Integrable branes in generalized $\lambda$-deformations

Georgios P. D. Pappas

Department of Nuclear and Particle Physics,
Faculty of Physics, National and Kapodistrian University of Athens,
15784 Athens, Greece

<EMAIL_ADDRESS>

Abstract

We search for integrable boundary conditions and their geometric
interpretation as $D$-branes, in models constructed as generalized
$\lambda$-deformations of products of group- and coset-spaces. Using the
sigma-model approach, we find that all the conformal brane geometries known in
the literature for a product of WZW models solve the corresponding boundary
conditions, thus persisting as integrable branes along the RG flows of our
sigma-models. They consist of the well known $G$-conjugacy classes, twisted
$G$-conjugacy classes by a permutation automorphism (permutation branes) and
generalized permutation branes. Subsequently, we study the properties of the
aforementioned brane geometries, especially of those embedded in the
backgrounds interpolating between the UV and IR fixed points.

###### Contents

  1. 1 Introduction
  2. 2 Integrable boundary conditions
    1. 2.1 The isotropic deformation at equal levels
    2. 2.2 The isotropic deformation at unequal levels
    3. 2.3 The isotropic deformation of $G_{k_{1}}\times G_{k_{2}}/G_{k_{1}+k_{2}}$ space
  3. 3 Identifying the integrable brane configurations
    1. 3.1 The isotropic deformation at equal levels
    2. 3.2 The isotropic deformation at unequal levels
    3. 3.3 The isotropic deformation of $G_{k_{1}}\times G_{k_{2}}/G_{k_{1}+k_{2}}$ space
    4. 3.4 The allowed set of integrable branes
    5. 3.5 Comments on the brane geometry
    6. 3.6 Some comments on integrable brane geometries for $N>2$
  4. 4 The integrable branes in the CFT limits
    1. 4.1 Integrable branes in the conformal limits of model (II)
    2. 4.2 Integrable branes in the conformal limits of model (III)
  5. 5 Generalized permutation branes in the integrable flows
    1. 5.1 Preliminaries on the generalized permutation branes
    2. 5.2 Generalized permutation branes in the deformed models
      1. 5.2.1 GPB in the deformed coset space
      2. 5.2.2 GPB in the deformed current algebra models
    3. 5.3 The lowest dimensional GPB for $G=SU(2)$
  6. 6 Conclusions
  7. A Integrable branes in the $\lambda$-model
  8. B Derivation of the boundary terms for model (I)-(III)
    1. B.1 Model (I)
    2. B.2 Model (II)
    3. B.3 Model (III)
      1. B.3.1 Boundary conditions (4.18)
      2. B.3.2 Boundary conditions (5.7)

##  1 Introduction

An extensive amount of literature devoted to the analysis of D-branes, with
approaches ranging from algebraic techniques [1, 2, 3, 4] to geometric
descriptions [5, 6, 7, 8, 9, 10, 11, 12, 13], provides us with their complete
picture in group spaces $G$ and coset spaces $G/H$. In the first case the
D-branes are wrapped around a finite set of allowed $G$-conjugacy classes,
denoted with $\mathcal{C}_{G}$, while in the second one, they are described by
the product $\mathcal{C}_{G}\mathcal{C}_{H}$ of conjugacy classes.

If we extend the WZW model to a product space $G\times G$ there is an
additional class of maximally symmetric branes that can be defined, which
include the permutation symmetry between the two manifolds and are known as
permutation branes. They have been constructed algebraically in [14] and
geometrically in [15, 16].

Since the construction of the permutation branes involves the permutation
symmetry between the two manifolds they can be defined only in the case where
the two WZW models have the same level $k$. In [17] the authors suggested a
generalization of permutation branes in products of Lie groups with different
levels, known as generalized permutation branes (GPB). Unlike the previous
ones they are not maximally symmetric, but the symmetry of the diagonally
embedded group $G$ in the product space is still conserved. The conformality
of these D-branes has been established geometrically in [17, 18]. An algebraic
construction has been presented in [19], but only in the case of a product of
$N=2$ minimal models with $k_{1}=1$, $k_{2}=4$.

All the brane geometries mentioned above are solutions of boundary conditions
in WZW and gauged WZW models, thus they are branes embedded in conformal
target spaces. For a general non-linear sigma-model finding consistent
boundary conditions and the corresponding brane geometries they describe is
both interesting and challenging. Nevertheless, progress can be made in the
case where the sigma-model is an integrable model. In this case one can define
boundary conditions preserving its integrable structure. These were introduced
in [20] and subsequently applied in several examples of integrable field
theories such as the affine Toda field theories [21], the Green-Schwarz sigma-
model [22] and the principal chiral model (PCM) [23].

Another well known class of integrable models are the single
$\lambda$-deformed sigma-models which appear as deformations of a group $G$ or
a coset $G/H$ CFT. A construction of these deformations has been done in [24,
25] and is based on the gauging of a WZW field with a principal chiral field.
Integrable boundary conditions have been found for these models in [26, 27],
where in particular the elegant geometric picture of the corresponding
integrable branes was presented.

Inspired by these works, we will search for integrable boundary conditions and
their geometric interpretation as $D$-branes, in a specific class of
generalized $\lambda$-deformed models. The first such class is constructed in
[28, 29, 30] and represents the effective action of coupled WZW models, all at
the same level $k$, with the characteristic that when the deviation from the
conformal point is small they mutually interact via current bilinears forming
a closed chain,111 The currents $J_{i\pm}$ are defined as
$J_{i+}=\partial_{+}g_{i}g_{i}^{-1},\quad J_{i-}=g_{i}^{-1}\partial_{-}g_{i}.$
i.e.

$S_{k;\lambda_{1},\dots,\lambda_{N}}=\sum_{i=1}^{N}S_{\text{WZW},k}(g_{i})-\frac{k}{\pi}\sum_{i=1}^{N}\int
d^{2}\sigma\,{\rm
Tr}(J_{i+1+}\lambda_{i+1}J_{i-})+\mathcal{O}(\lambda^{2})\,,$ (1.1)

where here and in the rest of the paper the index $i$ is defined mod$(N)$.
Note that the operators driving the theory from the UV point couple the
currents of adjacent copies of the Lie Group $G$. A Hamiltonian analysis of
the model revealed it’s canonical equivalence to $N$ independent single
$\lambda$-deformed models with couplings $\lambda_{1},\dots,\lambda_{N}$. Thus
the RG flow equations of each of the coupling matrices $\lambda_{i}$ are the
same as that of a single $\lambda$-deformed model [30]. The second class, for
which we will search for integrable branes, was introduced in [31] and appears
as a double deformation of a product of WZW models defined at different
levels. Compared to (1.1), the effective action for the $N=2$ case, is
slightly modified and takes the form

$S_{k_{1},k_{2};\lambda_{1},\lambda_{2}}=S_{k_{1}}(g_{1})+S_{k_{2}}(g_{2})-\frac{\sqrt{k_{1}k_{2}}}{\pi}\int
d^{2}\sigma{\rm
Tr}(J_{1+}\lambda_{1}J_{2-}+J_{2+}\lambda_{2}J_{1-})+\mathcal{O}(\lambda^{2})$
(1.2)

It was shown in [31] that due to the different levels the RG flow acquires a
fixed point in the IR, in which the CFT is given as a product of current- and
coset-type symmetries (see (4.2)). Generalizations of (1.2) in the case of an
arbitrary number of WZW models has been done in [32], where a detailed
analysis of the IR CFTs showed that although they are characterized by an
asymmetry between their holomorphic (right) and antiholomorphic (left)
symmetry algebras the left and right sectors posses the same central charge,
i.e. $c_{L}=c_{R}$. Finally, the third class of models was constructed in [33]
and further studied in [34] and is given as a deformation of a diagonal coset
space CFT $G_{k_{1}}\times G_{k_{2}}/G_{k_{1}+k_{2}}$. In these works it has
been shown that the operator driving the theory from the UV point is a
parafermion bilinear. Additionally the theory smoothly flows to an IR unitary
CFT given as $G_{k_{1}-k_{2}}\times G_{k_{2}}/G_{k_{1}}$ [34].

Since the models presented above appear as integrable deformations of product
spaces, they have richer mathematical and physical structures compared to the
$\lambda$-deformed cases, a fact that is expected to be reflected in the
variety of integrable branes that can be defined in these theories.
Specifically, we will see that these models admit as integrable configurations
all the consistent brane geometries presented above for the WZW model.

The paper is organized as follows: In section 2, we will apply the method of
finding integrable boundary conditions in the sigma-models of our interest. In
section 3, we will determine the integrable brane configurations, i.e. the
brane geometry and its gauge invariant two-form, corresponding to the boundary
conditions found in section 2. On the contrary to [26] and [27], the task of
determining the configurations directly from the integrable conditions is more
complicated due to the boundary equations being more involved. Towards this,
we will use the sigma-model approach, e.g. [8] [10] [35] [36]. Having
determined the integrable brane geometries, in section 4 we will investigate
the symmetries they preserve in the two conformal points of the RG flows
described in [31] and [34]. In section 5, we will study the generalized
permutation branes embedded in the aforementioned RG flows [31] and [34]. For
the particular case with one of the couplings set to zero, where the model
interpolates between two current algebra CFTs [31], and for $G=SU(2)$ we will
present explicitly the induced background fields (metric and $H$-field) on the
lowest dimensional GPB. Finally, in Appendix A we apply the sigma-model
approach in the group and coset space $\lambda$-models and in the Appendix (B)
we present the detailed computations of the boundary conditions to which our
branes correspond.

