# Bethe ansatz equations
for quantum $\mathcal{N}=2$ KdV systems

Dmitry Kolyaskin1 and Alexey Litvinov2,3
1\. Australian National University, Canberra ACT 2601, Australia
2\. Krichever Center for Advanced Studies, Skolkovo Institute of Science and
Technology, 143026 Moscow, Russia
3\. Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia

###### Abstract

Based on our previous studies of affine Yangian of
$\widehat{\mathfrak{gl}}(1|1)$ we propose Bethe ansatz equations for the
spectrum of $\mathcal{N}=2$ quantum KdV systems.

##  1 Introduction

The systematic study of integrable models derived from corresponding conformal
field theories (CFTs) was initiated by A. B. Zamolodchikov in [1]. It was
shown that for particular perturbing operators, an infinite number of mutually
commuting integrals of motion (IMs) are preserved. This phenomenon can be
illustrated by a large class of 2D CFTs, which are described by Toda action

$\displaystyle
S_{0}=\int\bigg{(}\dfrac{1}{8\pi}(\partial_{\mu}\bm{\varphi},\partial_{\mu}\bm{\varphi})+\Lambda\sum\limits_{r=1}^{N}e^{b(\bm{\alpha}_{r},\bm{\varphi})}\bigg{)}d^{2}x,$
(1)

where $\bm{\varphi}=(\varphi_{1},...,\varphi_{N})$ is the $N$-component
bosonic field and $\\{\bm{\alpha}_{1},...,\bm{\alpha}_{N}\\}$ is the set of
linearly independent vectors. It is known that under certain conditions on
these vectors the model contains a larger symmetry. Namely, it turns out that
this theory has an extended conformal symmetry, which is also called $W$ \-
algebra [2], provided that $\bm{\alpha}_{r}$ are the simple roots of a semi-
simple Lie algebra $\mathfrak{g}$ of rank $N$. Corresponding integrable QFT is
described by the perturbation

$\displaystyle S_{0}\rightarrow S=S_{0}+\lambda\int
e^{b(\bm{\alpha}_{0},\bm{\varphi})}d^{2}x$ (2)

where the additional vector $\bm{\alpha}_{0}$ is either untwisted or twisted
affine root of the corresponding affine system. Integrability implies that the
resulting theory survives an infinite set of mutually commuting Integrals of
Motion

$\displaystyle\bm{I}_{m}(\lambda)=\bm{I}_{m}+O(\lambda),\quad\bar{\bm{I}}_{m}(\lambda)=\bar{\bm{I}}_{m}+O(\lambda)$
(3)

perturbative in $\lambda$, where $\bm{I}_{m},\bar{\bm{I}}_{m}$ are the
Integrals of Motion related to $W-$algebra in CFT.

Another example of integrable perturbation from CFT and the main object of the
current work is the supersymmetric generalization of Toda system whose action
is of the form

$S_{0}^{SUSY}=\dfrac{1}{2\pi}\int\bigg{(}(D\mathbf{\Phi},\bar{D}\mathbf{\Phi})+\dfrac{2\pi\Lambda}{b^{2}}\sum\limits_{r=1}^{N}e^{b(\bm{\alpha}_{r},\mathbf{\Phi})}\bigg{)}d^{2}zd^{2}\theta,$
(4)

where $\bm{\Phi}=(\Phi_{1},\dots,\Phi_{N})$ is the bosonic $N-$component
superfield

$\bm{\Phi}(z,\bar{z},\theta,\bar{\theta})=\bm{\varphi}(z,\bar{z})+\theta\bm{\psi}+\bar{\theta}\bar{\bm{\psi}}+\theta\bar{\theta}\bm{F}$
(5)

and superderivatives are defined as

$D=\frac{\partial}{\partial\theta}-\theta\partial,\qquad\bar{D}=\frac{\partial}{\partial\bar{\theta}}-\bar{\theta}\bar{\partial}.$
(6)

In this case, integrability implies that the set of vectors
$\\{\bm{\alpha}_{r}\\}$ is the completely odd system of simple roots of some
Lie superalgebra (which admits such a set). This model is both classical and
quantum integrable ultimately invariant under $\mathcal{N}=1$ supersymmetric
transformations [3, 4]. In particular, in this work we focus on the theories
based on $A(n|n)$ Lie superalgebras (for which one can choose a purely odd
system of simple roots). Actually, in addition to explicit $N=1$
supersymmetry, $A(n|n)$ Toda theory is $\mathcal{N}=2$ supersymmetric [4]. The
integrable perturbation of the action (4) is realized by the affine untwisted
extension $\hat{A}(n|n)$

$\displaystyle S_{0}^{SUSY}\rightarrow S^{SUSY}=S_{0}^{SUSY}+\lambda\int
e^{b(\bm{\alpha_{0}},\bm{\Phi})}d^{2}x,$ (7)

where $\bm{\alpha}_{0}$ is the lowest fermionic root and the perturbed theory
is still $N=2$ supersymmetric [3].

A separate problem of the integrable quantum field theories arises regarding
simultaneous diagonalization of corresponding IMs. The systematic study of
this question was initiated in the series of work by Bazhanov, Lukyanov and
Zamolodchikov [5, 6, 7]. Their research was devoted to the basic example of
the integrable model in CFT - the quantum KdV system, which corresponds to
$\mathfrak{sl}_{2}$ case. Among the main results of the study was the
construction of the so-called $\bm{T}$-operator, which plays the role of the
generating function for the local and nonlocal IMs, and obtaining functional
relations on it. And finally, the work [8] of the same authors drew a line in
the study of the spectrum of IMs for $\mathfrak{sl}_{2}$ model. Their approach
was based on the connection between integrable systems of CFT and spectral
theory of particular differential equation [9, 10, 11], and the eigenvalues of
IMs were expressed through the solutions to a particular system of algebraic
equations.

There are also different approaches to the diagonalization problem including
those based on the affine Yangian Symmetry [12], which, in particular, we
apply to finding the spectra of $\mathcal{N}=2$ KdV system in the current
work. This approach allows to obtain Bethe ansatz equations, and it is
expected that the corresponding eigenvalues of local and quasi-local IMs are
expected to be symmetric polynomials in Bethe roots (see also [13, 14]). So,
for example, based on Yangian symmetry, Bethe ansatz equations were derived,
and diagonalization problem was solved for the $Y(\widehat{\mathfrak{gl}}(1))$
case [15] and for $Y(\widehat{\mathfrak{gl}}(2))$ case [16]. Following similar
approach, we find the spectra for local and quasi-local IMs of quantum
$\mathcal{N}=2$ KdV hierarchy. Our present note will heavily draws on our
recent paper [17], where the current realization of affine Yangian
$Y(\widehat{\mathfrak{gl}}(1|1))$ was obtained, and can be considered as
sequel.

The content of the paper is organized as follows. In section 2 we introduce
quantum $\mathcal{N}=2$ KdV systems, as they appear in field theory. In
section 3 we formulate the spectral problem for the simplest model with
$\mathcal{N}=2$ Virasoro symmetry and provide Bethe ansatz equations for its
spectrum. In section 4 we sketch the relation between $\mathcal{N}=2$ KdV
systems and $\textrm{Y}(\widehat{\mathfrak{gl}}(1|1))$ introduced in [17]. We
also sketch the derivation of Bethe ansatz equations developed in [15, 16].

