f(\bm{u})\overset{\text{def}}{=}\langle\varnothing|\psi_{\frac{1}{2}}\mathcal{L}(\bm{u})|\varnothing\rangle
h^{-1}(\bm{u}),\\\
e^{*}(\bm{u})\overset{\text{def}}{=}\frac{1}{2u^{*}-b^{-1}}h^{-1}(\bm{u})\langle\varnothing|\mathcal{L}(\bm{u})\psi^{*}_{-\frac{1}{2}}|\varnothing\rangle,\quad
f^{*}(\bm{u})\overset{\text{def}}{=}\frac{1}{2u^{*}+b^{-1}}\langle\varnothing|\psi_{\frac{1}{2}}^{*}\mathcal{L}(\bm{u})|\varnothing\rangle
h^{-1}(\bm{u}).\end{gathered}$ (5.3)

Taking vacuum average of $RLL=LLR$ relation, one finds

$[h(\bm{u}),h(\bm{v})]=0.$ (5.4)

Similarly, using (4.18), (4.19), (4.21) and (4.22), we find

$\big{(}(u+u^{*})-(v-v^{*})+b^{-1}\big{)}h(\bm{u})e(\bm{v})=\big{(}(u-u^{*})-(v-v^{*})\big{)}\,e(\bm{v})h(\bm{u}){\color[rgb]{0,0,1}+\big{(}2u^{*}+b^{-1}\big{)}h(\bm{u})e(\bm{u})},$
(5.5)
$\big{(}(u-u^{*})-(v+v^{*})+b^{-1}\big{)}h(\bm{u})e^{*}(\bm{v})=\big{(}(u+u^{*})-(v+v^{*})\big{)}\,e^{*}(\bm{v})h(\bm{u}){\color[rgb]{0,0,1}-\big{(}2u^{*}-b^{-1}\big{)}h(\bm{u})e^{*}(\bm{u})},$
(5.6)
$\big{(}(u-u^{*})-(v+v^{*})+b^{-1}\big{)}f(\bm{v})h(\bm{u})=\big{(}(u+u^{*})-(v+v^{*})\big{)}\,h(\bm{u})f(\bm{v}){\color[rgb]{0,0,1}-\big{(}2u^{*}-b^{-1}\big{)}f(\bm{u})h(\bm{u})},$
(5.7)

and

$\big{(}(u+u^{*})-(v-v^{*})+b^{-1}\big{)}f^{*}(\bm{v})h(\bm{u})=\big{(}(u-u^{*})-(v-v^{*})\big{)}\,h(\bm{u})f^{*}(\bm{v}){\color[rgb]{0,0,1}+\big{(}2u^{*}+b^{-1}\big{)}f^{*}(\bm{u})h(\bm{u})}.$
(5.8)

The terms shown in blue are called local terms, since they depend only on
$\bm{u}$, but not on $\bm{v}$.

Consider for example the relation (5.5), set $v-v^{*}=u-u^{*}$ and multiply it
by $h^{-1}(\bm{u})$ from the left. Then the first term in the r.h.s. of (5.5)
vanishes and one obtains

$e(v+v^{*},u-u^{*})=e(u+u^{*},u-u^{*})\implies e=e(u-u^{*}).$ (5.9)

Similarly one finds

$f=f(u+u^{*}),\quad e^{*}=e^{*}(u+u^{*}),\quad\text{and}\quad
f^{*}=f^{*}(u-u^{*}).$ (5.10)

The relations similar to (5.9)–(5.10) persist at higher levels as well. Let us
consider the basic currents at level $1$

$\begin{gathered}a(\bm{u})\overset{\text{def}}{=}h^{-1}(\bm{u})\langle\varnothing|\mathcal{L}(\bm{u})a_{-1}|\varnothing\rangle,\quad
a^{*}(\bm{u})\overset{\text{def}}{=}\frac{1}{2u^{*}-b^{-1}}h^{-1}(\bm{u})\langle\varnothing|\mathcal{L}(\bm{u})a^{*}_{-1}|\varnothing\rangle,\\\
k(\bm{u})\overset{\text{def}}{=}\frac{1}{2u^{*}-b^{-1}}h^{-1}(\bm{u})\langle\varnothing|\mathcal{L}(\bm{u})\psi_{-\frac{1}{2}}\psi^{*}_{-\frac{1}{2}}|\varnothing\rangle,\end{gathered}$
(5.11)

and their linear combinations

$\rho(\bm{u})\overset{\text{def}}{=}a(\bm{u})-(2u^{*}-b^{-1})a^{*}(\bm{u})\quad\text{and}\quad\rho^{*}(\bm{u})\overset{\text{def}}{=}a(\bm{u})+(2u^{*}-2b-b^{-1})a^{*}(\bm{u})+2ik(\bm{u}).$
(5.12)

Using commutation relations (A.1) and (A.2) presented in the appendix, one
finds

$\rho=\rho\left(u-u^{*}\right),\quad\rho^{*}=\rho^{*}\left(u+u^{*}\right)$
(5.13)

Also the currents at level $-1$

$\begin{gathered}\bar{a}(\bm{u})\overset{\text{def}}{=}\langle\varnothing|a_{1}\mathcal{L}(\bm{u})|\varnothing\rangle
h^{-1}(\bm{u}),\quad\bar{a}^{*}(\bm{u})\overset{\text{def}}{=}\frac{1}{2u^{*}+b^{-1}}\langle\varnothing|a_{1}^{*}\mathcal{L}(\bm{u})|\varnothing\rangle
h^{-1}(\bm{u}),\\\
\bar{k}(\bm{u})\overset{\text{def}}{=}\frac{1}{2u^{*}+b^{-1}}\langle\varnothing|\psi_{\frac{1}{2}}\psi^{*}_{\frac{1}{2}}\mathcal{L}(\bm{u})|\varnothing\rangle
h^{-1}(\bm{u})\end{gathered}$ (5.14)

and

$\mu(\bm{u})\overset{\text{def}}{=}\bar{a}(\bm{u})+\left(2u^{*}+b^{-1}\right)\bar{a}^{*}(\bm{u})\quad\text{and}\quad\mu^{*}(\bm{u})\overset{\text{def}}{=}\bar{a}(\bm{u})-\left(2u^{*}-2b+b^{-1}\right)\bar{a}^{*}(\bm{u})-2i\bar{k}(\bm{u}),$
(5.15)

satisfy similar relations with $h(\bm{u})$, which imply

$\mu=\mu\left(u+u^{*}\right),\quad\mu^{*}=\mu^{*}\left(u-u^{*}\right).$ (5.16)