##  2 Integrable boundary conditions

In this section we will closely follow [22, 23, 26, 27] for obtaining
integrability preserving boundary conditions for a class of generalized
$\lambda$-deformed models.

###  2.1 The isotropic deformation at equal levels

Let us consider the $\lambda$-deformation of two WZW models, at the same level
$k$. This model, referred from now on as model (I), is described through the
action [28]

$\begin{split}&S_{k;\lambda_{1},\lambda_{2}}=S_{\text{WZW},k}(g_{1})+S_{\text{WZW},k}(g_{2})-\\\
&-\frac{k}{\pi}\int_{\Sigma}d^{2}\sigma{\rm
Tr}\left[\begin{pmatrix}J_{1+}&J_{2+}\end{pmatrix}\begin{pmatrix}\lambda_{1}\lambda_{2}\mathcal{O}_{21}D_{2}^{T}&\lambda_{1}\mathcal{O}_{21}\\\
\lambda_{2}\mathcal{O}_{12}&\lambda_{1}\lambda_{2}\mathcal{O}_{12}D_{1}^{T}\end{pmatrix}\begin{pmatrix}J_{1-}\\\
J_{2-}\end{pmatrix}\right],\end{split}$ (2.1)

where we assumed isotropic couplings $\lambda_{1},\lambda_{2}$ and we
introduced the operator

$\mathcal{O}_{12}=(\mathbb{1}-\lambda_{1}\lambda_{2}D^{T}_{1}D^{T}_{2})^{-1}\,,$
(2.2)

which is given in terms of the adjoint operator, $D_{i}(X)=g_{i}(X)g_{i}^{-1}$
for $i=1,2$ and $X\in\mathfrak{g}$. Its equations of motion can be encoded in
two independent sets of first order differential equations

$\partial_{\pm}A_{i\mp}=\pm\frac{1}{1+\lambda_{i}}[A_{i+},A_{i-}]\,,\quad
i=1,2\,,$ (2.3)

where the algebra valued fields $A_{1\pm},A_{2\pm}\in\mathfrak{g}$ are given
in terms of the group elements $g_{1},g_{2}\in G$ as

$\begin{split}&A_{1+}=\lambda_{1}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{1}D_{2})^{-1}(J_{1+}+\lambda_{2}D_{1}J_{2+})\,,\\\
&A_{1-}=-\lambda_{1}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{2}^{T}D_{1}^{T})^{-1}(J_{2-}+\lambda_{2}D_{2}^{T}J_{1-})\,,\\\
&A_{2+}=\lambda_{2}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{2}D_{1})^{-1}(J_{2+}+\lambda_{1}D_{2}J_{1+})\,,\\\
&A_{2-}=-\lambda_{2}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{1}^{T}D_{2}^{T})^{-1}(J_{1-}+\lambda_{1}D_{1}^{T}J_{2-})\,,\end{split}$
(2.4)

and its energy-momentum tensor takes the simple form

$T_{\pm\pm}=k\sum_{i=1}^{2}\frac{1-\lambda_{i}^{2}}{\lambda_{i}^{2}}{\rm
Tr}(A_{i\pm},A_{i\pm})\,.$ (2.5)

Integrability of the model utilizes the fact that (2.3) can be put in the form
of the zero curvature condition [28]

$\partial_{+}\mathcal{L}_{-}(z)-\partial_{-}\mathcal{L}_{+}(z)=[\mathcal{L}_{+}(z),\mathcal{L}_{-}(z)],\quad
z\in\mathbb{C}\,,$ (2.6)

for the Lax matrices

$\mathcal{L}_{i\pm}(z_{i})=\frac{2z_{i}}{z_{i}\mp
1}\frac{1}{1+\lambda_{i}}A_{i\pm}\,,\quad z_{i}\in\mathbb{C}\,,\quad i=1,2\,.$
(2.7)

Thus, one can construct two transport matrices

$\displaystyle T_{i}(b,a;z_{i})=P\exp\left(-\int_{a}^{b}d\sigma
L_{i\sigma}(\tau,\sigma,z_{i})\right)\,,\quad
L_{i\sigma}=\frac{1}{2}(L_{i+}-L_{i-})\,,\quad i=1,2\,,$ (2.8)

which in the case of a closed string, generate two independent infinite sets
of conserved charges.

Keeping the discussion general, the integrable structure of a sigma-model
defined on an open string, i.e. $\sigma\in[0,\pi]$, is generically broken,
since the underlying monodromy matrix, denoted with $T(z)$, could cease to be
conserved. As has been argued in the literature, see e.g. [21, 22, 23], in
this case the correct object to use as a generating function for integrals of
motion, reads

$T_{b}(z)=T^{\Omega}_{R}(2\pi,\pi;z)T(\pi,0;z)\,,$ (2.9)

and is known as the boundary monodromy matrix. This involves an integral from
one endpoint of the string to the other and then back into the opposite
direction. The matrix $T^{\Omega}_{R}$ is defined in the region
$\sigma_{R}\in[\pi,2\pi]$ and is constructed from the reflected values of the
Lax pair, $R:\mathcal{L}(\sigma)\to\mathcal{L}^{R}(\sigma)$. The superscript
$\Omega$ denotes the possibility of including the action of a constant algebra
automorphism. Being specific

$T^{\Omega}_{R}(b,a;z)=P\exp\left(-\int_{a}^{b}d\sigma\,\Omega
L^{R}_{\sigma}(\tau,\sigma;z)\right)\,.$ (2.10)

Then imposing integrability, i.e. demanding that

$\partial_{\tau}T_{b}(z)=[T_{b}(z),N]\,,$ (2.11)

for some matrix $N(z)$ one finds appropriate boundary conditions such that
(2.11) is true. In fact, one may construct more involved boundary matrices as
long as their time derivative can be put in the form (2.11), (see e.g. (3.19)
in [26]).

Returning to our case we will see that two different definitions of $R$, will
lead to integrable boundary conditions which will prove to describe two
distinct brane configurations.

To proceed let us consider the following action of the reflection

$R:\sigma\to 2\pi-\sigma,\quad g_{i}\to g_{i+1}^{-1}\,,\quad i=1,2\,,$ (2.12)

Under (2.12) the WZW currents transform as

$J_{i\pm}(\sigma)\to J_{iR\pm}(\sigma)=-J_{i+1\mp}(2\pi-\sigma)\,,\quad
i=1,2\,,$ (2.13)

which, with the aid of (2.4), lead to

$A_{i\pm}(\sigma)\to A_{iR\pm}(\sigma)=A_{i\mp}(2\pi-\sigma)\,,\quad i=1,2\,.$
(2.14)

Using (2.10) and (2.14) one can easily show that the reflected transport
matrices (2.8) satisfy the relation

$T^{\Omega_{i}}_{iR}(2\pi,\pi;z)=T^{\Omega_{i}}_{i}(0,\pi;-z),\quad i=1,2\,,$
(2.15)

with $\Omega_{i}$ an automorphism acting on
$\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$ and the boundary monodromy matrices
(2.9) are given as

$T_{ib}=T^{\Omega_{i}}_{i}(0,\pi;-z)T_{i}(\pi,0;z)\,,\quad i=1,2\,.$ (2.16)

Differentiating (2.16) leads to222To derive (2.18) we used the identity
$\partial_{\tau}T^{\Omega}(b,a;z)=T^{\Omega}(b,a;z)L^{\Omega}_{\tau}(a;z)-L^{\Omega}_{\tau}(b;z)T^{\Omega}(b,a;z)$
(2.17) where for simplicity we defined $L^{\Omega}(z)=\Omega L(z)$.

$\begin{split}\partial_{\tau}T_{ib}=&[T_{i}^{\Omega_{i}}(0,\pi;-z)\mathcal{L}^{\Omega_{i}}_{i\tau}(\pi;-z)-\mathcal{L}^{\Omega_{i}}_{i\tau}(0;-z)T_{i+1}^{\Omega_{i}}(0,\pi;-z)]T_{i}(\pi,0;z)\\\
&+T^{\Omega_{i}}_{i}(0,\pi;-z)[T_{i}(\pi,0;z)\mathcal{L}_{i\tau}(0;z)-\mathcal{L}_{i\tau}(\pi;z)T_{i}(\pi,0;z)]\,,\quad
i=1,2\,.\end{split}$ (2.18)

It is a matter of a simple algebra to show that (2.18) can be written in the
form (2.11) for the matrices $N_{i}(z)=\mathcal{L}_{i\tau}(z)$ and the
boundary conditions

$\mathcal{L}_{i\tau}(z)|_{\partial\Sigma}=\Omega_{i}\mathcal{L}_{i\tau}(-z)|_{\partial\Sigma}\,,\quad
i=1,2\,,$ (2.19)

in both string ends. Substituting in (2.19) the form of the time component of
the Lax333The time components of the Lax pairs are given as
$\mathcal{L}_{i\tau}=\frac{z}{z^{2}-1}\frac{1}{1+\lambda_{i}}\left((z+1)A_{i+}+(z-1)A_{i-}\right),\quad
i=1,2$ (2.20) and comparing terms of the same power in $z$ we find that

$A_{1+}|_{\partial\Sigma}=\Omega_{1}A_{1-}|_{\partial\Sigma}\,,\quad
A_{2+}|_{\partial\Sigma}=\Omega_{2}A_{2-}|_{\partial\Sigma}$ (2.21)

As in [26], for consistency reasons the automorphisms $\Omega_{i}$ must be
constant involutive matrices, i.e. $\Omega^{2}_{i}=\mathbb{1}$. Demanding
additionally that the conditions (2.21) lead to 444For a general field theory
the condition of the absence of momentum flow across the boundary, translates
to the boundary conditions $T_{\tau\sigma}|_{\partial\Sigma}=0\to
T_{++}-T_{--}|_{\partial\Sigma}=0$, known as conformal boundary conditions. In
the case where the theory is conformal they preserve one copy of the Virassoro
algebra.