##  2 Quantum $\mathcal{N}=2$ KdV systems

The most natural way to treat the affine supersymmetric $\mathfrak{sl}(n|n)$
Toda field theory ($A(n-1|n-1)$ Toda) is to consider it as an integrable
perturbation of the conformal $\mathfrak{sl}(n|n-1)$ Toda field theory by two
exponential operators corresponding to auxiliary $\mathfrak{sl}(n|n)$ root and
to affine root of $\mathfrak{sl}(n|n)$. It is well known that
$\mathfrak{sl}(n|n-1)$ Toda field theory is invariant under the action of
$\mathcal{N}=2$ $W_{n}$-algebra [18]. This class of algebras contain one
$\mathcal{N}=2$ multiplet of holomorphic currents for each integers spin
$s=1,\dots,n-1$. Perturbing $\mathfrak{sl}(n|n-1)$ by two exponential
operators breaks down the $W_{n}$ symmetry, but survives an infinite tower of
Integrals of Motion. The ultraviolet limit of these Integrals of Motion splits
into two sets holomorphic and antiholomorphic conformal IM’s. Each of these
sets corresponds to integrable system that we will call quantum
$\mathcal{N}=2$ $\text{KdV}_{n}$ system with $n\geq 2$. In the following we
will consider in details the case of $n=2$.

Consider $N=2$ Virasoro algebra, which is generated by the currents of spins
$(1,\frac{3}{2},\frac{3}{2},2)$

$J(x)=\sum_{n\in\mathbb{Z}}J_{n}e^{-inx},\quad
G^{\pm}(x)=\sum_{r}G^{\pm}_{r}e^{-irx}\quad\text{and}\quad
T(x)=\sum_{n\in\mathbb{Z}}L_{n}e^{-inx}-\frac{c}{24},$ (8)

with commutation relations

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c}{12}(m^{3}-m)\delta_{m,-n},$
(9) $\displaystyle[J_{m},J_{n}]=\frac{c}{3}m\delta_{m,-n},$
$\displaystyle[L_{m},J_{n}]=-nJ_{m+n},$
$\displaystyle\\{G^{+}_{r},G^{-}_{s}\\}=L_{r+s}+\frac{r-s}{2}J_{r+s}+\frac{c}{6}\left(r^{2}-\frac{1}{4}\right)\delta_{r,-s},$
$\displaystyle[L_{m},G^{\pm}_{r}]=\left(\frac{m}{2}-r\right)G^{\pm}_{r+m},$
$\displaystyle[J_{m},G^{\pm}_{r}]=\pm G^{\pm}_{r+m}.$

This algebra acts in the Hilbert space of some $N=2$ superconformal field
theory which splits into direct sum of irreducible highest weight
representations. There are two types of representations

$\text{NS}:\,r\in\mathbb{Z}+\frac{1}{2},\qquad\text{R}:\,r\in\mathbb{Z}.$ (10)

In the following we will consider NS representations.

Quantum $\mathcal{N}=2$ KdV system is a direct analog of the bosonic one [19].
It can be defined as an infinite dimensional abelian subalgebra in universal
enveloping of $\mathcal{N}=2$ Virasoro algebra (9). This subalgebra is spanned
by local and quasilocal integrals of odd spins

$\mathbf{I}_{2k-1}=\frac{1}{2\pi}\int_{0}^{2\pi}G_{2k}(x)dx,\quad\tilde{\mathbf{I}}_{2k-1}=\frac{1}{2\pi}\int_{0}^{2\pi}\tilde{G}_{2k}(x)dx.$
(11)

Local densities are analytically regularized polynomials in currents (8) and
their derivatives. In particular

$G_{2}(x)=T(x)\implies\mathbf{I}_{1}=L_{0}-\frac{c}{24}.$ (12)

Conversely, the quasilocal densities (not to be confused to non-local ones)
might include negative degrees of derivative (see below).

In order to make contact with the supersymmetric $\mathfrak{sl}(2|1)$ Toda
field theory, it is convenient to use the free field representation of the
algebra (9). Namely, the generating currents (8) can be rewritten in terms of
holomorphic free fields

$X(z)X^{*}(w)=-2\log(z-w)+\dots,\quad\psi(z)\psi^{*}(w)=\frac{2}{z-w}+\dots$
(13)

as

$\begin{gathered}J=\frac{1}{2}\psi\psi^{*}-\frac{1}{b}\partial\bigl{(}X-X^{*}\bigr{)},\\\
G^{+}=\frac{i}{2}\left(\partial X^{*}\psi-\frac{2}{b}\partial\psi\right),\quad
G^{-}=\frac{i}{2}\left(\partial
X\psi^{*}-\frac{2}{b}\partial\psi^{*}\right),\\\ T=-\frac{1}{2}\partial
X\partial
X^{*}-\frac{1}{4}\bigl{(}\psi\partial\psi^{*}+\psi^{*}\partial\psi\bigr{)}+\frac{1}{2b}\partial^{2}\bigl{(}X+X^{*}\bigr{)},\end{gathered}$
(14)

where the Wick ordering is assumed. The modes of these currents obey the
relations of $\mathcal{N}=2$ Virasoro algebra (9) with the central charge

$c=3\left(1+\dfrac{4}{b^{2}}\right).$ (15)

One can check that (14) commute with the screening fields

$\mathcal{S}_{1}=\oint\psi^{*}e^{\frac{b}{2}X^{*}}dz,\qquad\mathcal{S}_{2}=\oint\psi
e^{\frac{b}{2}X}dz,$ (16)

in the following sense

$\oint_{\mathcal{C}_{z}}\psi^{*}(\xi)e^{\frac{b}{2}X^{*}(\xi)}J(z)d\xi=\oint_{\mathcal{C}_{z}}\psi(\xi)e^{\frac{b}{2}X(\xi)}J(z)d\xi=0\quad\text{etc}.$
(17)

The screening operators $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are the
holomorphic counterparts of the exponential fields in the conformal action (4)
with $N=2$ and

$\bm{\alpha}_{1}=\frac{1}{2}(1,-i),\quad\bm{\alpha}_{2}=\frac{1}{2}(1,i)\implies\big{(}\bm{\alpha}_{i}\cdot\bm{\alpha}_{j}\big{)}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},$ (18)

which correspond to the superalgebra $\mathfrak{sl}(2|1)$. This theory is
nothing else, but $\mathcal{N}=2$ Liouville theory. It is known to admit an
integrable perturbation that correspond to $\mathcal{N}=2$ sinh-Gordon model,
or affine $sl(2|2)$ Toda field theory in the terminology used above. On the
level of screening fields one has to add

$\mathcal{S}_{3}=\oint\psi^{*}e^{-\frac{b}{2}X^{*}}dz,\qquad\mathcal{S}_{4}=\oint\psi
e^{-\frac{b}{2}X}dz.$ (19)

The resulting theory corresponds to the affine root system
$\widehat{\mathfrak{sl}}(2|2)$.