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

$\displaystyle\sigma(\bm{u})\overset{\text{def}}{=}\left(2u^{*}-b^{-1}\right)\left(e^{*}(\bm{u})e(\bm{u})+k(\bm{u})\right):\quad\sigma=\sigma(u-u^{*}),$
(5.17)
$\displaystyle\sigma^{*}(\bm{u})\overset{\text{def}}{=}(2u^{*}+b^{-1})\big{(}e(\bm{u})e^{*}(\bm{u})-k(\bm{u})\big{)}-2ia^{*}(\bm{u}):\quad\sigma^{*}=\sigma^{*}(u+u^{*}),$

and two level $-1$ currents

$\displaystyle\nu(\bm{u})\overset{\text{def}}{=}\left(2u^{*}+b^{-1}\right)\left(f(\bm{u})f^{*}(\bm{u})-\bar{k}(\bm{u})\right):\quad\nu=\nu(u+u^{*}),$
(5.18)
$\displaystyle\nu^{*}(\bm{u})\overset{\text{def}}{=}(2u^{*}-b^{-1})(f^{*}(\bm{u})f(\bm{u})+\bar{k}(\bm{u}))-2i\bar{a}^{*}(\bm{u}):\quad\nu^{*}=\nu^{*}(u-u^{*}).$

The derivation of (5.17) and (5.18) is straightforward, yet tedious task and
is based on commutation relation with $h(\bm{u})$.

We will also need the following currents

$\uppsi(\bm{u})\overset{\text{def}}{=}\langle\varnothing\psi^{*}_{\frac{1}{2}}\mathcal{L}(\bm{u})\psi_{-\frac{1}{2}}|\varnothing\rangle
h^{-1}(\bm{u})-\langle\varnothing|\psi^{*}_{\frac{1}{2}}\mathcal{L}(\bm{u})|\varnothing\rangle
h^{-1}(\bm{u})\langle\varnothing|\mathcal{L}(\bm{u})\psi_{-\frac{1}{2}}|\varnothing\rangle
h^{-1}(\bm{u}),$ (5.19)

and

$\uppsi^{*}(\bm{u})\overset{\text{def}}{=}\langle\varnothing|\psi_{\frac{1}{2}}\mathcal{L}(\bm{u})|\varnothing\rangle
h^{-1}(\bm{u})\langle\varnothing|\mathcal{L}(\bm{u})\psi^{*}_{-\frac{1}{2}}|\varnothing\rangle
h^{-1}(\bm{u})-\langle\varnothing|\psi_{\frac{1}{2}}\mathcal{L}(\bm{u})\psi^{*}_{-\frac{1}{2}}|\varnothing\rangle
h^{-1}(\bm{u}),$ (5.20)

which can be shown to satisfy

$\uppsi(\bm{u})=\uppsi(u-u^{*})\quad\text{and}\quad\uppsi^{*}(\bm{u})=\uppsi(u+u^{*}).$
(5.21)

##  6 Current relations

We define the parameters

$h_{1}\overset{\text{def}}{=}\frac{b}{2}+b^{-1},\quad
h_{2}\overset{\text{def}}{=}\frac{b}{2}-b^{-1}$ (6.1)

and the currents

$e_{1}(u)\overset{\text{def}}{=}e(u),\quad
e_{2}(u)\overset{\text{def}}{=}e^{*}\left(u+\frac{b}{2}\right),\quad
f_{1}(u)\overset{\text{def}}{=}f^{*}(u)\quad\text{and}\quad
f_{2}(u)\overset{\text{def}}{=}f\left(u+\frac{b}{2}\right),$ (6.2)
$\uppsi_{1}(u)\overset{\text{def}}{=}\uppsi(u),\quad\uppsi_{2}(u)\overset{\text{def}}{=}\uppsi^{*}(u)$
(6.3)

It is also convenient to define the currents labeled by pyramid partitions
(see (5.12), (5.15), (5.17) and (5.18))444The analog of MacMahon for
$Y(\widehat{\mathfrak{gl}}(1|1))$ is naturally associated with the space of
pyramid partitions (see [24]). It induces natural notations for the currents.
There are two pyramid partitions with one stone which correspond to $e_{1}(u)$
and $e_{2}(u)$
$e_{1}(u)=e_{\includegraphics[scale={0.05}]{6.eps}}(u)\quad\text{and}\quad
e_{2}(u)=e_{\includegraphics[scale={0.05}]{5.eps}}(u).$ (6.4) Also there are
$4$ pyramid partitions with $2$ stones which correspond to the currents (6.5).
Similar identification holds for $f$ currents as well.