$T_{++}|_{\partial\Sigma}=T_{--}|_{\partial\Sigma}\,,$ (2.22)

and using (2.5), we find that $\Omega_{i}$ must preserve the trace in the
$\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$ algebra, i.e.
$\Omega_{i}^{T}\Omega_{i}=\mathbb{1}$, $i=1,2$.

Let us now define the reflection operator to be

$R:\sigma\to 2\pi-\sigma,\quad g_{i}\to
g_{i}^{-1},\quad\lambda_{i}\to\lambda_{i+1}\,.$ (2.23)

In this case the fields $A_{i\pm}$ transform as

$A_{i\pm}(\sigma)\to A^{R}_{i\pm}(\sigma)=A_{i+1\mp}(2\pi-\sigma)\,,\quad
i=1,2$ (2.24)

and the reflected transport matrix satisfies the relation

$T^{\Omega_{i}}_{iR}(2\pi,\pi;z)=T^{\Omega_{i}}_{i+1}(0,\pi;-z)\,,\quad
i=1,2\,,$ (2.25)

where we remind to the reader that the index $i$ is defined mod$(2)$.
Following the same steps as before we end up with

$\mathcal{L}_{i\tau}(z)|_{\partial\Sigma}=\Omega_{i}\mathcal{L}_{i+1\tau}(-z)|_{\partial\Sigma}\,,\quad
i=1,2\,.$ (2.26)

For consistency reasons (2.26) leads to $\Omega_{1}=\Omega_{2}^{-1}=\Omega$,
where $\Omega$ in comparison with the previous case (see below (2.22)), does
not need to be involutive. Substituting now (2.20) in (2.26) we find the
boundary conditions

$\begin{split}\frac{1}{1+\lambda_{1}}A_{1+}|_{\partial\Sigma}=\frac{1}{1+\lambda_{2}}\Omega
A_{2-}|_{\partial\Sigma}\,,\quad\frac{1}{1+\lambda_{2}}A_{2+}|_{\partial\Sigma}=\frac{1}{1+\lambda_{1}}\Omega^{-1}A_{1-}|_{\partial\Sigma}\,.\end{split}$
(2.27)

Further, demanding that (2.27) are conformal boundary conditions, i.e.
satisfying (2.22), we find that $\Omega$ is metric preserving and additionally
that the couplings are equal $\lambda_{1}=\lambda_{2}$. Thus we find the
integrable boundary conditions

$A_{1+}|_{\partial\Sigma}=\Omega A_{2-}|_{\partial\Sigma}\,,\quad
A_{2+}|_{\partial\Sigma}=\Omega^{-1}A_{1-}|_{\partial\Sigma}\,.$ (2.28)

For the reader’s convenience we gather the results of the model (I). It admits
two sets of integrable boundary conditions, namely

$\begin{pmatrix}A_{1+}\\\
A_{2+}\end{pmatrix}_{\partial\Sigma}=\begin{pmatrix}\Omega_{1}&0\\\
0&\Omega_{2}\end{pmatrix}\begin{pmatrix}A_{1-}\\\
A_{2-}\end{pmatrix}_{\partial\Sigma}\,,$ (2.29)

and

$\begin{pmatrix}A_{1+}\\\
A_{2+}\end{pmatrix}_{\partial\Sigma}=\begin{pmatrix}0&\Omega\\\
\Omega^{-1}&0\end{pmatrix}\begin{pmatrix}A_{1-}\\\
A_{2-}\end{pmatrix}_{\partial\Sigma}\,.$ (2.30)

The former set of boundary conditions do not require equality of the
couplings, while in the latter, one has to impose them equal. Note that this
constraint does not contradict with their beta functions [29].

Already at this level, we notice that in the UV limit the model (I) describes
a product of WZW models $G_{k}\times G_{k}$ and the integrable boundary
conditions (2.29), (2.30), for trivial acting automorphisms, reduce to the
well known maximally symmetric boundary conditions

$J_{1+}|_{\partial\Sigma}=-J_{2-}|_{\partial\Sigma}\,,\quad
J_{2+}|_{\partial\Sigma}=-J_{1-}|_{\partial\Sigma}\,,$ (2.31)

and

$J_{1+}|_{\partial\Sigma}=-J_{1-}|_{\partial\Sigma}\,,\quad
J_{2+}|_{\partial\Sigma}=-J_{2-}|_{\partial\Sigma}\,,$ (2.32)

respectively. Equation (2.32) describes branes wrapping around a product of
conjugacy classes
$\mathcal{C}_{f_{1}f_{2}}=\mathcal{C}_{f_{1}}\times\mathcal{C}_{f_{2}}$, with
$\mathcal{C}_{f_{i}}=\\{hf_{i}h_{i}^{-1}|\,\forall h\in G\\}$, while (2.31)
describe conjugacy classes twisted by a permutation automorphism (permutation
branes) [15, 16], which we will denote as
${\mathcal{C}^{\pi}}_{f_{1}f_{2}}={\mathcal{C}^{\pi}}_{f_{1}}\times\big{(}{\mathcal{C}^{\pi}}_{f^{-1}_{2}}\big{)}^{-1}$,
with
$\pi(\mathfrak{g}_{1},\mathfrak{g}_{2})=(\mathfrak{g}_{2},\mathfrak{g}_{1})$.
555Here ${\mathcal{C}^{\pi}}_{f_{i}}=\\{hf_{i}\Pi(h_{i}^{-1})|\,\forall h\in
G\\}$ and $\Pi$ is defined as $\Pi(h)=\exp(\pi(X))=\exp(X^{A}\pi(T^{A}))$. In
later sections, we will realize (2.29), (2.30) geometrically and we will see
that the above picture of the maximally symmetric branes survive the
deformation, as also shown in [26] for the case of the single $\lambda$-model.

###  2.2 The isotropic deformation at unequal levels

The $\lambda$-deformation of two WZW models defined at different levels
$k_{1},k_{2}$, to be referred from now on as model (II), is described by the
action [31]

$\begin{split}&S_{k_{1},k_{2};\lambda_{1},\lambda_{2}}=S_{\text{WZW},k_{1}}(g_{1})+S_{\text{WZW},k_{2}}(g_{2})-\\\
&-\frac{1}{\pi}\int_{\Sigma}d^{2}\sigma{\rm
Tr}\left[\begin{pmatrix}J_{1+}&J_{2+}\end{pmatrix}\begin{pmatrix}k_{1}\lambda_{1}\lambda_{2}\mathcal{O}_{21}D_{2}^{T}&k_{2}\lambda_{0}\lambda_{1}\mathcal{O}_{21}\\\
k_{1}\lambda_{0}^{-1}\lambda_{2}\mathcal{O}_{12}&k_{2}\lambda_{1}\lambda_{2}\mathcal{O}_{12}D_{1}^{T}\end{pmatrix}\begin{pmatrix}J_{1-}\\\
J_{2-}\end{pmatrix}\right].\end{split}$ (2.33)

As has been analyzed in [31, 32], the difference in the levels results to an
RG flow which acquires a new fixed point in the IR for specific values of the
couplings, $(\lambda_{1},\lambda_{2})=(\lambda_{0},\lambda_{0})$ with
$\lambda_{0}=\sqrt{k_{1}/k_{2}}$. This level inequality leads to several
interesting features for the integrable branes embedded in (2.33). These
comprise its surviving symmetries, their quantization condition and the
consistent definition of the so-called generalized permutation branes in the
RG fixed points [17].