Quantum $\mathcal{N}=2$ KdV system is most naturally defined in terms of the
free fields (14). Namely, the Integrals of Motion $\mathbf{I}_{2k-1}$ and
$\tilde{\mathbf{I}}_{2k-1}$ are defined as quantities that commute with all
the screening charges $\mathcal{S}_{k}$ with $k=1,2,3,4$. The commutativity
condition with $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ imply that the
Integrals of Motion are local (or quasilocal) in terms of the basic currents
(14), while commutativity with $\mathcal{S}_{3}$ and $\mathcal{S}_{4}$ is
equivalent to the following condition for the corresponding densities (11)

$\displaystyle\oint_{\mathcal{C}_{z}}\psi(\xi)e^{-\frac{b}{2}X(\xi)}G_{2k-1}(z)d\xi=\partial\mathcal{V}_{2k-1},\quad\oint_{\mathcal{C}_{z}}\psi^{*}(\xi)e^{-\frac{b}{2}X^{*}(\xi)}G_{2k-1}(z)d\xi=\partial\mathcal{V}^{*}_{2k-1},$
(20)
$\displaystyle\oint_{\mathcal{C}_{z}}\psi(\xi)e^{-\frac{b}{2}X(\xi)}\tilde{G}_{2k-1}(z)d\xi=\partial\tilde{\mathcal{V}}_{2k-1},\quad\oint_{\mathcal{C}_{z}}\psi^{*}(\xi)e^{-\frac{b}{2}X^{*}(\xi)}\tilde{G}_{2k-1}(z)d\xi=\partial\tilde{\mathcal{V}}^{*}_{2k-1}.$

The set of equations (20) can be used to construct local and quasilocal
Integrals of Motion explicitly. This calculation, being very tedious,
nevertheless allows to obtain explicit expressions for IM’s at lower levels.
The first two nontrivial representatives of both series have the form

$\tilde{\mathbf{I}}_{1}=2\sum\limits_{n=1}^{\infty}J_{-n}J_{n}+2\sum\limits_{r>0}\frac{1}{r}\left(G_{-r}^{+}G_{r}^{-}+G_{-r}^{-}G_{r}^{+}\right),$
(21)

and

$\mathbf{I}_{3}=\sum_{r>0}\left(3(-1)^{r+\frac{1}{2}}+r\Big{(}9-\dfrac{12}{b^{2}}\Big{)}\right)\left(G_{-r}^{+}G^{-}_{r}+G^{-}_{-r}G^{+}_{r}\right)+\sum_{k=1}^{\infty}\left(\dfrac{3}{2}(-1)^{k+1}-\dfrac{c}{4}+\dfrac{k^{2}}{2}\Big{(}9-\dfrac{12}{b^{2}}\Big{)}\right)J_{-k}J_{k}-\\\
-\Big{(}\dfrac{3}{4}+\dfrac{c}{8}\Big{)}J_{0}^{2}+2\Big{(}9-\dfrac{12}{b^{2}}\Big{)}\sum_{k=1}^{\infty}L_{-k}L_{k}+\bigg{(}9-\dfrac{12}{b^{2}}\bigg{)}L_{0}^{2}+3\sum_{k=1}^{\infty}(L_{-(2k-1)}J_{2k-1}+J_{-(2k-1)}L_{2k-1})+\\\
+6\sum_{k+r+s=0}:J_{k}G^{+}_{r}G^{-}_{s}:+6\sum_{k+n+m=0}:L_{k}J_{n}J_{m}:+3L_{0}J_{0}^{2}+\dfrac{576(5+2c)-48b^{2}(39+20c)}{1152b^{2}}L_{0}+\\\
+\dfrac{-12c(13+2c)+b^{2}(123c+22c^{2})}{1152b^{2}}.$ (22)

Here we write the Integrals of Motion in terms of modes. The corresponding
expression for the densities can be found in appendix A. In principle, using
the condition of commutativity with screening charges, one can compute higher
Integrals of Motion.

##  3 Spectrum and Bethe ansatz equations

In this section we consider the spectral problem for the quantum
$\mathcal{N}=2$ $\text{KdV}_{2}$ system and provide the set of algebraic
equations that describe the eigenvalues of Integrals of Motion (the Bethe
ansatz equations). For simplicity we consider NS representation of
$\mathcal{N}=2$ Virasoro algebra spanned by the states

$\mathcal{V}^{\scriptscriptstyle{\textrm{NS}}}_{\Delta,q}=\text{span}\left(L_{-\bm{\lambda}}J_{-\bm{\mu}}G^{+}_{-\bm{r}}G^{-}_{-\bm{s}}|\Delta,q\rangle\right)\quad
L_{-\bm{\lambda}}\overset{\text{def}}{=}L_{-\lambda_{1}}L_{-\lambda_{2}}\dots\quad\text{for}\quad\bm{\lambda}=(\lambda_{1}\geq\lambda_{2}\geq\dots)\quad\text{etc}$
(23)

where the highest weight state $|\Delta,q\rangle$ is defined as follows

$\begin{gathered}L_{n}|\Delta,q\rangle=J_{n}|\Delta,q\rangle=G^{\pm}_{r}|\Delta,q\rangle=0\quad\text{for}\quad
n,r>0,\\\ L_{0}|\Delta,q\rangle=\Delta|\Delta,q\rangle,\quad
J_{0}|\Delta,q\rangle=q|\Delta,q\rangle.\end{gathered}$ (24)

For generic values of $\Delta$ and $q$ the representation
$\mathcal{V}^{\scriptscriptstyle{\textrm{NS}}}_{\Delta,q}$ is irreducible. It
can be decomposed into direct sum of finite dimensional subspaces with the
fixed eigenvalues of the operators $L_{0}$ and $J_{0}$

$\mathcal{V}^{\scriptscriptstyle{\textrm{NS}}}_{\Delta,q}=\bigoplus_{N_{1}=0}^{\infty}\bigoplus_{N_{2}=0}^{\infty}\mathcal{V}^{\scriptscriptstyle{\textrm{NS}},N_{1},N_{2}}_{\Delta,q},$
(25)

where

$\mathcal{V}^{\scriptscriptstyle{\textrm{NS}},N_{1},N_{2}}_{\Delta,q}=\text{span}\left[|k\rangle\in\mathcal{V}^{\scriptscriptstyle{\textrm{NS}}}_{\Delta,q}:L_{0}|k\rangle=\left(\Delta+\frac{1}{2}\big{(}N_{1}+N_{2}\big{)}\right)|k\rangle,\,J_{0}|k\rangle=\big{(}N_{1}-N_{2}\big{)}|k\rangle\right]$
(26)

The number of states in
$\mathcal{V}^{\scriptscriptstyle{\textrm{NS}},N_{1},N_{2}}_{\Delta,q}$ can be
computed from the generating function

$\chi(q,t)\overset{\text{def}}{=}\sum_{N_{1}=0}^{\infty}\sum_{N_{2}=0}^{\infty}p(N_{1},N_{2})q^{\frac{N_{1}+N_{2}}{2}}t^{N_{1}-N_{2}}=\prod_{k=1}^{\infty}\frac{\left(1+tq^{k-\frac{1}{2}}\right)\left(1+t^{-1}q^{k-\frac{1}{2}}\right)}{(1-q^{k})^{2}}.$
(27)

