YITP-21-144

FZZ-triality and large $\mathcal{N}=4$ super Liouville theory

Thomas Creutziga***E-mail<EMAIL_ADDRESS>and Yasuaki Hikidab†††E-mail:
<EMAIL_ADDRESS>

aDepartment of Mathematical and Statistical Sciences, University of Alberta,

Edmonton, Alberta T6G 2G1, Canada

bCenter for Gravitational Physics, Yukawa Institute for Theoretical Physics,

Kyoto University, Kyoto 606-8502, Japan

We examine dualities of two dimensional conformal field theories by applying
the methods developed in previous works. We first derive the duality between
$SL(2|1)_{k}/(SL(2)_{k}\otimes U(1))$ coset and Witten’s cigar model or sine-
Liouville theory. The latter two models are Fateev-Zamolodchikov-Zamolodchikov
(FZZ-)dual to each other, hence the relation of the three models is named FZZ-
triality. These results are used to study correlator correspondences between
large $\mathcal{N}=4$ super Liouville theory and a coset of the form
$Y(k_{1},k_{2})/SL(2)_{k_{1}+k_{2}}$, where $Y(k_{1},k_{2})$ consists of two
$SL(2|1)_{k_{i}}$ and free bosons or equivalently two $U(1)$ cosets of
$D(2,1;k_{i}-1)$ at level one. These correspondences are a main result of this
paper. The FZZ-triality acts as a seed of the correspondence, which in
particular implies a hidden $SL(2)_{k^{\prime}}$ in $SL(2|1)_{k}$ or
$D(2,1;k-1)_{1}$. The relation of levels is $k^{\prime}-1=1/(k-1)$. We also
construct boundary actions in sine-Liouville theory as another use of the FZZ-
triality. Furthermore, we generalize the FZZ-triality to the case with
$SL(n|1)_{k}/(SL(n)_{k}\otimes U(1))$ for arbitrary $n>2$.

###### Contents

  1. 1 Introduction and summary
    1. 1.1 Organization of the paper
  2. 2 Dualities
  3. 3 First order formulation of coset models
  4. 4 FZZ-triality
    1. 4.1 A realization of the coset
    2. 4.2 Another realization of the coset
    3. 4.3 Fermionic dualities
  5. 5 Theory with a $\mathfrak{d}(2,1;-\psi)$-structure from a coset
  6. 6 Large $\mathcal{N}=4$ Liouville from a coset
    1. 6.1 $D(2,1;k-1)_{1}$ from $SL(2|1)_{k}$
    2. 6.2 Reduction of a coset
    3. 6.3 Large $\mathcal{N}=4$ super Liouville theory
    4. 6.4 Further reduction of the coset
  7. 7 Boundary FZZ-triality
  8. A Convention for Lie superalgebras
    1. A.1 Lie superalgebra $\mathfrak{sl}(2|1)$
    2. A.2 Lie superalgebra $\mathfrak{sl}(n|1)$
  9. B Large $\mathcal{N}=4$ superconformal algebra
  10. C Boundary $SL(2|1)$ WZNW model
    1. C.1 Boundary actions for B-branes
    2. C.2 Boundary actions for A-branes
  11. D Generalized FZZ-triality
    1. D.1 Bosonic triality
    2. D.2 Fermionic triality

##  1 Introduction and summary

We examine dualities in two dimensional conformal field theories by deriving
correlator correspondences of primary operators. Combined with the match of
symmetry algebras, we can deduce the equivalence of dual theories. In general,
dualities are quite useful since it often happens that some features of a
theory are easily captured by dual one. In particular, a strong/weak duality
helps us to examine strong coupling phenomena from a dual theory in a
tractable regime. In previous works [1, 2, 3, 4, 5], correlator
correspondences for several important dualities have been derived by applying
the reduction method from $SL(2)$ Wess-Zumino-Novikov-Witten (WZNW) model to
Liouville field theory [6, 7, 8] and its generalizations [9, 10, 11, 12]. The
results in the previous works may be regarded as conformal field theory
realizations of dualities of corner vertex operator algebras (VOAs)
conjectured by Gaiotto and Rapčák via brane junctions in superstring theory
[13] and proven in [14, 15]. In this paper, we examine more dualities, which
are not be directly related to the Gaiotto-Rapčák dualities. In particular, we
derive correlator correspondences between the large $\mathcal{N}=4$ super
Liouville theory and a coset model, where the agreement of symmetry algebra
was proven in [16].

Let us put this new duality into context: An important (strong/weak) duality
is the one conjectured by Fateev-Zamolodchikov-Zamolodchikov (FZZ) [17] and
proven in [1]. The FZZ-duality is an equivalence between two dimensional cigar
model described by the coset [18]

$\displaystyle\frac{SL(2)_{k}}{U(1)}$ (1.1)

and sine-Liouville theory, see, e.g., [19] for details. Its supersymmetric
version was proven in [20] as a mirror symmetry and in [2] by the method
similar to [1]. The supersymmetric version is useful to examine, e.g.,
singular Calabi-Yau geometry [21]. Two series of generalization of the FZZ-
duality have been analyzed in [3, 4, 5]. A generalization is given by
replacing the coset (1.1) by a higher rank one [3, 5]

$\displaystyle\frac{SL(n+1)_{k}}{SL(n)_{k}\otimes U(1)}\,.$ (1.2)

The coset appears as conformal field theory dual of higher spin
(super-)gravity [22, 23], thus the duality should be useful in that context as
well. Another generalization is the free boson, i.e. $U(1)$, coset of
subregular W-algebras of $\mathfrak{sl}(n)$ [4]. Arguably the most important
example is the duality between $\mathfrak{sl}(n)$ Toda field theory and the
diagonal coset

$\displaystyle\frac{SL(n)_{k}\otimes SL(n)_{-1}}{SL(n)_{k-1}}$ (1.3)

analyzed in [5]. The duality may be regarded as an analytic continuation of
coset realization of Wn-minimal model proven only rather recently in [24].

We have derived correlator correspondences by applying the reduction methods
from $SL(n)$ WZNW model to $\mathfrak{sl}(n)$ Toda field theory developed
particularly in [8, 12]. An important process is to adopt a simple first order
formulation of coset models. Such a formulation was proposed in [25, 26] and
established in our recent paper [5]. A main idea is to express the denominator
and numerator algebras of coset models by Wakimoto free field realizations
with free bosons and $(\beta,\gamma)$-systems [27]. The proposal of [25, 26]
is to construct field space by using orthogonal free bosons and removing some
sets of $(\beta,\gamma)$-systems. We can indeed show that the central charge
of the reduced theory matches with that of the original coset model. In [5],
we have not only derived the procedure but also established a way to obtain
proper interaction terms by applying the BRST formulation of coset models [28,
29, 30, 31] and Kugo-Ogima method [32]. In particular, it can be shown that
the coset algebra obtained in this way is isomorphic to the one by Goddard-
Kent-Olive (GKO) construction [33]. Fortunately, the first order formulation
of coset models is general enough to apply to more theories, in particular the
problem of this work.

In this paper, we first examine correlation functions of the coset

$\displaystyle\frac{SL(2|1)_{k}}{SL(2)_{k}\otimes U(1)}\,.$ (1.4)

As suggested by Gaiotto-Rapčák dualities [13] (see also [34]), the coset is
conjecturally dual to the cigar model described by (1.1) with level
$k^{\prime}$

$\displaystyle k^{\prime}-1=\frac{1}{k-1}$ (1.5)

and sine-Liouville theory as well. We derive correlator correspondences of the
“FZZ-triality.” We next derive correlator correspondences between a diagonal
coset

$\displaystyle\frac{SL(2)_{k_{1}}\otimes SL(2)_{k_{2}}}{SL(2)_{k_{1}+k_{2}}}$
(1.6)

and a theory with a $\mathfrak{d}(2,1;-\psi)$-structure [35]. This can be
regarded as a simplified version of our main problem with large
$\mathcal{N}=4$ super Liouville theory. In fact this theory also appears as a
coset of large $\mathcal{N}=4$ super Liouville theory [16].

We then move to the main problem of this paper. The coset realization of large
$\mathcal{N}=4$ superconformal algebra [36, 37]111There are two types of large
$\mathcal{N}=4$ superconformal algebra, which are often called as linear and
non-linear ones. In this paper, we only deal with non-linear one. was provided
in [16]. In terms of conformal field theory, it corresponds to a duality
between a coset model and large $\mathcal{N}=4$ super Liouville theory. The
coset model takes the form [16]

$\displaystyle\frac{Y(k_{1},k_{2})}{SL(2)_{k_{1}+k_{2}}}\,,$ (1.7)

where the numerator $Y(k_{1},k_{2})$ consists of
$SL(2|1)_{k_{1}},SL(2|1)_{k_{2}}$ and free bosons or two $U(1)$-cosets of
$D(2,1;-\psi)$ with $\psi=1-k_{1},1-k_{2}$ at level one. The two affine
superalgebras have subalgebras
$\mathfrak{sl}(2)_{k_{1}},\mathfrak{sl}(2)_{k_{2}}$ and the coset is
constructed by gauging the diagonal $\mathfrak{sl}(2)_{k_{1}+k_{2}}$. The
large $\mathcal{N}=4$ superconformal algebra includes two affine
$\mathfrak{sl}(2)$ subalgebras with levels $k_{1}^{\prime},k_{2}^{\prime}$,
where the relation of levels is
$k_{1}^{\prime}-1=1/(k_{1}-1),k_{2}^{\prime}-1=1/(k_{2}-1)$ as in (1.5). These
two affine $\mathfrak{sl}(2)_{k^{\prime}}$ are hidden in
$\mathfrak{sl}(2|1)_{k}$ or $\mathfrak{d}(2,1;k-1)$ [38, 39, 40]. This implies
that the FZZ-triality is a key relation for the duality between the coset
(1.7) and the large $\mathcal{N}=4$ super Liouville theory.

The large $\mathcal{N}=4$ superconformal algebra is important particularly in
the context of holography. For instance, conformal field theory dual of
superstrings on AdS${}_{3}\times$S${}^{3}\times$S${}^{3}\times$S1 has the
large $\mathcal{N}=4$ superconformal algebra as its symmetry [41, 42, 43, 44,
45], see also [46]. Moreover, the Wolf space model possesses the large
$\mathcal{N}=4$ superconformal symmetry [47, 48, 49] and it was proposed to be
dual to a higher spin gravity in [50]. We expect that our analysis helps us to
reveal some important aspects of holography. The large $\mathcal{N}=4$
superconformal algebra itself deserves further study as well. Unitary
representations were examined in [51, 52, 53], and it would be interesting to
investigate the spectrum from the coset viewpoints as well.

In this paper, we mainly consider correlation functions on worldsheet of
sphere topology. It is an important problem to extend the analysis to Riemann
surfaces of higher genus $g>0$. The original FZZ-duality was extended to
higher genus worldsheet in [1], and it might not be so difficult to do so more
generally. However, it would be more involved to examine Riemann surfaces with
boundaries. In [2], we have shown the equivalence of boundary correlation
functions for D1-branes in the cigar model (1.1) and those for D2-branes in
sine-Liouville theory. We have also analyzed fermionic FZZ-duality. In this
paper, we extend the FZZ-triality to the case with boundary. The point is that
there is a somehow understood procedure to obtain actions of boundary WZNW
theories on supergroups [54, 55, 56, 57]. Using in particular [55] we show
that the FZZ-triality is useful to obtain boundary actions of sine-Liouville
theory even for D1-branes and its supersymmetric counterparts form the coset
(1.4).

###  1.1 Organization of the paper

The paper is organized as follows. In the next section, we collect
mathematical facts underlying the current works on dualities of two
dimensional conformal field theories. In section 3, we review the first order
formulation of coset models developed in our previous paper [5]. In
particular, we closely examine the simplest but non-trivial example of the
diagonal coset (1.3) with $n=2$ and relate it to Liouville field theory. In
section 4, we derive correlator correspondences between the coset (1.4) and
sine-Liouville theory or the cigar model (1.1) by applying the first order
formulation explained in section 3. We further consider its supersymmetric
versions by adding a complex fermion. In section 5, we derive correlator
correspondences between a diagonal coset (1.6) and a theory with a
$\mathfrak{d}(2,1;-\psi)$-structure [35]. In section 6, we investigate the
coset (1.7). We first realize $D(2,1;-\psi)$ at level one as a coset of
$SL(2|1)$ and free bosons. We then construct $Y(k_{1},k_{2})$ in the numerator
of the coset (1.7) by making use of the expressions. Applying the first order
formulation of [5] and the reduction methods of [1], we rewrite the $N$-point
functions of the coset (1.7) by those of a different theory. Applying the
fermionic FZZ-duality, we show that the theory obtained in this way is indeed
the large $\mathcal{N}=4$ super Liouville theory, which corresponds to a free
field realization of large $\mathcal{N}=4$ superconformal algebra in [58]. We
also check that a coset of the theory reduces to the model analyzed in section
5. We consider the case with additional fermions as well. In section 7, we
construct boundary actions for branes in sine-Liouville theory and
$\mathcal{N}=2$ super Liouville theory by making use of boundary FZZ-triality.
In appendix A, we summarize our conventions for generators of Lie
superalgebras $\mathfrak{sl}(2|1)$ and $\mathfrak{sl}(n|1)$ with $n>2$. In
appendix B, we explicitly write down the operator product expansions (OPEs)
for generators of large $\mathcal{N}=4$ superconformal algebra. In appendix C,
we obtain classical boundary actions for B-branes and A-branes in $SL(2|1)$
WZNW model by following the analysis for $OSP(1|2)$ WZNW model in [55]. In
appendix D, we generalize the FZZ-triality analyzed in section 4 and reduce
the coset

$\displaystyle\frac{SL(n|1)_{k}}{SL(n)_{k}\otimes U(1)}$ (1.8)

with $n>2$ to a theory with an $\mathfrak{sl}(n|1)$-structure by applying the
first order formulation reviewed in section 3.

##  2 Dualities

$S$-duality of four dimensional $\mathcal{N}=4$ supersymmetric GL-twisted
theories is closely related to the quantum geometric Langlands correspondence
as well as dualities of vertex algebras. Both are mathematical cousins of
dualities in two dimensional conformal field theory. The quantum geometric
Langlands correspondence is concerned with twisted $D$-modules, which in turn
arise as spaces of conformal blocks. Vertex algebras are the symmetry algebras
of conformal field theory.

We recall some results of [39]. Fix a compact Lie group $G$, the gauge group.
Let $\psi$ be the coupling of the gauge theory, a complex number. One then
considers three dimensional boundary conditions. If two such boundary
conditions, say $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ intersect in a two
dimensional corner, then this corner usually supports a vertex operator
algebra. This VOA depends on the choice of boundary condition and on $G$. Line
defects can end on vertex operators associated to modules of the VOA at the
corner. The type of modules is determined by the boundary conditions
$\mathcal{B}_{1}$ and $\mathcal{B}_{2}$. The duality group is the modular
group $PSL(2,\mathbb{Z})$. Let $g$ be an element of $PSL(2,\mathbb{Z})$. It
acts on $\psi$ via Möbius transformation and acts on boundary conditions in a
certain way, see section 2.3 of [39]. It acts on VOAs via a certain
convolution, that is it maps the VOA $V$ to a certain BRST-cohomology of
$V\otimes K_{g}$ with $K_{g}$ a very special kernel VOA. In particular duality
is not an isomorphism of VOAs but only a correspondence. The idea however is
that this correspondence respects line defects. This should be visible in the
VOA setting that VOAs that are related via duality have equivalent categories
of modules. Geometrically this means that one expects to have equivalent
spaces of conformal blocks, exactly as the quantum geometric Langlands
correspondence predicts. From the conformal field theory point of view this
means that one would like to have dualities between correlation functions of
two theories whose underlying VOAs are dual. The idea is that both dualities
of conformal field theories and equivalences of vertex algebra tensor
categories can be explained using the kernel VOAs. The FZZ-trialities exactly
correspond to such duality operations, while our coset realization of
$\mathcal{N}=4$ super Liouville theory is a low rank example of a somehow more
complicated story that is under current investigation.

The kernel VOAs are of the form

$A^{n}[\mathfrak{g},\mathfrak{g}^{\prime},\psi]=\bigoplus_{\lambda\in
P^{+}\cap L}V^{\psi}(\lambda)\otimes V^{\phi}(\tau(\lambda))$ (2.1)

with

$\frac{1}{\ell_{\mathfrak{g}}\psi}+\frac{1}{\ell_{\mathfrak{g}^{\prime}}\phi}=n$
(2.2)

and $\mathfrak{g}^{\prime}$ is a Lie (super)algebra related to $\mathfrak{g}$.
In most cases, the two coincide, but for example if $\mathfrak{g}$ is of type
$\mathfrak{so}_{2m+1}$ and $n$ is odd, then
$\mathfrak{g}^{\prime}=\mathfrak{osp}_{1|2m}$. $L$ is a suitable sublattice of
the weight lattice $P$ of $\mathfrak{g}$. $P^{+}$ is the set of dominant
weights, $n$ is an integer (usually positive), and $\psi,\phi$ are the levels
shifted by the respective dual Coxeter numbers
$h^{\vee}_{\mathfrak{g}},h^{\vee}_{\mathfrak{g}^{\prime}}$. $\tau$ is a
suitable map from weights of $\mathfrak{g}$ to $\mathfrak{g}^{\prime}$.
Finally $\ell_{\mathfrak{g}}$ is the lacety of $\mathfrak{g}$ (and for
$\mathfrak{osp}_{1|2n}$ it is two). The existence of these VOAs has been
conjectured in [39] and proven in [59, 60].

The simplest example is

$A^{n}[\mathfrak{gl}_{1},\mathfrak{gl}_{1},\psi]=\bigoplus_{m\in\mathbb{Z}}\pi^{\psi}_{m}\otimes\pi^{\phi}_{m}\cong
V_{\sqrt{n}\mathbb{Z}}\otimes\pi$ (2.3)

which is just the lattice VOA $V_{\sqrt{n}\mathbb{Z}}$ times a free boson
$\pi$. Here $\pi^{\psi}_{n}$ denotes the Fock module of highest-weight $n$ of
a free boson of level $\psi$.

The general idea is that one has two VOAs, $V^{-\psi}$ and $W^{\phi}$, such
that $V^{-\psi}$ has an affine VOA
$\mathfrak{g}_{-\psi-h^{\vee}_{\mathfrak{g}}}$ of $\mathfrak{g}$ at level
$-\psi-h^{\vee}_{\mathfrak{g}}$ as subalgebra and $W^{\phi}$ has an affine VOA
$\mathfrak{g^{\prime}}_{\phi-h^{\vee}_{\mathfrak{g}^{\prime}}}$ of
$\mathfrak{g}^{\prime}$ at level $-\phi-h^{\vee}_{\mathfrak{g}^{\prime}}$ as
subalgebra. Assume that these two VOAs are dual in the sense that their cosets
coincide

$\frac{V^{-\psi}}{\mathfrak{g}_{-\psi-h^{\vee}_{\mathfrak{g}}}}=\frac{W^{\phi}}{\mathfrak{g^{\prime}}_{\phi-h^{\vee}_{\mathfrak{g}^{\prime}}}}\,,$
(2.4)

the main examples of isomorphisms of such cosets are provided by Gaiotto-
Rapčák triality [13] proven in [14, 15]. Let $H^{\text{BRST}}_{\mathfrak{g}}(\
\bullet\ )$ denote the semi-infinite Lie algebra cohomology of
$\widehat{\mathfrak{g}}$, relative to $\mathfrak{g}$ [61]. It satisfies

$H^{\text{BRST}}_{\mathfrak{g}}(V^{-\psi}(\lambda)\otimes
V^{\psi}(\mu))=\begin{cases}\mathbb{C}&\quad\text{if}\ \mu=\lambda^{*}\,,\\\
0&\quad\text{else}\,.\end{cases}$ (2.5)

The proposal of [14, 15] is that the Gaiotto-Rapčák dualities extend to the
relation

$H^{\text{BRST}}_{\mathfrak{g}}(V^{-\psi}\otimes
A^{1}[\mathfrak{g},\mathfrak{g}^{\prime},\psi])=W^{\phi}.$ (2.6)

See [62] for the mathematics of this statement for
$\mathfrak{g}=\mathfrak{gl}_{1}$ and [63] for the general case. The simplest
example is $\mathfrak{g}=\mathfrak{g^{\prime}}=\mathfrak{gl}_{1}$ and
$V^{-\psi}$ the WZNW theory of $SL(2)$ at level $-\psi-2$ and $W^{\phi}$ the
$\mathcal{N}=2$ superconformal algebra at same central charge as the WZNW
theory. In this case the kernel VOA is just a pair of free fermions times a
free boson and our procedure is exactly the Kazama-Suzuki coset realization of
the super conformal algebra [64, 65]. The corresponding duality of conformal
field theories is the well established $H_{3}^{+}$ model to $\mathcal{N}=2$
super Liouville duality [20, 2]. The natural generalization is Feigin-
Semikhatov’s duality between subregular $W$-algebras and principal
$W$-superalgebras [66], proven in [67].

This means if we take a module $M$ of $V^{-\psi}$, then
$H^{\text{BRST}}_{\mathfrak{g}}(M\otimes
A^{1}[\mathfrak{g},\mathfrak{g}^{\prime},\psi])$ is automatically a
$W^{\phi}$-module. The idea is that there is a blockwise equivalence of
categories of $V^{-\psi}$ and $W^{\phi}$ whose underlying functor is the BRST-
cohomology against the kernel VOA. This functor should also provide
isomorphisms on spaces of intertwining algebras, which is indeed true for
$\mathfrak{g}=\mathfrak{gl}_{1}$ by [62]. From a conformal field theory
perspective one prefers a precise matching of correlation functions and
similarly from the quantum geometric Langlands point of view one is interested
in an equivalence of spaces of conformal blocks. Our aim is to provide this
conformal field theory perspective. If $\mathfrak{g}=\mathfrak{gl}_{1}$, then
we already succeeded [4].