As before its equations of motion can be recast in the form of a zero
curvature condition (2.6) for the Lax matrices

$\begin{split}\mathcal{L}_{1\pm}=\frac{2z}{z^{2}-1}\frac{1-(\lambda_{0})^{\mp
1}\lambda_{1}}{1-\lambda_{1}^{2}}A_{1\pm}\,,\quad\mathcal{L}_{2\pm}=\frac{2z}{z^{2}-1}\frac{1-(\lambda_{0})^{\pm
1}\lambda_{2}}{1-\lambda_{2}^{2}}A_{2\pm}\,,\end{split}$ (2.34)

with

$\begin{split}&A_{1+}=\lambda_{1}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{1}D_{2})^{-1}(\lambda_{0}J_{1+}+\lambda_{2}D_{1}J_{2+})\,,\\\
&A_{1-}=-\lambda_{1}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{2}^{T}D_{1}^{T})^{-1}(\lambda_{0}^{-1}J_{2-}+\lambda_{2}D_{2}^{T}J_{1-})\,,\\\
&A_{2+}=\lambda_{2}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{2}D_{1})^{-1}(\lambda_{0}^{-1}J_{2+}+\lambda_{1}D_{2}J_{1+})\,,\\\
&A_{2-}=-\lambda_{2}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{1}^{T}D_{2}^{T})^{-1}(\lambda_{0}J_{1-}+\lambda_{1}D_{1}^{T}J_{2-})\end{split}$
(2.35)

and its energy-momentum tensor reads

$\begin{split}&T_{++}=k_{2}\frac{1-\lambda_{1}^{2}}{\lambda_{1}^{2}}{\rm
Tr}(A_{1+},A_{1+})+k_{1}\frac{1-\lambda_{2}^{2}}{\lambda_{2}^{2}}{\rm
Tr}(A_{2+},A_{2+})\,,\\\
&T_{--}=k_{1}\frac{1-\lambda_{1}^{2}}{\lambda_{1}^{2}}{\rm
Tr}(A_{1-},A_{1-})+k_{2}\frac{1-\lambda_{2}^{2}}{\lambda_{2}^{2}}{\rm
Tr}(A_{2-},A_{2-})\,.\end{split}$ (2.36)

If we define the reflection transformation as

$R:\sigma\to 2\pi-\sigma\,,\quad g_{i}\to
g_{i+1}^{-1}\,,\quad\lambda_{0}\to\lambda_{0}^{-1}\,,\quad i=1,2\,,$ (2.37)

and proceed along the same lines as before we arrive at the boundary
conditions

$\begin{split}&\frac{1-\lambda_{0}^{-1}\lambda_{1}}{1-\lambda_{1}^{2}}A_{1+}|_{\partial\Sigma}=\frac{1-\lambda_{0}\lambda_{1}}{1-\lambda_{1}^{2}}\Omega_{1}A_{1-}|_{\partial\Sigma}\,,\\\
&\frac{1-\lambda_{0}\lambda_{2}}{1-\lambda_{2}^{2}}A_{2+}|_{\partial\Sigma}=\frac{1-\lambda_{0}^{-1}\lambda_{2}}{1-\lambda_{2}^{2}}\Omega_{2}A_{2-}|_{\partial\Sigma}\,.\end{split}$
(2.38)

If we instead define that

$R:\sigma\to 2\pi-\sigma\,,\quad g_{i}\to
g_{i}^{-1}\,,\quad\lambda_{i}\to\lambda_{i+1}\,,\quad i=1,2\,,$ (2.39)

we find the boundary conditions

$\begin{split}&\frac{1-\lambda_{0}^{-1}\lambda_{1}}{1-\lambda_{1}^{2}}A_{1+}|_{\partial\Sigma}=\frac{1-\lambda_{0}^{-1}\lambda_{2}}{1-\lambda_{2}^{2}}\Omega
A_{2-}|_{\partial\Sigma}\,,\\\
&\frac{1-\lambda_{0}\lambda_{2}}{1-\lambda_{2}^{2}}A_{2+}|_{\partial\Sigma}=\frac{1-\lambda_{0}\lambda_{1}}{1-\lambda_{1}^{2}}\Omega
A_{1-}|_{\partial\Sigma}\,,\end{split}$ (2.40)

where, as in the case of equal levels, $\Omega$ does not need to be
involutive.

Using (2.36) and demanding that (2.40) satisfy (2.22), we find that $\Omega$
preserves the trace $k_{1}{\rm Tr}(,)+k_{2}{\rm Tr}(,)$ and that
$\lambda_{1}=\lambda_{2}=\lambda$ along the whole deformation line.666As in
the case of equal levels the restriction on the running of the couplings is
compatible with their beta fuction. Additionally, one can see that in the UV
limit where model (II) describes a $G_{k_{1}}\times G_{k_{2}}$ CFT, the
boundary conditions (2.40) describe the product $\mathcal{C}_{f_{1}f_{2}}$.
Turning our attention to the other set of boundary conditions (2.38) we see
that they do not satisfy (2.22) 777Unless $\lambda_{0}=1$ which is the case
already studied and they do not have a geometric realization in terms of
Dirichlet and Neumann conditions. However they comprise consistent integrable
boundary conditions.

Thus model (II) admits one kind of integrable boundary conditions compatible
with the vanishing of the momentum flow through the boundary

$\begin{split}\begin{pmatrix}A_{1+}\\\
A_{2+}\end{pmatrix}_{\partial\Sigma}=\begin{pmatrix}0&\Omega\\\
\Omega^{-1}&0\end{pmatrix}\begin{pmatrix}A_{1-}\\\
A_{2-}\end{pmatrix}_{\partial\Sigma},\quad\lambda_{1}=\lambda_{2}\,.\end{split}$
(2.41)

###  2.3 The isotropic deformation of $G_{k_{1}}\times
G_{k_{2}}/G_{k_{1}+k_{2}}$ space

As a last example we consider the isotropic $\lambda$-deformed
$G_{k_{1}}\times G_{k_{2}}/G_{k_{1}+k_{2}}$ coset space [34]. As it has been
shown it has a $G$-gauge invariance given as

$(g_{1},g_{2})\sim(Lg_{1}L^{-1},Lg_{2}L^{-1}),\quad
L(\sigma_{+},\sigma_{-})\in G$ (2.42)

and it flows to a coset space CFT in the IR. Its action reads as

$\begin{split}S&{}_{k_{1},k_{2},\lambda}=S_{WZW,k_{1}}(g_{1})+S_{WZW,k_{2}}(g_{2})-\\\
&-\frac{k_{1}}{\pi}\int_{\Sigma}{\rm
Tr}\left[J_{1+}\Lambda_{12}^{-T}\left((1-\lambda)(s_{1}J_{1-}+s_{2}J_{2-})-4s_{1}s_{2}\lambda(D_{2}^{T}-1)J_{1-}\right)\right]\\\
&-\frac{k_{2}}{\pi}\int_{\Sigma}{\rm
Tr}\left[J_{2+}\Lambda_{21}^{-T}\left((1-\lambda)(s_{1}J_{1-}+s_{2}J_{2-})-4s_{1}s_{2}\lambda(D_{1}^{T}-1)J_{2-}\right)\right]\,,\end{split}$
(2.43)

where $s_{i}=k_{i}/k$, $i=1,2$ with $k=k_{1}+k_{2}$. It has been proven to be
integrable as it admits the flat Lax connection

$L_{\pm}=\mathcal{A}_{\pm}+(a+z^{\pm
1}\sqrt{a^{2}+\beta})\mathcal{B}_{\pm}\,,\quad z\in\mathbb{C}\,,$ (2.44)

with the coefficients given by

$\begin{split}\alpha=-\frac{(s_{1}-s_{2})(1-\lambda)}{1-\lambda(1-8s_{1}s_{2})}\,,\quad\beta=\frac{1+\lambda-2\lambda^{2}(1-4s_{1}s_{2})}{\lambda(1-\lambda(1-8s_{1}s_{2}))}\end{split}$
(2.45)

and

$\mathcal{A}_{\pm}=\frac{1}{2}(A_{1\pm}+A_{2\pm})\,,\quad\mathcal{B}_{\pm}=\frac{1}{2}(A_{1\pm}-A_{2\pm})\,.$
(2.46)

The gauge fields $A_{i\pm}$ are given as

$\begin{split}&A_{1+}=\Lambda^{-1}_{21}((1-\lambda)(s_{1}J_{1+}+s_{2}J_{1+})-4s_{1}s_{2}\lambda(D_{2}-1)J_{1+})\,,\\\
&A_{2+}=\Lambda^{-1}_{12}((1-\lambda)(s_{1}J_{1+}+s_{2}J_{2+})-4s_{1}s_{2}\lambda(D_{1}-1)J_{2+})\,,\\\
&A_{1-}=-\Lambda^{-T}_{12}((1-\lambda)(s_{1}J_{1-}+s_{2}J_{2-})-4s_{1}s_{2}\lambda(D_{2}^{T}-1)J_{1-})\,,\\\
&A_{2-}=-\Lambda^{-T}_{21}((1-\lambda)(s_{1}J_{1-}+s_{2}J_{2-})-4s_{1}s_{2}\lambda(D_{1}^{T}-1)J_{2-})\,,\end{split}$
(2.47)

with

$\begin{split}&\Lambda_{12}=4\lambda
s_{1}s_{2}(D_{1}-1)(D_{2}-1)+(\lambda-1)(s_{1}D_{1}+s_{2}D_{2}-1)\,.\end{split}$
(2.48)

As before we built the boundary monodromy matrix

$T_{b}(z)=T_{R}(2\pi,\pi;z)T(\pi,0;z)\,,$ (2.49)

where we considered $\Omega=\mathbb{1}$ and $T_{R}(2\pi,\pi;z)$ is constructed
from the Lax (2.44) under the reflection $\sigma\to 2\pi-\sigma$ so that

$T_{R}(2\pi,\pi;z)=T(0,\pi;z^{-1})\,.$ (2.50)

Demanding then integrability for (2.49) we find that (2.11) is satisfied for
$N(z)=L_{\tau}(0,z)$ and the boundary conditions [27]

$L_{\tau}(z)|_{\partial\Sigma}=L_{\tau}(z^{-1})|_{\partial\Sigma}\,,$ (2.51)

on both the end points of the string. Using the form of the time component of
the Lax pair and expanding order by order in the spectral parameter $z$ we
find the following integrable boundary conditions

$\mathcal{B}_{+}|_{\partial\Sigma}=\mathcal{B}_{-}|_{\partial\Sigma}\,.$
(2.52)

##  3 Identifying the integrable brane configurations

D-branes are submanifolds which are endowed with a gauge invariant two-form
$\omega$. Usually in the literature the starting point to determine a D-brane
configuration, i.e. the submanifold and the two-form, is directly from the
boundary conditions which it corresponds. This project has been carried out in
[5][7][8][15][37], where it was shown that the maximally symmetric boundary
conditions

$J_{+}=\Omega J_{-}\,,\quad\Omega\in\text{Aut}(G)\,,$ (3.1)

in the WZW model, describe branes which lie in conjugacy classes
$\mathcal{C}_{\omega}(f)=\omega(h)fh^{-1}$, with $\omega$ defined as
$\omega(e^{tX})=e^{t\Omega[X]}$. In the same line, in [26] the authors showed
that the integrable boundary conditions

$A_{+}=\Omega A_{-}\,,\quad\Omega\in\text{Aut}(G)\,,$ (3.2)

for the $\lambda$-deformed sigma-model, describe branes which differ only in
terms of size compared to the ones described by (3.1). Furthermore, they
derived the boundary two-form for $G=SU(2)$ and $SL(2,\mathbb{R})$, which is
$\lambda$-dependent. Using this form they showed that the points in which the
stable D-branes are located are independent of the deformation parameter.