$\begin{gathered}e_{\includegraphics[scale={0.05}]{4b.eps}}(u)\overset{\text{def}}{=}\frac{i\rho(u)}{2h_{2}},\quad
e_{\includegraphics[scale={0.05}]{3b.eps}}(u)\overset{\text{def}}{=}\sigma(u)-\frac{i\rho(u)}{2h_{2}},\\\
e_{\includegraphics[scale={0.05}]{2b.eps}}(u)\overset{\text{def}}{=}\sigma^{*}\left(u+\frac{b}{2}\right)-\frac{i\rho^{*}\left(u+\frac{b}{2}\right)}{2h_{1}},\quad
e_{\includegraphics[scale={0.05}]{1b.eps}}(u)\overset{\text{def}}{=}\frac{i\rho^{*}\left(u+\frac{b}{2}\right)}{2h_{1}},\end{gathered}$
(6.5)

and

$\begin{gathered}f_{\includegraphics[scale={0.05}]{4b.eps}}(u)\overset{\text{def}}{=}\frac{i\mu\left(u+\frac{b}{2}\right)}{2h_{2}},\quad
f_{\includegraphics[scale={0.05}]{3b.eps}}(u)\overset{\text{def}}{=}\nu\left(u+\frac{b}{2}\right)-\frac{i\mu\left(u+\frac{b}{2}\right)}{2h_{2}},\\\
f_{\includegraphics[scale={0.05}]{2b.eps}}(u)\overset{\text{def}}{=}\nu^{*}(u)-\frac{i\mu^{*}(u)}{2h_{1}},\quad
f_{\includegraphics[scale={0.05}]{1b.eps}}(u)\overset{\text{def}}{=}\frac{i\mu^{*}(u)}{2h_{1}}.\end{gathered}$
(6.6)

One can derive the following commutation relations (see appendix A for
details)

$\left[\uppsi_{i}(u),\uppsi_{j}(v)\right]=0,\quad\left\\{e_{i}(u),e_{i}(v)\right\\}=0,\quad\left\\{f_{i}(u),f_{i}(v)\right\\}=0,$
(6.7)
$\left\\{e_{i}(u),f_{j}(v)\right\\}=\delta_{ij}\frac{\uppsi_{i}(u)-\uppsi_{i}(v)}{u-v},$
(6.8) $(u-v-
h_{1})(u-v+h_{1})\left(e_{1}(u)e_{2}(v){\color[rgb]{0,0,1}+\frac{e_{\includegraphics[scale={0.05}]{2.eps}}(v)}{u-v-
h_{1}}+\frac{e_{\includegraphics[scale={0.05}]{1.eps}}(v)}{u-v+h_{1}}}\right)=\\\
=-(v-u-
h_{2})(v-u+h_{2})\left(e_{2}(v)e_{1}(u){\color[rgb]{0,0,1}+\frac{e_{\includegraphics[scale={0.05}]{4.eps}}(u)}{v-u-
h_{2}}+\frac{e_{\includegraphics[scale={0.05}]{3.eps}}(u)}{v-u+h_{2}}}\right)$
(6.9) $(u-v-
h_{1})(u-v+h_{1})\left(f_{2}(u)f_{1}(v){\color[rgb]{0,0,1}+\frac{f_{\includegraphics[scale={0.05}]{2.eps}}(u)}{v-u-
h_{1}}+\frac{f_{\includegraphics[scale={0.05}]{1.eps}}(u)}{v-u+h_{1}}}\right)=\\\
=-(v-u-
h_{2})(v-u+h_{2})\left(f_{1}(v)f_{2}(u){\color[rgb]{0,0,1}+\frac{f_{\includegraphics[scale={0.05}]{4.eps}}(u)}{u-v-
h_{2}}+\frac{f_{\includegraphics[scale={0.05}]{3.eps}}(v)}{u-v+h_{2}}}\right)$
(6.10) $\uppsi_{i}(u)e_{j}(v)=\frac{(u-v-h_{j})(u-v+h_{j})}{(u-v-
h_{i})(u-v+h_{i})}e_{j}(v)\uppsi_{i}(u)+\\\
{\color[rgb]{0,0,1}+\frac{h_{i}^{2}-h_{j}^{2}}{2h_{i}(u-v-
h_{i})}e_{j}(u-h_{i})\uppsi_{i}(u)-\frac{h_{i}^{2}-h_{j}^{2}}{2h_{i}(u-v+h_{i})}e_{j}(u+h_{i})\uppsi_{i}(u)}$
(6.11) $\uppsi_{i}(u)f_{j}(v)=\frac{(u-v-h_{i})(u-v+h_{i})}{(u-v-
h_{j})(u-v+h_{j})}f_{j}(v)\uppsi_{i}(u)+\\\
{\color[rgb]{0,0,1}+\frac{h_{j}^{2}-h_{i}^{2}}{2h_{j}(u-v-
h_{j})}f_{j}(u-h_{j})\uppsi_{i}(u)-\frac{h_{j}^{2}-h_{i}^{2}}{2h_{j}(u-v+h_{j})}f_{j}(u+h_{j})\uppsi_{i}(u)}$
(6.12)

The relations (6.7)–(6.12) should be understood in terms of modes of basic
currents expanded at $u=\infty$

$e_{k}(u)=\frac{e_{k}^{(0)}}{u}+\frac{e_{k}^{(1)}}{u^{2}}+\dots,\quad
f_{k}(u)=\frac{f_{k}^{(0)}}{u}+\frac{f_{k}^{(1)}}{u^{2}}+\dots,\quad\uppsi_{k}(u)=1+\frac{\uppsi_{k}^{(0)}}{u}+\frac{\uppsi_{k}^{(1)}}{u^{2}}+\dots$
(6.13)

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 $i\leq 1$ or $j\leq 1$ can be used to express higher
currents in terms of basic ones. In particular, one finds from (6.9)