We now want to turn to non-abelian underlying Lie algebras. The most
interesting case is the large $\mathcal{N}=4$ superconformal algebra that has
been realized as a BRST-cohomology of two algebras associated to the
exceptional Lie superalgebra $\mathfrak{d}(2,1;1-\psi)$ at level one [16]. In
fact $\mathfrak{d}(2,1;1-\psi)$ at level one is exactly the kernel VOA
$A^{1}[\mathfrak{sl}_{2},\mathfrak{sl}_{2},\psi])$ [39] and the
$\mathcal{N}=4$ superconformal algebra is a simple current extension of
$H^{\text{BRST}}_{\mathfrak{sl}_{2}}(A^{1}[\mathfrak{sl}_{2},\mathfrak{sl}_{2},-\psi])\otimes
A^{1}[\mathfrak{sl}_{2},\mathfrak{sl}_{2},\psi]))$.

In order to lift dualities of VOAs of this type to correspondences of
conformal field theory we have to understand how to implement the BRST-
cohomology (2.6) on the level of correlation functions. The first step for
this is a suitable first order formulation of the conformal field theory.

##  3 First order formulation of coset models

In the analysis of this paper, we utilize the first order formulation of coset
models in [5] which refines the proposal of [25, 26] along with the reduction
methods of [8, 1]. In this section, we review the first order formulation of
cosets in [5] using the BRST formulation [28, 29, 30, 31] and the Kugo-Ogima
method [32]. In order to illustrate the formulation, we analyze the simplest
and non-trivial example the coset (1.3) with $n=2$ and relate it with
Liouville field theory.

In the BRST formulation, the effective action of the coset theory (1.3) is
given by

$\displaystyle
S=S_{k}^{\text{WZNW}}[\phi,\beta,\gamma]+S_{\psi}[\psi]+S_{-k+5}^{\text{WZNW}}[\tilde{\phi},\tilde{\beta},\tilde{\gamma}]+S_{bc}[b^{a},c_{a}]\,.$
(3.1)

For the action of $SL(2)$ WZNW model, we use the first order formulation as

$\displaystyle S_{k}^{\text{WZNW}}[\phi,\beta,\gamma]=\frac{1}{2\pi}\int
d^{2}w\left[\partial\phi\bar{\partial}\phi+\beta\bar{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}+\frac{b}{4}\sqrt{g}\mathcal{R}\phi+\lambda\beta\bar{\beta}e^{2b\phi}\right]$
(3.2)

with $b=1/\sqrt{k-2}$. Here $g_{\mu\nu}$ represents the worldsheet metric and
$g=\det g_{\mu\nu}$. Furthermore, $\mathcal{R}$ is the Ricci curvature with
respect to the worldsheet metric. We denote the $\mathfrak{sl}(2)$ currents at
level $k$ by $J^{a}$ with $a=\pm,3$, which are expressed as

$\displaystyle J^{+}(z)=\beta\,,\quad
J^{3}(z)=b^{-1}\partial\phi+\beta\gamma\,,\quad
J^{-}(z)=\beta\gamma\gamma+2b^{-1}\gamma\partial\phi-k\partial\gamma\,.$ (3.3)

For the third action, the level $k$ is replaced by $-k+1-2c_{SL(2)}=-k+5$,
where $c_{SL(2)}=-2$ is the dual Coxeter number of $SL(2)$. We also use
$\tilde{b}=1/\sqrt{-k+3}$. The $\mathfrak{sl}(2)$ currents at level
$\tilde{k}=-k+5$ is denoted by $\tilde{J}^{a}$ with $a=\pm,3$. The second
action $S_{\psi}[\psi]$ is for complex free fermions
$\psi^{\pm},\bar{\psi}^{\pm}$ and given by

$\displaystyle S_{\psi}[\psi]=\frac{1}{2\pi}\int
d^{2}w\left[\psi^{+}\bar{\partial}\psi^{-}+\bar{\psi}^{+}\partial\bar{\psi}^{-}\right]\,.$
(3.4)

The conformal weights of the fermions are $1/2$. We frequently use its
bosonized formulation as

$\displaystyle\psi^{\pm}=e^{\pm i\sqrt{2}Y^{L}}\,,\quad
Y^{L}(z)Y^{L}(0)\sim-\frac{1}{2}\ln z\,.$ (3.5)

We also introduce $Y^{R}(\bar{z})$ in a similar manner and set
$Y=Y^{L}+Y^{R}$. This model has the symmetry of $\mathfrak{sl}(2)$ current
algebra with level $-1$, and the generators are

$\displaystyle J^{+}_{\psi}=-e^{2iY^{L}}\,,\quad J^{3}_{\psi}=i\partial
Y^{L}\,,\quad J^{-}_{\psi}=e^{-2iY^{L}}\,.$ (3.6)

The other action $S_{bc}[b^{a},c_{a}]$ is for the BRST ghosts and given by

$\displaystyle S_{bc}[b^{a},c_{a}]=\frac{1}{2\pi}\int
d^{2}w\sum_{a=\pm,3}\left[b^{a}\bar{\partial}c_{a}+\bar{b}^{a}\partial\bar{c}_{a}\right]\,,$
(3.7)

where the conformal weights of $(b^{a},c_{a})$ are $(1,0)$. We can construct
$\mathfrak{sl}(2)$ currents at level $2c_{SL(2)}=-4$, which are denoted by
$J^{a}_{bc}$ with $a=\pm,3$. The BRST charge for the holomorphic part is then
given by

$\displaystyle Q=\oint\frac{dz}{2\pi
i}\left[c_{a}(z)\left(J^{a}(z)+J^{a}_{\psi}(z)+\tilde{J}^{a}(z)+\frac{1}{2}J_{bc}^{a}(z)\right)\right]\,,$
(3.8)

which satisfies the nilpotency condition as $Q^{2}=0$. The physical states are
given by elements of $Q$-cohomology in the BRST formulation.

We consider the correlation functions of vertex operators of the form

$\displaystyle
V=\mathcal{P}(\gamma,\tilde{\gamma})e^{2bj\phi}e^{2i(sY^{L}+\bar{s}Y^{R})}e^{2\tilde{b}\tilde{\jmath}\tilde{\phi}}\,,$
(3.9)

where $\mathcal{P}$ is a function of $\gamma,\tilde{\gamma}$ (and
$\bar{\gamma},\bar{\tilde{\gamma}}$). We define an operator

$\displaystyle N^{\beta\gamma}=\sum_{m=-\infty}^{\infty}\beta_{-m}\gamma_{m}$
(3.10)

and decompose the BRST charge as $Q=Q_{1}+Q_{0}+Q_{-1}$ by its eigenvalue. In
particular, we have $Q_{1}=\sum_{m}\beta_{-m}c_{+,m}$, which satisfy
$(Q_{1})^{2}=0$ as well. Defining $R=\sum_{m}\gamma_{-m}b^{+}_{m}$, we
introduce

$\displaystyle
S=\\{Q_{1},R\\}=\sum_{m=1}^{\infty}\beta_{-m}\gamma_{m}+\sum_{m=0}^{\infty}\gamma_{-m}\beta_{m}+\sum_{m=1}^{\infty}b^{+}_{-m}c_{+,m}-\sum_{m=0}^{\infty}c_{+,-m}b^{+}_{m}\,.$
(3.11)

The terms including the BRST ghosts will be neglected in the current analysis.
Since $S$ commute with $Q_{1}$, we can deal with eigenstates of $S$ in
$Q_{1}$-cohomology as $S|s\rangle=s|s\rangle$. If $s$ is non-zero, then we can
rewrite the eigenstate as

$\displaystyle|s\rangle=\frac{1}{s}\\{Q_{1},R\\}|s\rangle=\frac{1}{s}Q_{1}R|s\rangle\,,$
(3.12)

which is $Q_{1}$-exact. Thus, non-trivial elements of $Q_{1}$-cohomology come
from the sector with zero eigenvalue. With a new operator
$U=\\{Q_{0}+Q_{-1},R\\}$, a new state can be defined by

$\displaystyle|s^{\prime}\rangle=(1-S^{-1}U+S^{-1}US^{-1}U-\cdots)|s\rangle\,.$
(3.13)

If $S|s\rangle=0$, then we can show that $(S+U)|s^{\prime}\rangle=0$. This
means that $|s^{\prime}\rangle$ can be an non-trivial element of
$Q$-cohomology as well. In other words, a non-trivial element of
$Q$-cohomology can be put in the form of (3.13).

In the effective action (3.1) with (3.2), an interaction term includes
$\beta$, which increases the eigenvalue of $N^{\beta\gamma}$ defined in
(3.10). The field can be removed by subtracting a BRST exact term as

$\displaystyle\beta(w)-\oint_{w}\frac{dz}{2\pi
i}b^{+}(z)Q(w)=e^{2iY^{L}}(w)-\tilde{\beta}(w)-J^{+}_{bc}(w)\,.$ (3.14)

After removing $\beta$ in the action, there is no operator increasing the
eigenvalue of $N^{\beta\gamma}$ defined in (3.10). On the other hand, we have
shown that elements of $Q$-cohomology can be put into the form (3.13) and the
terms except for the first one in the right hand side decrease the eigenvalue
of $N^{\beta\gamma}$. Therefore, we can conclude that $\beta,\gamma$ can be
removed from the system once $\beta$ in the interaction term is replaced by
(3.14).

We can restrict the form of vertex operators furthermore. We notice that the
operators

$\displaystyle
J^{\text{tot},3}_{0}\equiv\\{Q,b_{0}^{3}\\}=J^{3}_{0}+J^{3}_{\psi,0}+\tilde{J}_{0}^{3}+J_{bc,0}^{3}\,,$
(3.15) $\displaystyle
L^{\text{tot}}_{0}\equiv\\{Q,\tfrac{1}{k-3}\sum_{n}(J^{a}_{n}+J^{a}_{\psi,n}-\tilde{J}^{a}_{n})b_{-n,a}\\}$
(3.16)

commute with $Q$, so we can take eigenfunctions of these operators in the
$Q$-cohomology. Moreover, from the arguments around (3.12), we can conclude
that non-trivial elements of $Q$-cohomology only come from the sector with
zero eigenvalues. Since $\gamma$-dependence (and $\bar{\gamma}$-dependence) is
removed, the vertex operators take the form

$\displaystyle
V=\tilde{\gamma}^{m}{\bar{\tilde{\gamma}}}^{\bar{m}}e^{2bj\phi}e^{2i(sY^{L}+\bar{s}Y^{R})}e^{2\tilde{b}\tilde{\jmath}\tilde{\phi}}\,.$
(3.17)

The conditions of zero eigenvalues for the operators (3.15) and (3.16)
become222There is a term $+1$ in the left hand side of the first equation.
This term can be explained by the definition of vacuum for the BRST ghosts,
see [31, 5].

$\displaystyle-
j-m+s-\tilde{\jmath}+1=0\,,\quad-\frac{(j-s)(j-s-1)}{k-3}+\frac{\tilde{\jmath}(\tilde{\jmath}-1)}{k-3}=0$
(3.18)

and that with $m,s$ replaced by $\bar{m},\bar{s}$. From these conditions, the
vertex operators can be put into the form

$\displaystyle V=e^{2bj\phi}e^{2isY}e^{2\tilde{b}(1-j+s)\tilde{\phi}}\,.$
(3.19)

Since there is no $\tilde{\gamma}$-dependence here, we can neglect
$\tilde{\beta}$ in the action as well.

We may rotate the fields as

$\displaystyle
b\phi+iY+b^{\prime}\phi^{\prime}\,,\quad-i\phi+bY=b^{\prime}Y^{\prime}\,,\quad
b^{\prime}=\sqrt{\frac{3-k}{k-2}}\,,$ (3.20)

then the vertex operators become

$\displaystyle
V=e^{2((b^{\prime}+1/b^{\prime})j-1/b^{\prime}s)\phi^{\prime}-2i\tilde{b}(j-s)Y^{\prime}+2\tilde{b}(1-j+s)\tilde{\phi}}\,.$
(3.21)

The background charges for $Y^{\prime},\tilde{\phi}$ are
$Q_{Y^{\prime}}=-i\tilde{b},Q_{\tilde{\phi}}=\tilde{b}$, respectively, and
moreover there is no interaction terms for $Y^{\prime},\tilde{\phi}$.
Therefore, after performing reflection relations, contributions from
$Y^{\prime},\tilde{\phi}$ to the correlation functions cancel out with each
other. In summary, the $N$-point function of primary operators in the coset
(1.3) can be evaluated as

$\displaystyle\left\langle\prod_{\nu=1}^{N}V_{\nu}(z_{\nu})\right\rangle\,,\quad
V_{\nu}(z_{\nu})=e^{2((b^{\prime}+1/b^{\prime})j_{\nu}-1/b^{\prime}s_{\nu})\phi^{\prime}(z_{\nu})}$
(3.22)

with the action

$\displaystyle S=\frac{1}{2\pi}\int
d^{2}w\left[\partial\phi^{\prime}\partial\phi^{\prime}+\frac{\sqrt{g}\mathcal{R}}{4}(b^{\prime}+1/b^{\prime})\phi^{\prime}+\lambda
e^{2b^{\prime}\phi^{\prime}}\right]\,.$ (3.23)

This is nothing but an $N$-point function of Liouville field theory.

Here we compare the analysis to that of [25, 26]. As they proposed,
$\beta,\gamma$ from the numerator are canceled with
$\tilde{\beta},\tilde{\gamma}$ from the denominator. Moreover, the free boson
corresponding to Cartan direction is taken to be orthogonal to $\tilde{\phi}$
from the denominator via the rotation (3.20). However, from their analysis, it
is not clear how the interaction term of Liouville field theory arises. In our
formulation, this comes from the replacement (3.14), which may be the most
important point in our approach. In the succeeding sections, we apply this
analysis to more involved examples.

##  4 FZZ-triality

Applying the method reviewed in the previous section, we examine correlation
functions of primary operators in the coset (1.4). We derive correlator
correspondences between the coset (1.4) and sine-Liouville theory in the next
subsection and those between the cosets (1.4) and (1.1) in subsection 4.2. For
the two cases, we use different first order formulations of $SL(2|1)_{k}$ in
the numerator of (1.4). In subsection 4.3, we extend the analysis by
introducing $\mathcal{N}=2$ superconformal symmetry.

###  4.1 A realization of the coset

In order to apply our method to the coset (1.4), we first need a first order
formulation of $SL(2|1)$ WZNW model. In this subsection, we use the action
obtained in [9, 10]. We express the element of Lie supergroup $SL(2|1)$ as

$\displaystyle g=g_{-1}g_{-\frac{1}{2}}g_{0}g_{\frac{1}{2}}g_{1}$ (4.1)

with

$\displaystyle g_{-1}=e^{\gamma E^{+}}\,,\quad
g_{-\frac{1}{2}}=e^{\theta_{1}F^{+}+\theta_{2}G^{+}}\,,\quad
g_{0}=e^{2\phi_{1}H+2\phi_{2}I}\,,\quad
g_{\frac{1}{2}}=e^{\bar{\theta}_{2}F^{-}+\bar{\theta}_{1}G^{-}}\,,\quad
g_{1}=e^{\bar{\gamma}E^{-}}\,.$ (4.2)

We use the notation for generators of $\mathfrak{sl}(2|1)$ given in appendix
A.1. The grading is made by $H$. With the help of Polyakov-Wiegmann identity,
we then obtain

$\displaystyle\begin{aligned} S=\frac{k}{2\pi}\int
d^{2}z\left[\bar{\partial}\phi_{1}\partial\phi_{1}-\bar{\partial}\phi_{2}\partial\phi_{2}-e^{-\phi_{1}-\phi_{2}}\bar{\partial}\theta_{1}\partial\bar{\theta}_{1}+e^{-\phi_{1}+\phi_{2}}\bar{\partial}\theta_{2}\partial\bar{\theta}_{2}\right.\\\
+\left.e^{-2\phi_{1}}\left(\partial\bar{\gamma}-\tfrac{1}{2}(\bar{\theta}_{2}\partial\bar{\theta}_{1}+\bar{\theta}_{1}\partial\bar{\theta}_{2})\right)\left(\bar{\partial}\gamma-\tfrac{1}{2}(\theta_{2}\bar{\partial}\theta_{1}+\theta_{1}\bar{\partial}\theta_{2})\right)\right]\,.\end{aligned}$
(4.3)

Introducing auxiliary fields, we can rewrite the action as

$\displaystyle\begin{aligned} S&=\frac{1}{2\pi}\int
d^{2}z\left[k\bar{\partial}\phi_{1}\partial\phi_{1}-k\bar{\partial}\phi_{2}\partial\phi_{2}+\beta\bar{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}+\sum_{a=1}^{2}(p_{a}\bar{\partial}\theta_{a}+\bar{p}_{a}\partial\bar{\theta}_{a})-\frac{1}{k}e^{2\phi_{1}}\beta\bar{\beta}\right]\\\
&\quad+\frac{1}{2k\pi}\int
d^{2}z\left[e^{\phi_{1}+\phi_{2}}(p_{1}+\tfrac{1}{2}\beta\theta_{2})(\bar{p}_{1}+\tfrac{1}{2}\bar{\beta}\bar{\theta}_{2})-e^{\phi_{1}-\phi_{2}}(p_{2}+\tfrac{1}{2}\beta\theta_{1})(\bar{p}_{2}+\tfrac{1}{2}\bar{\beta}\bar{\theta}_{1})\right]\,.\end{aligned}$
(4.4)

The equations of motion lead to

$\displaystyle\begin{aligned}
&\beta=ke^{-2\phi_{1}}(\partial\bar{\gamma}-\tfrac{1}{2}(\bar{\theta}_{2}\partial\bar{\theta}_{1}+\bar{\theta}_{1}\partial\bar{\theta}_{2}))\,,\\\
&p_{1}+\tfrac{1}{2}\beta\theta_{2}=ke^{-\phi_{1}-\phi_{2}}\partial\bar{\theta}_{1}\,,\quad
p_{2}+\tfrac{1}{2}\beta\theta_{1}=-ke^{-\phi_{1}+\phi_{2}}\partial\bar{\theta}_{2}\,,\\\
&\bar{\beta}=ke^{-2\phi_{1}}(\bar{\partial}\gamma-\tfrac{1}{2}(\theta_{2}\bar{\partial}\theta_{1}+\theta_{1}\bar{\partial}\theta_{2}))\,,\\\
&\bar{p}_{1}+\tfrac{1}{2}\bar{\beta}\bar{\theta}_{2}=-ke^{-\phi_{1}-\phi_{2}}\bar{\partial}\theta_{1}\,,\quad\bar{p}_{2}+\tfrac{1}{2}\bar{\beta}\bar{\theta}_{1}=ke^{-\phi_{1}+\phi_{2}}\bar{\partial}\theta_{2}\,.\end{aligned}$
(4.5)

Taking into account quantum corrections due to the change of variables (see,
e.g., (2.8) and (2.9) of [9]), we have

$\displaystyle\begin{aligned} S&=\frac{1}{2\pi}\int
d^{2}z\left[\bar{\partial}\phi_{1}\partial\phi_{1}-\bar{\partial}\phi_{2}\partial\phi_{2}+\beta\bar{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}+\sum_{a=1}^{2}(p_{a}\bar{\partial}\theta_{a}+\bar{p}_{a}\partial\bar{\theta}_{a})-\frac{1}{k}e^{2b\phi_{1}}\beta\bar{\beta}\right]\\\
&\quad+\frac{1}{2k\pi}\int
d^{2}z\left[e^{b(\phi_{1}+\phi_{2})}(p_{1}+\tfrac{1}{2}\beta\theta_{2})(\bar{p}_{1}+\tfrac{1}{2}\bar{\beta}\bar{\theta}_{2})-e^{b(\phi_{1}-\phi_{2})}(p_{2}+\tfrac{1}{2}\beta\theta_{1})(\bar{p}_{2}+\tfrac{1}{2}\bar{\beta}\bar{\theta}_{1})\right]\end{aligned}$
(4.6)

with $b=1/\sqrt{k-1}$. Since the interaction term with
$e^{2b\phi}\beta\bar{\beta}$ can be induced from the rest two, we simply
neglect the term.