In our cases the task of determining the brane configurations directly from
the integrable conditions (2.29), (2.30), (2.41), (2.52) is more complicated.
Thus using a different approach based on [8] [10] [35] [36], we will show that
the corresponding boundary conditions for the three models under study
describe branes whose Dirichlet directions remain invariant when we turn on
the deformation parameters. Demanding a well defined action we will determine
the induced boundary two-form for an arbitrary group $G$. In this framework
the reason of the stable branes being located at points that are independent
of the deformation parameters will become more transparent. Finally, let us
mention that the brane geometries we will derive correspond to the cases where
the automorphisms appearing in the boundary equations act trivially, i.e.
$\Omega=\mathbb{1}$. Furthermore, in Appendix A we show how one can include
the case where $\Omega$ acts non trivially.

###  3.1 The isotropic deformation at equal levels

In the case of a worldsheet with no boundaries the action of model (I) is
given in (2.1). To proceed we recall this action in the form

$\begin{split}S_{k;\lambda_{1},\lambda_{2}}=\int_{\Sigma}L_{k;\lambda_{1},\lambda_{2}}+\int_{M}H_{k;\lambda_{1},\lambda_{2}}\,,\end{split}$
(3.3)

where $M$ is a three manifold bounded by the worldsheet $\Sigma$, i.e.
$\partial M=\Sigma$ and 888For the worldsheet conventions see (A.5).

$\displaystyle L_{k,\lambda_{1},\lambda_{2}}=-\frac{k}{8\pi}\sum_{i=1}^{2}$
$\displaystyle{\rm
Tr}(g_{i}^{-1}\partial_{\mu}g_{i},g_{i}^{-1}\partial^{\mu}g_{i})+2\lambda_{i}{\rm
Tr}(\partial^{\mu}g_{i}g_{i}^{-1},\mathcal{O}_{i+1,i}g_{i+1}^{-1}\partial_{\mu}g_{i+1})$
$\displaystyle+2\lambda_{i}\lambda_{i+1}{\rm
Tr}(\partial^{\mu}g_{i}g_{i}^{-1},\mathcal{O}_{i+1,i}D_{i+1}^{T}g_{i}^{-1}\partial_{\mu}g_{i})\,,$
(3.4) $\displaystyle
H_{k,\lambda_{1},\lambda_{2}}=\frac{k}{4\pi}\sum_{i=1}^{2}$
$\displaystyle(H_{WZ}(g_{i})+\lambda_{i}d{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}_{i+1,i}(g_{i+1}^{-1}dg_{i+1}+\lambda_{i+1}D_{i+1}^{T}g_{i}^{-1}dg_{i})))\,.$

As it stands, action (3.3) is not well defined for a worldsheet with
boundaries. The ill defined term is the last one as there is no region $M$
bounded by $\Sigma$ when $\Sigma$ itself has boundaries. To fix that, action
(3.3) is modified as

$\begin{split}S_{k;\lambda_{1},\lambda_{2}}&=\int_{\Sigma}L_{k;\lambda_{1},\lambda_{2}}+\int_{M^{\prime}}H_{k;\lambda_{1},\lambda_{2}}-\int_{D}\omega_{k;\lambda_{1},\lambda_{2}}\,,\end{split}$
(3.5)

where $M^{\prime}$ is a three-dimensional manifold bounded by $\Sigma\cup D$,
with $D$ a two dimensional disc embedded in the brane, and the boundary two-
form $\omega_{k;\lambda_{1},\lambda_{2}}$ is such that [5], [8], [36], [38]

$H_{k;\lambda_{1},\lambda_{2}}|_{\text{brane}}=d\omega_{k;\lambda_{1},\lambda_{2}}\,,$
(3.6)

where one restricts to the brane surface. At this point the form of the brane
geometry enters into consideration. We will embed in the model a brane whose
worldvolume lies in a product of two $G$-conjugacy classes, i.e.

$\mathcal{C}_{f_{1}f_{2}}=\mathcal{C}_{f_{1}}\times\mathcal{C}_{f_{2}}=\\{(h_{1}f_{1}h_{1}^{-1},h_{2}f_{2}h_{2}^{-1}),|\forall
h_{1},h_{2}\in G\\}\,,$ (3.7)

where $f_{1},f_{2}$ are fixed elements of the group chosen from the Cartan
torus of $G$. The dimension of (3.7) is $2(d_{G}-r_{G})$, where
$d_{G}=\dim(G)$ and $r_{G}=\text{rank}(G)$.

Having specified the brane geometry we are in the position to compute the
boundary two form. Using (3.6), (3.7) we find that999Notice that we can add an
arbitrary exact two form in (3.8) $F=dA$, which in principle can be
$\lambda$-dependent. However, such a choice would not lead to the desired
integrable boundary conditions below in (3.12).

$\begin{split}\omega_{k;\lambda_{1},\lambda_{2}}=\frac{k}{4\pi}&\sum_{i=1}^{2}\Big{(}\omega_{WZ}(h_{i})+\lambda_{i}{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}_{i+1,i}g_{i+1}^{-1}dg_{i+1})|_{\mathcal{C}_{f_{1}f_{2}}}\\\
&+\lambda_{i}\lambda_{i+1}{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}_{i+1,i}D_{i+1}^{T}g_{i}^{-1}dg_{i})|_{\mathcal{C}_{f_{1}f_{2}}}\Big{)}\,,\end{split}$
(3.8)

where $\omega_{\text{WZ}}(f_{i})$ reads [8][10][36]

$\omega_{\text{WZ}}(f_{i})={\rm Tr}(h^{-1}_{i}dh_{i}\wedge
f_{i}h_{i}^{-1}dh_{i}f_{i}^{-1}),\quad i=1,2$ (3.9)

and is such that it satisfies the relation,
$H_{WZ}(g_{i})|_{\mathcal{C}_{f_{1}f_{2}}}=d\omega_{WZ}(h_{i})$. The second
term in (3.8) is just the deformation dependent two form in (3.1) restricted
on the brane (3.7), i.e. we just replace the group elements $g_{i}$ with the
boundary values, $g_{i}|_{\partial\Sigma}=h_{i}f_{i}h_{i}^{-1}$. Having
determined the boundary two-form we proceed by computing the boundary
contribution in the variation of (3.5), where we present the technical details
in the appendix B.1. The result is

$\displaystyle\delta S|_{\partial\Sigma}$
$\displaystyle=\frac{k}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{1}h_{1}^{-1},\nabla_{+}g_{1}g_{1}^{-1}+g_{1}^{-1}\nabla_{-}g_{1}+(A_{1+}-A_{2+})+(A_{1-}-A_{2-}))$
$\displaystyle+\frac{k}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{2}h_{2}^{-1},\nabla_{+}g_{2}g_{2}^{-1}+g_{2}^{-1}\nabla_{-}g_{2}+(A_{2+}-A_{1+})+(A_{2-}-A_{1-}))$
$\displaystyle=\frac{k}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{1}h_{1}^{-1},\lambda_{1}^{-1}A_{1+}-\lambda_{2}^{-1}A_{2-}-(A_{2+}-A_{1-}))$
(3.10) $\displaystyle+\frac{k}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{2}h_{2}^{-1},\lambda_{2}^{-1}A_{2+}-\lambda_{1}^{-1}A_{1-}-(A_{1+}-A_{2-}))\,.$

To pass from the first to the second equality we used the fact the the fields
$A_{i\pm}$ are solutions of the constraint equations

$\nabla_{+}g_{i}g_{i}^{-1}=(\lambda_{i}^{-1}-1)A_{i+},\quad
g^{-1}_{i}\nabla_{-}g_{i}=-(\lambda_{i+1}^{-1}-1)A_{i+1-},\quad i=1,2\,.$
(3.11)

It is straightforward to see that setting $\lambda_{1}=\lambda_{2}=\lambda$,
the vanishing of the boundary contribution leads exactly to the integrable
conditions in (2.29) for trivial automorphisms

$A_{1+}|_{\partial\Sigma}=A_{2-}|_{\partial\Sigma},\quad
A_{2+}|_{\partial\Sigma}=A_{1-}|_{\partial\Sigma}\,.$ (3.12)