$\displaystyle
e_{\includegraphics[scale={0.05}]{2b.eps}}(u)=-\frac{1}{2h_{1}}\left(\big{\\{}e_{1}^{(0)},e_{2}^{(0)}\big{\\}}-(u-h_{1})\big{\\{}e_{1}^{(0)},e_{2}(u)\big{\\}}+\big{\\{}e_{1}^{(1)},e_{2}(u)\big{\\}}\right),$
(6.14) $\displaystyle
e_{\includegraphics[scale={0.05}]{1b.eps}}(u)=\frac{1}{2h_{1}}\left(\big{\\{}e_{1}^{(0)},e_{2}^{(0)}\big{\\}}-(u+h_{1})\big{\\{}e_{1}^{(0)},e_{2}(u)\big{\\}}+\big{\\{}e_{1}^{(1)},e_{2}(u)\big{\\}}\right),$
$\displaystyle
e_{\includegraphics[scale={0.05}]{4b.eps}}(u)=-\frac{1}{2h_{2}}\left(\big{\\{}e_{2}^{(0)},e_{1}^{(0)}\big{\\}}-(u-h_{2})\big{\\{}e_{2}^{(0)},e_{1}(u)\big{\\}}+\big{\\{}e_{2}^{(1)},e_{1}(u)\big{\\}}\right),$
$\displaystyle
e_{\includegraphics[scale={0.05}]{3b.eps}}(u)=\frac{1}{2h_{2}}\left(\big{\\{}e_{2}^{(0)},e_{1}^{(0)}\big{\\}}-(u+h_{2})\big{\\{}e_{2}^{(0)},e_{1}(u)\big{\\}}+\big{\\{}e_{2}^{(1)},e_{1}(u)\big{\\}}\right).$

Similar relations hold for higher currents (6.6) as well.

We note that (6.7)–(6.12) coincide exactly with commutation relations for
$Y(\widehat{\mathfrak{gl}}(1|1))$ listed in [11]. It is important to note that
$RLL$-algebra contains more currents than $Y(\widehat{\mathfrak{gl}}(1|1))$.
In particular, it has the current $h(\bm{u})$, which is not in
$Y(\widehat{\mathfrak{gl}}(1|1))$. Moreover this current depends on two
spectral parameters rather than one. The precise relation between two
algebras, $RLL$ and $Y(\widehat{\mathfrak{gl}}(1|1))$, 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 $Y(\widehat{\mathfrak{gl}}(1|1))$ 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
$Y(\widehat{\mathfrak{gl}}(1))$).

##  7 Conclusion

In this paper we derived current relations of
$Y(\widehat{\mathfrak{gl}}(1|1))$ (6.7)–(6.12) starting from $RLL$
realization. First, we defined the $R$-matrix, which serves as an intertwining
operator for certain $\mathcal{N}=2$ 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 $\uppsi_{i}(u)$
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)).

  * •

It is interesting to describe explicitly the embedding
$Y(\widehat{\mathfrak{gl}}(1))\oplus Y(\widehat{\mathfrak{gl}}(1))\in
Y(\widehat{\mathfrak{gl}}(1|1))$, which is suggested by the results of [7, 8].
Namely, one expects to construct two copies of $Y(\widehat{\mathfrak{gl}}(1))$
currents $(e(u),f(u),\uppsi(u))$ and
$(\hat{e}(u),\hat{f}(u),\hat{\uppsi}(u))$, as well as auxiliary fermionic
currents $x(u)$, $\bar{x}(u)$, $y(u)$ and $\bar{y}(u)$ (see [7, 8]).

  * •

Using current realization of $Y(\widehat{\mathfrak{gl}}(1|1))$ 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 $\mathcal{N}=2$ KdV hierarchy and its $W_{n}$ generalizations. We
plan to provide more details elsewhere.

## Acknowledgements

We acknowledge the discussions with Mikhail Bershtein, Boris Feigin, Mikhail
Lashkevich, Andrii Liashyk and Sergei Parkhomenko.

### Funding information:

this work has been supported by the Russian Science Foundation under the grant
22-22-00991 and by Basis foundation.

##  Appendix A Current commutation relations

Here we provide some details on derivation of commutation relations of
$Y(\widehat{\mathfrak{gl}}(1|1))$.

###  $h\rho$ and $h\rho^{*}$ commutation relations.

Using $RLL$ relations on level $1$ and charge $0$ one finds:

$h(\bm{u})\rho(\bm{v})=\dfrac{((u-u^{*})-(v-v^{*}))((u+u^{*})-(v-v^{*})-b+b^{-1})}{((u-u^{*})-(v-v^{*})-b+2b^{-1})((u+u^{*})-(v-v^{*})+b^{-1})}\rho(\bm{v})h(\bm{u})+\\\
+\dfrac{2(u^{*}+\frac{1}{2b})}{(u+u^{*})-(v-v^{*})+b^{-1}}h(\bm{u})\rho(\bm{u})-\\\
-\dfrac{2((u-u^{*})-(v-v^{*}))(u^{*}-\frac{1}{2b})}{((u-u^{*})-(v-v^{*})-b+2b^{-1})((u+u^{*})-(v-v^{*})+b^{-1})}h(\bm{u})\rho^{*}(\bm{u})+\\\
+\frac{4((u-u^{*})-(v-v^{*}))(u^{*}-\frac{1}{2b})}{((u-u^{*})-(v-v^{*})-b+2b^{-1})((u+u^{*})-(v-v^{*})+b^{-1})}h(\bm{u})a^{*}(\bm{u})+\\\
+\frac{4i((u-u^{*})-(v-v^{*}))(u^{*}-\frac{1}{2b})}{((u-u^{*})-(v-v^{*})-b+2b^{-1})((u+u^{*})-(v-v^{*})+b^{-1})}e(\bm{v})h(\bm{u})e^{*}(\bm{u})$
(A.1)

and

$h(\bm{u})\rho^{*}(\bm{v})=\dfrac{((u+u^{*})-(v+v^{*}))((u-u^{*})-(v+v^{*})+b+b^{-1})}{((u-u^{*})-(v+v^{*})+b^{-1})((u+u^{*})-(v+v^{*})+b+2b^{-1})}\rho^{*}(\bm{v})h(\bm{u})-\\\
-\dfrac{2(u^{*}-\frac{1}{2b})}{(u-u^{*})-(v+v^{*})+b^{-1}}h(\bm{u})\rho^{*}(\bm{u})+\\\
+\dfrac{2((u+u^{*})-(v+v^{*}))(u^{*}+\frac{1}{2b}))}{((u-u^{*})-(v+v^{*})+b^{-1})((u+u^{*})-(v+v^{*})+b+2b^{-1})}h(\bm{u})(a(\bm{u})+a^{*}(\bm{u}))-\\\
-\dfrac{4i((u+u^{*})-(v+v^{*}))(u^{*}+\frac{1}{2b}))}{((u-u^{*})-(v+v^{*})+b^{-1})((u+u^{*})-(v+v^{*})+b+2b^{-1})}e^{*}(\bm{v})h(\bm{u})e(\bm{u})$
(A.2)