The symmetry of $SL(2|1)$ WZNW model is the $\mathfrak{sl}(2|1)$ current
algebra. In the first order formulation, the generators are written in terms
of fields and commute with interaction terms. We find that they are given by

$\displaystyle F^{+}=p_{1}-\frac{1}{2}\beta\theta_{2}\,,\quad
G^{+}=-p_{2}+\frac{1}{2}\beta\theta_{1}\,,\quad E^{+}=\beta\,,$ $\displaystyle
H=b^{-1}\partial\phi_{1}+\gamma\beta+\frac{1}{2}(p_{1}\theta_{1}+p_{2}\theta_{2})\,,\quad
I=-b^{-1}\partial\phi_{2}+\frac{1}{2}(p_{1}\theta_{1}-p_{2}\theta_{2})\,,$
$\displaystyle
F^{-}=-b^{-1}\partial\phi_{1}\theta_{2}+b^{-1}\partial\phi_{2}\theta_{2}-\frac{1}{2}\gamma\beta\theta_{2}-\frac{1}{2}p_{1}\theta_{1}\theta_{2}-\left(\frac{1}{2}-k\right)\partial\theta_{2}+\gamma
p_{1}\,,$ (4.7) $\displaystyle
G^{-}=b^{-1}\partial\phi_{1}\theta_{1}+b^{-1}\partial\phi_{2}\theta_{1}+\frac{1}{2}\gamma\beta\theta_{1}-\frac{1}{2}p_{2}\theta_{1}\theta_{2}+\left(\frac{1}{2}-k\right)\partial\theta_{1}-\gamma
p_{2}\,,$ $\displaystyle E^{-}=-\gamma\gamma\beta-\gamma
p_{1}\theta_{1}-\gamma
p_{2}\theta_{2}-2b^{-1}\partial\phi_{1}\gamma+b^{-1}\partial\phi_{2}\theta_{1}\theta_{2}-\frac{1}{2}(1-k)(\theta_{1}\partial\theta_{2}-\partial\theta_{1}\theta_{2})+k\partial\gamma\,.$

Here the normal ordering prescription is assumed for the products of fields.
It will be convenient to bosonize the fermionic systems $(p_{a},\theta_{a})$
as

$\displaystyle p_{a}=e^{iX_{a}^{L}}\,,\quad\theta_{a}=e^{-iX_{a}^{L}}\,,\quad
X_{a}^{L}(z)X_{b}^{L}(0)\sim-\delta_{a,b}\ln z\,.$ (4.8)

We also define $X^{R}_{a}$ analogously from $(\bar{p}_{a},\bar{\theta}_{a})$
and consider linear combinations $X_{a}=X_{a}^{L}+X_{a}^{R}$. With these
bosons, the Cartan generators are expressed as

$\displaystyle H=b^{-1}\partial\phi_{1}+\gamma\beta+\frac{i}{2}(\partial
X_{1}+\partial X_{2})\,,\quad I=-b^{-1}\partial\phi_{2}+\frac{i}{2}(\partial
X_{1}-\partial X_{2})$ (4.9)

in particular.

Now that we have the first order formulation of $SL(2|1)$ WZNW model, we can
apply the procedure of [5] reviewed in the previous section. Firstly, two new
bosons $\hat{\phi}_{1},\hat{\phi}_{2}$ are introduced by

$\displaystyle H=\sqrt{k-2}\partial\hat{\phi}_{1}+\beta\gamma\,,\quad
I=\sqrt{k}\partial\hat{\phi}_{2}\,.$ (4.10)

Then the field space of the coset model (1.4) is spanned by the fields
$\phi_{1},\phi_{2},X_{1},X_{2}$ but orthogonal to the new bosons
$\hat{\phi}_{1},\hat{\phi}_{2}$. The $(\beta,\gamma)$-system for the free
filed realization of $\mathfrak{sl}(2|1)$ are removed as explained above. In
other words, the orthogonal space can be generated by $\phi,\chi$, where we
have defined as

$\displaystyle\sqrt{k-2}\phi=-i\phi_{1}+\frac{1}{2b}(X_{1}+X_{2})\,,\quad\sqrt{k}\chi=-i\phi_{2}+\frac{1}{2b}(X_{1}-X_{2})\,.$
(4.11)

With the new variables, the correlation functions are written as

$\displaystyle\left\langle\prod_{\nu=1}^{N}\Psi_{\nu}(z_{\nu})\right\rangle\,,\quad\Psi_{\nu}(z_{\nu})=e^{2\hat{b}j\phi+i\frac{2}{\sqrt{\kappa}}(m\chi_{L}+\bar{m}\chi_{R})}$
(4.12)

with

$\displaystyle\kappa=\frac{k}{k-1}\,.$ (4.13)

The correlation function is evaluated with the action333The coefficients in
front of interaction terms are modified by shifting fields. The same procedure
will be performed in later analysis as well.

$\displaystyle\begin{aligned} S&=\frac{1}{2\pi}\int
d^{2}z\left[\bar{\partial}\phi\partial\phi+\bar{\partial}\chi\partial\chi+\frac{\hat{b}}{4}\sqrt{g}\mathcal{R}\phi+2\lambda
e^{\phi/\hat{b}}\cos\left(\sqrt{\kappa}\chi\right)\right]\,,\end{aligned}$
(4.14)

where we set $\hat{b}=1/\sqrt{\kappa-2}$. The background charge for $\phi$
arises from those for $X_{1}$ and $X_{2}$. This is nothing but the action of
sine-Liouville theory. In this way, we can write down the $N$-point function
of the coset (1.4) in terms of sine-Liouville theory as a direct consequence
of the first order formulation reviewed in the previous section.

###  4.2 Another realization of the coset

In the previous subsection, we have shown that correlation functions of the
coset (1.4) directly reduce to those of sine-Liouville theory. Combined with
the FZZ-duality proven in [1], we can derive the correlator correspondences
between the two cosets (1.4) and (1.1). Instead of doing so, we directly
relate the correlation functions of these two cosets by choosing a different
first order formulation of $SL(2|1)$ WZNW model in this subsection.

It is known that there are several different free field realizations of the
same affine Lie algebra. In terms of WZNW model, this corresponds to the fact
that there are several ways to parameterize group elements. In [9, 10] or
(4.1), we use the grading corresponding to the Cartan subalgebra generated by
$H$. This is a natural choice, but here we adopt a different one with the
grading of linear combination $H+I$. Then the element of Lie supergroup
$SL(2|1)$ can be expressed as

$\displaystyle g=g_{+}g_{0}g_{-}$ (4.15)

with

$\displaystyle g_{+}=e^{\gamma E^{+}}e^{\theta_{1}F^{+}}\,,\quad
g_{0}=e^{\theta_{2}F^{-}}e^{2\phi_{1}H+2\phi_{2}I}e^{\bar{\theta}_{2}G^{+}}\,,\quad
g_{-}=e^{\bar{\theta}_{1}G^{-}}e^{\bar{\gamma}E^{-}}\,.$ (4.16)

Note that $g_{+}$ ($g_{-}$) is put on the left (right) side of $g$ but a
factor including $G^{+}$ ($F^{-}$) is put on the right (left) side inside
$g_{0}$. Otherwise, the expression of Lie superalgebra becomes identical to
the one in the previous subsection. Using the Polyakov-Wiegmann identity
again, we obtain

$\displaystyle\begin{aligned} S&=\frac{k}{2\pi}\int
d^{2}z\left[\bar{\partial}\phi_{1}\partial\phi_{1}-\bar{\partial}\phi_{2}\partial\phi_{2}+e^{-2\phi_{1}}\partial\bar{\gamma}\bar{\partial}\gamma\right.\\\
&\quad-\left.e^{-\phi_{1}-\phi_{2}}(\bar{\partial}\theta_{1}+\theta_{2}\bar{\partial}\gamma)(\partial\bar{\theta}_{1}+\bar{\theta}_{2}\partial\bar{\gamma})-e^{\phi_{1}-\phi_{2}}\bar{\partial}\theta_{2}\partial\bar{\theta}_{2}\right]\,.\end{aligned}$
(4.17)

Introducing auxiliary fields, we rewrite the action as

$\displaystyle S$ $\displaystyle=\frac{1}{2\pi}\int
d^{2}z\left[\bar{\partial}\phi_{1}\partial\phi_{1}-\bar{\partial}\phi_{2}\partial\phi_{2}+\frac{b}{4}\sqrt{g}\mathcal{R}(\phi_{1}-\phi_{2})+\beta\bar{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}+\sum_{a=1}^{2}(p_{a}\bar{\partial}\theta_{a}+\bar{p}_{a}\partial\bar{\theta}_{a})\right]$
$\displaystyle\quad-\frac{1}{2k\pi}\int d^{2}z\left[e^{2b\phi_{1}}(\beta-
p_{1}\theta_{2})(\bar{\beta}-\bar{p}_{1}\bar{\theta}_{2})-e^{b(\phi_{1}+\phi_{2})}p_{1}\bar{p}_{1}-e^{b(-\phi_{1}+\phi_{2})}p_{2}\bar{p}_{2}\right]$
(4.18)

with $b=1/\sqrt{k-1}$ as above. Here we have already included quantum
corrections. We will neglect the interaction term
$e^{b(\phi_{1}+\phi_{2})}p_{1}\bar{p}_{1}$ since it can be generated by the
rest two.

As mentioned above, a different parameterization of group element in WZNW
model corresponds to a different free field realization of the symmetry
algebra. The corresponding free field realization of affine Lie superalgebra
$\mathfrak{sl}(2|1)$ is given by fields in the kinetic terms and generators
made of the free fields should commute with the interaction terms. We find
these generators as

$\displaystyle\begin{aligned} &F^{+}=p_{1}\,,\quad
G^{+}=\beta\theta_{1}-b^{-1}\partial\phi_{1}\theta_{2}+b^{-1}\partial\phi_{2}\theta_{2}+(1-k)\partial\theta_{2}\,,\quad
E^{+}=\beta\,,\\\
&H=b^{-1}\partial\phi_{1}+\gamma\beta+\frac{1}{2}(p_{1}\theta_{1}-p_{2}\theta_{2})\,,\quad
I=-b^{-1}\partial\phi_{2}+\frac{1}{2}(p_{1}\theta_{1}+p_{2}\theta_{2})\,,\\\
&F^{-}=-p_{2}+\gamma p_{1}\,,\quad
G^{-}=b^{-1}\partial\phi_{1}\theta_{1}+b^{-1}\partial\phi_{2}\theta_{1}+\gamma\beta\theta_{1}+p_{2}\theta_{1}\theta_{2}-k\partial\theta_{1}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad-b^{-1}\partial\phi_{1}\gamma\theta_{2}+b^{-1}\partial\phi_{2}\gamma\theta_{2}-(k-1)\gamma\partial\theta_{2}\,,\\\
&E^{-}=-\gamma\gamma\beta-\gamma p_{1}\theta_{1}+\gamma
p_{2}\theta_{2}-2b^{-1}\partial\phi_{1}\gamma+p_{2}\theta_{1}+k\partial\gamma\,.\end{aligned}$
(4.19)

The Cartan generators can be expressed as

$\displaystyle H=b^{-1}\partial\phi_{1}+\gamma\beta+\frac{i}{2}(\partial
X_{1}-\partial X_{2})\,,\quad I=-b^{-1}\partial\phi_{2}+\frac{i}{2}(\partial
X_{1}+\partial X_{2})$ (4.20)

in terms of $X_{a}$ introduced in (4.8).

Let us apply the first order formulation reviewed in the previous section to
this case. As before we define new bosons $\hat{\phi}_{1},\hat{\phi}_{2}$ by

$\displaystyle H=\sqrt{k-2}\partial\hat{\phi}_{1}+\beta\gamma\,,\quad
I=\sqrt{k}\partial\hat{\phi}_{2}\,.$ (4.21)

The field space for the numerator is generated by
$\phi_{1},\phi_{2},X_{1},X_{2}$ if $\beta,\gamma$ are neglected, and the field
space orthogonal to $\hat{\phi}_{1},\hat{\phi}_{2}$ may be spanned by
$\phi,\chi$ with

$\displaystyle\sqrt{k-2}\phi=-i\phi_{1}+\frac{1}{2b}(X_{1}-X_{2})\,,\quad\sqrt{k}\chi=-i\phi_{2}+\frac{1}{2b}(X_{1}+X_{2})\,.$
(4.22)

The correlation functions are of the form as (4.12) but now the action is

$\displaystyle\begin{aligned} S&=\frac{1}{2\pi}\int
d^{2}z\left[\bar{\partial}\phi\partial\phi+\bar{\partial}\chi\partial\chi+\frac{1}{4}\sqrt{g}\mathcal{R}(Q_{\phi}\phi+Q_{\chi}\chi)-\frac{1}{k}(e^{2\phi/\hat{b}}-e^{-\phi/\hat{b}+i\sqrt{\kappa}\chi})\right]\,.\end{aligned}$
(4.23)

Here we set

$\displaystyle Q_{\phi}=\hat{b}+\frac{1}{\hat{b}}\,,\quad
Q_{\chi}=-i\sqrt{\kappa}$ (4.24)

with $\kappa$ given in (4.13) and $\hat{b}=1/\sqrt{\kappa-2}$. The action
coincides with (3.14) of [1], which directly arises from the reduction of
$SL(2)\otimes U(1)\,(\otimes\text{BRST ghosts})$ description for the coset
$SL(2)/U(1)$. The first term of the interaction comes from (the dual of) the
interaction term in the first order formulation of $SL(2)$ WZNW model and the
second one stems from the extra insertions of degenerate operators in the
Ribault-Teschner relation [6, 7, 8].

###  4.3 Fermionic dualities

In this subsection, we extend the analysis by introducing $\mathcal{N}=2$
supersymmetry. The fermionic version of the coset is given by

$\displaystyle\frac{SL(2|1)_{k}\otimes SO(2)_{1}}{SL(2)_{k}\otimes U(1)}\,.$
(4.25)

The factor $SO(2)_{1}$ can be generated by a complex fermion $\psi^{\pm}$ with
conformal weight $1/2$, and the bosonization formula

$\displaystyle\psi^{\pm}=e^{\pm iY^{L}}\,,\quad Y^{L}(z)Y^{L}(0)\sim-\ln z$
(4.26)

is used. We define $Y^{R}$ in a similar way and introduce $Y=Y^{L}+Y^{R}$.
Notice that the Cartan directions of the denominator algebra are generated by

$\displaystyle H\,,\quad I+\tfrac{1}{2}\psi^{+}\psi^{-}=I+\frac{i}{2}\partial
Y\,.$ (4.27)

We first use the free field realization of affine Lie superalgebra
$\mathfrak{sl}(2|1)$ in subsection 4.1. In this case, the orthogonal space is
generated by $\hat{Y}$ as well as $\phi,\chi$ defined in (4.11). Here
$\hat{Y}$ may be given by

$\displaystyle\sqrt{\frac{k(1+2k)}{2}}\hat{Y}=ib^{-1}\phi_{2}+\frac{1}{2}(X_{1}-X_{2})-kY\,.$
(4.28)

We further rotate the fields as

$\displaystyle\sqrt{\frac{k}{k-1}}\chi\to\sqrt{\frac{2-k}{k-1}}\chi+\hat{Y}\,,\quad\sqrt{\frac{k}{k-1}}\hat{Y}\to-2\chi+\sqrt{\frac{2-k}{k-1}}\hat{Y}\,.$
(4.29)

The correlation functions are of the form as

$\displaystyle\left\langle\prod_{\nu=1}^{N}\Psi_{\nu}(z_{\nu})\right\rangle\,,\quad\Psi_{\nu}(z_{\nu})=e^{2\hat{b}j\phi+i(sY^{L}+\bar{s}Y^{R})+i\frac{2}{\sqrt{\kappa}}(m\chi_{L}+\bar{m}\chi_{R})}$
(4.30)

and the action is

$\displaystyle\begin{aligned} S&=\frac{1}{2\pi}\int
d^{2}z\left[\bar{\partial}\phi\partial\phi+\bar{\partial}\chi\partial\chi+\psi_{+}\bar{\partial}\psi_{-}+\bar{\psi}_{+}\partial\bar{\psi}_{-}+\frac{\hat{b}}{4}\sqrt{g}\mathcal{R}\phi\right]\\\
&\quad+\frac{\lambda}{2\pi}\int
d^{2}z\left[\psi_{+}\bar{\psi}_{+}e^{\hat{b}{}^{-1}(\phi+i\chi)}+\psi_{-}\bar{\psi}_{-}e^{\hat{b}{}^{-1}(\phi-i\chi)}\right]\,.\end{aligned}$
(4.31)

Here we have introduced new complex fermions $\psi^{\pm}$ using (4.26) but
with $Y$ replaced by $\hat{Y}$. This is nothing but the action of
$\mathcal{N}=2$ super Liouville theory. In this way, we have shown that
$N$-point functions of the super coset (4.25) can be reduced to those of
$\mathcal{N}=2$ super Liouville theory.

We next use another free field realization of affine Lie superalgebra
$\mathfrak{sl}(2|1)$ in subsection 4.2. The field space orthogonal to the
denominator algebra with Cartan directions (4.27) is generated by $\phi,\chi$
in (4.22) and $\hat{Y}$ defined via

$\displaystyle\sqrt{\frac{k(1+2k)}{2}}\hat{Y}=ib^{-1}\phi_{2}+\frac{1}{2}(X_{1}+X_{2})-kY\,.$
(4.32)

We also perform the rotation of fields in (4.29). The correlation functions
are of the form as (4.30) and the action is now

$\displaystyle\begin{aligned} S&=\frac{1}{2\pi}\int
d^{2}z\left[\bar{\partial}\phi\partial\phi+\bar{\partial}\chi\partial\chi+\frac{1}{2}\bar{\partial}\hat{Y}\partial\hat{Y}+\frac{1}{4}\sqrt{g}\mathcal{R}\left(Q_{\phi}\phi+Q_{\chi}\chi+Q_{Y}\hat{Y}\right)\right]\\\
&\quad-\frac{1}{2\pi k}\int
d^{2}z\left[e^{2\hat{b}{}^{-1}\phi}-e^{-\hat{b}{}^{-1}(\phi-i\chi)+i\hat{Y}}\right]\,.\end{aligned}$
(4.33)

The background charges are

$\displaystyle Q_{\phi}=\hat{b}+\hat{b}^{-1}\,,\quad
Q_{\chi}=-i\hat{b}\,,\quad Q_{Y}=-i\,.$ (4.34)

This kinetic terms of the action are the same as those in (3.11) and the
interaction terms are (3.12) in [2].444The notation of $Y$ is different from
$\tilde{h}$ in [2] by factor $-\sqrt{2}$. Applying the analysis of the paper,
we can thus map the correlation functions of (4.25) to those of the super
cigar model described by

$\displaystyle\frac{SL(2)_{\kappa}\otimes SO(2)_{1}}{U(1)}\,.$ (4.35)

In the BRST formulation, the coset can be described by $SL(2)_{\kappa}\otimes
SO(2)_{1}\otimes U(1)\,(\otimes\text{BRST ghosts})$. The first term of the
interaction directly comes from bosonic $SL(2)_{\kappa}$ WZNW model and the
second one stems from an interpretation of extra insertions of degenerate
operators in the Ribault-Teschner relation as in the bosonic case.

##  5 Theory with a $\mathfrak{d}(2,1;-\psi)$-structure from a coset

A main purpose of this paper is to express the large $\mathcal{N}=4$ super
Liouville theory in terms of a coset model (1.7). Before going to it, we would
like to consider a related but different problem. Namely, we study $N$-point
functions of the coset (1.6) and reduce them to those of a theory with a
$\mathfrak{d}(2,1;-\psi)$-structure. The relation between the two models was
examined in [35].

In the examples analyzed in the previous section, we actually needed only the
first order formulation of the coset proposed in [25, 26]. However, in the
example with coset (1.6), we have to use the first order formulation
elaborated in [5] and also the reduction methods utilized in the proof of
original FZZ-duality in [1]. We begin by describing the coset theory (1.6) in
the BRST formulation [28, 29, 30, 31]. The effective action is given by

$\displaystyle
S=\sum_{i=1}^{3}S_{k_{i}}^{\text{WZNW}}[\phi_{i},\beta_{i},\gamma_{i}]+S_{bc}[b^{a},c_{a}]$
(5.1)

with $k_{3}=-k_{1}-k_{2}-2c_{SL(2)}=-k_{1}-k_{2}+4$. For the action of $SL(2)$
WZNW model, we use the first order formulation given in (3.2). We denote the
$\mathfrak{sl}(2)$ currents at level $k_{i}$ by $J^{a}_{i}$ with $a=\pm,3$. In
particular, we have $J^{+}_{i}=\beta_{i}$. The other action
$S_{bc}[b^{a},c_{a}]$ is for the BRST ghosts given in (3.7) and
$\mathfrak{sl}(2)$ currents at level $2c_{SL(2)}=-4$ are denoted by
$J^{a}_{bc}$ with $a=\pm,3$ as before. The BRST charge is

$\displaystyle Q=\oint\frac{dz}{2\pi
i}\left[c_{a}(z)\left(\sum_{i=1}^{3}J^{a}_{i}(z)+\frac{1}{2}J_{bc}^{a}(z)\right)\right]\,,$
(5.2)

and physical states are given by elements of the $Q$-cohomology in the BRST
formulation.

As explained in section 3, we can remove one of pairs of
$(\beta_{i},\gamma_{i})$, say, $(\beta_{1},\gamma_{1})$. Namely, we replace
$\beta_{1}$ by

$\displaystyle\beta_{1}(w)-\oint\frac{dz}{2\pi
i}b^{+}(z)Q(w)=-\beta_{2}(w)-\beta_{3}(w)-J_{bc}^{+}(w)\,.$ (5.3)

Furthermore, we decompose the BRST-charge by
$N^{\beta_{1}\gamma_{1}}=\sum_{m}\beta_{1,-m}\gamma_{1,m}$ and pick up non-
trivial elements of $Q_{1}$-cohomology.