Recall that in section 2.1 we derived the above equations using the boundary
monodromy method and were led to the same condition $\lambda_{1}=\lambda_{2}$,
demanding that (2.27) satisfy (2.22). Turning our attention to the other set
of integrable boundary conditions in (2.30) we embed in (3.5) a different
brane geometry, known in the literature as permutation branes, which is
defined as

${\mathcal{C}^{\pi}}_{f_{1}f_{2}}={\mathcal{C}^{\pi}}_{f_{1}}\times\left({\mathcal{C}^{\pi}}_{f^{-1}_{2}}\right)^{-1}=\\{(h_{1}f_{1}h_{2}^{-1},h_{2}f_{2}h_{1}^{-1}),|\forall
h_{1},h_{2}\in G\\}\,.$ (3.13)

In general the dimension of (3.13) is $2d_{G}-r_{G}$, but for certain values
of $f_{1},f_{2}$ it will have a lower value [16]. In particular, if
$f_{1}f_{2}=\mathbb{1}$ then we have
$g_{1}|_{\partial\Sigma}=g_{2}^{-1}|_{\partial\Sigma}$ on the boundary. In
this case the dimension is given by $d_{G}$. The boundary two form induced in
(3.13) is given as in (3.8), but with the difference now
that101010$\omega_{\text{WZ}}$ is defined such that [16]
$(H_{\text{WZ}}(g_{1})+H_{\text{WZ}}(g_{2}))|_{{\mathcal{C}^{\pi}}_{f_{1}f_{2}}}=d(\omega_{\text{WZ}}(f_{1})+\omega_{\text{WZ}}(f_{2})).$
(3.14)

$\omega_{\text{WZ}}(f_{i})={\rm Tr}(h_{i}^{-1}dh_{i}\wedge
f_{i}h_{i+1}^{-1}dh_{i+1}f_{i}^{-1})\,,\quad i=1,2$ (3.15)

and the two-form induced by the deformation is restricted on the brane surface
(3.13). As before we are now in position to determine the boundary
contribution (see again appendix B.1) which reads

$\begin{split}\delta
S|_{\partial\Sigma}&=\frac{k}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{1}h_{1}^{-1},\nabla_{+}g_{1}g_{1}^{-1}+g_{2}^{-1}\nabla_{-}g_{2})+{\rm
Tr}(\delta
h_{2}h_{2}^{-1},\nabla_{+}g_{2}g_{2}^{-1}+g_{1}^{-1}\nabla_{-}g_{1})\\\
&=\frac{k}{2\pi}\int_{\partial\Sigma}(\lambda_{1}^{-1}-1){\rm Tr}(\delta
h_{1}h_{1}^{-1},A_{1+}-A_{1-})+(\lambda_{2}^{-1}-1){\rm Tr}(\delta
h_{2}h_{2}^{-1},A_{2+}-A_{2-})\,,\end{split}$ (3.16)

where we used (3.11). Its vanishing leads to the boundary conditions

$A_{1+}|_{\partial\Sigma}=A_{1-}|_{\partial\Sigma}\,,\quad
A_{2+}|_{\partial\Sigma}=A_{2-}|_{\partial\Sigma}\,,$ (3.17)

which exactly correspond to the other set of integrable conditions in (2.29).

Thus we found that model (I) admits two kinds of brane configurations along
its RG flow, namely that

$\begin{split}&\mathcal{C}_{f_{1}f_{2}}=\\{(h_{1}f_{1}h_{1}^{-1},h_{2}f_{2}h_{2}^{-1},|\forall
h_{1},h_{2}\in G)\\}\,,\\\
\omega_{k;\lambda_{1},\lambda_{2}}=&\frac{k}{4\pi}\sum_{i=1}^{2}\Big{(}\omega_{WZ}(h_{i})+\lambda{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}_{i+1,i}g_{i+1}^{-1}dg_{i+1})|_{\mathcal{C}_{f_{1}f_{2}}}\\\
&+\lambda^{2}{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}_{i+1,i}D_{i+1}^{T}g_{i}^{-1}dg_{i})|_{\mathcal{C}_{f_{1}f_{2}}}\Big{)}\,,\\\
&\omega_{WZ}(f_{i})={\rm Tr}(h^{-1}_{i}dh_{i}\wedge
f_{i}h_{i}^{-1}dh_{i}f_{i}^{-1})\,,\end{split}$ (3.18)

and that

$\begin{split}&{{\mathcal{C}^{\pi}}}_{f_{1}f_{2}}=\\{(h_{1}f_{1}h_{2}^{-1},h_{2}f_{2}h_{1}^{-1},|\forall
h_{1},h_{2}\in G)\\}\,,\\\
\omega_{k;\lambda_{1},\lambda_{2}}=&\frac{k}{4\pi}\sum_{i=1}^{2}\Big{(}\omega_{WZ}(h_{i})+\lambda_{i}{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}_{i+1,i}g_{i+1}^{-1}dg_{i+1})|_{\mathcal{C}_{\pi}}\\\
&+\lambda_{i}\lambda_{i+1}{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}_{i+1,i}D_{i+1}^{T}g_{i}^{-1}dg_{i})|_{\mathcal{C}_{\pi}}\Big{)}\,,\\\
&\omega_{WZ}(f_{i})={\rm Tr}(h^{-1}_{i}dh_{i}\wedge
f_{i}h_{i+1}^{-1}dh_{i+1}f_{i}^{-1})\,,\end{split}$ (3.19)

As we have said (3.18) require $\lambda_{1}=\lambda_{2}=\lambda$, while (3.19)
do not. Moreover, they are solutions of the boundary conditions (3.12) and
(3.17) respectively, thus proving that they preserve integrability of the
model for arbitrary values of the couplings.

###  3.2 The isotropic deformation at unequal levels

The action of model (II) (2.33), which describes the deformation of the direct
product of two current algebras, in the precence of boundaries can be written
as

$\begin{split}S_{k_{1},k_{2};\lambda_{1},\lambda_{2}}&=\int_{\Sigma}L_{k_{1},k_{2};\lambda_{1},\lambda_{2}}+\int_{M^{\prime}}H_{k_{1},k_{2};\lambda_{1},\lambda_{2}}-\int_{D}\omega_{k_{1},k_{2};\lambda_{1},\lambda_{2}}\,,\end{split}$
(3.20)

where

$\displaystyle
L_{k_{1},k_{2},\lambda_{1},\lambda_{2}}=-\frac{1}{8\pi}\sum_{i=1}^{2}$
$\displaystyle k_{i}{\rm
Tr}(g_{i}^{-1}\partial_{\mu}g_{i},g_{i}^{-1}\partial^{\mu}g_{i})+2\lambda_{i}k^{(i+1)}{\rm
Tr}(\partial^{\mu}g_{i}g_{i}^{-1},\mathcal{O}_{i+1,i}g_{i+1}^{-1}\partial_{\mu}g_{i+1})$
$\displaystyle+2\lambda_{i}\lambda_{i+1}k_{i}{\rm
Tr}(\partial^{\mu}g_{i}g_{i}^{-1},\mathcal{O}_{i+1,i}D_{i+1}^{T}g_{i}^{-1}\partial_{\mu}g_{i}))$
(3.21) $\displaystyle
H_{k_{1},k_{2},\lambda_{1},\lambda_{2}}=\frac{1}{4\pi}\sum_{i=1}^{2}(k_{i}$
$\displaystyle H_{WZ}(g_{i})+\lambda_{i}d{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}_{i+1,i}(k^{(i+1)}g_{i+1}^{-1}dg_{i+1}+k_{i}\lambda_{i+1}D_{i+1}^{T}g_{i}^{-1}dg_{i})))\,.$

with $k^{(i)}=\sqrt{k_{i}k_{i+1}}$. The integrability preserving boundary
conditions consistent with the model are of the form (2.41), thus the brane we
will consider is (3.7). As before the boundary two-form trivializing the three
form $H_{k_{1},k_{2};\lambda_{1},\lambda_{2}}$ on the brane
$\mathcal{C}_{f_{1}f_{2}}$ is

$\begin{split}\omega_{k_{1},k_{2};\lambda_{1},\lambda_{2}}=\frac{1}{4\pi}\sum_{i=1}^{2}&\Big{(}k_{i}\omega_{WZ}(h_{i})+\lambda_{i}k^{(i+1)}{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}_{i+1,i}g_{i+1}^{-1}dg_{i+1})|_{\mathcal{C}_{f_{1}f_{2}}}\\\
&+\lambda_{i}\lambda_{i+1}k_{i}{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}^{T}_{i+1,i}D_{i+1}^{T}g_{i}^{-1}dg_{i})|_{\mathcal{C}_{f_{1}f_{2}}}\Big{)}\,.\end{split}$
(3.22)

where $\omega(f_{i})$ is given in (3.9). Since the technical details are the
same with the previous case we will just write down the result