The Integrals of Motion $\mathbf{I}_{2k-1}$ and $\tilde{\mathbf{I}}_{2k-1}$
commute with the zero modes $L_{0}$ and $J_{0}$. It implies that they can be
diagonalized in
$\mathcal{V}^{\scriptscriptstyle{\textrm{NS}},N_{1},N_{2}}_{\Delta,q}$. For
given $N_{1}$ and $N_{2}$ $\mathbf{I}_{2k-1}$ and $\tilde{\mathbf{I}}_{2k-1}$
reduce to finite matrices. Let us fix the pair $N_{1}$ and $N_{2}$ and
associate to it the set of Bethe roots

$x_{1},\dots,x_{N_{1}}\quad\text{and}\quad y_{1},\dots,y_{N_{2}},$ (28)

and the functions

$S(x)=\frac{x^{2}-h_{1}^{2}}{x^{2}-h_{2}^{2}},\quad
A(x)=\frac{\left(x-\frac{h_{2}}{2}\right)^{2}-\left(\frac{u-u^{*}}{2}\right)^{2}}{\left(x-\frac{h_{1}}{2}\right)^{2}-\left(\frac{u+u^{*}}{2}\right)^{2}},\quad
B(x)=\frac{\left(x+\frac{h_{1}}{2}\right)^{2}-\left(\frac{u+u^{*}}{2}\right)^{2}}{\left(x+\frac{h_{2}}{2}\right)^{2}-\left(\frac{u-u^{*}}{2}\right)^{2}},$
(29)

where

$h_{1}=\frac{b}{2}+b^{-1},\quad h_{2}=\frac{b}{2}-b^{-1}.$ (30)

Our main result states that the following Bethe ansatz equations

$\displaystyle A(x_{i})\prod_{j=1}^{N_{2}}S(x_{i}-y_{j})=1,\quad$
$\displaystyle\text{for}\quad i=1,\dots,N_{1},$ (31) $\displaystyle
B(y_{i})\prod_{j=1}^{N_{1}}S^{-1}(y_{i}-x_{j})=1,\quad$
$\displaystyle\text{for}\quad i=1,\dots,N_{2},$

describe the spectrum of quantum $\mathcal{N}=2$ $\text{KdV}_{2}$ system with
the central charge $c$ and the highest weight parameters $\Delta$ and $q$
given by

$c=3\left(1+\frac{4}{b^{2}}\right)\quad\text{and}\quad
q=\frac{u-u^{*}}{b},\quad\Delta=\frac{1}{2b^{2}}-\frac{uu^{*}}{2}.$ (32)

The eigenvalues of local and semi-local integrals of motion are symmetric
polynomials in Bethe roots. In particular, we have found that

$\tilde{\mathbf{I}}_{1}\sim
8b^{-1}\left(\sum_{i=1}^{N_{1}}x_{i}-b\big{(}N_{1}-N_{2}\big{)}\left(\frac{\Delta+N_{1}}{2}+\frac{q}{4}\right)+\frac{N_{1}(2+b^{2})}{4b}\right),$
(33)

where the sign $\sim$ means that the corresponding eigenvalues are given by
the expression in the right hand side with Bethe roots solving (31).

One might wounder that the expression (33) does include the roots $x_{k}$, but
not $y_{k}$. However, it is well known that symmetric polynomials of Bethe
roots are algebraically related. In particular, based on explicit calculations
on lower levels we have empirically established the constraint

$\sum_{i=1}^{N_{1}}x_{i}+\sum_{j=1}^{N_{2}}y_{j}=-\frac{b}{4}\big{(}N_{1}-N_{2}\big{)}\Big{(}2uu^{*}-2\big{(}N_{1}+N_{2}\big{)}+1\Big{)}.$
(34)

The fact that the spectrum of $\mathcal{N}=2$ $\text{KdV}_{2}$ system is
described by Bethe equations of the form (31) implicitly follow from the
results of [17]. Namely, in [17] it has been show that $\mathcal{N}=2$ $W_{n}$
algebras appear naturally as representations of affine Yangian of
$\widehat{\mathfrak{gl}}(1|1)$ (it has been already noticed in [20, 21, 22,
23]). This fact provides natural definition of the transfer-matrix whose
expansion at large values of the spectral parameter gives local Integrals of
Motion of some integrable system. It can be shown, that in the case of
untwisted transfer-matrix this integrable system coincides with
$\mathcal{N}=2$ $\text{KdV}_{n}$ system for a class of representations of
$\textrm{Y}\left(\widehat{\mathfrak{gl}}(1|1)\right)$ on $n$ sites. The
spectrum of this integrable system is governed by Bethe ansatz equations that
can be derived using a version of algebraic Bethe ansatz approach developed in
[15, 24, 16]. In the next section we will outline this derivation.

##  4 Affine Yangian $Y(\widehat{\mathfrak{gl}}(1|1))$ and Bethe ansatz
equations

The commutation relations of $Y(\widehat{\mathfrak{gl}}(1|1))$ can be
compactly written in terms of modes of the currents admitting the expansion at
$\infty$

$e_{k}(u)=\frac{e_{k}^{(0)}}{u}+\frac{e_{k}^{(1)}}{u^{2}}+\dots,\quad
f_{k}(u)=\frac{e_{k}^{(0)}}{u}+\frac{e_{k}^{(1)}}{u^{2}}+\dots,\quad\uppsi_{k}(u)=\frac{e_{k}^{(0)}}{u}+\frac{e_{k}^{(1)}}{u^{2}}+\dots,\quad
k=1,2,$ (35)

as

$\left[\uppsi_{i}(u),\uppsi_{j}(v)\right]=0,\quad\left\\{e_{i}(u),e_{i}(v)\right\\}=0,\quad\left\\{f_{i}(u),f_{i}(v)\right\\}=0,$
(36)
$\left\\{e_{i}(u),f_{j}(v)\right\\}=\delta_{ij}\frac{\uppsi_{i}(u)-\uppsi_{i}(v)}{u-v},$
(37) $e_{1}(u)e_{2}(v)=-\frac{(v-u-h_{2})(v-u+h_{2})}{(u-v-
h_{1})(u-v+h_{1})}e_{2}(v)e_{1}(u)+\text{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}local}},$
(38) $f_{2}(u)f_{1}(v)=-\frac{(v-u-h_{2})(v-u+h_{2})}{(u-v-
h_{1})(u-v+h_{1})}f_{1}(v)f_{2}(u)+\text{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}local}},$
(39) $\uppsi_{i}(u)e_{j}(v)=\frac{(u-v-h_{j})(u-v+h_{j})}{(u-v-
h_{i})(u-v+h_{i})}e_{j}(v)\uppsi_{i}(u)+\text{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}local}},$
(40) $\uppsi_{i}(u)f_{j}(v)=\frac{(u-v-h_{i})(u-v+h_{i})}{(u-v-
h_{j})(u-v+h_{j})}f_{j}(v)\uppsi_{i}(u)+\text{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}local}},$
(41)

where by local one denotes operators that depend only on one variable either
$u$ or $v$ (see [17]). These terms do not contribute to the commutation
relations of modes with indexes higher than $1$. Note that despite the
similarity of the relations (38)–(41) to OPE in CFT, the product of the
currents is regular. So the visible poles in the relations (38)–(41) are
canceled by the local terms.