###  $ee$ commutation relation.

We introduce the following notations

$\begin{gathered}\mathcal{L}_{\varnothing\psi}(\bm{u})\overset{\text{def}}{=}\langle\varnothing|\mathcal{L}(\bm{u})\psi_{-\frac{1}{2}}|\varnothing\rangle,\quad\mathcal{L}_{\varnothing\psi^{*}}(\bm{u})\overset{\text{def}}{=}\langle\varnothing|\mathcal{L}(\bm{u})\psi^{*}_{-\frac{1}{2}}|\varnothing\rangle,\\\
\mathcal{L}_{\psi\varnothing}(\bm{u})\overset{\text{def}}{=}\langle\varnothing|\psi_{\frac{1}{2}}\mathcal{L}(\bm{u})|\varnothing\rangle,\quad\mathcal{L}_{\psi^{*}\varnothing}(\bm{u})\overset{\text{def}}{=}\langle\varnothing|\psi^{*}_{\frac{1}{2}}\mathcal{L}(\bm{u})|\varnothing\rangle,\\\
\mathcal{L}_{\psi\psi^{*}}(\bm{u})\overset{\text{def}}{=}\langle\varnothing|\psi_{\frac{1}{2}}\mathcal{L}(\bm{u})\psi^{*}_{-\frac{1}{2}}|\varnothing\rangle.\end{gathered}$
(A.3)

Then equation

$\langle\varnothing|\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})(\psi_{1})_{-\frac{1}{2}}(\psi_{2})_{-\frac{1}{2}}|\varnothing\rangle=\langle\varnothing|\mathcal{L}_{2}(\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{R}_{12}(\bm{u},\bm{v})(\psi_{1})_{-\frac{1}{2}}(\psi_{2})_{-\frac{1}{2}}|\varnothing\rangle$
(A.4)

can be rewritten in the explicit form

$\mathcal{L}_{\varnothing\psi}(\bm{u})\mathcal{L}_{\varnothing\psi}(\bm{v})=-\dfrac{(u-u^{*})-(v+v^{*})-b^{-1}}{(u+u^{*})-(v-v^{*})+b^{-1}}\mathcal{L}_{\varnothing\psi}(\bm{v})\mathcal{L}_{\varnothing\psi}(\bm{u}),$
(A.5)

where we considered the action of $R$-matrix on
$\mathcal{R}_{12}(\bm{u},\bm{v})(\psi_{1})_{-\frac{1}{2}}(\psi_{2})_{-\frac{1}{2}}|\varnothing\rangle$.
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

$\langle\varnothing|\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})(\psi_{1})_{-\frac{1}{2}}|\varnothing\rangle=\langle\varnothing|\mathcal{L}_{2}(\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{R}_{12}(\bm{u},\bm{v})(\psi_{1})_{-\frac{1}{2}}|\varnothing\rangle,$
(A.6)

which takes the form (compare to (4.21))

$\mathcal{L}_{\varnothing\psi}(\bm{u})h(\bm{v})=\dfrac{(u+u^{*})-(v+v^{*})}{(u+u^{*})-(v-v^{*})+b^{-1}}h(\bm{v})\mathcal{L}_{\varnothing\psi}(\bm{u})+\dfrac{2v^{*}+b^{-1}}{(u+u^{*})-(v-v^{*})+b^{-1}}\mathcal{L}_{\varnothing\psi}(\bm{v})h(\bm{u}).$
(A.7)

Then multiplying (A.5) by $h^{-1}(\bm{u})h^{-1}(\bm{v})$ on the left and using
(A.7), we get the anticommutation relation

$\\{e(\bm{u}),e(\bm{v})\\}=\dfrac{2u^{*}+b^{-1}}{(u+u^{*})-(v-v^{*})+b^{-1}}e^{2}(\bm{u})-\dfrac{2v^{*}+b^{-1}}{(u-u^{*})-(v+v^{*})-b^{-1}}e^{2}(\bm{v}).$
(A.8)

The relations (A.8) should not have any poles. It implies that
$e^{2}(\bm{u})=0$ and $\\{e(\bm{u}),e(\bm{v})\\}=0$.

###  $ee^{*}$ commutation relation.

We consider the equation

$\langle\varnothing|\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})(\psi_{1})_{-\frac{1}{2}}(\psi^{*}_{2})_{-\frac{1}{2}}|\varnothing\rangle=\langle\varnothing|\mathcal{L}_{2}(\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{R}_{12}(\bm{u},\bm{v})(\psi_{1})_{-\frac{1}{2}}(\psi^{*}_{2})_{-\frac{1}{2}}|\varnothing\rangle.$
(A.9)

In principle, one can add to this formula an arbitrary linear combination of
$RLL$-relations at level 1 with charge 0:

$\langle\varnothing|\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})(\psi_{1})_{-\frac{1}{2}}(\psi^{*}_{2})_{-\frac{1}{2}}|\varnothing\rangle+\alpha_{1}\langle\varnothing|\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})(a_{1})_{-1}|\varnothing\rangle+\\\
+\alpha_{2}\langle\varnothing|\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})(\psi_{1})_{-\frac{1}{2}}(\psi^{*}_{1})_{-\frac{1}{2}}|\varnothing\rangle+\dots{}$
(A.10)

but we leave only those terms which, after multiplying by
$h^{-1}(\bm{u})h^{-1}(\bm{v})$ from the left, turn out to be local:

$\langle\varnothing|\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})(\psi_{1})_{-\frac{1}{2}}(\psi^{*}_{2})_{-\frac{1}{2}}|\varnothing\rangle+\alpha\langle\varnothing|\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})(a_{2})_{-1}|\varnothing\rangle+\\\
+\beta\langle\varnothing|\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})(a^{*}_{2})_{-1}|\varnothing\rangle+\gamma\langle\varnothing|\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})(\psi_{2})_{-\frac{1}{2}}(\psi^{*}_{2})_{-\frac{1}{2}}|\varnothing\rangle.$
(A.11)

The coefficients $\alpha$, $\beta$, $\gamma$ are found from the condition of
cancellation of the terms

$\langle\varnothing|\mathcal{L}_{2}(\bm{v})\mathcal{L}_{1}(\bm{u})(a_{2})_{-1}|\varnothing\rangle,\quad\langle\varnothing|\mathcal{L}_{2}(\bm{v})\mathcal{L}_{1}(\bm{u})(a^{*}_{2})_{-1}|\varnothing\rangle,\quad\langle\varnothing|\mathcal{L}_{2}(\bm{v})\mathcal{L}_{1}(\bm{u})(\psi_{2})_{-\frac{1}{2}}(\psi^{*}_{2})_{-\frac{1}{2}}|\varnothing\rangle,$
(A.12)

in the right side of the $RLL$-relation (A.11). Their explicit expressions are

$\begin{gathered}\alpha=-\dfrac{i(2v^{*}-b^{-1})}{((u-u^{*})-(v-v^{*}))(\Delta+h_{1}+\frac{b}{2}))},\quad\beta=-\dfrac{2i((u-u^{*})-v+\frac{1}{2b})}{((u-u^{*})-(v-v^{*}))(\Delta+h_{1}+\frac{b}{2}))},\\\
\gamma=\dfrac{2}{((u-u^{*})-(v-v^{*}))(\Delta+h_{1}+\frac{b}{2}))}-\dfrac{2v^{*}+b^{-1}}{(u-u^{*})-(v-v^{*})}.\end{gathered}$
(A.13)

Here and below, we use the following notation

$\Delta=(u-u^{*})-(v+v^{*}),\quad h_{1}=\dfrac{b}{2}+b^{-1},\quad
h_{2}=\dfrac{b}{2}-b^{-1}.$ (A.14)

Then, taking into account the action of the $R$-matrix on the states of level
$1$ with charge $0$, we get

$\mathcal{L}_{\varnothing\psi}(\bm{u})\mathcal{L}_{\varnothing\psi^{*}}(\bm{v})+\alpha
h(\bm{u})\mathcal{L}_{\varnothing a}(\bm{v})+\beta
h(\bm{u})\mathcal{L}_{\varnothing
a^{*}}(\bm{v})+\gamma\mathcal{L}_{\varnothing k}(\bm{v})=\\\
=-\dfrac{((u+u^{*})-(v+v^{*}))(\Delta+h_{2}+\frac{b}{2})}{((u-u^{*})-(v-v^{*}))(\Delta+h_{1}+\frac{b}{2})}\mathcal{L}_{\varnothing\psi^{*}}(\bm{v})\mathcal{L}_{\varnothing\psi}(\bm{u})-\\\
-\dfrac{(2v^{*}-b^{-1})(\Delta+h_{2}+\frac{b}{2})}{((u-u^{*})-(v-v^{*}))(\Delta+h_{1}+\frac{b}{2})}h(\bm{v})\mathcal{L}_{\varnothing
k}(\bm{u})+\\\
+\dfrac{i(2v^{*}-b^{-1})}{((u-u^{*})-(v-v^{*}))(\Delta+h_{1}+\frac{b}{2})}h(\bm{v})(\mathcal{L}_{\varnothing
a^{*}}(\bm{u})-\mathcal{L}_{\varnothing a}(\bm{u})).$ (A.15)

Multiplying this expression by $h^{-1}(\bm{u})h^{-1}(\bm{v})$ on the left and
using commutation relations
$h^{-1}(\bm{v})\mathcal{L}_{\varnothing\psi}(\bm{u})$ and
$h^{-1}(\bm{u})\mathcal{L}_{\varnothing\psi^{*}}(\bm{v})$, one obtains

$e(\bm{u})e^{*}(\bm{v}){\color[rgb]{0,0,1}-\frac{i\rho^{*}(\bm{v})}{(\Delta+h_{1}+\frac{b}{2})(\Delta-
h_{1}+\frac{b}{2})}}{\color[rgb]{0,0,1}+\frac{\overbrace{(2v^{*}+b^{-1})\big{(}e(\bm{v})e^{*}(\bm{v})-k(\bm{v})\big{)}-2ia^{*}(\bm{v})}^{\sigma^{*}(\bm{v})}}{(\Delta-
h_{1}+\frac{b}{2})}}=\\\ =-\dfrac{(\Delta+h_{2}+\frac{b}{2})(\Delta-
h_{2}+\frac{b}{2})}{(\Delta+h_{1}+\frac{b}{2})(\Delta-
h_{1}+\frac{b}{2})}e^{*}(\bm{v})e(\bm{u}){\color[rgb]{0,0,1}-\frac{i\rho(\bm{u})}{(\Delta+h_{1}+\frac{b}{2})(\Delta-
h_{1}+\frac{b}{2})}}{\color[rgb]{0,0,1}-}\\\
{\color[rgb]{0,0,1}-\frac{(\Delta+h_{2}+\frac{b}{2})}{(\Delta+h_{1}+\frac{b}{2})(\Delta-
h_{1}+\frac{b}{2})}\underbrace{(2u^{*}-b^{-1})\left(e^{*}(\bm{u})e(\bm{u})+k(\bm{u})\right)}_{\sigma(\bm{u})}},$
(A.16)

We see that, this commutation relation corresponds to (6.9).