In this way, we can put the vertex operators of the form

$\displaystyle
V=\gamma_{2}^{m_{2}}\bar{\gamma}_{2}^{\bar{m}_{2}}\gamma_{3}^{m_{3}}\bar{\gamma}_{3}^{\bar{m}_{3}}e^{2(b_{1}j_{1}\phi_{1}+b_{2}j_{2}\phi_{2}+b_{3}j_{3}\phi_{3})}$
(5.4)

with $b_{i}=1/\sqrt{k_{i}-2}$. The conditions corresponding to zero
eigenvalues of (3.15) and (3.16) are now given by

$\displaystyle\begin{aligned} &-j_{1}-j_{2}-m_{2}-j_{3}-m_{3}+1=0\,,\\\
&-\frac{(j_{1}+j_{2}+m)(j_{1}+j_{2}+m-1)}{k_{1}+k_{2}-2}+\frac{j_{3}(j_{3}-1)}{k_{1}+k_{2}-2}=0\end{aligned}$
(5.5)

with some integer $m$. There is also a similar condition with $m_{2},m_{3},m$
replaced by $\bar{m}_{2},\bar{m}_{3},\bar{m}$. We have shown that a map can be
constructed between elements of $Q_{1}$-cohomology and $Q$-cohomology for our
restricted form of vertex operators. Thus, we can pick up an element of
$Q_{1}$-cohomology with our preferred choice of $m_{2},m_{3},m$ as long as
they satisfy (5.5). Our choice here is

$\displaystyle m_{2}=-j_{1}-j_{2}-j_{3}+1\,,\quad m_{3}=0\,,\quad m=m_{2}\,.$
(5.6)

It is convenient to perform a reflection relation to $\phi_{3}$-direction as

$\displaystyle
V=\gamma_{2}^{m_{2}}\bar{\gamma}_{2}^{\bar{m}_{2}}e^{2(b_{1}j_{1}\phi_{1}+b_{2}j_{2}\phi_{2}+b_{3}(1-j_{3})\phi_{3})}$
(5.7)

with

$\displaystyle m_{2}=\bar{m}_{2}=-j_{1}-j_{2}-j_{3}+1\,.$ (5.8)

This choice is useful since vertex operators do not depend on $\gamma_{3}$ so
we can neglect $\beta_{3}$ in the action as well.

The action and vertex operators still depend on
$(\beta_{2},\gamma_{2})$-system. We deal with them by applying the reduction
method of [8, 1]. We rewrite the vertex operators as

$\displaystyle\Psi_{\nu}(z_{\nu})=\int\frac{d^{2}\mu_{\nu}}{|\mu_{\nu}|^{2}}V_{\nu}(z_{\nu})\,,$
(5.9) $\displaystyle
V_{\nu}(z_{\nu})=|\mu_{\nu}|^{2j_{1}+2j_{2}+2j_{3}-2}e^{\mu_{\nu}\gamma_{2}-\bar{\mu}_{\nu}\bar{\gamma}_{2}}e^{2(b_{1}j^{\nu}_{1}\phi_{1}+b_{2}j^{\nu}_{2}\phi_{2}+b_{3}(1-j^{\nu}_{3})\phi_{3})}\,.$
(5.10)

We may insert an identity operator in the coset theory as in, e.g., [1, 5].
Now we perform $s$ spectral flow operations to $SL(2)_{k_{1}},SL(2)_{k_{2}}$
in the numerator of (1.6) and the operation is undone in the coset theory by
performing $s$ spectral flow to $SL(2)_{k_{1}+k_{2}}$ in the denominator. In
the current first order formulation, the identity operator is expressed as

$\displaystyle\mathbbm{1}=v^{(s)}(\xi)e^{s(\phi_{1}/b_{1}-\phi_{3}/b_{3})}(\xi)\,,$
(5.11)

where the spectral flow operator $v^{(s)}(\xi)$ restricts the domain of
integral over $\beta_{2}$ such as to have a zero of order $s$ and effectively
inserts $e^{s\phi_{2}/b_{2}}$ at $w=\xi$. We then compute the correlation
function of the coset model (1.6),

$\displaystyle\begin{aligned} &\left\langle
v^{(s)}(\xi)e^{s(\phi_{1}/b_{1}-\phi_{3}/b_{3})}(\xi)\prod_{\nu=1}^{N}V_{\nu}(z_{\nu})\right\rangle\\\
&\quad=\int_{s}\left[\prod_{i=1}^{3}\mathcal{D}\phi_{i}\right]\mathcal{D}\gamma_{2}\mathcal{D}\beta_{2}e^{-S}\prod_{\nu=1}^{N}V_{\nu}(z_{\nu})e^{s(\phi_{1}/b_{1}+\phi_{2}/b_{2}-\phi_{3}/b_{3})}(\xi)\,.\end{aligned}$
(5.12)

Here the effective action is

$\displaystyle S=\frac{1}{2\pi}\int
d^{2}w\left[\sum_{i=1}^{3}\partial\phi_{i}\bar{\partial}\phi_{i}+\beta_{2}\bar{\partial}\gamma_{2}+\bar{\beta}_{2}\partial\bar{\gamma}_{2}+\frac{\sqrt{g}\mathcal{R}}{4}\left(\sum_{i=1}^{3}b_{i}\phi_{i}\right)+\lambda\beta_{2}\bar{\beta}_{2}\left(e^{2b_{1}\phi_{1}}+e^{2b_{2}\phi_{2}}\right)\right]\,.$
(5.13)

Notice that $\beta_{1}$ in the interaction term is replaced by (5.3) and the
terms except for $\beta_{2}$ are neglected. The subscript $s$ in the integral
symbol represents the restriction of integral domain for $\beta_{2}$.
Integrating $\gamma_{2}$ out, we obtain delta functions for $\mu_{\nu}$ and a
delta functional for $\beta_{2}$ as explained in [8, 1]. After further
integrating $\beta_{2}$ out, it is replaced by a function as

$\displaystyle-\beta_{2}(w)=\sum_{\nu=1}^{N}\frac{\mu_{\nu}}{w-z_{\nu}}=u\frac{(w-\xi)^{s}\prod_{i=1}^{N-2-s}(w-y_{i})}{\prod_{\nu=1}^{N}(w-z_{\nu})}\,=u\mathcal{B}(w;z_{\nu},y_{i})\,.$
(5.14)

The second equality defines a map of variables from $\mu_{\nu}$ to $y_{i}$,
which is possible only when the conditions coming from the delta functions,

$\displaystyle\sum_{\nu=1}^{N}\frac{\mu_{\nu}}{(w-\xi)^{a}}=0$ (5.15)

are satisfied for $a=0,1,\ldots,s$. For more details, see, e.g., [1].

In order to remove the functions in the interaction terms, we shift $\phi_{i}$
as

$\displaystyle\phi_{1}+\frac{1}{2b_{1}}\ln|u\mathcal{B}|^{2}\to\phi_{1}\,,\quad\phi_{2}+\frac{1}{2b_{2}}\ln|u\mathcal{B}|^{2}\to\phi_{2}\,,\quad\phi_{3}-\frac{1}{2b_{3}}\ln|u\mathcal{B}|^{2}\to\phi_{3}\,.$
(5.16)

The third one is chosen such that there would be no extra factor with
$\mu_{\nu}$ in the vertex operator of the form (5.10). Now the correlation
function becomes

$\displaystyle\left\langle\prod_{\nu=1}^{N}V_{\nu}(z_{\nu})\right\rangle=\left\langle\prod_{\nu=1}^{N}\tilde{V}_{\nu}(z_{\nu})\prod_{i=1}^{N-2-s}\tilde{V}_{b}(y_{i})\right\rangle$
(5.17)

with the vertex operators

$\displaystyle\tilde{V}_{\nu}(z_{\nu})=e^{2(b_{1}(j^{\nu}_{1}+1/2b_{1}^{2})\phi^{\nu}_{1}+b_{2}(j^{\nu}_{2}+1/2b_{2}^{2})\phi_{2}+b_{3}(1-j^{\nu}_{3}-1/2b_{3}^{2})\phi_{3})}\,,$
(5.18)
$\displaystyle\tilde{V}_{b}(y_{i})=e^{-\phi_{1}/b_{1}-\phi_{2}/b_{2}+\phi_{3}/b_{3}}\,.$
(5.19)

Notice that the insertion at $w=\xi$ is canceled out. The right hand side is
evaluated with the action

$\displaystyle S=\frac{1}{2\pi}\int
d^{2}w\left[\sum_{i=1}^{3}\partial\phi_{i}\bar{\partial}\phi_{i}+\frac{\sqrt{g}\mathcal{R}}{4}\left(\sum_{i=1}^{3}Q_{i}\phi_{i}\right)-\lambda\sum_{i=1}^{2}V_{i}\right]$
(5.20)

with

$\displaystyle Q_{1}=b_{1}+1/b_{1}\,,\quad Q_{2}=b_{2}+1/b_{2}\,,\quad
Q_{3}=b_{3}-1/b_{3}$ (5.21)

and

$\displaystyle
V_{i}=e^{2\beta_{i}\cdot\phi}\,,\quad\beta_{1}=(b_{1},0,0)\,,\quad\beta_{2}=(0,b_{2},0)\,.$
(5.22)

The relation between the correlation functions in (5.17) is up to a factor
that is a function of $z_{\nu},y_{i}$.

We then move to correlation functions of vertex operators of the form (5.9).
As in [8], the $\mu_{\nu}$-integration in (5.9) can be mapped to the
$y_{i}$-integration, and the extra insertions of vertex operators at $y_{i}$
can be interpreted as an interaction term. Thus we can write

$\displaystyle\left\langle\prod_{\nu=1}^{N}\Psi_{\nu}(z_{\nu})\right\rangle=\left\langle\prod_{\nu=1}^{N}\tilde{V}_{\nu}(z_{\nu})\right\rangle\,.$
(5.23)

The relative factor is canceled with the Jacobian due to the change of
variables from $\mu_{\nu}$ to $y_{i}$, see [6, 7, 1]. The action for the right
hand side is given by (5.20) but now there are three interaction terms as

$\displaystyle
V_{i}=e^{2\beta_{i}\cdot\phi}\,,\quad\beta_{1}=\left(\frac{1}{b_{1}},0,0\right)\,,\quad\beta_{2}=\left(0,\frac{1}{b_{2}},0\right)\,,\quad\beta_{3}=\left(-\frac{1}{2b_{1}},-\frac{1}{2b_{2}},\frac{1}{2b_{3}}\right)\,.$
(5.24)

For $V_{1},V_{2}$, we have performed the self-duality of Liouville field
theory. As explained in [1], we may perform reflection relations to
$V_{1},V_{2}$ with respect to $V_{3}$, which lead to

$\displaystyle\beta_{1}=\left(\frac{k_{1}-1/2}{b_{1}},\frac{k_{1}-3/2}{b_{2}},\frac{3/2-k_{1}}{b_{3}}\right)\,,\quad\beta_{2}=\left(\frac{k_{2}-3/2}{b_{1}},\frac{k_{2}-1/2}{b_{2}},\frac{3/2-k_{2}}{b_{3}}\right)\,.$
(5.25)

Computing the Gram matrix, we find

$\displaystyle-2\beta_{i}\beta_{j}=\begin{pmatrix}1&k_{1}+k_{2}-3&-k_{1}+1\\\
k_{1}+k_{2}-3&1&-k_{2}+1\\\ -k_{1}+1&-k_{2}+1&1\end{pmatrix}\,,$ (5.26)

which reproduces (2.11)-(2.14) in [35].555In order to match the convention, we
need to replace $k_{i}$ by $-k_{i}$ and set $n=-1$. This means that the theory
with the action (5.20) corresponds to the free field realization for the coset
(1.6) given in [35]. In this way, we have shown that the $N$-point functions
of the coset (1.6) can be reduced to those of the theory with a
$\mathfrak{d}(2,1,-\psi)$-structure.

##  6 Large $\mathcal{N}=4$ Liouville from a coset

In this section, we examine the correlation functions of primary operators in
the coset (1.7) and relate them to those of the large $\mathcal{N}=4$ super
Liouville theory, see [16]. The factor $Y(k_{1},k_{2})$ in the numerator is
made of two $\mathfrak{d}(2,1;-\psi)$ with $\psi=1-k_{1},1-k_{2}$ at level
one. The large $\mathcal{N}=4$ superconformal algebra has two
$\mathfrak{sl}(2)$ current algebras with levels
$k_{1}^{\prime},k_{2}^{\prime}$ satisfying the relations like (1.5). This
duality structure is hidden in $\mathfrak{d}(2,1;-\psi)_{1}$ as explained in
section 2, see [39] for more details. In the next subsection, we find out a
description of $D(2,1;-\psi)_{1}$. Since the action in the WZNW model should
be quite complicated, we use a description as a coset of $SL(2|1)$ and free
bosons [39]. In subsection 6.2, we examine the $N$-point functions of primary
operators in the coset (1.7) and relate them to those of a different theory.
In subsection 6.3, we identify the theory as the large $\mathcal{N}=4$ super
Liouville theory, which corresponds to the free field realization of large
$\mathcal{N}=4$ superconformal algebra in [58]. In subsection 6.4, we further
gauge $SU(2)_{k_{1}^{\prime}+k_{2}^{\prime}}$ subsector and reproduce the
theory with a $\mathfrak{d}(2,1;-\psi)$-structure obtained in section 5.

###  6.1 $D(2,1;k-1)_{1}$ from $SL(2|1)_{k}$

The factor $Y(k_{1},k_{2})$ in the numerator of the coset (1.7) is made of two
$D(2,1|-\psi)$ at level one. As a simple description of $D(2,1|-\psi)$ at
level one, we consider the coset

$\displaystyle D(2,1;k-1)_{1}\simeq\frac{SL(2|1)_{k}\otimes U(1)_{\partial
Y}\otimes U(1)_{\partial Z}}{U(1)_{J}}$ (6.1)

as in [39]. Here we set $\psi=1-k$. We introduce free bosons $Y,Z$ with OPEs
as

$\displaystyle Y(z)Y(0)\sim-\frac{1}{2}\ln|z|^{2}\,,\quad
Z(z)Z(0)\sim\frac{1}{2}\ln|z|^{2}\,.$ (6.2)

We will later determine the action of $U(1)$-generator $J$ in the denominator
of (6.1). We have already examined the $SL(2|1)$ WZNW model to some extend in
section 4. It will be convenient to adopt the first order formulation of
$SL(2|1)$ WZNW model at the level $k$ given in (4.6) as

$\displaystyle\begin{aligned} S^{\text{WZNW}}_{k}[g]&=\frac{1}{2\pi}\int
d^{2}z\left[\bar{\partial}\phi\partial\phi-\bar{\partial}\varphi\partial\varphi+\beta\bar{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}+p\bar{\partial}\theta+\bar{p}\partial\bar{\theta}+q\bar{\partial}\eta+\bar{q}\partial\bar{\eta}\right]\\\
&\quad+\frac{\lambda}{2\pi}\int
d^{2}z\left[e^{b(\phi+\varphi)}(p+\tfrac{1}{2}\beta\eta)(\bar{p}+\tfrac{1}{2}\bar{\beta}\bar{\eta})+e^{b(\phi-\varphi)}(q+\tfrac{1}{2}\beta\theta)(\bar{q}+\tfrac{1}{2}\bar{\beta}\bar{\theta})\right]\end{aligned}$
(6.3)

with $b=1/\sqrt{k-1}$. We have slightly changed the notation such as

$\displaystyle\phi_{1}\to\phi\,,\quad\phi_{2}\to\varphi,\quad p_{1}\to
p\,,\quad\theta_{1}\to\theta\,,\quad p_{2}\to q\,,\quad\theta_{2}\to\eta$
(6.4)

and similarly for $(\bar{p}_{i},\bar{\theta}_{i})$. Here we use a bosonization
formulation as

$\displaystyle p=e^{ig^{L}}\,,\quad\theta=e^{-ig^{L}}\,,\quad
q=e^{ih^{L}}\,,\quad\theta=e^{-ih^{L}}$ (6.5)

with OPEs

$\displaystyle g^{L}(z)g^{L}(0)\sim-\ln z\,,\quad h^{L}(z)h^{L}(0)\sim-\ln
z\,.$ (6.6)

We bosonize $\bar{p},\bar{\theta},\bar{q},\bar{\eta}$ in terms of
$g^{R},h^{R}$ in a similar way and define $g=g^{L}+g^{R},h=h^{L}+h^{R}$.

The affine Lie superalgebra $\mathfrak{sl}(2|1)_{k}$ has obvious subalgebra
$\mathfrak{sl}(2)_{k}$ generated by $E_{1}^{\pm}\equiv\pm E^{\pm}$ and
$H_{1}\equiv H$. There is actually hidden $\mathfrak{sl}(2)_{k^{\prime}}$ as
shown in [38, 39]. The relation of levels is given in (1.5). Generators of the
hidden $\mathfrak{sl}(2)_{k^{\prime}}$ may be expressed as

$\displaystyle E^{+}_{2}=e^{2Y}F^{+}F^{-}\,,\quad
E^{-}_{2}=-e^{-2Y}G^{+}G^{-}\,,\quad
H_{2}=\left(\frac{k}{1-k}\right)\left(\partial Y+\frac{1}{k}I\right)\,.$ (6.7)

With the additional free boson $Z$, the fermionic generators of
$\mathfrak{d}(2,1;k-1)$ can be expressed as

$\displaystyle F^{+}e^{Y\pm Z}\,,\quad F^{-}e^{Y\pm Z}\,,\quad G^{+}e^{-Y\pm
Z}\,,\quad G^{-}e^{-Y\pm Z}\,.$ (6.8)

These generators should survive under the gauging of $U(1)_{J}$. This
determines the action of $J$ as

$\displaystyle J=I+\partial Y=-b^{-1}\partial\varphi+\frac{i}{2}(\partial
g-\partial h)+\partial Y\,.$ (6.9)

In order to cancel the contribution, we may introduce a boson $W$ with OPE
$W(z)W(0)\sim-\frac{1}{2}\ln|z|^{2}$ and use the vertex operators of the form

$\displaystyle
V=e^{-\frac{i}{2}(g+h)}\gamma^{m}\bar{\gamma}^{\bar{m}}\bar{e}^{2bj\phi+2bl\varphi+2tY+2uZ+2b(l+t)W}\,.$
(6.10)

It will be convenient in working on the Ramond sector indicated by the first
factor. Moreover, it will be enough to deal with $t=0$ sector for our purpose.

###  6.2 Reduction of a coset

According to [16], the factor $Y(k_{1},k_{2})$ is given by a sum of two
$\mathfrak{d}(2,1|-\psi)$ at level one in the coset description (1.7) but
removing $Z$-fields. In the BRST formulation of the coset (1.7), we use the
sum of actions as

$\displaystyle
S=S^{1}_{k_{1}}+S^{2}_{k_{2}}+S^{\mathfrak{sl}(2)}_{-k_{1}-k_{2}+4}[\phi_{3},\gamma_{3},\beta_{3}]+S_{bc}[b^{a},c_{a}]\,.$
(6.11)

For the action $S^{1}_{k_{1}}$, we use

$\displaystyle S^{1}_{k_{1}}=\frac{1}{2\pi}\int
d^{2}z\left[\bar{\partial}\phi_{1}\partial\phi_{1}-\bar{\partial}\varphi_{1}\partial\varphi_{1}+\bar{\partial}W_{1}\partial
W_{1}+\beta_{1}\bar{\partial}\gamma_{1}+\bar{\beta}_{1}\partial\bar{\gamma}_{1}\right]$
$\displaystyle+\frac{1}{2\pi}\int
d^{2}z\left[p_{1}\bar{\partial}\theta_{1}+\bar{p}_{1}\partial\bar{\theta}_{1}+q_{1}\bar{\partial}\eta_{1}+\bar{q}_{1}\partial\bar{\eta}_{1}\right]$
(6.12) $\displaystyle+\frac{\lambda}{2\pi}\int
d^{2}z\left[e^{b_{1}(\phi_{1}+\varphi_{1})}(p_{1}+\tfrac{1}{2}\beta_{1}\eta_{1})(\bar{p}_{1}+\tfrac{1}{2}\bar{\beta}_{1}\bar{\eta}_{1})+e^{b_{1}(\phi_{1}-\varphi_{1})}(q_{1}+\tfrac{1}{2}\beta_{1}\theta_{1})(\bar{q}_{1}+\tfrac{1}{2}\bar{\beta}_{1}\bar{\theta}_{1})\right]\,,$

where $b_{1}=1/\sqrt{k_{1}-1}$. Now that we are interested in the correlation
functions among primary operators of the form (6.10) with $t=0$, we have
neglected a free boson $Y_{1}$. For $S^{2}_{k_{2}}$, we replace the subscript
$1$ by $2$. The action
$S^{\mathfrak{sl}(2)}_{-k_{1}-k_{2}+4}[\phi_{3},\gamma_{3},\beta_{3}]$ is for
the $\mathfrak{sl}(2)$ WZNW model as in (3.2) and $S_{bc}[b^{a},c_{a}]$ is for
the BRST ghosts as in (3.7).