Affine Yangian of $\mathfrak{gl}(1|1)$ appears naturally in conformal field
theory as a subalgebra of $RLL$ algebra. Namely consider the following Lax
operator [17]

$\mathcal{L}^{(0)}=-b^{-2}\partial+\frac{1}{2}\left(b^{-1}\partial
X-\psi\psi^{*}-b^{-1}\partial
X^{*}\right)+\frac{1}{2b}\left(\partial\psi-b\psi\partial
X^{*}\right)\theta-b^{-1}\psi^{*}\frac{\partial}{\partial\theta}+b^{-1}\partial
X^{*}\theta\frac{\partial}{\partial\theta},$ (42)

where $(X,X^{*},\psi,\psi^{*})$ are the fields satisfying (13), but in
different normalization, with the factor $1$ instead of $2$ in the right hand
side. For simplicity we consider NS representation for fermions, i.e. in terms
of the modes on the cylinder $z=e^{-ix}$, $x\sim x+2\pi$ one has

$\begin{gathered}\partial
X(x)=\sum_{k\in\mathbb{Z}}X_{k}e^{-ikx},\quad\partial
X^{*}(x)=\sum_{k\in\mathbb{Z}}X^{*}_{k}e^{-ikx},\quad\psi(x)=i^{\frac{1}{2}}\sum_{r\in\mathbb{Z}+\frac{1}{2}}\psi_{r}e^{-irx},\quad\psi^{*}(x)=i^{\frac{1}{2}}\sum_{r\in\mathbb{Z}+\frac{1}{2}}\psi^{*}_{r}e^{-irx},\end{gathered}$
(43)

with commutation relations (other commutation relations are trivial)

$[X_{m},X_{n}^{*}]=m\delta_{m,-n},\quad\\{\psi_{r},\psi_{s}^{*}\\}=\delta_{r,-s}.$
(44)

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

$\begin{gathered}a_{n}|\bm{u}\rangle=a^{*}_{n}|\bm{u}\rangle=\psi_{r}|\bm{u}\rangle=\psi^{*}_{r}|\bm{u}\rangle=0\quad\text{for}\quad
n>0,\,r>0,\\\ a_{0}|\bm{u}\rangle=-iu|\bm{u}\rangle,\quad
a_{0}^{*}|\bm{u}\rangle=-iu^{*}|\bm{u}\rangle,\end{gathered}$ (45)

by the action of the creation operators

$\mathcal{F}_{\bm{u}}=\textrm{Span}\left(\psi_{-\bm{s}^{*}}^{*}\psi_{-\bm{s}}a^{*}_{-\bm{\lambda}^{*}}a_{-\bm{\lambda}}|\bm{u}\rangle\right),\quad\text{where}\quad
a_{-\bm{\lambda}}=a_{-\lambda_{1}}a_{-\lambda_{2}}\dots\,\text{etc}.,$ (46)

where

$\bm{\lambda}=\\{\lambda_{1}\geq\lambda_{2}\geq\dots\\},\quad\bm{\lambda}^{*}=\\{\lambda_{1}^{*}\geq\lambda_{2}^{*}\geq\dots\\},\quad\bm{s}=\\{s_{1}>s_{2}>\dots\\},\quad\bm{s}^{*}=\\{s_{1}^{*}>s_{2}^{*}>\dots\\}.$
(47)

Thus in given representation one has $\mathcal{L}=\mathcal{L}(\bm{u})$. The
Fock representation $\mathcal{F}_{\bm{u}}$ admits the action of the current
algebra $\widehat{\mathfrak{gl}}(1|1)$ in the so-called typical
representation. The components of the Lax operator (42) correspond to the
current $\bm{E}\in\widehat{\mathfrak{gl}}(1|1)$ as

$\bm{E}=\begin{pmatrix}b^{-1}\partial X-\psi\psi^{*}&\psi^{*}\\\
\partial\psi-b\psi\partial X^{*}&\psi\psi^{*}-b^{-1}\partial X-b\partial
X^{*}\end{pmatrix}.$ (48)

Having defined the Lax operator (42), the $R-$matrix of
$\textrm{Y}(\widehat{\mathfrak{gl}}(1|1))$ is defined as an intertwining
operator acting in the tensor product of two Fock modules
$\mathcal{F}_{\bm{u}_{1}}$ and $\mathcal{F}_{\bm{u}_{2}}$

$\mathcal{R}_{12}(\bm{u}_{1}|\bm{u}_{2})\mathcal{L}_{1}^{(0)}(\bm{u}_{1})\mathcal{L}^{(0)}_{2}(\bm{u}_{2})=\mathcal{L}_{2}^{(0)}(\bm{u}_{2})\mathcal{L}_{1}^{(0)}(\bm{u}_{1})\mathcal{R}_{12}(\bm{u}_{1}|\bm{u}_{2}),$
(49)

where by $\mathcal{L}_{k}^{(0)}(\bm{u}_{k})$ one denotes (42) with
$X(z)\rightarrow X_{k}(z)$. Using (49) one can compute the matrix of the
operator $\mathcal{R}_{12}(\bm{u}_{1}|\bm{u}_{2})$ at any given level, but the
computations become very complicated as the level grows (see [17] for
details). We note that (49) is the special case of $RLL$ algebra with the Lax
operator $\mathcal{L}^{(0)}(\bm{u})$ whose quantum and auxiliary spaces (in
the standard terminology of the spin chains) are

$\displaystyle\text{quantum space}:\quad$ $\displaystyle\text{functions of $x$
and $\theta$, where $\partial$ and $\frac{\partial}{\partial\theta}$ act},$
(50) $\displaystyle\text{auxiliary space}:\quad$ $\displaystyle\text{Fock
module}\quad\mathcal{F}_{\bm{u}}.$

On the other hand with quantum space being unspecified one has the algebra

$\mathcal{R}_{12}(\bm{u}_{1}|\bm{u}_{2})\mathcal{L}_{1}(\bm{u}_{1})\mathcal{L}_{2}(\bm{u}_{2})=\mathcal{L}_{2}(\bm{u}_{2})\mathcal{L}_{1}(\bm{u}_{1})\mathcal{R}_{12}(\bm{u}_{1}|\bm{u}_{2}),$
(51)

which we call $RLL$ or the Yang-Baxter algebra of
$\widehat{\mathfrak{gl}}(1|1)$. It has been shown in [17] that (51) contains
$Y(\widehat{\mathfrak{gl}}(1|1))$ as subalgebra with parameters $h_{k}$ given
by (30).