###  $e^{*}f$ commutation relation.

Let us consider the equation

$\langle\varnothing|(\psi_{2})_{\frac{1}{2}}\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})(\psi_{1}^{*})_{-\frac{1}{2}}|\varnothing\rangle=\langle\varnothing|(\psi_{2})_{\frac{1}{2}}\mathcal{L}_{2}(\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{R}_{12}(\bm{u},\bm{v})(\psi_{1}^{*})_{-\frac{1}{2}}|\varnothing\rangle.$
(A.17)

Then from the definition of the $R$-matrix one finds

$-\dfrac{2u^{*}-b^{-1}}{(u-u^{*})-(v+v^{*})+b^{-1}}\mathcal{L}_{\psi\psi^{*}}(\bm{u})h(\bm{v})-\dfrac{(u+u^{*})-(v+v^{*})}{(u-u^{*})-(v+v^{*})+b^{-1}}\mathcal{L}_{\varnothing\psi^{*}}(\bm{u})\mathcal{L}_{\psi\varnothing}(\bm{v})=\\\
=\dfrac{(u-u^{*})-(v-v^{*})}{(u-u^{*})-(v+v^{*})+b^{-1}}\mathcal{L}_{\psi\varnothing}(\bm{v})\mathcal{L}_{\varnothing\psi^{*}}(\bm{u})-\dfrac{2u^{*}-b^{-1}}{(u-u^{*})-(v+v^{*})+b^{-1}}\mathcal{L}_{\psi\psi^{*}}(\bm{v})h(\bm{u}).$
(A.18)

Let us also consider two auxiliary equations

$\displaystyle\langle\varnothing|\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})(\psi_{1}^{*})_{-\frac{1}{2}}|\varnothing\rangle=\langle\varnothing|\mathcal{L}_{2}(\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{R}_{12}(\bm{u},\bm{v})(\psi_{1}^{*})_{-\frac{1}{2}}|\varnothing\rangle,$
(A.19)
$\displaystyle\langle\varnothing|(\psi_{2})_{\frac{1}{2}}\mathcal{R}_{12}(\bm{u},\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{L}_{2}(\bm{v})|\varnothing\rangle=\langle\varnothing|(\psi_{2})_{\frac{1}{2}}\mathcal{L}_{2}(\bm{v})\mathcal{L}_{1}(\bm{u})\mathcal{R}_{12}(\bm{u},\bm{v})|\varnothing\rangle,$
(A.20)

which can be rewritten explicitly as

$\mathcal{L}_{\varnothing\psi^{*}}(\bm{u})h(\bm{v})=\dfrac{(u-u^{*})-(v-v^{*})}{(u-u^{*})-(v+v^{*})+b^{-1}}h(\bm{v})\mathcal{L}_{\varnothing\psi^{*}}(\bm{u})-\dfrac{2u^{*}-b^{-1}}{(u-u^{*})-(v+v^{*})+b^{-1}}\mathcal{L}_{\varnothing\psi^{*}}(\bm{v})h(\bm{u}),$
(A.21)

and

$\mathcal{L}_{\psi\varnothing}(\bm{v})h(\bm{u})=\dfrac{(u+u^{*})-(v+v^{*})}{(u-u^{*})-(v+v^{*})+b^{-1}}h(\bm{u})\mathcal{L}_{\psi\varnothing}(\bm{v})-\dfrac{2u^{*}-b^{-1}}{(u-u^{*})-(v+v^{*})+b^{-1}}\mathcal{L}_{\psi\varnothing}(\bm{u})h(\bm{v}),$
(A.22)

Multiplying (A.18) by $h^{-1}(\bm{u})h^{-1}(\bm{v})$ from the right and using
these relations, one finds

$-\mathcal{L}_{\varnothing\psi^{*}}(\bm{u})h^{-1}(\bm{u})\mathcal{L}_{\psi\varnothing}(\bm{v})h^{-1}(\bm{v})-\dfrac{2u^{*}-b^{-1}}{(u-u^{*})-(v+v^{*})+b^{-1}}\mathcal{L}_{\psi\psi^{*}}(\bm{u})h^{-1}(\bm{u})-\\\
-\dfrac{2u^{*}-b^{-1}}{(u-u^{*})-(v+v^{*})+b^{-1}}\mathcal{L}_{\varnothing\psi^{*}}(\bm{u})h^{-1}(\bm{u})\mathcal{L}_{\psi\varnothing}(\bm{u})h^{-1}(\bm{u})=\\\
=\mathcal{L}_{\psi\varnothing}(\bm{v})h^{-1}(\bm{v})\mathcal{L}_{\varnothing\psi^{*}}(\bm{u})h^{-1}(\bm{u})-\dfrac{2u^{*}-b^{-1}}{(u-u^{*})-(v+v^{*})+b^{-1}}\mathcal{L}_{\psi\psi^{*}}(\bm{v})h^{-1}(\bm{v})+\\\
+\dfrac{2u^{*}-b^{-1}}{(u-u^{*})-(v+v^{*})+b^{-1}}\mathcal{L}_{\psi\varnothing}(\bm{v})h^{-1}(\bm{v})\mathcal{L}_{\varnothing\psi^{*}}(\bm{v})h^{-1}(\bm{v}).$
(A.23)

Equation (5.6) implies

$e^{*}(u-u^{*}+b^{-1})=\dfrac{1}{2u^{*}-b^{-1}}\mathcal{L}_{\varnothing\psi^{*}}(\bm{u})h^{-1}(\bm{u}),$
(A.24)

from which one can obtain

$\\{e^{*}(u-u^{*}+b^{-1}),f(\bm{v})\\}=\dfrac{\uppsi^{*}(u,-u^{*}+b^{-1})-\uppsi^{*}(\bm{v})}{(u-u^{*})-(v+v^{*})+b^{-1}},$
(A.25)

where (compare to (5.20))