The coset can be now analyzed in a way similar to (1.6). We neglect
$\beta_{1},\gamma_{1}$ and $\beta_{1}$ in the action is replaced by (5.3). The
vertex operators are considered of the form

$\displaystyle
V=\gamma_{2}^{m_{2}}\bar{\gamma}_{2}^{\bar{m}_{2}}\gamma_{3}^{m_{3}}\bar{\gamma}_{3}^{\bar{m}_{3}}\left[\prod_{i=1}^{2}e^{-\frac{i}{2}(g_{i}+h_{i})+2b_{i}(j_{i}\phi_{i}+l_{i}\varphi_{i}+l_{i}W_{i})}\right]e^{2b_{3}j_{3}\phi_{3}}\,.$
(6.13)

Here we set $b_{3}=1/\sqrt{k_{3}-2}$ with $k_{3}=-k_{1}-k_{2}+4$. The
conditions corresponding to (5.5) are

$\displaystyle\begin{aligned}
&-j_{1}-1/2-j_{2}-1/2-m_{2}-j_{3}-m_{3}+1=0\,,\\\
&\frac{(-j_{1}-j_{2}-1+m)(2+j_{1}+j_{2}-m)}{k_{1}+k_{2}-2}-\frac{j_{3}(1-j_{3})}{k_{1}+k_{2}-2}=0\end{aligned}$
(6.14)

with some integer $m$. There are also similar conditions with
$\bar{m}_{2},\bar{m}_{3},\bar{m}$. We may set

$\displaystyle m_{2}=-j_{1}-j_{2}-j_{3}\,,\quad m_{3}=0\,,\quad m=m_{2}+1$
(6.15)

such that $\beta_{3},\gamma_{3}$ can be neglected. We thus consider the vertex
operators of the form

$\displaystyle\Psi_{\nu}(z_{\nu})=\int\frac{d\mu_{\nu}^{2}}{|\mu_{\mu}|^{2}}V_{\nu}(z_{\nu})\,,$
(6.16) $\displaystyle
V_{\nu}(z_{\nu})=e^{\mu_{\nu}\gamma_{2}-\bar{\mu}_{\nu}\bar{\gamma}_{2}}\left[\prod_{i=1}^{2}|\mu_{\nu}|^{2j_{i}+1}e^{-\frac{i}{2}(g_{i}+h_{i})+2b_{i}(j_{i}^{\nu}\phi_{i}+l_{i}^{\nu}\varphi+l_{i}^{\nu}W_{i})}\right]|\mu_{\nu}|^{2j_{3}^{\nu}-2}e^{2b_{3}(1-j_{3}^{\nu})\phi_{3}}\,.$
(6.17)

We have performed a reflection relation to $\phi_{3}$-direction. An identity
operator can be inserted as in the previous case. Here, it is given by

$\displaystyle\mathbbm{1}=v^{(s)}(\xi)e^{s(\frac{i}{2}(g_{1}+h_{1})+\phi_{1}/b_{1}-\phi_{3}/b_{3})}(\xi)\,,$
(6.18)

where the spectral flow operator $v^{(s)}(\xi)$ restricts the domain of
integral over $\beta_{2}$ such as to have a zero of order $s$ and effectively
inserts $e^{s(\frac{i}{2}(g_{2}+h_{2})+\phi_{2}/b_{2})}$ at $w=\xi$ as before.

We first examine the correlation functions of the form

$\displaystyle\begin{aligned} &\left\langle
v^{(s)}(\xi)e^{s(\frac{i}{2}(g_{1}+h_{1})+\phi_{1}/b_{1}-\phi_{3}/b_{3})}(\xi)\prod_{\nu=1}^{N}V_{\nu}(z_{\nu})\right\rangle\\\
&\quad=\int_{s}\mathcal{D}ge^{-S}\prod_{\nu=1}^{N}V_{\nu}(z_{\nu})e^{s(\sum_{j=1}^{2}\frac{i}{2}(g_{j}+h_{j})+\phi_{1}/b_{1}+\phi_{2}/b_{2}-\phi_{3}/b_{3})}(\xi)\,.\end{aligned}$
(6.19)

The path integral measure is

$\displaystyle\mathcal{D}g=\left[\prod_{i=1}^{3}\mathcal{D}\phi_{i}\right]\left[\prod_{i=1}^{2}\mathcal{D}\varphi_{i}\mathcal{D}p_{i}\mathcal{D}\theta_{i}\mathcal{D}q_{i}\mathcal{D}\eta_{i}\mathcal{D}W_{i}\right]\mathcal{D}\gamma_{2}\mathcal{D}\beta_{2}$
(6.20)

and the effective action is

$\displaystyle S$ $\displaystyle=\frac{1}{2\pi}\int
d^{2}z\left[\sum_{i=1}^{3}\bar{\partial}\phi_{i}\partial\phi_{i}+\frac{1}{4}\sqrt{g}\mathcal{R}b_{3}\phi_{3}+\sum_{i=1}^{2}\left(-\bar{\partial}\varphi_{i}\partial\varphi_{i}+\bar{\partial}W_{i}\partial
W_{i}\right)\right]$ (6.21) $\displaystyle\quad+\frac{1}{2\pi}\int
d^{2}z\left[\sum_{i=1}^{2}(p_{i}\bar{\partial}\theta_{i}+\bar{p}_{i}\partial\bar{\theta}_{i}+q_{i}\bar{\partial}\eta_{i}+\bar{q}_{i}\partial\bar{\eta}_{i})+\beta_{2}\bar{\partial}\gamma_{2}+\bar{\beta}_{2}\partial\bar{\gamma}_{2}+\lambda\sum_{i=1}^{4}V_{i}\right]\,.$

Here the interaction terms are

$\displaystyle\begin{aligned}
&V_{1}=e^{b_{1}(\phi_{1}+\varphi_{1})}(p_{1}+\tfrac{1}{2}\beta_{1}\eta_{1})(\bar{p}_{1}+\tfrac{1}{2}\bar{\beta}_{1}\bar{\eta}_{1})\,,\quad
V_{2}=e^{b_{1}(\phi_{1}-\varphi_{1})}(q_{1}+\tfrac{1}{2}\beta_{1}\theta_{1})(\bar{q}_{1}+\tfrac{1}{2}\bar{\beta}_{1}\bar{\theta}_{1})\,,\\\
&V_{3}=e^{b_{2}(\phi_{2}+\varphi_{2})}(p_{2}+\tfrac{1}{2}\beta_{2}\eta_{2})(\bar{p}_{1}+\tfrac{1}{2}\bar{\beta}_{2}\bar{\eta}_{2})\,,\quad
V_{4}=e^{b_{2}(\phi_{2}-\varphi_{2})}(q_{2}+\tfrac{1}{2}\beta_{2}\theta_{2})(\bar{q}_{2}+\tfrac{1}{2}\bar{\beta}_{2}\bar{\theta}_{2})\,.\end{aligned}$
(6.22)

As before, integration over zero mode of $\gamma_{2}$ gives delta functions
for $\mu_{\nu}$ and integration over non-zero modes of $\gamma_{2}$ leads to a
delta functional for $\beta_{2}$. After integrating $\beta_{2}$ out,
$\beta_{2}$ is replaced by a function

$\displaystyle-\beta_{2}(w)=\sum_{\nu=1}^{N}\frac{\mu_{\nu}}{w-z_{\nu}}=u\frac{(w-\xi)^{s}\prod_{i=1}^{N-2-s}(w-y_{i})}{\prod_{\nu=1}^{N}(w-z_{\nu})}=u\mathcal{B}(w;z_{\nu},y_{i})\,.$
(6.23)

The restrictions coming from delta functions are

$\displaystyle\sum_{\nu=1}^{N}\frac{\mu_{\nu}}{(w-\xi)^{a}}=0$ (6.24)

for $a=0,1,\ldots,s$.

In order to remove the function appearing in the action, we shift the fields
as

$\displaystyle\begin{aligned}
&\phi_{1}+\frac{1}{2b_{1}}\ln|u\mathcal{B}|^{2}\to\phi_{1}\,,\quad\phi_{2}+\frac{1}{2b_{2}}\ln|u\mathcal{B}|^{2}\to\phi_{2}\,,\quad\phi_{3}-\frac{1}{2b_{3}}\ln|u\mathcal{B}|^{2}\to\phi_{3}\,,\\\
&g_{i}+\frac{i}{2}\ln|u\mathcal{B}|^{2}\to g_{i}\,,\quad
h_{i}+\frac{i}{2}\ln|u\mathcal{B}|^{2}\to h_{i}\,.\end{aligned}$ (6.25)

The correlation function becomes

$\displaystyle\left\langle\prod_{\nu=1}^{N}V_{\nu}(z_{\nu})\right\rangle=\left\langle\prod_{\nu=1}^{N}\tilde{V}_{\nu}(z_{\nu})\prod_{i=1}^{N-2-s}\tilde{V}_{b}(y_{i})\right\rangle$
(6.26)

up to a factor that is a function of $z_{\nu},y_{i}$. The vertex operators are

$\displaystyle\begin{aligned}
V_{\nu}(z_{\nu})&=\left[\prod_{i=1}^{2}e^{2b_{i}((j_{i}^{\nu}+1/2b_{i}^{2})\phi_{i}+l_{i}^{\nu}\varphi_{i}+l_{i}^{\nu}W_{i})}\right]e^{2b_{3}(1-j_{3}^{\nu}-1/2b_{3}^{2})\phi_{3}}\end{aligned}$
(6.27)

and

$\displaystyle\tilde{V}_{b}(y_{i})=e^{-\frac{i}{2}(g_{1}+h_{1}+g_{2}+h_{2})-\phi_{1}/b_{1}-\phi_{2}/b_{2}+\phi_{3}/b_{3}}\,.$
(6.28)

If we consider $\Psi_{\nu}(z_{\nu})$ in (6.16) instead of $V_{\nu}(z_{\nu})$
in (6.17), then the vertex operators $\tilde{V}_{b}(y_{i})$ can be regarded as
an interaction term as above. The correlation functions are now written as

$\displaystyle\left\langle\prod_{\nu=1}^{N}\Psi_{\nu}(z_{\nu})\right\rangle=\left\langle\prod_{\nu=1}^{N}\tilde{V}_{\nu}(z_{\nu})\right\rangle\,,$
(6.29)

where the factor is canceled by the Jacobian due to the change of variables
$\mu_{\nu}$ to $y_{i}$. The action for the right hand side becomes

$\displaystyle\begin{aligned} S&=\frac{1}{2\pi}\int
d^{2}z\left[\sum_{i=1}^{3}\bar{\partial}\phi_{i}\partial\phi_{i}+\frac{1}{4}\sqrt{g}\mathcal{R}\left(\sum
Q_{\phi_{i}}\phi_{i}\right)+\sum_{i=1}^{2}\left(-\bar{\partial}\varphi_{i}\partial\varphi_{i}+\bar{\partial}W_{i}\partial
W_{i}\right)\right]\\\ &\quad+\frac{1}{2\pi}\int
d^{2}z\left[\sum_{i=1}^{2}(p_{i}\bar{\partial}\theta_{i}+\bar{p}_{i}\partial\bar{\theta}_{i}+q_{i}\bar{\partial}\eta_{i}+\bar{q}_{i}\partial\bar{\eta}_{i})+\lambda\sum_{i=1}^{5}V_{i}\right]\,,\end{aligned}$
(6.30)

where the interaction terms are

$\displaystyle\begin{aligned}
&V_{1}=e^{b_{1}(\phi_{1}+\varphi_{1})}(p_{1}-\tfrac{1}{2}\eta_{1})(\bar{p}_{1}+\tfrac{1}{2}\bar{\eta}_{1})\,,\quad
V_{2}=e^{b_{1}(\phi_{1}-\varphi_{1})}(q_{1}-\tfrac{1}{2}\theta_{1})(\bar{q}_{1}+\tfrac{1}{2}\bar{\theta}_{1})\,,\\\
&V_{3}=e^{b_{2}(\phi_{2}+\varphi_{2})}(p_{2}-\tfrac{1}{2}\eta_{2})(\bar{p}_{1}+\tfrac{1}{2}\bar{\eta}_{2})\,,\quad
V_{4}=e^{b_{2}(\phi_{2}-\varphi_{2})}(q_{2}-\tfrac{1}{2}\theta_{2})(\bar{q}_{2}+\tfrac{1}{2}\bar{\theta}_{2})\,,\\\
&V_{5}=e^{-\frac{i}{2}(g_{1}+h_{1}+g_{2}+h_{2})-\phi_{1}/b_{1}-\phi_{2}/b_{2}+\phi_{3}/b_{3}}\,.\end{aligned}$
(6.31)

The background charges are

$\displaystyle Q_{\phi_{1}}=\frac{1}{b_{1}}\,,\quad
Q_{\phi_{2}}=\frac{1}{b_{2}}\,,\quad Q_{\phi_{3}}=b_{3}-\frac{1}{b_{3}}$
(6.32)

and the conformal weights of $p_{i},\theta_{i},q_{i},\eta_{i}$ are $1/2$ and
similarly for $\bar{p}_{i},\bar{\theta}_{i},\bar{q}_{i},\bar{\eta}_{i}$.

###  6.3 Large $\mathcal{N}=4$ super Liouville theory

In the previous subsection, we have rewritten the correlation functions of the
coset (1.7) in terms of a different theory with the action (6.30). In this
subsection, we identify the theory as the $\mathcal{N}=4$ super Liouville
theory. Let us first observe that if we change the fermionic variables as

$\displaystyle\tfrac{1}{2}\theta_{i}+q_{i}\to\theta_{i}\,,\quad
p_{i}+\tfrac{1}{2}\eta_{i}\to
p_{i}\,,\quad\tfrac{1}{2}\eta_{i}-p_{i}\to\eta_{i}\,,\quad
q_{i}-\tfrac{1}{2}\theta_{i}\to q_{i}\,,$ (6.33)

then the theory with interaction terms $V_{1},V_{2}$ (or $V_{3},V_{4}$) can be
identified with those of the $\mathcal{N}=2$ super Liouville theory.
Therefore, treating an interaction term $V_{5}$ perturbatively, the theory can
be regarded as a sum of two $\mathcal{N}=2$ super Liouville theories with
parameters $k_{i}-1$.

It is known that $\mathcal{N}=2$ super Liouville theory is dual to a super
cigar model [20, 2]

$\displaystyle\frac{SL(2)_{k^{\prime}_{i}+1}\otimes SO(2)_{1}}{U(1)}\,,$
(6.34)

where $k^{\prime}_{i}-1=1/(k_{i}-1)$ as in (1.5). We apply the fermionic FZZ-
duality to the two $\mathcal{N}=2$ super Liouville theories. Then the
correlation functions of the coset (1.7) become

$\displaystyle\left\langle\prod_{\nu=1}^{N}\Psi_{\nu}(z_{\nu})\right\rangle=\left\langle\prod_{\nu=1}^{N}\tilde{\Psi}_{\nu}(z_{\nu})\right\rangle\,,$
(6.35)

where the right hand side is evaluated with the action

$\displaystyle\begin{aligned} S&=\frac{1}{2\pi}\int
d^{2}z\left[\sum_{i=1}^{3}\bar{\partial}\phi_{i}\partial\phi_{i}+\frac{1}{4}\sqrt{g}\mathcal{R}\left(\sum
Q_{\phi_{i}}\phi_{i}\right)+\sum_{i=1}^{2}\left(-\bar{\partial}\varphi_{i}\partial\varphi_{i}+\bar{\partial}W_{i}\partial
W_{i}\right)\right]\\\ &\quad+\frac{1}{2\pi}\int
d^{2}z\left[\sum_{i=1}^{2}(p_{i}\bar{\partial}\theta_{i}+\bar{p}_{i}\partial\bar{\theta}_{i}+q_{i}\bar{\partial}\eta_{i}+\bar{q}_{i}\partial\bar{\eta}_{i}+\beta_{i}\bar{\partial}\gamma_{i}+\bar{\beta}_{i}\partial\bar{\gamma}_{i})+\lambda\sum_{i=1}^{3}V_{i}\right]\,.\end{aligned}$
(6.36)

The interaction terms are

$\displaystyle V_{1}=e^{2\phi_{1}/b_{1}}\beta_{1}\bar{\beta}_{1}\,,\quad
V_{2}=e^{2\phi_{2}/b_{2}}\beta_{2}\bar{\beta}_{2}$ (6.37)

and $V_{3}$ obtained by the duality map from $V_{5}$ as analyzed later. The
background charges are the same as those in (6.32). The vertex operators are
now

$\displaystyle\begin{aligned}
\tilde{\Psi}_{\nu}(z_{\nu})&=\left[\prod_{i=1}^{2}|\gamma_{i}|^{-2b_{i}^{2}(j_{i}^{\nu}+1/2b_{i}^{2}+l_{i}^{\nu})}e^{2b_{i}((j_{i}^{\nu}+1/2b_{i}^{2})\phi_{i}+l_{i}^{\nu}\varphi_{i}+l_{i}^{\nu}W_{i})}\right]e^{2b_{3}(1-j_{3}^{\nu}-1/2b_{3}^{2})\phi_{3}}\,.\end{aligned}$
(6.38)

Notice that $\varphi_{i}$ plays the role of $U(1)$-factor in the denominator
of $\eqref{scigar2}$.

Since $\varphi_{1},\varphi_{2}$ (and $W_{1},W_{2}$) do not appear in the
interaction terms (6.37) anymore (see also $V_{5}$ in (6.31)), we can
integrate them out. Due to the form of vertex operators (6.38), the
contributions cancel out with each other. Thus we can use the vertex operators

$\displaystyle\begin{aligned}
\tilde{\Psi}_{\nu}(z_{\nu})&=\left[\prod_{i=1}^{2}|\gamma_{i}|^{-2b_{i}^{2}(j_{i}^{\nu}+1/2b_{i}^{2}+l_{i}^{\nu})}e^{2b_{i}(j_{i}^{\nu}+1/2b_{i}^{2})\phi_{i}}\right]e^{2b_{3}(1-j_{3}^{\nu}-1/2b_{3}^{2})\phi_{3}}\end{aligned}$
(6.39)

and the effective action

$\displaystyle\begin{aligned} S&=\frac{1}{2\pi}\int
d^{2}z\left[\sum_{i=1}^{3}\bar{\partial}\phi_{i}\partial\phi_{i}+\frac{1}{4}\sqrt{g}\mathcal{R}\left(\sum
Q_{\phi_{i}}\phi_{i}\right)\right]\\\ &\quad+\frac{1}{2\pi}\int
d^{2}z\left[\sum_{i=1}^{2}(p_{i}\bar{\partial}\theta_{i}+\bar{p}_{i}\partial\bar{\theta}_{i}+q_{i}\bar{\partial}\eta_{i}+\bar{q}_{i}\partial\bar{\eta}_{i}+\beta_{i}\bar{\partial}\gamma_{i}+\bar{\beta}_{i}\partial\bar{\gamma}_{i})+\lambda\sum_{i=1}^{3}V_{i}\right]\,,\end{aligned}$
(6.40)

where interaction terms and background charges are the same as before.

It is time to think about $V_{3}$, which involves a spin field. It would be
useful to move to another description of fermions as

$\displaystyle\begin{aligned}
&\psi^{\epsilon_{1}\epsilon_{2}\epsilon_{3}\epsilon_{4}}=e^{\frac{i}{2}(\epsilon_{1}g_{1}+\epsilon_{2}h_{1}+\epsilon_{3}g_{2}+\epsilon_{4}h_{2})}\end{aligned}$
(6.41)

by utilizing the triality relation of $SO(8)$ representations. Here
$\epsilon_{i}=\pm$ and the mutually locality requires that
$\prod_{i=1}^{4}\epsilon_{i}=+1$. We may decouple the half of fermions
furthermore and left

$\displaystyle\psi^{\epsilon_{1}\epsilon_{2}}\equiv\psi^{\epsilon_{1}\epsilon_{1}\epsilon_{2}\epsilon_{2}}\,,\quad\psi^{++}(z)\psi^{--}(0)\sim\frac{1}{z}\,,\quad\psi^{+-}(z)\psi^{-+}(0)\sim\frac{1}{z}\,.$
(6.42)

Before decoupling the half of fermions, the interaction term $V_{3}$ is
neutral under the $\mathfrak{sl}(2)_{k_{i}-1}$ currents, which are the
diagonal sum of $\mathfrak{sl}(2)_{k_{i}+1}$ and $\mathfrak{sl}(2)_{-2}$ made
of fermions. The $\mathfrak{sl}(2)_{k_{i}+1}$ currents may be expressed as

$\displaystyle J^{+}_{b,i}=\beta_{i}\,,\quad
J^{3}_{b,i}=b_{i}\partial\phi_{i}+\beta_{i}\gamma_{i}\,,\quad
J^{-}_{b,i}=\beta_{i}\gamma_{i}\gamma_{i}+2b_{i}\gamma_{i}\partial\phi_{i}-(k_{i}^{\prime}+1)\partial\gamma_{i}\,.$
(6.43)

After decoupling the half of fermions, the currents made of fermions are
modified as $\mathfrak{sl}(2)_{-1}$. We may express them as

$\displaystyle\begin{aligned}
J_{f,1}^{3}=\tfrac{1}{2}(\psi^{++}\psi^{--}+\psi^{+-}\psi^{-+})\,,\quad
J_{f,1}^{+}=\psi^{+-}\psi^{++}\,,\quad J_{f,1}^{-}=\psi^{-+}\psi^{--}\,,\\\
J_{f,2}^{3}=\tfrac{1}{2}(\psi^{++}\psi^{--}-\psi^{+-}\psi^{-+})\,,\quad
J_{f,2}^{+}=\psi^{-+}\psi^{++}\,,\quad
J_{f,3}^{-}=\psi^{+-}\psi^{--}\,.\end{aligned}$ (6.44)

The interaction term $V_{3}$ can be fixed up to overall factor by the singlet
condition with respect to the symmetry of $\mathfrak{sl}(2)_{k_{1}}$ and
$\mathfrak{sl}(2)_{k_{2}}$, which are the diagonal sum of
$\mathfrak{sl}(2)_{k_{i}+1}$ and $\mathfrak{sl}(2)_{-1}$ made of fermions. The
singlet is then given by

$\displaystyle
V_{3}=|\gamma_{1}\gamma_{2}\psi^{++}-\gamma_{1}\psi^{+-}-\gamma_{2}\psi^{-+}-\psi^{--}|^{2}e^{-\phi_{1}/b_{1}-\phi_{2}/b_{2}+\phi_{3}/b_{3}}\,,$
(6.45)

which is the same as the one in (27) of [58].