Apart from $e_{k}(u)$, $f_{k}(u)$ and $\uppsi_{k}(u)$ the algebra (51) also
contains auxiliary Cartan current $h(\bm{u})=h(u,u^{*})$ that depend on two
spectral parameters rather than one. This current satisfies the following
commutation relations with $\textrm{Y}(\widehat{\mathfrak{gl}}(1|1))$ currents
[17]

$[h(\bm{u}),h(\bm{v})]=[h(\bm{u}),\uppsi_{k}(v)]=0.$ (52)
$h(\bm{u})e_{1}(v)=\frac{(u-u^{*})-v}{(u+u^{*})-v+b^{-1}}\,e_{1}(v)h(\bm{u})+\text{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}local}},$
(53)
$h(\bm{u})e_{2}(v)=\frac{(u+u^{*})-v-b/2}{(u-u^{*})-v-h_{2}}\,e_{2}(v)h(\bm{u})+\text{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}local}},$
(54)
$f_{1}(v)h(\bm{u})=\frac{(u-u^{*})-v}{(u+u^{*})-v+b^{-1}}\,h(\bm{u})f_{1}(v)+\text{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}local}}.$
(55)
$f_{2}(v)h(\bm{u})=\frac{(u+u^{*})-v-b/2}{(u-u^{*})-v-h_{2}}\,h(\bm{u})f_{2}(v)+\text{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}local}}.$
(56)

Different representations of (51) correspond to different CFT’s. We will
consider the most natural representation

$\mathcal{L}(\bm{u})=\mathcal{R}_{01}(\mathbf{u}|\mathbf{u}_{1})\dots\mathcal{R}_{0n}(\mathbf{u}|\mathbf{u}_{n}),$
(57)

which can be thought as a generalization of the spin-chain representation on
$n-$sites. The corresponding transfer matrix has the form

$\bm{T}(\bm{u})=\textrm{Tr}^{\prime}\left(\mathcal{R}_{01}(\mathbf{u}|\mathbf{u}_{1})\dots\mathcal{R}_{0n}(\mathbf{u}|\mathbf{u}_{n})\right),$
(58)

where ′ means certain regularization of the trace (see [15, 16] for details).
This transfer matrix corresponds to quantum $\mathcal{N}=2$ $\textrm{KdV}_{n}$
system in the certain limit. In order to see it, let us consider the case
$n=2$. In this case an immediate consequence of (51) shows

$[\bm{T}(u),\mathcal{R}_{12}(\bm{u}_{1}|\bm{u}_{2})]=0\quad\text{for}\quad
n=2.$ (59)

The operator $\mathcal{R}_{12}(\bm{u}_{1}|\bm{u}_{2})$ serves as an
intertwining operator (49) between two $W-$algebras. It has been shown in [17]
that the currents appearing in the product
$\mathcal{L}_{1}^{(0)}\cdot\mathcal{L}_{2}^{(0)}$ can be defined as a
commutant of the set of screening fields

$\hat{\mathcal{S}}_{1}=\int\psi_{1}^{*}e^{bX_{1}^{*}}dz\quad\hat{\mathcal{S}}_{1,2}=\int(\psi_{1}-\psi_{2})e^{\frac{b}{2}\left(X_{1}-X_{2}-X_{1}^{*}-X_{2}^{*}\right)}dz,\quad\hat{\mathcal{S}}_{2}=\int\psi_{2}^{*}e^{bX_{2}^{*}}dz$
(60)

There are eight currents, including
$\left(J_{12},G_{12},G_{12}^{*},T_{12}\right)$, that form $\mathcal{N}=2$
Virasoro algebra with the central charge $c=6.$ The screenings
$\mathcal{S}_{1}$, $\mathcal{S}_{2}$ are the Wakimoto type operators that
define two copies of $\widehat{\mathfrak{gl}}(1|1)$ current algebra, and
$\mathcal{S}_{1,2}$ provides the interaction between the two. The integrable
perturbation of this model is implemented by the additional screening field
[17]

$\hat{\mathcal{S}}_{2,1}=\int(\psi_{1}-\psi_{2})e^{\frac{b}{2}\left(X_{2}-X_{1}-X_{1}^{*}-X_{2}^{*}\right)}dz.$
(61)

If we drop this exponent, the symmetry algebra of the initial theory with the
screenings ($\mathcal{S}_{1}$, $\mathcal{S}_{1,2}$, $\mathcal{S}_{2}$) is
described by the Lax operator
$\mathcal{L}_{1}^{(0)}\cdot\mathcal{L}_{2}^{(0)}$. On the other hand, we can
consider different but isomorphic $W-$algebra defined by the set of screenings
($\mathcal{S}_{2}$, $\mathcal{S}_{2,1}$, $\mathcal{S}_{1}$) with the Lax
operator $\mathcal{L}_{2}^{(0)}\cdot\mathcal{L}_{1}^{(0)}$, which are
connected by $RLL$-algebra (49). It is clear, that the Integrals of Motion
should belong to the intersection of these two $W$-algebras and commute with
$\mathcal{R}_{12}(\bm{u}_{1}|\bm{u}_{2})$.

In order to see how the integrable system provided by the screening fields
(60) and (61) is related to $\mathcal{N}=2$ quantum $\text{KdV}_{2}$ hierarchy
it is convenient to transform to new coordinates. Namely, we introduce complex
bosons ($X,X^{*},\widetilde{X},\widetilde{X}^{*}$) as

$X_{1}=\dfrac{X+\lambda^{-1}\widetilde{X}}{2},\quad
X_{2}=\dfrac{-X+\lambda^{-1}\widetilde{X}}{2},\quad
X_{1}^{*}=\dfrac{X^{*}+\lambda\widetilde{X}^{*}}{2},\quad
X_{2}^{*}=\dfrac{-X^{*}+\lambda\widetilde{X}^{*}}{2},$ (62)

and complex fermions ($\psi,\psi^{*},\widetilde{\psi},\widetilde{\psi}^{*}$):

$\psi_{1}=\dfrac{\psi+\lambda^{-1}\widetilde{\psi}}{2},\quad\psi_{2}=\dfrac{-\psi+\lambda^{-1}\widetilde{\psi}}{2},\quad\psi_{1}^{*}=\dfrac{\psi^{*}+\lambda\widetilde{\psi}^{*}}{2},\quad\psi_{2}^{*}=\dfrac{-\psi^{*}+\lambda\widetilde{\psi}^{*}}{2}.$
(63)

Here $\lambda$ is a formal parameter that does not alter commutation relations
of the fields $(\tilde{X},\tilde{X}^{*},\tilde{\psi},\tilde{\psi}^{*})$. We
notice that in the limit $\lambda\rightarrow 0$ these fields decouple from
(60) and (61) and the resulting screening fields take the form

$\mathcal{S}_{1}=\oint\psi^{*}e^{\frac{b}{2}X^{*}}dz,\qquad\mathcal{S}_{2}=\oint\psi
e^{\frac{b}{2}X}dz,\qquad\mathcal{S}_{3}=\oint\psi^{*}e^{-\frac{b}{2}X^{*}}dz,\qquad\mathcal{S}_{4}=\oint\psi
e^{-\frac{b}{2}X}dz,$ (64)

that is precisely the one for $\mathcal{N}=2$ $\text{KdV}_{2}$ systems (16)
and (19). Similar logic allows to decouple the ”center of mass” degrees of
freedom for the theory with $n-$degrees of freedom.