$\displaystyle\delta S|_{\partial\Sigma}$
$\displaystyle=\frac{k_{1}}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{1}h_{1}^{-1},\nabla_{+}g_{1}g_{1}^{-1}+g_{1}^{-1}\nabla_{-}g_{1}+(A_{1+}-A_{2+})+(A_{1-}-A_{2-}))$
$\displaystyle+\frac{k_{2}}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{2}h_{2}^{-1},\nabla_{+}g_{2}g_{2}^{-1}+g_{2}^{-1}\nabla_{-}g_{2}+(A_{2+}-A_{1+})+(A_{2-}-A_{1-}))$
$\displaystyle=\frac{k_{1}}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{1}h_{1}^{-1},\lambda_{0}^{-1}(\lambda_{1}^{-1}A_{1+}-\lambda_{2}^{-1}A_{2-})-(A_{2+}-A_{1-}))$
(3.23) $\displaystyle+\frac{k_{2}}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{2}h_{2}^{-1},\lambda_{0}(\lambda_{2}^{-1}A_{2+}-\lambda_{1}^{-1}A_{1-})-(A_{1+}-A_{2-}))\,.$

As before to pass from the first to second equality we used the constraints

$\begin{split}&\nabla_{+}g_{1}g_{1}^{-1}=(\lambda_{0}^{-1}\lambda_{1}^{-1}-1)A_{1+}\,,\quad
g^{-1}_{2}\nabla_{-}g_{2}=-(\lambda_{0}\lambda_{1}^{-1}-1)A_{1-}\,,\\\
&\nabla_{+}g_{2}g_{2}^{-1}=(\lambda_{0}\lambda_{2}^{-1}-1)A_{2+}\,,\quad
g^{-1}_{1}\nabla_{-}g_{1}=-(\lambda_{0}^{-1}\lambda_{2}^{-1}-1)A_{2-}\,.\end{split}$
(3.24)

Setting $\lambda_{1}=\lambda_{2}=\lambda$ one can see that the vanishing of
(3.2) leads to the desired integrable boundary conditions found in (2.41)

$A_{1+}|_{\partial\Sigma}=A_{2-}|_{\partial\Sigma}\,,\quad
A_{2+}|_{\partial\Sigma}=A_{1-}|_{\partial\Sigma}\,.$ (3.25)

Summarizing our results, the model (II) admits the following integrable brane
configuration along its RG flow

$\begin{split}&\mathcal{C}_{f_{1}f_{2}}=\\{(h_{1}f_{1}h_{1}^{-1},h_{2}f_{2}h_{2}^{-1},|\forall
h_{1},h_{2}\in G)\\}\,,\\\
\omega_{k_{1},k_{2};\lambda_{1},\lambda_{2}}=\frac{1}{4\pi}&\sum_{i=1}^{2}\Big{(}k_{i}\omega_{WZ}(h_{i})+\lambda
k^{(i+1)}{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}_{i,i+1}g_{i+1}^{-1}dg_{i+1})|_{\mathcal{C}_{f_{1}f_{2}}}\\\
&+\lambda^{2}k_{i}{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}_{i,i+1}D_{i+1}^{T}g_{i}^{-1}dg_{i})|_{\mathcal{C}_{f_{1}f_{2}}}\Big{)}\,,\\\
&\omega_{WZ}(f_{i})={\rm Tr}(h^{-1}_{i}dh_{i}\wedge
f_{i}h_{i}^{-1}dh_{i}f_{i}^{-1})\,.\end{split}$ (3.26)

We remind to the reader that the model (II) interpolates between two exact
CFTs in the UV and IR fixed points for $\lambda=0$ and $\lambda=\lambda_{0}$
respectively, and that the integrable branes $\eqref{B19'}$ interpolate
between two conformal configurations at the same points, preserving one copy
of the Virassoro algbera respectively.111111Recall that the boundary
conditions (3.25) satisfy (2.22) for arbitrary values of the couplings. Thus
at the conformal points the two copies of the Virassoro algebra are reduced to
one.

Turning our attention to the other set of integrable boundary conditions
(2.38) we notice that they suggest that the corresponding brane geometry
contains a permutation between the two groups like in (3.13).121212To see this
note that in the UV limit the boundary conditions (3.13) become
$k_{1}J_{1+}|_{\partial\Sigma}=-k_{2}J_{2-}|_{\partial\Sigma}\,,\quad
k_{2}J_{2+}|_{\partial\Sigma}=-k_{1}J_{1-}|_{\partial\Sigma}\,.$ (3.27)
However, permutation branes can not be defined consistently in the case of
$k_{1}\neq k_{2}$, in the sense that there is no two-form trivializing the
three-form $H_{k_{1},k_{2};\lambda_{1},\lambda_{2}}$ on the brane, leading to
the conclusion that (2.38) do not admit a geometric solution.

###  3.3 The isotropic deformation of $G_{k_{1}}\times
G_{k_{2}}/G_{k_{1}+k_{2}}$ space

As a final example we consider the $\lambda$-deformed of a $G_{k_{1}}\times
G_{k_{2}}/G_{k_{1}+k_{2}}$ coset space (2.43). Repeating the same steps as
before we rewrite its action as

$S_{k_{1},k_{2};\lambda}=\int_{\Sigma}L_{k_{1},k_{2};\lambda}+\int_{M}H_{k_{1},k_{2};\lambda}-\int_{D}\omega_{k_{1},k_{2};\lambda}\,,$
(3.28)

with

$\displaystyle L_{k_{1},k_{2},\lambda}=-\frac{1}{8\pi}\sum_{i=1}^{2}k_{i}{\rm
Tr}(g_{i}^{-1}\partial_{\mu}g_{i},g_{i}^{-1}\partial^{\mu}g_{i})-2\lambda
s_{i}s_{i+1}k_{i}{\rm
Tr}(\partial^{\mu}g_{i}g_{i}^{-1},\Lambda_{i,i+1}^{-T}(D_{i+1}^{T}-\mathbb{1})g_{i}^{-1}\partial_{\mu}g_{i})$
$\displaystyle\hskip 51.21495pt+2(1-\lambda)k_{i}{\rm
Tr}(\partial^{\mu}g_{i}g_{i}^{-1},\Lambda^{-T}_{i,i+1}(s_{i}g^{-1}_{i}\partial_{\mu}g_{i}+s_{i+1}g^{-1}_{i+1}\partial_{\mu}g_{i+1}))\,,$
$\displaystyle
H_{k_{1},k_{2},\lambda}=\frac{1}{4\pi}\sum_{i=1}^{2}(k_{i}H_{WZ}(g_{i})-2\lambda
s_{i}s_{i+1}k_{i}d{\rm
Tr}(dg_{i}g_{i}^{-1},\Lambda_{i,i+1}^{-T}(D_{i+1}^{T}-\mathbb{1})g_{i}^{-1}dg_{i}))$
$\displaystyle\hskip 51.21495pt+2(1-\lambda)k_{i}d{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\Lambda^{-T}_{i,i+1}(s_{i}g^{-1}_{i}dg_{i}+s_{i+1}g^{-1}_{i+1}dg_{i+1}))\,.$
(3.29)

If we consider the brane (3.7) and the boundary two-form trivializing
$H_{k_{1},k_{2};\lambda}$ on the brane, we find that the boundary contribution
to the variation of (3.28) is

$\displaystyle\delta S|_{\partial\Sigma}$
$\displaystyle=\frac{1}{2\pi}\int_{\partial\Sigma}k_{1}{\rm Tr}(\delta
h_{1}h_{1}^{-1},\nabla_{+}g_{1}g_{1}^{-1}+g^{-1}_{1}\nabla_{-}g_{1})+k_{2}{\rm
Tr}(\delta
h_{2}h_{2}^{-1},\nabla_{+}g_{2}g_{2}^{-1}+g^{-1}_{2}\nabla_{-}g_{2})$
$\displaystyle=\frac{k(\lambda^{-1}-1)}{4\pi}\int_{\partial\Sigma}{\rm
Tr}(\delta h_{1}h_{1}^{-1},\mathcal{B}_{+}-\mathcal{B}_{-})-{\rm Tr}(\delta
h_{2}h_{2}^{-1},\mathcal{B}_{+}-\mathcal{B}_{-})\,,$ (3.30)

where recall that $k=k_{1}+k_{2}$ and $s_{i}=k_{i}/k$, $i=1,2$. We used also
the constraints

$\begin{split}&s_{1}\nabla_{+}g_{1}g_{1}^{-1}=\frac{1}{2}(\lambda^{-1}-1)\mathcal{B}_{+}\,,\quad
s_{2}\nabla_{+}g_{2}g_{2}^{-1}=-\frac{1}{2}(\lambda^{-1}-1)\mathcal{B}_{+}\,,\\\
&s_{1}g_{1}^{-1}\nabla_{-}g_{1}=-\frac{1}{2}(\lambda^{-1}-1)\mathcal{B}_{-}\,,\quad
s_{2}g_{2}^{-1}\nabla_{-}g_{2}=\frac{1}{2}(\lambda^{-1}-1)\mathcal{B}_{-}\,.\end{split}$
(3.31)

The vanishing of (3.3) leads to the integrable boundary conditions (2.52).
Thus we find that the brane configuration

$\displaystyle\mathcal{C}_{f_{1}f_{2}}=\\{(h_{1}f_{1}h_{1}^{-1},h_{2}f_{2}h_{2}^{-1},|\forall
h_{1},h_{2}\in G)\\}\,,$
$\displaystyle\omega_{k_{1},k_{2};\lambda}=\frac{1}{4\pi}\sum_{i=1}^{2}$
$\displaystyle\Big{(}k_{i}\omega_{WZ}(h_{i})+2k_{i}(1-\lambda){\rm
Tr}(dg_{i}g_{i}^{-1},\Lambda^{-T}_{i,i+1}(s_{i}g_{i}^{-1}dg_{i}+s_{i+1}g_{i+1}^{-1}dg_{i+1}))|_{\mathcal{C}_{f_{1}f_{2}}}$
$\displaystyle-2\lambda s_{i}s_{i+1}k_{i}{\rm
Tr}(dg_{i}g_{i}^{-1},\Lambda_{i,i+1}^{-T}(D_{i+1}^{T}-\mathbb{1})g_{i}^{-1}dg_{i})|_{\mathcal{C}_{f_{1}f_{2}}})\,,$
(3.32) $\displaystyle\omega_{WZ}(f_{i})={\rm Tr}(h^{-1}_{i}dh_{i}\wedge
f_{i}h_{i}^{-1}dh_{i}f_{i}^{-1})\,,$

preserves integrability of the deformed coset space. In the following sections
we will consider a different brane geometry, known in the literature as
generalized permutation branes [17], and we will show that it also solves the
boundary conditions (2.52).