The energy momentum tensor of the theory is given by

$\displaystyle
T=-\sum_{i=1}^{3}\left[\partial\phi_{i}\partial\phi_{i}+Q_{\phi_{i}}\partial^{2}\phi_{i}\right]+\tfrac{1}{2}\sum_{\epsilon=\pm}(\partial\psi^{+,\epsilon}\psi^{-,-\epsilon}-\psi^{+,\epsilon}\partial\psi^{-,-\epsilon})-\sum_{i=1}^{2}\beta_{i}\partial\gamma_{i}\,,$
(6.46)

whose central charge is

$\displaystyle
c=9+6\left(\sum_{i=1}^{3}Q_{i}^{2}\right)=-3-\frac{6}{k_{1}+k_{2}-2}\,.$
(6.47)

We except that the theory has the symmetry of the large $\mathcal{N}=4$
superconformal algebra including $\mathfrak{sl}(2)_{k_{1}^{\prime}}$ and
$\mathfrak{sl}(2)_{k_{2}^{\prime}}$ subalgebras with (1.5). Indeed we can
rewrite as

$\displaystyle
c=\frac{3(k_{1}^{\prime}+k_{2}^{\prime})-6k_{1}^{\prime}k_{2}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}\,,$
(6.48)

which is the expected value of central charge, see, e.g., eq. (B.37) of [50]
(up to $k_{i}^{\prime}\to-\hat{k}_{i}^{\pm}$). Fermionic currents are
constructed such as to commute with interaction terms as well. From this
condition, we find

$\displaystyle
b_{3}^{-1}G^{++}=b_{2}^{-1}\partial\phi_{2}\psi^{++}+b_{1}^{-1}\partial\phi_{1}\psi^{++}-b_{3}^{-1}\partial\phi_{3}\psi^{++}+(1-k_{1})\beta_{1}\psi^{-+}+(k_{1}-1)\gamma_{1}\beta_{1}\psi^{++}$
$\displaystyle\quad+(k_{2}-1)\gamma_{2}\beta_{2}\psi^{++}+(1-k_{2})\beta_{2}\psi^{+-}+(k_{2}-k_{1})\psi^{+-}\psi^{++}\psi^{-+}+(k_{1}+k_{2}-1)\partial\psi^{++}\,,$
$\displaystyle
b_{3}^{-1}G^{-+}=b_{2}^{-1}\partial\phi_{2}\psi^{-+}-b_{1}^{-1}\partial\phi_{1}\psi^{-+}+2b_{1}^{-1}\partial\phi_{1}\gamma_{1}\psi^{++}-b_{3}^{-1}\partial\phi_{3}\psi^{-+}+(1-k_{1})\gamma_{1}\beta_{1}\psi^{-+}$
$\displaystyle\quad+(k_{2}-1)\gamma_{2}\beta_{2}\psi^{-+}+(k_{2}-1)\beta_{2}\psi^{--}+(k_{1}-1)\gamma_{1}\gamma_{1}\beta_{1}\psi^{++}+(1-2k_{1})\partial\gamma_{1}\psi^{++}$
$\displaystyle\quad+(k_{1}-k_{2})\psi^{++}\psi^{-+}\psi^{--}+(k_{1}+k_{2}-1)\partial\psi^{-+}\,,$
$\displaystyle
b_{3}^{-1}G^{+-}=2b_{2}^{-1}\partial\phi_{2}\gamma_{2}\psi^{++}-b_{2}^{-1}\partial\phi_{2}\psi^{+-}+b_{1}^{-1}\partial\phi_{1}\psi^{+-}-b_{3}^{-1}\partial\phi_{3}\psi^{+-}+(k_{1}-1)\beta_{1}\psi^{--}$
(6.49)
$\displaystyle\quad+(k_{1}-1)\gamma_{1}\beta_{1}\psi^{+-}+(1-k_{2})\gamma_{2}\beta_{2}\psi^{+-}+(k_{2}-1)\gamma_{2}\gamma_{2}\beta_{2}\psi^{++}+(1-2k_{2})\partial\gamma_{2}\psi^{++}$
$\displaystyle\quad+(k_{1}-k_{2})\psi^{+-}\psi^{++}\psi^{--}+(k_{1}+k_{2}-1)\partial\psi^{+-}\,,$
$\displaystyle
b_{3}^{-1}G^{--}=2b_{2}^{-1}\partial\phi_{2}\gamma_{2}\psi^{-+}+b_{2}^{-1}\partial\phi_{2}\psi^{--}+b_{1}^{-1}\partial\phi_{1}\psi^{--}+2b_{1}^{-1}\partial\phi_{1}\gamma_{1}\psi^{+-}+b_{3}^{-1}\partial\phi_{3}\psi^{--}$
$\displaystyle\quad+(k_{1}-1)\gamma_{1}\beta_{1}\psi^{--}+(k_{2}-1)\gamma_{2}\beta_{2}\psi^{--}+(k_{2}-1)\gamma_{2}\gamma_{2}\beta_{2}\psi^{-+}+(1-2k_{2})\partial\gamma_{2}\psi^{-+}$
$\displaystyle\quad+(k_{1}-1)\gamma_{1}\gamma_{1}\beta_{1}\psi^{+-}+(1-2k_{1})\partial\gamma_{1}\psi^{+-}+(k_{1}-k_{2})\psi^{+-}\psi^{-+}\psi^{--}-(k_{1}+k_{2}-1)\partial\psi^{--}\,.$

The OPEs among the generators of large $\mathcal{N}=4$ superconformal algebra
are summarized in appendix B.

###  6.4 Further reduction of the coset

In [16], it was also shown that the coset algebra of (1.6) can be reproduced
from a further coset of (1.7) given by

$\displaystyle\frac{Y(k_{1},k_{2})}{SU(2)_{k_{1}^{\prime}}\otimes
SU(2)_{k_{2}^{\prime}}\otimes SU(2)_{k_{1}+k_{2}}}\,.$ (6.50)

We describe it as a $SU(2)_{k_{1}^{\prime}}\otimes SU(2)_{k_{2}^{\prime}}$
coset of the large $\mathcal{N}=4$ super Liouville theory obtained above. Here
we derive correlator correspondences for the duality. In the BRST formulation,
the action for the coset is given by

$\displaystyle
S=S^{\mathcal{N}=4}_{k_{1}^{\prime}k_{2}^{\prime}}+\sum_{i=1}^{2}S^{\mathfrak{sl}(2)}_{-k_{i}^{\prime}-4}[\tilde{\phi}_{i},\tilde{\beta}_{i},\tilde{\gamma}_{i}]+S_{bc}[b^{a},c_{a}]\,.$
(6.51)

The first action is for the large $\mathcal{N}=4$ super Liouville theory given
by (6.40) with (6.37) and (6.45). We bosonize the fermions as

$\displaystyle\psi^{++}=e^{iY_{1}^{L}}\,,\quad\psi^{--}=e^{-iY_{1}^{L}}\,,\quad\psi^{+-}=e^{iY_{2}^{L}}\,,\quad\psi^{-+}=e^{-iY_{2}^{L}}$
(6.52)

with $Y^{L}_{i}(z)Y^{L}_{j}(0)\sim-\delta_{i,j}\ln z$ and similarly for
$\bar{\psi}^{\epsilon_{1},\epsilon_{2}}$ with $Y_{i}^{R}$. We further define
$Y_{i}=Y_{i}^{L}+Y_{i}^{R}$. The second and third actions are for
$\mathfrak{sl}(2)$ WZNW models with levels $k_{1}^{\prime}.k_{2}^{\prime}$,
respectively, given in (3.2). The fourth action is for the BRST ghosts, which
can be found in (3.7).

We apply the first order formulation of the coset as usual. Namely, in the
action of large $\mathcal{N}=4$ super Liouville theory, we remove
$\gamma_{1},\gamma_{2}$ and replace $\beta_{1},\beta_{2}$ in the interaction
terms $V_{1},V_{2}$ in (6.37) by $J^{+}_{f,1},J^{+}_{f,2}$ given in (6.44),
see, e.g., (3.14). Now the interaction terms becomes

$\displaystyle V_{1}=e^{2\phi_{1}/b_{1}}e^{i(Y_{1}+Y_{2})}\,,\quad
V_{2}=e^{2\phi_{1}/b_{2}}e^{i(Y_{1}-Y_{2})}\,,\quad
V_{3}=e^{-\phi_{1}/b_{1}-\phi_{2}/b_{2}+\phi_{3}/b_{3}}e^{-iY_{1}}\,.$ (6.53)

In order to express the field space orthogonal to that of denominator of the
coset (6.50), we may redefine

$\displaystyle 2\phi_{1}/b_{1}+i(Y_{1}+Y_{2})\to
2\phi_{1}/b_{1}^{\prime}\,,\quad 2\phi_{1}/b_{2}+i(Y_{1}-Y_{2})\to
2\phi_{2}/b_{2}^{\prime}$ (6.54)

with $b_{1}^{\prime}=1/\sqrt{k_{1}-2}$ and $b_{2}^{\prime}=1/\sqrt{k_{2}-2}$.
The interaction terms are now

$\displaystyle V_{1}=e^{2\phi_{1}/b_{1}^{\prime}}\,,\quad
V_{2}=e^{2\phi_{1}/b_{2}^{\prime}}\,,\quad
V_{3}=e^{-\phi_{1}/b_{1}^{\prime}-\phi_{2}/b_{2}^{\prime}+\phi_{3}/b_{3}}\,,$
(6.55)

where reproduces (5.24). In this way, we have shown that the coset (6.50)
reduces to the theory with a $\mathfrak{d}(2,1;-\psi)$-structure dual to
another coset (1.6).

##  7 Boundary FZZ-triality

Up to now, we have examined correlation functions on a worldsheet of sphere
topology. In this section, we examine the FZZ-triality in the presence of
boundary, i.e, D-branes. D-branes in cigar model are classified in [68], and
it was shown that there are D1-branes and D2-branes (and somehow degenerate
D0-branes). Boundary actions for D2-branes in sine-Liouville theory can be
derived from those for D1-branes in cigar model as in [2]. However, boundary
actions for D2-branes in cigar model have not been obtained yet, and due to
this fact, boundary actions for D1-branes in sine-Liouville theory cannot be
obtained in the way. In appendix C, we construct boundary actions in $SL(2|1)$
WZNW model by following the analysis for branes in $OSP(1|2)$ WZNW model in
[55]. From them, we can obtain boundary actions in sine-Liouville theory both
for D2-branes and D1-branes. We repeat the same analysis for supersymmetric
setup, and our results reproduce boundary actions for B-branes and A-branes in
$\mathcal{N}=2$ super Liouville theory found in [69].

In appendix C, we obtain boundary actions in $SL(2|1)$ WZNW model at the
classical level, and it is straightforward to include quantum corrections in
the first order formulation. Applying the first order formulation to the coset
(1.4), we can read off boundary actions in sine-Liouville theory. However, the
classical relations for parameters $\mu_{i}$ receive quantum corrections and
hence the values of parameters $\mu_{i}$ should be modified. We do not try to
determine the parameters $\mu_{i}$ in this paper, but in principal it could be
done by repeating the analysis of [69].

In section 4, we obtained the bulk action of sine-Liouville theory from the
coset (1.4) as in (4.14). For D2-branes, the boundary actions can be read off
as

$\displaystyle\begin{aligned} S_{\text{boundary}}=\frac{1}{2\pi}\int
du\left[\hat{b}\mathcal{K}\phi+(\mu_{1}\sigma^{+}+\mu_{2}\sigma^{-})e^{\phi^{L}/\hat{b}+i\sqrt{\kappa}\chi^{L}}+(\mu_{3}\sigma_{+}+\mu_{4}\sigma^{-})e^{\phi^{L}/\hat{b}-i\sqrt{\kappa}\chi^{L}}\right]\end{aligned}$
(7.1)

from those for B-branes in $SL(2|1)$ WZNW model in (C.13).666In a quantum
treatment, the boundary interaction term $\beta e^{\phi_{1}}$ can be neglected
since it can be generated by the other boundary interaction terms. Similar
arguments hold also for the other cases below. Here we have decomposed
$\phi=\phi^{L}(z)+\phi^{R}(\bar{z})$ and $\chi=\chi^{L}(z)+\chi^{R}(\bar{z})$.
Moreover, $\mathcal{K}$ represents the curvature of boundary and the Neumann
boundary conditions are assigned both for $\phi$ and $\chi$. The boundary
fermions $\eta,\bar{\eta}$ are replaced by $2\times 2$ matrices

$\displaystyle\sigma^{+}=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\,,\quad\sigma^{-}=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}$
(7.2)

due to the Grassmann even property of exponential type potentials. The
boundary actions (7.1) reproduces (2.54) in [2]. For D1-branes, the boundary
actions are

$\displaystyle\begin{aligned} S_{\text{boundary}}=\frac{1}{2\pi}\int
du\left[\hat{b}\mathcal{K}\phi+\mu_{1}\sigma^{+}e^{\phi^{L}/\hat{b}+i\sqrt{\kappa}\chi^{L}}+\mu_{2}\sigma^{-}e^{\phi_{L}/\hat{b}-i\sqrt{\kappa}\chi^{L}}\right]\,,\end{aligned}$
(7.3)

which is obtained from those for A-branes in $SL(2|1)$ WZNW model in (C.23).
Here we assign the Neumann boundary condition for $\phi$ and the Dirichlet
boundary condition for $\chi$. The form was already anticipated as in (E.2) of
[2].

We can consider the coset (4.25) with an additional complex fermion. In this
case, we have obtained $\mathcal{N}=2$ super Liouville theory with the bulk
action (4.31). The boundary actions for B-branes are

$\displaystyle S_{\text{boundary}}$ $\displaystyle=\int
du\left[\eta(\partial+\bar{\partial})\bar{\eta}+\frac{\hat{b}}{2\pi}\mathcal{K}\phi\right]$
(7.4) $\displaystyle\quad+\frac{1}{2\pi}\int
du\left[(\mu_{1}\eta+\mu_{2}\bar{\eta})e^{\hat{b}{}^{-1}(\phi^{L}+i\chi^{L})+iY^{L}}+(\mu_{3}\eta+\mu_{4}\bar{\eta})e^{\hat{b}{}^{-1}(\phi^{L}-i\chi^{L})-iY^{L}}\right]\,,$

which come from those for B-branes in $SL(2|1)$ WZNW model in (C.13). They
reproduce (5.8) in [69] and (3.28) in [2]. Here we assign the Neumann boundary
conditions for $\phi,\chi$ and $Y$. The boundary actions for A-brane are

$\displaystyle\begin{aligned} S_{\text{boundary}}&=\int
du\left[\eta(\partial+\bar{\partial})\bar{\eta}+\frac{\hat{b}}{2\pi}\mathcal{K}\phi\right]\\\
&\quad+\frac{1}{2\pi}\int du\left[\mu_{1}\eta
e^{\hat{b}{}^{-1}(\phi^{L}+i\chi^{L})+iY^{L}}+\mu_{2}\bar{\eta}e^{\hat{b}{}^{-1}(\phi^{L}-i\chi^{L})-iY^{L}}\right]\,,\end{aligned}$
(7.5)

which come from those for A-branes in $SL(2|1)$ WZNW model in (C.23). They
reproduce (5.28) of [69]. Here we assign the Neumann boundary condition for
$\phi$ and the Dirichlet boundary conditions for $\chi$ and $Y$.

### Acknowledgements

The work of TC is supported by NSERC Grant Number RES0048511. The work of YH
is supported by JSPS KAKENHI Grant Number 19H01896, 21H04469, 21H05187.

##  Appendix A Convention for Lie superalgebras

In this appendix, we introduce generators of Lie superalgebras, which may be
expressed in terms of supermatrix of the form

$\displaystyle M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\,.$ (A.1)

Here $A,B,C$ and $D$ are $n\times n$, $n\times m$, $m\times n$ and $m\times m$
matrices, respectively. We call it as $n|m$-dimensional representation of
corresponding Lie superalgebra. The supertrace is defined by

$\displaystyle\text{str}\,M=\text{tr}A-\text{tr}D$ (A.2)

and the norm is fixed by $\langle X,Y\rangle=\text{str}\,(XY)$. See, e.g.,
[70] for more details of Lie superalgebras.

###  A.1 Lie superalgebra $\mathfrak{sl}(2|1)$

The Lie superalgebra is generated by four bosonic generators $E^{\pm},H,I$ and
four fermionic ones $F^{\pm},G^{\pm}$. In the $2|1$-dimensional
representation, they are expressed by

$\displaystyle\begin{aligned} &H=\begin{pmatrix}\frac{1}{2}&0&0\\\
0&-\frac{1}{2}&0\\\ 0&0&0\end{pmatrix}\,,\quad
I=\begin{pmatrix}\frac{1}{2}&0&0\\\ 0&\frac{1}{2}&0\\\
0&0&1\end{pmatrix}\,,\quad E^{+}=\begin{pmatrix}0&1&0\\\ 0&0&0\\\
0&0&0\end{pmatrix}\,,\quad E^{-}=\begin{pmatrix}0&0&0\\\ 1&0&0\\\
0&0&0\end{pmatrix}\,,\\\ &F^{+}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\
0&1&0\end{pmatrix}\,,\quad F^{-}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
0&0&0\end{pmatrix}\,,\quad G^{+}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\
1&0&0\end{pmatrix}\,,\quad G^{-}=\begin{pmatrix}0&0&0\\\ 0&0&-1\\\
0&0&0\end{pmatrix}\,.\end{aligned}$ (A.3)

Non-trivial (anti-)commutation relations are given by

$\displaystyle\begin{aligned} &[H,E^{\pm}]=\pm
E^{\pm}\,,\quad[H,F^{\pm}]=\pm\frac{1}{2}F^{\pm}\,,\quad[H,G^{\pm}]=\pm\frac{1}{2}G^{\pm}\,,\quad[I,F^{\pm}]=\frac{1}{2}F^{\pm}\,,\\\
&[I,G^{\pm}]=-\frac{1}{2}G^{\pm}\,,\quad[E^{+},E^{-}]=2H\,,\quad[E^{\pm},F^{\mp}]=-F^{\pm}\,,\quad[E^{\pm},G^{\mp}]=-G^{\pm}\,,\\\
&\\{F^{\pm},G^{\mp}\\}=\mp I+H\,,\quad\\{F^{\pm},G^{\pm}\\}=\pm
E^{\pm}\,.\end{aligned}$ (A.4)

Their norms are

$\displaystyle\begin{aligned} &\langle H,H\rangle=\frac{1}{2}\,,\quad\langle
I,I\rangle=-\frac{1}{2}\,,\quad\langle E^{+},E^{-}\rangle=\langle
E^{-},E^{+}\rangle=1\,,\\\ &\langle F^{+},G^{-}\rangle=\langle
F^{-},G^{+}\rangle=-1\,,\quad\langle G^{-},F^{+}\rangle=\langle
G^{+},F^{-}\rangle=1\end{aligned}$ (A.5)

and others are zero.

###  A.2 Lie superalgebra $\mathfrak{sl}(n|1)$

The Lie superalgebra $\mathfrak{sl}(n|1)$ is generated by the Cartan
directions $H_{i},I$ with $i=1,2,\ldots,n-1$ and bosonic generators $J_{i,j}$
with $i\neq j$ and $i,j=1,2,\ldots,n$. Moreover, there are fermionic
generators $F_{i},G_{i}$ with $i=1,2,\ldots n$. The generators may be
expressed by matrices in the $n|1$-dimensional representation. We introduce
elementary matrices $(e_{I,J})_{K,L}=\delta_{I,K}\delta_{J,L}$. The Cartan
generators may be expressed as

$\displaystyle H_{i}=e_{i,i}-e_{i+1,i+1}\,,\quad
I=\frac{1}{n-1}\left(\sum_{i=1}^{n}e_{i,i}+ne_{n+1,n+1}\right)$ (A.6)

with $i=1,2,\ldots,n-1$. Other bosonic generators are

$\displaystyle J_{i,j}=e_{i,j}\quad(i\neq j)\,.$ (A.7)

The non-trivial commutation relations among them are

$\displaystyle\begin{aligned}
&[H_{i},J_{k,l}]=(\delta_{i,k}-\delta_{i+1,k}-\delta_{i,l}+\delta_{i+1,l})J_{k,l}\,,\\\
&[J_{ij},J_{k,l}]=\delta_{j,k}J_{i,l}-\delta_{i,l}J_{k,j}\,.\end{aligned}$
(A.8)

The fermionic generators are

$\displaystyle F_{i}=e_{n+1,i}\,,\quad G_{i}=e_{i,n+1}$ (A.9)

and the commutation relations

$\displaystyle\begin{aligned}
&[H_{i},F_{j}]=-\delta_{i,j}F_{j}+\delta_{i+1,j}F_{j}\,,\quad[H_{i},G_{j}]=\delta_{i,j}G_{j}-\delta_{i+1,j}G_{j}\,,\\\
&[I,F_{j}]=F_{j}\,,\quad[I,G_{j}]=-G_{j}\,,\quad[J_{i,j},G_{k}]=\delta_{j,k}G_{i}\,,\quad[J_{i,j},F_{k}]=-\delta_{i,k}F_{j}\end{aligned}$
(A.10)

might be useful. With this definition, non-trivial norms are given by

$\displaystyle\begin{aligned} &\langle
H_{i},H_{j}\rangle=G^{(n)}_{ij}\,,\quad\langle I,I\rangle=-\frac{n}{n-1}\,,\\\
&\langle J_{i,j},J_{k,l}\rangle=\delta_{i,l}\delta_{j,k}\,,\quad\langle
G_{i},F_{j}\rangle=-\langle F_{i},G_{j}\rangle=\delta_{i,j}\,.\end{aligned}$
(A.11)

Here $G^{(n)}_{ij}$ is the Cartan matrix of $\mathfrak{sl}(n)$.

##  Appendix B Large $\mathcal{N}=4$ superconformal algebra

The large $\mathcal{N}=4$ superconformal algebra is generated by the energy
momentum tensor $T$, four spin $3/2$ generators $G^{\pm\pm}$ and two set of
affine $\mathfrak{sl}(2)$ currents $J^{a}_{i}$ with $a=\pm,3$ and $i=1,2$ [36,
37]. We may denote the two levels of $\mathfrak{sl}(2)$ current algebras by
$k^{\prime}_{1}$ and $k_{2}^{\prime}$, then the central charge is given by

$\displaystyle
c=\frac{3(k_{1}^{\prime}+k_{2}^{\prime})-6k_{1}^{\prime}k_{2}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}$
(B.1)

as in (6.47). A free field realization of the algebra is given in subsection
6.3. In this appendix, we summarize the OPEs among the generators.