Having defined the integrable system, Bethe ansatz equations for its spectrum
follow from the results of [15, 16]. We comment on the simplest case $n=2$.
The key idea is to diagonalize the operator
$\mathcal{R}_{12}(\bm{u}_{1}|\bm{u}_{2})$ instead of local Integrals of
Motion. In order to do this one defines the off-shell Bethe vector

$|B(\bm{x},\bm{y})\rangle\in\mathcal{F}_{\bm{u}_{1}}\otimes\mathcal{F}_{\bm{u}_{2}}.$
(65)

The vector $|B(\bm{x},\bm{y})\rangle$ has been constructed in such a way that
its projection on arbitrary state
$|\psi_{1}\rangle\otimes|\psi_{2}\rangle=|\psi_{1},\psi_{2}\rangle$ can be
expressed as a matrix element in the auxiliary space

$\langle\psi_{1},\psi_{2}|B(\bm{x},\bm{y})\rangle=\,_{\scriptscriptstyle{\text{aux}}}\langle\varnothing|\mathcal{L}_{\psi_{1},\varnothing}(\bm{u}_{1})\mathcal{L}_{\psi_{2},\varnothing}(\bm{u}_{2})|\chi(\bm{x},\bm{y})\rangle_{\scriptscriptstyle{\text{aux}}},$
(66)

where
$\mathcal{L}_{\psi,\varnothing}(\bm{u})=\langle\psi|\mathcal{L}(\bm{u})|\bm{u}\rangle$
is the element of the matrix $\mathcal{L}(\bm{u})$ and
$|\chi(\bm{x},\bm{y})\rangle_{\scriptscriptstyle{\text{aux}}}$ is the special
state in the auxiliary space which depends on variables
$\bm{x}=(x_{1},\dots,x_{N_{1}})$ and $\bm{y}=(y_{1},\dots,y_{N_{2}})$. In
these terms the action of the operator
$\mathcal{R}_{12}(\bm{u}_{1}|\bm{u}_{2})$ is very simple

$\langle\psi_{1},\psi_{2}|\mathcal{R}_{12}(\bm{u}_{1}|\bm{u}_{2})|B(\bm{x},\bm{y})\rangle=\,_{\scriptscriptstyle{\text{aux}}}\langle\varnothing|\mathcal{L}_{\psi_{2},\varnothing}(\bm{u}_{2})\mathcal{L}_{\psi_{1},\varnothing}(\bm{u}_{1})|\chi(\bm{x},\bm{y})\rangle_{\scriptscriptstyle{\text{aux}}},$
(67)

In order to compare (66) and (67) it is useful to know how to commute
$\mathcal{L}_{\psi_{2},\varnothing}(\bm{u}_{2})$ and
$\mathcal{L}_{\psi_{1},\varnothing}(\bm{u}_{1})$. It is done using the fact
that $\mathcal{L}_{\psi,\varnothing}(\bm{u})$ can be represented by the
contour integral [15, 16]

$\mathcal{L}_{\psi,\varnothing}(\bm{u})=\oint
F_{\psi}(\bm{z},\bm{w})f_{1}(z_{1})\dots f_{1}(z_{n_{1}})f_{2}(w_{1})\dots
f_{2}(w_{n_{2}})h(\bm{u})d\bm{z}d\bm{w},$ (68)

where $n_{1}$ and $n_{2}$ are related to conformal dimension and charge of the
states $|\psi\rangle$. Integration in (68) goes around infinity and includes
all singularities of the rational function $F_{\psi}(\bm{z},\bm{w})$ which
encodes information about the state $|\psi\rangle$. The explicit form of this
function is important for actual form of the Bethe vector (65), but not for
Bethe ansatz equations. Using the Cauchy theorem the integral in (68)
localizes to the poles coming from the state
$|\chi(\bm{x},\bm{y})\rangle_{\scriptscriptstyle{\text{aux}}}$, the position
of these poles are the Bethe roots. On the other hand permutation of two
$\mathcal{L}(\bm{u})$ operators in (67) will produce extra factors coming from
the commutation relations (55), (56) and (39). These factors are nothing but
the functions $S(x)$, $A(x)$ and $B(x)$ in (29) after the scaling provided by
(62) and in the limit $\lambda\rightarrow 0$.

We should stress that while the derivation of Bethe ansatz equations is rather
straightforward and follows exactly [15, 16], the problem of expressing the
eigenvalues of local and semi-local IM’s through the Bethe roots is tough (see
[15]). Of course the eigenvalues are some symmetric functions of the roots.
Moreover, from explicit examples [13, 14] it is known that they are in fact
symmetric polynomials. Admitting this conjecture in current case we have found
(33) by direct diagonalization and comparing to the solution of BAE up to
level $4$. Moreover (33) can be considered as an independent of of the whole
machinery used in [13, 14, 17].

##  5 Concluding remarks

In these notes we have found Bethe ansatz equations (31) for the spectrum of
$\mathcal{N}=2$ KdV system. Equations (31) constitute main our main results.
Here we provide some remarks and open questions.

  * •

Our results admits straightforward generalization for $\mathcal{N}=2$
$\text{KdV}_{n}$ system with $n>2$ which correspond to integrable
perturbations of Kazama-Suzuki models [25]. The Bethe ansatz equations in this
case are given by (31), but with with the source terms $A(x)$ and $B(x)$ being
the ratio of polynomials of degree $n$ (see [15, 16]).

  * •

The approach to integrability in CFT based on affine Yangian symmetry
naturally defines twisted integrable system [24, 16]. Namely, instead of
regularized transfer-matrix (58) one considers twisted transfer matrix

$\bm{T}(\bm{u})=\textrm{Tr}\left(q^{L_{0}^{(0)}}t^{h_{0}^{(0)}}\mathcal{R}_{01}(\mathbf{u}|\mathbf{u}_{1})\dots\mathcal{R}_{0n}(\mathbf{u}|\mathbf{u}_{n})\right),$
(69)

where $L_{0}^{(0)}$ and $h_{0}^{(0)}$ act in the auxiliary space (here $h_{0}$
is the zero mode of $\psi\psi^{*}$). It would be interesting to identify Toda
field theory that corresponds to this integrable system.

  * •

The quantum $\mathcal{N}=2$ $\text{KdV}_{2}$ system is a particular case of
the Fateev model [26] (see also [27]). This model is purely bosonic and
corresponds to the exceptional root system $\mathrm{D}(2,1|\alpha)$. In
particular, its screening operators do not include fermions. They can be
obtained from (64) by boson-fermion correspondence. Thus our BA equations
provide the spectrum of the Fateev model in the supersymmetric point. However,
we do not expect that the full Fateev model is related to
$\mathrm{Y}(\widehat{\mathfrak{gl}}(1|1))$. It rather corresponds to
$\mathrm{Y}(\widehat{\mathfrak{gl}}(1))$ with the ”boundary” [24], but it
requires further study. In particular, it is still a challenging problem to
find BAE in this case.

## Acknowledgments

This work has been supported by the Russian Science Foundation under the grant
22-22-00991.

##  Appendix A Quantum $\mathcal{N}=2$ KdV integrals and analytic ordering

In this appendix we will comment on construction of quantum $\mathcal{N}=2$
KdV integrals and on analytic regularization of the densities. We will work in
$\mathcal{N}=2$ notations. We define the holomorphic part of supercoordinates
$Z=(z,\theta,\bar{\theta}).$ and introduce the holomorphic superfield

$\mathbb{T}=J+\theta^{+}G^{+}+\theta^{-}G^{-}+\theta^{+}\theta^{-}T$ (70)

and super derivatives

$D^{+}=\frac{\partial}{\partial\theta^{+}}-\dfrac{1}{2}\theta^{-}\partial,\qquad
D^{-}=\frac{\partial}{\partial\theta^{-}}-\dfrac{1}{2}\theta^{+}\partial.$
(71)

Then in these notations the Integrals of Motion can be written in
supersymmetric form and satisfy quantum $\mathcal{N}=2$ KdV hierarchy