$\uppsi^{*}(u+u^{*})\overset{\text{def}}{=}\mathcal{L}_{\psi\varnothing}(\bm{u})h^{-1}(\bm{u})\mathcal{L}_{\varnothing\psi^{*}}(\bm{u})h^{-1}(\bm{u})-\mathcal{L}_{\psi\psi^{*}}(\bm{u})h^{-1}(\bm{u}).$
(A.26)

So, this formula corresponds to (6.8). Similarly, one can get the commutation
relation for $e(\bm{u})$ and $f^{*}(\bm{v})$:

$\\{e(u+u^{*}+b^{-1}),f^{*}(\bm{v})\\}=\dfrac{\uppsi(u,-u^{*}-b^{-1})-\uppsi(\bm{v})}{(u-u^{*})-(v-v^{*})}$
(A.27)

with

$\uppsi(u-u^{*})\overset{\text{def}}{=}\mathcal{L}_{\psi^{*}\psi}(\bm{u})h^{-1}(\bm{u})-\mathcal{L}_{\psi^{*}\varnothing}(\bm{u})h^{-1}(\bm{u})\mathcal{L}_{\varnothing\psi}(\bm{u})h^{-1}(\bm{u}).$
(A.28)

The remaining relations from (6.7)–(6.12) can be derived in a similar way.

## References

  * [1] D. Maulik and A. Okounkov, _Quantum Groups and Quantum Cohomology_ , _Astérisque_ 408 (2019)  [1211.1287].
  * [2] A. B. Zamolodchikov and A. B. Zamolodchikov, _Structure constants and conformal bootstrap in Liouville field theory_ , _Nucl. Phys. B_ 477 (1996) 577 [hep-th/9506136].
  * [3] A. Tsymbaliuk, _The affine Yangian of $\mathfrak{gl}_{1}$ revisited_, _Adv. Math._ 304 (2017) 583 [1404.5240].
  * [4] A. Litvinov and I. Vilkoviskiy, _Liouville reflection operator, affine Yangian and Bethe ansatz_ , _JHEP_ 12 (2020) 100 [2007.00535].
  * [5] T. Procházka, _$\mathcal{W}$ -symmetry, topological vertex and affine Yangian_, _JHEP_ 10 (2016) 077 [1512.07178].
  * [6] M. R. Gaberdiel, R. Gopakumar, W. Li and C. Peng, _Higher spins and Yangian symmetries_ , _JHEP_ 04 (2017) 152 [1702.05100].
  * [7] M. R. Gaberdiel, W. Li, C. Peng and H. Zhang, _The supersymmetric affine Yangian_ , _JHEP_ 05 (2018) 200 [1711.07449].
  * [8] M. R. Gaberdiel, W. Li and C. Peng, _Twin-plane-partitions and $\mathcal{N}=2$ affine Yangian_, _JHEP_ 11 (2018) 192 [1807.11304].
  * [9] W. Li and P. Longhi, _Gluing two affine Yangians of $\mathfrak{gl}_{1}$_, _JHEP_ 10 (2019) 131 [1905.03076].
  * [10] W. Li, _Gluing affine Yangians with bi-fundamentals_ , _JHEP_ 06 (2020) 182 [1910.10129].
  * [11] W. Li and M. Yamazaki, _Quiver Yangian from Crystal Melting_ , _JHEP_ 11 (2020) 035 [2003.08909].
  * [12] D. Galakhov and M. Yamazaki, _Quiver Yangian and Supersymmetric Quantum Mechanics_ , 2008.07006.
  * [13] D. Galakhov, W. Li and M. Yamazaki, _Shifted quiver Yangians and representations from BPS crystals_ , _JHEP_ 08 (2021) 146 [2106.01230].
  * [14] Y. Kazama and H. Suzuki, _New $N=2$ superconformal field theories and Superstring compactification_, _Nucl. Phys. B_ 321 (1989) 232.
  * [15] J. Evans and T. J. Hollowood, _Supersymmetric Toda field theories_ , _Nucl. Phys. B_ 352 (1991) 723.
  * [16] S. Komata, K. Mohri and H. Nohara, _Classical and quantum extended superconformal algebra_ , _Nucl. Phys. B_ 359 (1991) 168.
  * [17] K. Ito, _$N=2$ superconformal $\mathbb{CP}(n)$ model_, _Nucl. Phys. B_ 370 (1992) 123.
  * [18] L. Rozansky and H. Saleur, _Quantum field theory for the multivariable Alexander-Conway polynomial_ , _Nucl. Phys. B_ 376 (1992) 461.
  * [19] E. Witten, _Nonabelian bosonization in two-dimensions_ , _Comm. Math. Phys._ 92 (1984) 455.
  * [20] M. Wakimoto, _Fock representations of the affine lie algebra $A1(1)$_, _Commun. Math. Phys._ 104 (1986) 605.
  * [21] E. Chistyakova, A. Litvinov and P. Orlov, _Affine Yangian of $\mathfrak{gl}(2)$ and integrable structures of superconformal field theory_, _JHEP_ 03 (2022) 102 [2110.05870].
  * [22] N. Jing, M. Liu and A. Molev, _Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: Type C_ , _J. Math. Phys._ 61 (2020) 031701 [1903.00204].
  * [23] B. S. Shastry, _Exact Integrability of the One-Dimensional Hubbard Model_ , _Phys. Rev. Lett._ 56 (1986) 2453.
  * [24] B. Szendroi, _Non-commutative Donaldson–Thomas invariants and the conifold_ , _Geom. Topol._ 12 (2008) 1171 [0705.3419].