We can check that the energy momentum tensor (6.46) satisfies the usual OPE
with the central charge (6.47). Moreover, we can show that the other
generators are primary with respect to the energy momentum tensor. The spin
one generators satisfy two affine $\mathfrak{sl}(2)$ current algebras with
levels $k^{\prime}_{1}$ and $k_{2}^{\prime}$ almost by construction. With our
convention, the spin $3/2$ generators transform under the action of
$\mathfrak{sl}(2)$ currents as

$\displaystyle\begin{aligned}
&J_{1}^{3}(z)G^{\pm\epsilon}(0)\sim\frac{\pm\frac{1}{2}G^{\pm\epsilon}(0)}{z}\,,\quad
J_{1}^{\mp}(z)G^{\pm\epsilon}(0)\sim\frac{\pm G^{\mp\epsilon}(0)}{z}\,,\\\
&J_{2}^{3}(z)G^{\epsilon\pm}(0)\sim\frac{\pm\frac{1}{2}G^{\epsilon\pm}(0)}{z}\,,\quad
J_{2}^{\mp}(z)G^{\epsilon\pm}(0)\sim\frac{\pm
G^{\epsilon\mp}(0)}{z}\end{aligned}$ (B.2)

with $\epsilon=\pm$.

The most non-trivial OPEs are among spin $3/2$ fermionic generators, which
produce composite operators consisting of two $\mathfrak{sl}(2)$ currents. We
find that

$\displaystyle
G^{++}(z)G^{++}(0)\sim\frac{-\frac{2}{k_{1}^{\prime}+k_{2}^{\prime}-2}J^{+}_{1}J^{+}_{2}(0)}{z}\,,$
$\displaystyle
G^{++}(z)G^{-+}(0)\sim\frac{\frac{2k_{1}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}J_{2}^{+}(0)}{z^{2}}+\frac{\frac{k_{2}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}\partial
J_{1}^{+}(0)-\frac{2}{k_{1}^{\prime}+k_{2}^{\prime}-2}J^{3}_{1}J^{+}_{2}(0)}{z}\,,$
$\displaystyle
G^{++}(z)G^{+-}(0)\sim\frac{\frac{2k_{2}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}J_{1}^{+}(0)}{z^{2}}+\frac{\frac{k_{2}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}\partial
J_{1}^{+}(0)-\frac{2}{k_{1}^{\prime}+k_{2}^{\prime}-2}J^{+}_{1}J^{3}_{2}(0)}{z}\,,$
$\displaystyle
G^{++}(z)G^{--}(0)\sim\frac{\frac{2k_{2}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}J_{1}^{3}(0)+\frac{2k_{1}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}J_{1}^{3}(0)}{z^{2}}+\frac{\frac{k_{2}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}\partial
J_{1}^{3}(0)+\frac{k_{2}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}\partial
J_{2}^{3}(0)+T(0)}{z}$
$\displaystyle\quad+\frac{\frac{1}{k_{1}^{\prime}+k_{2}^{\prime}-2}(J^{3}_{1}J^{3}_{1}(0)-\frac{1}{2}J_{1}^{+}J_{1}^{-}(0)-\frac{1}{2}J_{1}^{-}J_{1}^{+}(0))}{z}$
$\displaystyle\quad+\frac{\frac{1}{k_{1}^{\prime}+k_{2}^{\prime}-2}(J^{3}_{2}J^{3}_{2}(0)-\frac{1}{2}J_{2}^{+}J_{2}^{-}(0)-\frac{1}{2}J_{2}^{-}J_{2}^{+}(0))-\frac{2}{k_{1}^{\prime}+k_{2}^{\prime}-2}J_{1}^{3}J_{2}^{3}(0)}{z}\,,$
$\displaystyle
G^{-+}(z)G^{-+}(0)\sim\frac{-\frac{2}{k_{1}^{\prime}+k_{2}^{\prime}-2}J^{-}_{1}J^{+}_{2}(0)}{z}\,,$
(B.3) $\displaystyle
G^{-+}(z)G^{+-}(0)\sim\frac{\frac{2k_{2}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}J_{1}^{3}(0)-\frac{2k_{1}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}J_{1}^{3}(0)}{z^{2}}+\frac{\frac{k_{2}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}\partial
J_{1}^{3}(0)-\frac{k_{1}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}\partial
J_{2}^{3}(0)-T(0)}{z}$
$\displaystyle\quad+\frac{-\frac{1}{k_{1}^{\prime}+k_{2}^{\prime}-2}(J^{3}_{1}J^{3}_{1}(0)-\frac{1}{2}J_{1}^{+}J_{1}^{-}(0)-\frac{1}{2}J_{1}^{-}J_{1}^{+}(0))}{z}$
$\displaystyle\quad+\frac{-\frac{1}{k_{1}^{\prime}+k_{2}^{\prime}-2}(J^{3}_{2}J^{3}_{2}(0)-\frac{1}{2}J_{2}^{+}J_{2}^{-}(0)-\frac{1}{2}J_{2}^{-}J_{2}^{+}(0))-\frac{2}{k_{1}^{\prime}+k_{2}^{\prime}-2}J_{1}^{3}J_{2}^{3}(0)}{z}\,,$
$\displaystyle
G^{+-}(z)G^{+-}(0)\sim\frac{-\frac{2}{k_{1}^{\prime}+k_{2}^{\prime}-2}J^{+}_{1}J^{-}_{2}(0)}{z}\,,$
$\displaystyle
G^{-+}(z)G^{--}(0)\sim\frac{\frac{2k_{2}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}J_{1}^{-}(0)}{z^{2}}+\frac{\frac{k_{2}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}\partial
J_{1}^{-}(0)-\frac{2}{k_{1}^{\prime}+k_{2}^{\prime}-2}J^{-}_{1}J^{3}_{2}(0)}{z}\,,$
$\displaystyle
G^{+-}(z)G^{--}(0)\sim\frac{\frac{2k_{1}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}J_{2}^{-}(0)}{z^{2}}+\frac{\frac{k_{1}^{\prime}}{k_{1}^{\prime}+k_{2}^{\prime}-2}\partial
J_{2}^{-}(0)-\frac{2}{k_{1}^{\prime}+k_{2}^{\prime}-2}J^{3}_{1}J^{-}_{2}(0)}{z}\,,$
$\displaystyle
G^{--}(z)G^{--}(0)\sim\frac{-\frac{2}{k_{1}^{\prime}+k_{2}^{\prime}-2}J^{-}_{1}J^{-}_{2}(0)}{z}\,.$

##  Appendix C Boundary $SL(2|1)$ WZNW model

In this appendix, we find boundary actions for branes in the $SL(2|1)$ WZNW
model. For this, we follow the strategy taken for the $OSP(1|2)$ WZNW model in
[55]. Assigning two different gluing conditions for $\mathfrak{sl}(2|1)$
currents at the boundary, we can construct two types of branes, which will be
called as B-branes and A-branes by following usual convention.

As in the case of $OSP(1|2)$ WZNW model, it is convenient to start from the
first order formulation only for the bosonic fields. This means that we use
the classical action

$\displaystyle\begin{aligned} S&=\frac{k}{2\pi}\int
d^{2}z\left[\bar{\partial}\phi_{1}\partial\phi_{1}-\bar{\partial}\phi_{2}\partial\phi_{2}-e^{-\phi_{1}-\phi_{2}}\bar{\partial}\theta_{1}\partial\bar{\theta}_{1}+e^{-\phi_{1}+\phi_{2}}\bar{\partial}\theta_{2}\partial\bar{\theta}_{2}\right]\\\
&\quad+\frac{1}{2\pi}\int
d^{2}z\left[\beta\left(\bar{\partial}\gamma-\tfrac{1}{2}(\theta_{2}\bar{\partial}\theta_{1}+\theta_{1}\bar{\partial}\theta_{2})\right)+\bar{\beta}\left(\partial\bar{\gamma}-\tfrac{1}{2}(\bar{\theta}_{2}\partial\bar{\theta}_{1}+\bar{\theta}_{1}\partial\bar{\theta}_{2})\right)\right]\\\
&\quad-\frac{1}{2\pi k}\int
d^{2}z\beta\bar{\beta}e^{2\phi_{1}}\,.\end{aligned}$ (C.1)

The equations of motion lead to

$\displaystyle\beta=ke^{-2\phi_{1}}(\partial\bar{\gamma}-\tfrac{1}{2}(\bar{\theta}_{2}\partial\bar{\theta}_{1}+\bar{\theta}_{1}\partial\bar{\theta}_{2}))\,,\quad\bar{\beta}=ke^{-2\phi_{1}}(\bar{\partial}\gamma-\tfrac{1}{2}(\theta_{2}\bar{\partial}\theta_{1}+\theta_{1}\bar{\partial}\theta_{2}))\,.$
(C.2)

We express the $\mathfrak{sl}(2|1)$ currents by

$\displaystyle\bar{J}(\bar{z})$ $\displaystyle=kg^{-1}\bar{\partial}g$ (C.3)
$\displaystyle=\bar{J}^{F^{+}}F^{+}-\bar{J}^{G^{+}}G^{+}+\bar{J}^{E^{+}}E^{+}+2\bar{J}^{H}H-2J^{I}I+\bar{J}^{F^{-}}F^{-}-\bar{J}^{G^{-}}G^{-}+\bar{J}^{E^{-}}E^{-}\,.$

With the help of equations of motion in (C.2), we find

$\displaystyle\begin{aligned}
\bar{J}^{F^{+}}&=ke^{-\phi_{1}-\phi_{2}}\bar{\partial}\theta_{1}-\bar{\theta}_{2}\bar{\beta}\,,\\\
\bar{J}^{G^{+}}&=-ke^{-\phi_{1}+\phi_{2}}\bar{\partial}\theta_{2}+\bar{\theta}_{1}\bar{\beta}\,,\\\
\bar{J}^{E^{+}}&=\bar{\beta}\,,\\\
\bar{J}^{H}&=k\bar{\partial}\phi_{1}+\bar{\gamma}\bar{\beta}-\tfrac{k}{2}e^{-\phi_{1}-\phi_{2}}\bar{\theta}_{1}\bar{\partial}\theta_{1}-\tfrac{k}{2}e^{-\phi_{1}+\phi_{2}}\bar{\theta}_{2}\bar{\partial}\theta_{2}\,,\\\
\bar{J}^{I}&=-k\bar{\partial}\phi_{2}-\tfrac{k}{2}e^{-\phi_{1}-\phi_{2}}\bar{\theta}_{1}\bar{\partial}\theta_{1}+\tfrac{k}{2}e^{-\phi_{1}+\phi_{2}}\bar{\theta}_{2}\bar{\partial}\theta_{2}+\tfrac{1}{2}\bar{\theta}_{1}\bar{\theta}_{2}\bar{\beta}\,,\\\
\bar{J}^{F^{-}}&=k\bar{\partial}\bar{\theta}_{2}-k\bar{\theta}_{2}(\bar{\partial}\phi_{1}-\bar{\partial}\phi_{2})-\bar{\theta}_{2}\bar{\gamma}\bar{\beta}+ke^{-\phi_{1}-\phi_{2}}\bar{\gamma}\bar{\partial}\theta_{1}-\tfrac{k}{2}e^{-\phi_{1}-\phi_{2}}\bar{\theta}_{1}\bar{\theta}_{2}\bar{\partial}\theta_{1}\,,\\\
\bar{J}^{G^{-}}&=-k\bar{\partial}\bar{\theta}_{1}+k\bar{\theta}_{1}(\bar{\partial}\phi_{1}+\bar{\partial}\phi_{2})+\bar{\theta}_{1}\bar{\gamma}\bar{\beta}-ke^{-\phi_{1}+\phi_{2}}\bar{\gamma}\bar{\partial}\theta_{2}-\tfrac{k}{2}e^{-\phi_{1}+\phi_{2}}\bar{\theta}_{1}\bar{\theta}_{2}\bar{\partial}\theta_{2}\,,\\\
\bar{J}^{E^{-}}&=\tfrac{k}{2}(\bar{\theta}_{1}\bar{\partial}\bar{\theta}_{2}+\bar{\theta}_{2}\bar{\partial}\bar{\theta}_{1})+k\bar{\partial}\bar{\gamma}-2k\bar{\gamma}\bar{\partial}\phi_{1}-\bar{\gamma}\bar{\gamma}\bar{\beta}\\\
&\quad+ke^{-\phi_{1}-\phi_{2}}\bar{\gamma}\bar{\theta}_{1}\bar{\partial}\theta_{1}+ke^{-\phi_{1}+\phi_{2}}\bar{\gamma}\bar{\theta}_{2}\bar{\partial}\theta_{2}+k\bar{\theta}_{1}\bar{\theta}_{2}\bar{\partial}\phi_{2}\,.\end{aligned}$
(C.4)

We define $J^{a}$ in an analogous way but with replacing
$\bar{\partial}\theta_{i}$ by $-\partial\bar{\theta}_{i}$, see (4.5).

###  C.1 Boundary actions for B-branes

We examine a B-brane corresponding to the gluing conditions

$\displaystyle J^{a}=\bar{J}^{a}$ (C.5)

at the boundary in the current convention. The conditions can be reproduced
from the fields satisfying

$\displaystyle\beta=\bar{\beta}\,,\quad\bar{\gamma}-\gamma=ce^{\phi_{1}}+\tfrac{1}{2}(\theta_{1}\bar{\theta}_{2}+\theta_{2}\bar{\theta}_{1})\,,$
$\displaystyle
ke^{-\phi_{1}-\phi_{2}}(\bar{\partial}\theta_{1}+\partial\bar{\theta}_{1})=\beta(\bar{\theta}_{2}-\theta_{2})\,,\quad
ke^{-\phi_{1}+\phi_{2}}(\bar{\partial}\theta_{2}+\partial\bar{\theta}_{2})=\beta(\bar{\theta}_{1}-\theta_{1})\,,$
$\displaystyle k(\bar{\partial}-\partial)\phi_{1}+c\beta
e^{\phi_{1}}-\frac{k}{4}\left[e^{-\phi_{1}-\phi_{2}}(\bar{\theta}_{1}-\theta_{1})(\bar{\partial}\theta_{1}-\partial\bar{\theta}_{1})+e^{-\phi_{1}+\phi_{2}}(\bar{\theta}_{2}-\theta_{2})(\bar{\partial}\theta_{2}-\partial\bar{\theta}_{2})\right]=0\,,$
$\displaystyle(\bar{\partial}-\partial)\phi_{2}+\frac{1}{4}\left[e^{-\phi_{1}-\phi_{2}}(\bar{\theta}_{1}-\theta_{1})(\bar{\partial}\theta_{1}-\partial\bar{\theta}_{1})-e^{-\phi_{1}+\phi_{2}}(\bar{\theta}_{2}-\theta_{2})(\bar{\partial}\theta_{2}-\partial\bar{\theta}_{2})\right]=0\,,$
(C.6) $\displaystyle
2(\bar{\partial}\bar{\theta}_{2}-\partial\theta_{2})-(\bar{\theta}_{2}-\theta_{2})(\bar{\partial}+\partial)(\phi_{1}-\phi_{2})-ce^{-\phi_{2}}(\partial\bar{\theta}_{1}-\bar{\partial}\theta_{1})=0\,,$
$\displaystyle
2(\bar{\partial}\bar{\theta}_{1}-\partial\theta_{1})-(\bar{\theta}_{1}-\theta_{1})(\bar{\partial}+\partial)(\phi_{1}+\phi_{2})-ce^{\phi_{2}}(\partial\bar{\theta}_{2}-\bar{\partial}\theta_{2})=0$

at the boundary.

We would like to construct boundary action which leads to the above
conditions. Using the last two equations, we obtain

$\displaystyle\begin{aligned}
&\frac{4-c^{2}}{4}(\bar{\theta}_{1}-\theta_{1})(\bar{\partial}\theta_{1}-\partial\bar{\theta}_{1})=-(\bar{\theta}_{1}-\theta_{1})(\bar{\partial}+\partial)(\bar{\theta}_{1}-\theta_{1})\\\
&\quad+\frac{c}{2}e^{\phi_{2}}(\bar{\theta}_{1}-\theta_{1})(\bar{\partial}+\partial)(\bar{\theta}_{2}-\theta_{2})-\frac{c}{4}e^{\phi_{2}}(\bar{\theta}_{1}-\theta_{1})(\bar{\theta}_{2}-\theta_{2})(\bar{\partial}+\partial)(\phi_{1}-\phi_{2})\,,\\\
&\frac{4-c^{2}}{4}(\bar{\theta}_{2}-\theta_{2})(\bar{\partial}\theta_{2}-\partial\bar{\theta}_{2})=-(\bar{\theta}_{2}-\theta_{2})(\bar{\partial}+\partial)(\bar{\theta}_{2}-\theta_{2})\\\
&\quad+\frac{c}{2}e^{-\phi_{2}}(\bar{\theta}_{2}-\theta_{2})(\bar{\partial}+\partial)(\bar{\theta}_{1}-\theta_{1})+\frac{c}{4}e^{-\phi_{2}}(\bar{\theta}_{1}-\theta_{1})(\bar{\theta}_{2}-\theta_{2})(\bar{\partial}+\partial)(\phi_{1}+\phi_{2})\,.\end{aligned}$
(C.7)

This implies that the boundary action is given by

$\displaystyle\begin{aligned} &S_{\text{boundary}}=\frac{1}{2\pi i}\int
du\beta(\bar{\gamma}-\gamma-
ce^{\phi_{1}}-\frac{1}{2}(\theta_{1}\bar{\theta}_{2}+\theta_{2}\bar{\theta}_{1}))\\\
&\quad+\frac{k}{2\pi i}\frac{1}{4-c^{2}}\int
due^{-\phi_{1}-\phi_{2}}(\bar{\theta}_{1}-\theta_{1})(\partial+\bar{\partial})(\bar{\theta}_{1}-\theta_{1})\\\
&\quad+\frac{k}{2\pi i}\frac{1}{4-c^{2}}\int
due^{-\phi_{1}+\phi_{2}}(\bar{\theta}_{2}-\theta_{2})(\partial+\bar{\partial})(\bar{\theta}_{2}-\theta_{2})\\\
&\quad-\frac{k}{2\pi i}\frac{1}{4-c^{2}}\frac{c}{2}\int
due^{-(\phi_{1}+\phi_{2})/2}(\bar{\theta}_{1}-\theta_{1})(\partial+\bar{\partial})\left[e^{-(\phi_{1}-\phi_{2})/2}(\bar{\theta}_{2}-\theta_{2})\right]\,.\end{aligned}$
(C.8)

We move to the first order formulation both for the bosonic and fermionic
fields by introducing boundary fermions. The bulk action is given by (4.4) and
the boundary action is

$\displaystyle S_{\text{boundary}}=\frac{1}{2\pi i}\int
du\left[\beta(\bar{\gamma}-\gamma-
ce^{\phi_{1}})-(\bar{\theta}_{1}-\theta_{1})p_{1}-(\bar{\theta}_{2}-\theta_{2})p_{2}\right]$
$\displaystyle\quad+\frac{k}{2\pi i}\int
du\eta(\partial+\bar{\partial})\bar{\eta}+\frac{1}{2\pi i}\int
du(\mu_{1}\eta+\mu_{2}\bar{\eta})e^{(\phi_{1}+\phi_{2})/2}(\tfrac{1}{4}\beta(\theta_{2}+\bar{\theta}_{2})+p_{1})$
(C.9) $\displaystyle\quad+\frac{1}{2\pi i}\int
du(\mu_{3}\eta+\mu_{4}\bar{\eta})e^{(\phi_{1}-\phi_{2})/2}(\tfrac{1}{4}\beta(\theta_{1}+\bar{\theta}_{1})+p_{2})\,.$

Here the parameters $\mu_{1},\mu_{2},\mu_{3},\mu_{4}$ should satisfy

$\displaystyle\mu_{3}\mu_{2}-\mu_{1}\mu_{4}=\frac{2(4-c^{2})}{kc}\,,\quad\mu_{3}\mu_{4}=\mu_{1}\mu_{2}=\frac{4(c^{2}-4)}{kc^{2}}\,.$
(C.10)

At the boundary we assign

$\displaystyle\beta=\bar{\beta}\,,\quad p_{1}=\bar{p}_{1}\,,\quad
p_{2}=\bar{p}_{2}\,.$ (C.11)

The first line of (C.9) may be included in the bulk action as

$\displaystyle\begin{aligned} S&=\frac{1}{2\pi}\int
d^{2}z\left[k\bar{\partial}\phi_{1}\partial\phi_{1}-k\bar{\partial}\phi_{2}\partial\phi_{2}-\gamma\bar{\partial}\beta-\bar{\gamma}\partial\bar{\beta}+\sum_{a=1}^{2}(\theta_{a}\bar{\partial}p_{a}+\bar{\theta}_{a}\partial\bar{p}_{a})-\frac{1}{k}e^{2\phi_{1}}\beta\bar{\beta}\right]\\\
&\quad+\frac{1}{2k\pi}\int
d^{2}z\left[e^{\phi_{1}+\phi_{2}}(p_{1}+\tfrac{1}{2}\beta\theta_{2})(\bar{p}_{1}+\tfrac{1}{2}\bar{\beta}\bar{\theta}_{2})-e^{\phi_{1}-\phi_{2}}(p_{2}+\tfrac{1}{2}\beta\theta_{1})(\bar{p}_{2}+\tfrac{1}{2}\bar{\beta}\bar{\theta}_{1})\right]\,,\end{aligned}$
(C.12)

see, e.g., [71, 56] as well. Then the boundary action becomes

$\displaystyle\begin{aligned} S_{\text{boundary}}&=\frac{k}{2\pi i}\int
du\eta(\partial+\bar{\partial})\bar{\eta}-\frac{c}{2\pi i}\int du\beta
e^{\phi_{1}}\\\ &\quad+\frac{1}{2\pi i}\int
du(\mu_{1}\eta+\mu_{2}\bar{\eta})e^{(\phi_{1}+\phi_{2})/2}(\tfrac{1}{4}\beta(\theta_{2}+\bar{\theta}_{2})+p_{1})\\\
&\quad+\frac{1}{2\pi i}\int
du(\mu_{3}\eta+\mu_{4}\bar{\eta})e^{(\phi_{1}-\phi_{2})/2}(\tfrac{1}{4}\beta(\theta_{1}+\bar{\theta}_{1})+p_{2})\,.\end{aligned}$
(C.13)