$\displaystyle\mathbf{I}_{1}=\int dzd\theta^{+}d\theta^{-}\mathbb{T},$ (72)
$\displaystyle\mathbf{I}_{3}=\int
dzd\theta^{+}d\theta^{-}\left((\mathbb{T}(\mathbb{T}\mathbb{T}))+\left(9-\frac{12}{b^{2}}\right)(D^{+}\mathbb{T}D^{-}\mathbb{T})\right),$
$\displaystyle\dots$

where by $(AB)$ we denoted analytic ordering of two operators $A,$ $B$ which
is defined as follows. Consider two local operators $A(z)$ and $B(z)$

$A(z)=\sum_{r}A_{r}e^{-irz},\quad B(z)=\sum_{r}B_{r}e^{-irz},$ (73)

where we assume that the fields can be either bosonic ($r\in\mathbb{Z}$) or
fermionic in NS sector ($r\in\mathbb{Z}+\frac{1}{2}$). Analytic ordering is
defined as follows

$(A(z)B(z))\stackrel{{\scriptstyle\text{ def }}}{{=}}\frac{1}{2\pi
i}\oint_{\mathcal{C}_{z}}\frac{\mathcal{T}(A(w)B(z))}{w-z}dw=\sum_{r}\Lambda_{r}e^{-irz},$
(74)

where summation goes over $r\in\mathbb{Z}$ if both currents are bosonic or
fermionic, and $r\in\mathbb{Z}+\frac{1}{2}$ otherwise. $\mathcal{T}$ stands
for chronological order

$\mathcal{T}(A(w)B(z))=\begin{cases}A(w)B(z),&\quad\Im(w)<\Im(z),\\\
B(z)A(w),&\quad\Im(w)>\Im(z),\end{cases}$ (75)

and by definition

$\Lambda_{r}\stackrel{{\scriptstyle\text{ def
}}}{{=}}\frac{1}{2\pi}\int_{0}^{2\pi}e^{irz}(A(z)B(z))dz=\frac{1}{4i\pi^{2}}\int_{0}^{2\pi}\oint_{\mathcal{C}_{z}}e^{irz}\frac{\mathcal{T}(A(w)B(z))}{w-z}dwdz.$
(76)

For what follows, it will be convenient to ”approximate” $\frac{1}{z}$ by
periodic

$\frac{1}{z}=\frac{1}{2}\cot\frac{z}{2}-2\sum_{q=1}^{N}c_{q}\sin
qz+O(z^{2N})=\chi_{N}^{(+)}(z)+O(z^{2N}),\quad N\in\mathbb{Z}_{>0}$ (77)

and antiperiodic functions

$\frac{1}{z}=\frac{1}{2\sin\frac{z}{2}}-2\sum_{s=\frac{1}{2}}^{N}c^{\prime}_{s}\sin
sz+O(z^{2N+1})=\chi_{N}^{(-)}(z)+O(z^{2N+1}),\quad
N\in\mathbb{Z}_{>0}-\frac{1}{2}$ (78)

In both cases it is enough to take $N$ in such a way that $2N\geq p$ for the
periodic function and $2N+1\geq p$ for the antiperiodic one, where $p$ is
equal to the degree of the maximal pole in the OPE of considered operators:

$A(w)B(z)=\frac{\mathcal{O}}{(w-z)^{p}}+\ldots$ (79)

Then we can replace the function $(w-z)^{-1}\rightarrow\chi_{N}^{(\pm)}(z)$ in
the integral (76)

$\Lambda_{r}=\frac{1}{4i\pi^{2}}\int_{0}^{2\pi}\oint_{\mathcal{C}_{z}}e^{irz}\chi_{N}^{(\pm)}(w-z)\mathcal{T}(A(w)B(z))dwdz.$
(80)

In this formula we take $\chi_{N}^{(+)}(w-z)$ in two cases: if $A(w)$ and
$B(z)$ are both bosonic or if $A(w)$-bosonic and $B(z)$-fermionic; and the
function $\chi_{N}^{(-)}(w-z)$ is taken otherwise. Transforming the
integration contour

$\int\limits_{0}^{2\pi}\oint_{\mathcal{C}_{z}}dwdz=\mathop{\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi}}_{\Im(w)<\Im(z)}dwdz-\mathop{\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi}}_{\Im(w)>\Im(z)}dwdz,$
(81)

we come to the following formula

$\Lambda_{r}=\frac{1}{4i\pi^{2}}\mathop{\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi}}_{\Im(w)<\Im(z)}\left(e^{irz}A(w)B(z)+e^{irw}B(w)A(z)\right)\chi^{(\pm)}_{N}(w-z)dwdz$
(82)

The integral in (82) can be computed if we move the integration contour over
$dw$ to the domain of large negative $\Im(w)$. In this case one can expand

$-i\chi^{+}_{N}(w-z)=\frac{1}{2}+\sum_{k=1}^{\infty}e^{ik(z-w)}+2i\sum_{q=1}^{N}c_{q}\sin
q(w-z),$ (83)
$-i\chi^{-}_{N}(w-z)=\sum_{s\in\mathbb{Z}_{>0}-\frac{1}{2}}e^{is(z-w)}+2i\sum_{s=\frac{1}{2}}^{N}c^{\prime}_{s}\sin
s(w-z).$ (84)

Plugging (83), (84) into (82) we come to the following formulae:

  * •

$A-$bosonic, $B-$bosonic

$(AB)_{n}=\frac{1}{2}\left(A_{0}B_{n}+B_{n}A_{0}\right)+\sum_{k>0}\left(A_{-k}B_{n+k}+B_{n-k}A_{k}\right)+\sum_{q>0}c_{q}\left(\big{[}A_{q},B_{n-q}\big{]}+\big{[}B_{n+q},A_{-q}\big{]}\right),$
(85)

  * •

$A-$fermionic, $B-$fermionic (here $r\in\mathbb{Z}+\frac{1}{2}$)

$(AB)_{n}=\sum_{r>0}\left(A_{-r}B_{n+r}-B_{n-r}A_{r}\right)+\sum_{s>0}c^{\prime}_{s}\left(\big{\\{}A_{s},B_{n-s}\big{\\}}-\big{\\{}B_{n+s},A_{-s}\big{\\}}\right),$
(86)

  * •

$A-$bosonic, $B-$fermionic

$(AB)_{r}=\sum_{k>0}\left(A_{-k}B_{r+k}+B_{r-k}A_{k}\right)+\sum_{q>0}c_{q}\left(\big{[}A_{q},B_{r-q}\big{]}+\big{[}B_{r+q},A_{-q}\big{]}\right),$
(87)

  * •

$A-$fermionic, $B-$bosonic (here $s\in\mathbb{Z}+\frac{1}{2}$)

$(AB)_{r}=\sum_{s>0}\left(A_{-s}B_{r+s}+B_{r-s}A_{s}\right)+\sum_{s>0}c^{\prime}_{s}\left(\big{[}A_{s},B_{r-s}\big{]}+\big{[}B_{r+s},A_{-r}\big{]}\right).$
(88)

For our purposes it is enough to take the first two terms of the expansions
(77), (78). In this case the coefficients are

$c_{1}=-\frac{41}{720},\quad c_{2}=\frac{11}{1440}\quad\text{and}\quad
c^{\prime}_{\frac{1}{2}}=\frac{97}{1920},\quad
c^{\prime}_{\frac{3}{2}}=-\frac{17}{5760}.$ (89)

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