###  C.2 Boundary actions for A-branes

In this subsection, we examine an A-brane corresponding to the gluing
conditions

$\displaystyle\begin{aligned} &J^{H}=\bar{J}^{H}\,,\quad
J^{I}=-\bar{J}^{I}\,,\quad J^{F^{+}}=-\bar{J}^{G^{+}}\,,\quad
J^{G^{+}}=-\bar{J}^{F^{+}}\,,\\\ &J^{F^{-}}=-\bar{J}^{G^{-}}\,,\quad
J^{G^{-}}=-\bar{J}^{F^{-}}\,,\quad J^{E^{+}}=\bar{J}^{E^{+}}\,,\quad
J^{E^{-}}=\bar{J}^{E^{-}}\end{aligned}$ (C.14)

at the boundary. The conditions can be reproduced from the fields satisfying

$\displaystyle\begin{aligned}
&\beta=\bar{\beta}\,,\quad\bar{\gamma}-\gamma=ce^{\phi_{1}}+\tfrac{1}{2}(\theta_{1}\bar{\theta}_{1}+\theta_{2}\bar{\theta}_{2})\,,\\\
&ke^{-\phi_{1}}(e^{\phi_{2}}\partial\bar{\theta}_{2}+e^{-\phi_{2}}\bar{\partial}\theta_{1})=\beta(\bar{\theta}_{2}-\theta_{1})\,,\quad
ke^{-\phi_{1}}(e^{-\phi_{2}}\partial\bar{\theta}_{1}+e^{\phi_{2}}\bar{\partial}\theta_{2})=\beta(\bar{\theta}_{1}-\theta_{2})\,,\\\
&k(\bar{\partial}-\partial)\phi_{1}+c\beta e^{\phi_{1}}\\\
&\quad-\frac{k}{4}e^{-\phi_{1}}\left[(\theta_{2}-\bar{\theta}_{1})(e^{\phi_{2}}\partial\bar{\theta}_{2}-e^{-\phi_{2}}\bar{\partial}\theta_{1})+(\theta_{1}-\bar{\theta}_{2})(e^{-\phi_{2}}\partial\bar{\theta}_{1}-e^{\phi_{2}}\bar{\partial}\theta_{2})\right]=0\,,\\\
&k(\bar{\partial}+\partial)\phi_{2}+\frac{k}{4}e^{-\phi_{1}}\left[(\theta_{2}-\bar{\theta}_{1})(e^{\phi_{2}}\partial\bar{\theta}_{2}-e^{-\phi_{2}}\bar{\partial}\theta_{1})-(\theta_{1}-\bar{\theta}_{2})(e^{-\phi_{2}}\partial\bar{\theta}_{1}-e^{\phi_{2}}\bar{\partial}\theta_{2})\right]=0\,,\\\
&2(\bar{\partial}\bar{\theta}_{1}-\partial\theta_{2})-(\bar{\theta}_{1}-\theta_{2})\left[(\bar{\partial}+\partial)\phi_{1}+(\bar{\partial}-\partial)\phi_{2}\right]-c(e^{\phi_{2}}\bar{\partial}\theta_{2}-e^{-\phi_{2}}\partial\bar{\theta}_{1})=0\,,\\\
&2(\bar{\partial}\bar{\theta}_{2}-\partial\theta_{1})-(\bar{\theta}_{2}-\theta_{1})\left[(\bar{\partial}+\partial)\phi_{1}-(\bar{\partial}-\partial)\phi_{2}\right]-c(e^{-\phi_{2}}\bar{\partial}\theta_{1}-e^{\phi_{2}}\partial\bar{\theta}_{2})=0\end{aligned}$
(C.15)

at the boundary.

We would like to realize these equations from a theory consisting with free
kinetic terms and interaction terms. For free parts, we should assign
Dirichlet boundary condition for $\phi_{2}$. In order to realize this, we may
set

$\displaystyle\phi_{2}=\phi^{L}_{2}(z)+\phi^{R}_{2}(\bar{z})=0$ (C.16)

at the boundary. We further introduce a dual variable
$\tilde{\phi}_{2}=\phi^{L}_{2}(z)-\phi^{R}_{2}(\bar{z})$. The above equations
are now given by

$\displaystyle\begin{aligned}
&\beta=\bar{\beta}\,,\quad\bar{\gamma}-\gamma=ce^{\phi_{1}}+\tfrac{1}{2}(\theta_{1}\bar{\theta}_{1}+\theta_{2}\bar{\theta}_{2})\,,\\\
&ke^{-\phi_{1}}(\partial\bar{\theta}_{2}+\bar{\partial}\theta_{1})=\beta(\bar{\theta}_{2}-\theta_{1})\,,\quad
ke^{-\phi_{1}}(\partial\bar{\theta}_{1}+\bar{\partial}\theta_{2})=\beta(\bar{\theta}_{1}-\theta_{2})\,,\\\
&k(\bar{\partial}-\partial)\phi_{1}+c\beta
e^{\phi_{1}}-\frac{k}{4}e^{-\phi_{1}}\left[(\theta_{2}-\bar{\theta}_{1})(\partial\bar{\theta}_{2}-\bar{\partial}\theta_{1})+(\theta_{1}-\bar{\theta}_{2})(\partial\bar{\theta}_{1}-\bar{\partial}\theta_{2})\right]=0\,,\\\
&k(\bar{\partial}-\partial)\tilde{\phi}_{2}-\frac{k}{4}e^{-\phi_{1}}\left[(\theta_{2}-\bar{\theta}_{1})(\partial\bar{\theta}_{2}-\bar{\partial}\theta_{1})-(\theta_{1}-\bar{\theta}_{2})(\partial\bar{\theta}_{1}-\bar{\partial}\theta_{2})\right]=0\,,\\\
&2(\bar{\partial}\bar{\theta}_{1}-\partial\theta_{2})-(\bar{\theta}_{1}-\theta_{2})(\bar{\partial}+\partial)(\phi_{1}-\tilde{\phi}_{2})-c(\bar{\partial}\theta_{2}-\partial\bar{\theta}_{1})=0\,,\\\
&2(\bar{\partial}\bar{\theta}_{2}-\partial\theta_{1})-(\bar{\theta}_{2}-\theta_{1})(\bar{\partial}+\partial)(\phi_{1}+\tilde{\phi}_{2})-c(\bar{\partial}\theta_{1}-\partial\bar{\theta}_{2})=0\,.\end{aligned}$
(C.17)

From the last two equations we find

$\displaystyle\begin{aligned}
&2(\theta_{2}-\bar{\theta}_{1})(\bar{\partial}+\partial)(\bar{\theta}_{2}-\theta_{1})-(\theta_{2}-\bar{\theta}_{1})(\bar{\theta}_{2}-\theta_{1})(\bar{\partial}+\partial)(\phi_{1}+\tilde{\phi}_{2})\\\
&\qquad\qquad\qquad\qquad\qquad\qquad-(c-2)(\theta_{2}-\bar{\theta}_{1})(\bar{\partial}\theta_{1}-\partial\bar{\theta}_{2})=0\,,\\\
&2(\theta_{1}-\bar{\theta}_{2})(\bar{\partial}+\partial)(\bar{\theta}_{1}-\theta_{2})-(\theta_{1}-\bar{\theta}_{2})(\bar{\theta}_{1}-\theta_{2})(\bar{\partial}+\partial)(\phi_{1}-\tilde{\phi}_{2})\\\
&\qquad\qquad\qquad\qquad\qquad\qquad-(c-2)(\theta_{1}-\bar{\theta}_{2})(\bar{\partial}\theta_{2}-\partial\bar{\theta}_{1})=0\,.\end{aligned}$
(C.18)

This implies that the boundary action is

$\displaystyle\begin{aligned} &S_{\text{boundary}}=\frac{1}{2\pi i}\int
du\beta(\bar{\gamma}-\gamma-
ce^{\phi}-\frac{1}{2}(\theta_{1}\bar{\theta}_{1}+\theta_{2}\bar{\theta}_{2}))\\\
&\quad+\frac{k}{2\pi i}\frac{1}{c-2}\int
due^{-(\phi_{1}+\tilde{\phi}_{2})/2}(\bar{\theta}_{1}-\theta_{2})(\partial+\bar{\partial})\left[e^{-(\phi_{1}-\tilde{\phi}_{2})/2}(\theta_{1}-\bar{\theta}_{2})\right]\,.\end{aligned}$
(C.19)

We move to the first order formulation also for the fermionic fields. With the
bulk action (4.4), the boundary action is

$\displaystyle S_{\text{boundary}}=\frac{1}{2\pi i}\int
du\left[\beta(\bar{\gamma}-\gamma-
ce^{\phi_{1}})-(\bar{\theta}_{2}-\theta_{1})p_{1}-(\bar{\theta}_{1}-\theta_{2})p_{2}\right]$
$\displaystyle\quad+\frac{k}{2\pi i}\int
du\eta(\partial+\bar{\partial})\bar{\eta}+\frac{\mu_{1}}{2\pi i}\int du\eta
e^{(\phi_{1}+\tilde{\phi}_{2})/2}(\tfrac{1}{4}\beta(\theta_{2}+\bar{\theta}_{1})+p_{1})$
(C.20) $\displaystyle\quad+\frac{\mu_{2}}{2\pi i}\int
du\bar{\eta}e^{(\phi_{1}-\tilde{\phi}_{2})/2}(\tfrac{1}{4}\beta(\theta_{1}+\bar{\theta}_{2})+p_{2})\,.$

Here the parameters should satisfy

$\displaystyle\mu_{1}\mu_{2}=\frac{1}{c-2}\,.$ (C.21)

The boundary conditions are now assigned as

$\displaystyle\beta=\bar{\beta}\,,\quad\phi_{2}=0\,,\quad
p_{1}=\bar{p}_{2}\,,\quad p_{2}=\bar{p}_{1}\,.$ (C.22)

In terms of the bulk action (C.12), the boundary action is

$\displaystyle S_{\text{boundary}}=\frac{k}{2\pi i}\int
du\eta(\partial+\bar{\partial})\bar{\eta}-\frac{c}{2\pi i}\int du\beta
e^{\phi_{1}}$ (C.23) $\displaystyle\quad+\frac{\mu_{1}}{2\pi i}\int du\eta
e^{(\phi_{1}+\tilde{\phi}_{2})/2}(\tfrac{1}{4}\beta(\theta_{2}+\bar{\theta}_{1})+p_{1})+\frac{\mu_{2}}{2\pi
i}\int
du\bar{\eta}e^{(\phi_{1}-\tilde{\phi}_{2})/2}(\tfrac{1}{4}\beta(\theta_{1}+\bar{\theta}_{2})+p_{2})\,.$

##  Appendix D Generalized FZZ-triality

In section 4, the coset (1.4) has been examined and its correlation functions
of primary operators have been shown to match with those of sine-Liouville
theory or cigar model described by (1.1). In succeeding sections, it was shown
how the properties are useful to derive other dualities, such as the one
between the coset (1.7) and large $\mathcal{N}=4$ super Liouville theory.
Recall that many trialities were conjectured by Gaiotto-Rapčák [13] and in
particular a series of generalized FZZ-dualities were derived in [4]. In this
appendix we examine its relation to a coset of the type (1.8), whose symmetry
algebra is $Y_{n,1,0}$-algebra in the notation of [13]. We name the relation
as generalized FZZ-triality and expect it to be useful for other dualities as
in the case of the original FZZ-triality.

###  D.1 Bosonic triality

We start from a free field realization of affine Lie algebra
$\mathfrak{sl}(n)$, see, e.g., [26]. We introduce $n$ free bosons $\phi_{a}$
and $n(n-1)/2$ pairs of $(\beta_{i,j},\gamma_{i,j})$-systems with $i>j$, where
the weights of $(\beta_{i,j},\gamma_{i,j})$ are $(1,0)$. A linear combination
of $n$ free bosons $\phi_{a}$ decouples from the rest. The non-trivial OPEs
are

$\displaystyle\phi_{a}(z)\phi_{b}(0)\sim-\delta_{a,b}\ln
z\,,\quad\gamma_{i,j}(z)\beta_{k,l}(0)\sim\frac{\delta_{i,k}\delta_{j,l}}{z}\,.$
(D.1)

The currents $J^{\mathfrak{sl}(n)}_{i,j}$ with $i<j$ are given by

$\displaystyle
J_{i,j}^{\mathfrak{sl}(n)}=\beta_{j,i}-\sum_{l=j+1}^{n}\gamma_{l,j}\beta_{l,i}$
(D.2)

and the Cartan subalgebra is generated by

$\displaystyle
H^{\mathfrak{sl}(n)}_{a}=\hat{H}^{\mathfrak{sl}(n)}_{a}-\hat{H}^{\mathfrak{sl}(n)}_{a+1}$
(D.3)

with

$\displaystyle\hat{H}^{\mathfrak{sl}(n)}_{a}=\sqrt{k-n}\partial\phi_{a}-\sum_{l=1}^{a-1}\gamma_{a,l}\beta_{a,l}+\sum_{l=a+1}^{n}\gamma_{l,a}\beta_{l,a}\,.$
(D.4)

The other generators $J^{\mathfrak{sl}(n)}_{i,j}$ $(i>j)$ can be fixed through
the OPEs with these currents.

We then construct a free field realization of affine Lie superalgebra
$\mathfrak{sl}(n|1)$. We prepare an additional free boson $\varphi$ and $n$
pairs of free fermions $(p_{j},\theta_{j})$ with $j=1,2,\ldots,n$ such that

$\displaystyle\varphi(z)\varphi(0)\sim\ln z\,,\quad
p_{i}(z)\theta_{j}(0)\sim\frac{\delta_{i,j}}{z}\,.$ (D.5)

The weights of $(p_{j},\theta_{j})$ are $(1,0)$, respectively. We look for a
free field realization such that $J_{i,j}=J^{\mathfrak{sl}(n)}_{i,j}$ for
$i<j$. From the consistency with $J_{i,j}=J^{\mathfrak{sl}(n)}_{i,j}$, we find

$\displaystyle F_{i}=p_{i}+\sum_{l=i}^{n}\gamma_{l,i}p_{l}\,.$ (D.6)

The Cartan generators are similarly obtained as

$\displaystyle
H_{a}=\hat{H}_{a}-\hat{H}_{a+1}\,,\quad\hat{H}_{a}=\sqrt{k-n+1}\partial\phi_{a}-\sum_{l=1}^{a-1}\gamma_{a,l}\beta_{a,l}+\sum_{l=a+1}^{n}\gamma_{l,a}\beta_{l,a}-p_{a}\theta_{a}$
(D.7)

and

$\displaystyle
I=\frac{\sqrt{k-n+1}}{n-1}\left(n\partial\varphi+\sum_{a=1}^{n}\partial\phi_{a}\right)+\sum_{j=1}^{n}p_{j}\theta_{j}\,.$
(D.8)

From the conditions to commute with these currents, we find out screening
operators as

$\displaystyle Q_{i}=\int dzV_{i}(z)$ (D.9)

with

$\displaystyle
V_{l}=\left(\beta_{l+1,l}-\sum_{j=1}^{l-1}\beta_{l+1,j}\gamma_{l,j}-p_{l+1}\theta_{l}\right)e^{\frac{1}{\sqrt{k-n+1}}(\phi_{l}-\phi_{l+1})}$
(D.10)

for $l=1,2,\ldots,n-1$ and

$\displaystyle V_{0}=p_{1}e^{-\frac{1}{\sqrt{k-n+1}}(\varphi+\phi_{1})}\,.$
(D.11)

We then move to find a free field realization of the symmetry algebra of the
coset (1.8) by applying the method reviewed in section 3. It is convenient to
bosonize the free fermions as

$\displaystyle p_{i}=e^{iX_{i}}\,,\quad\theta_{i}=e^{-iX_{i}}\,,\quad
X_{i}(z)X_{j}(0)\sim-\ln z\,.$ (D.12)

We then introduce new bosons by

$\displaystyle\hat{H}^{a}=\sqrt{k-n}\partial\hat{\phi}_{a}-\sum_{l=1}^{a-1}\gamma_{a,l}\beta_{a,l}+\sum_{l=a+1}^{n}\gamma_{l,a}\beta_{k,a}\,,\quad
I=\sqrt{\frac{nk}{n-1}}\partial\hat{\varphi}\,.$ (D.13)

The field space of the coset (1.8) should be orthogonal to
$\hat{\phi}_{a}-\hat{\phi}_{a+1}$ and $\hat{\eta}$. Furthermore,
$(\beta_{i,j},\gamma_{i,j})$ with all $i>j$ are ignored. We define new bosons
by

$\displaystyle\sqrt{k-n+1}X_{a}+i\phi_{a}=-\sqrt{k-n}\hat{X}_{a}\,,\quad\varphi+i\frac{\sqrt{k-n}}{n}\sum_{a=1}^{n}\hat{X}_{a}=-i\sqrt{\frac{k}{n}}\hat{\chi}\,.$
(D.14)

The combinations $\hat{x}_{a}=\hat{X}_{a}-\hat{X}_{a+1}$ with
$a=1,2,\ldots,n-1$ and $\hat{\chi}$ are regular with respect to
$\hat{\phi}_{a}-\hat{\phi}_{a+1}$ and $\hat{\varphi}$. The screening operators
are now written as

$\displaystyle
V_{l}=-e^{i\sqrt{\frac{k-n}{k-n+1}}(\hat{X}_{l}-\hat{X}_{l+1})}\,,\quad
V_{0}=e^{-i\sqrt{\frac{k-n}{k-n+1}}(\hat{X}_{1}-\frac{1}{n}\sum_{a=1}^{n}\hat{X}_{a})+i\sqrt{\frac{k}{n(k-n+1)}}\hat{\chi}}\,.$
(D.15)

Note that

$\displaystyle\hat{X}_{1}-\frac{1}{n}\sum_{a=1}^{n}\hat{X}_{a}=\sum_{l=1}^{n}G^{(n)1l}\hat{x}_{l}\,.$
(D.16)

Moreover, $G^{(n)}_{ij}$ is the Cartan matrix of $\mathfrak{sl}(n)$ and
$G^{(n)ij}$ is the inverse matrix of $G^{(n)}_{ij}$. These are the same as
those for $Y_{1,0,n}$-algebra obtained in appendix B of [4] once the level $k$
is identified with that of [4] (denoted $\kappa$ here) as

$\displaystyle\frac{n-k}{k-n+1}=\kappa-n\,.$ (D.17)

###  D.2 Fermionic triality

In this subsection, we examine the coset

$\displaystyle\frac{SL(n|1)_{k}\otimes SO(2)_{1}}{SL(n)_{k}\otimes U(1)}\,,$
(D.18)

where $SO(2)_{1}$ represents a complex fermion $\psi^{\pm}$ with weight $1/2$.
We may bosonize the fermions as

$\displaystyle\psi^{\pm}=e^{\pm iY}\,,\quad Y(z)Y(0)\sim-\ln z\,.$ (D.19)

The generators of $\mathfrak{sl}(n)_{k}$ in the denominator is the same as
before. However, the generator of $\mathfrak{u}(1)$ is shifted as

$\displaystyle\tilde{I}=I+\psi^{+}\psi^{-}=I+i\partial Y\,.$ (D.20)

In order to generate the orthogonal space, we need another field $\hat{Y}$ in
addition to $\hat{X}_{a},\hat{\chi}$. We may choose the field as

$\displaystyle
i\sqrt{\frac{kn}{n-1}\left(1+\frac{kn}{n-1}\right)}\hat{Y}=\frac{\sqrt{k-n+1}}{n-1}\left(n\varphi+\sum_{a=1}^{n}\phi_{a}\right)+i\sum_{i=1}^{n}X_{a}-i\frac{kn}{n-1}Y\,.$
(D.21)

Further rotating the fields as

$\displaystyle\sqrt{\frac{k}{n(k-n+1)}}\hat{\chi}\to\sqrt{\frac{k}{n(k-n+1)}-1}\hat{\chi}-\hat{Y}\,,$
(D.22)
$\displaystyle\sqrt{\frac{k}{n(k-n+1)}}\hat{Y}\to\hat{\chi}+\sqrt{\frac{k}{n(k-n+1)}-1}\hat{Y}\,,$
(D.23)

the screening operators become

$\displaystyle
V_{l}=-e^{i\sqrt{\frac{k-n}{k-n+1}}(\hat{X}_{l+1}-\hat{X}_{l})}\,,\quad
V_{0}=e^{-i\sqrt{\frac{k-n}{k-n+1}}(\hat{X}_{1}-\frac{1}{n}\sum_{a=1}^{n}\hat{X}_{a})+i\sqrt{\frac{k}{n(k-n+1)}-1}\hat{\chi}-i\hat{Y}}\,.$
(D.24)

They reproduce (4.20) with (4.19) in [4].

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