###  3.4 The allowed set of integrable branes

As discussed in the literature [5][8][10][11][36], in the Lagrangian
description of boundary sigma-models the absence of topological obstructions,
originating from WZ-type terms, requires that the relative cohomology class
$[(H,\omega)]$ should be integral, that is

$\int_{S^{3}}H-\int_{S^{2}}\omega\in 2\pi\mathbb{Z}\,,$ (3.33)

where $S^{3}$ is a three dimensional ball, whose boundary $S^{2}$ is mapped
into the brane. Picking the specific form of the three-form $H$ for each
model, given in (3.1), (3.2), (3.3) and the corresponding boundary two-forms
$\omega$ in (3.18), (3.19), (3.26), (3.3), we find that (3.33) is independent
of the deformation parameters and specifically equals to

$\sum_{i=1}^{2}\frac{k_{i}}{4\pi}\left(\int_{S^{3}}H_{\text{WZ}}(g_{i})-\int_{S^{2}}\omega_{\text{WZ}}(f_{i})\right)\in
2\pi\mathbb{Z}\,.$ (3.34)

In other words, the flux of the $F$ field, $F=dA=\omega-B$, through any two
sphere $S^{2}$ is quantized and independent of the deformation. Equation
(3.34) leads to the quantization of the elements $f$ chosen from the Cartan
torus of the group $G$. This project has been carried out for the branes under
study for compact simply connected groups, see e.g. [10, 11, 12].

To be more specific let us consider the case where $G=SU(2)$, and we choose
the convenient parametrization

$g(\psi,\theta,\phi)=\begin{pmatrix}&\cos(\psi)+i\cos(\theta)\sin(\psi)&\sin(\psi)\sin(\theta)e^{i\phi}\\\
&-\sin(\psi)\sin(\theta)e^{-i\phi}&\cos(\psi)-i\cos(\theta)\sin(\psi)\end{pmatrix}\,,$
(3.35)

with $\theta\in[0,\pi]$, $\phi\in[0,2\pi]$ and $\psi\in[0,\pi]$.

Due to the previous analysis, the model (I) admits two kinds of stable
integrable branes along its RG flow given in (3.18) and (3.19). In the
parametrization (3.35) and using (3.34) we find that the first set of branes
are $(k+1)^{2}$ hypersurfaces located at [5], [8], [9], [11],

$\psi_{i}=\frac{n_{i}\pi}{k}\,,\quad n_{i}=0,1\,,\dots,k,\quad i=1,2\,,$
(3.36)

Turning our attention to the other set of integrable branes, (3.19), we know
from the analysis in [16] that the number of stable permutation branes in a
product of WZW models, $G_{k}\times G_{k}$, equals the number of maximally
symmetric branes in one copy $G_{k}$. Since this picture persists along the
whole deformation line, model (I) admits additionally $(k+1)$ stable
permutation branes embedded in its $\lambda$-dependent target space.

Similarly, model (II) admits $(k_{1}+1)(k_{2}+1)$ integrable branes described
by (3.26), which compared to (3.36) are located at

$\psi_{i}=\frac{n_{i}\pi}{k_{i}}\,,\quad n_{i}=0,1,\dots,k_{i}\,,\quad
i=1,2\,.$ (3.37)

The same result (3.37) holds for the integrable branes of model (III) given in
(3.3).

It is important to notice, that even though the bulk degrees of freedom of a
field theory reduce in number along its RG flow, from the UV towards the IR
CFT, the allowed branes remain invariant.

###  3.5 Comments on the brane geometry

The geometry of the target spaces of the models under study is
$\lambda$-dependent. However, we saw that the integrable branes defined in
each model, have some interesting $\lambda$-independent features. They are
hypersurfaces located at points in the transverse space which remain fixed
along the corresponding RG flows, i.e. their Dirichlet directions are
independent of the deformation. This feature were also found in [26], where
additionally the authors showed that the deformation affects only the size of
the integrable branes. Here we will give a plausible argument that this simple
action of the deformation does not hold for the highest dimensional integrable
branes that the corresponding target spaces admit.

We will choose to demonstrate our arguments for the branes (3.7) embedded in
model (II) for $G=SU(2)$. In the UV they are presented in table (1).131313The
conjugacy classes of $SU(2)$ are two points at $\psi=0,\pi$ and $k-1$
$S^{2}$-spheres located at $\psi=n\pi/k$ for $n=1,\dots,k-1$.

Number $(N)$ |  Geometry |  Dimension
---|---|---
$N_{1}=4$ |  $\\{\pm\mathbb{1}\\}\times\\{\pm\mathbb{1}\\}$ |  $0$
$N_{2}=2(k_{1}+k_{2}-2)$ |  $S^{2}\times\\{\pm\mathbb{1}\\}$, $\\{\pm\mathbb{1}\\}\times S^{2}$ |  $2$
$N_{3}=(k_{1}-1)(k_{2}-1)$ |  $S^{2}\times S^{2}$ |  $4$
Table 1:  The product of conjugacy classes in $SU(2)_{k_{1}}\times
SU(2)_{k_{2}}$. Note that $N_{1}+N_{2}+N_{3}=(k_{1}+1)(k_{2}+1)$.

Turning on the deformation, the topology of the $D2$-branes is unaffected.
Specifically the $D2$-branes change only in terms of size as in [26]. This can
be seen by restricting the background metric to $\psi_{i}=n_{i}\pi/k_{i}$ with
$n_{i}=1,\dots,k_{i}-1$ and $\psi_{i+1}=0,\pi$,

$d\hat{s}^{2}_{i}=\frac{k_{i}(1-\lambda^{4})\sin^{2}\psi_{i}(d\theta_{i}^{2}+\sin^{2}\theta_{i}d\phi_{i}^{2})}{1+\lambda^{4}-2\lambda^{2}\cos
2\psi_{i}}\,,\quad g_{i+1}=\pm\mathbb{1}\,,\quad i=1,2\,,$ (3.38)

What about the rest $N_{3}=(k_{1}-1)(k_{2}-1)$ integrable branes? Setting
$\psi_{i}=n_{i}\pi/k_{i}$ with $n_{i}\neq 0,\pi$ for both $i=1,2$, we find
that they are four dimensional hypersurfaces,
$\mathcal{S}_{\psi_{1}\psi_{2}}=\mathcal{S}_{n_{1}n_{2}}(\theta_{1},\phi_{1},\theta_{2},\phi_{2})$,
with the coordinates parameterizing them coupled among themselves in a non
trivial $\lambda$-dependent way. Due to the complexity of the induced metric
we do not present it here. However, calculating the Gauss-Bonnet term we found
that the topology of the corresponding branes is invariant and equals that of
a product of two-spheres.

###  3.6 Some comments on integrable brane geometries for $N>2$

Let us consider the $\lambda$-deformation of $N$ WZW models all at the same
level $k$ [30]. In that work it was shown that it admits $N$ independent Lax
pairs of the form (2.7) where the fields $A_{i\pm}$ are given in terms of
$g_{i}$, $i=1,\dots,N$ as

$\begin{split}&A_{i+}=(\mathbb{1}-\hat{D}_{i}\dots\hat{D}_{N+i-1})^{-1}\sum_{j=i}^{N+i-1}\hat{D}_{j}\dots\hat{D}_{j-1}\lambda_{j}J_{j+}\,,\\\
&A_{i-}=-(\mathbb{1}-\hat{D}^{T}_{i-1}\dots\hat{D}_{N+i-2})^{-1}\sum_{j=i-1}^{N+i-2}\hat{D}^{T}_{j}\dots\hat{D}^{T}_{j-1}\lambda_{j}J_{j-}\,,\end{split}$
(3.39)

where $\hat{D}_{i}=\lambda_{i}D_{i}$. The boundary conditions preserving its
integrable structure can be put in the form

$\mathbb{A}_{+}|_{\partial\Sigma}=\mathbb{R}_{N}\mathbb{A}_{-}|_{\partial\Sigma}\,,$
(3.40)

where $\mathbb{A}_{\pm}^{T}=(A_{1\pm},A_{2\pm},\dots,A_{N\pm})$ and
$\mathbb{R}_{N}$ is the gluing matrix describing all the possible conditions
between the $A_{i\pm}$’s on the boundary.

The simplest case for $N=3$ suffices to deduce the general structure.
Following the same procedure as in section 2 one can show that all possible
integrable conditions are described by gluing matrices of the form

$\mathbb{R}_{3}=\begin{pmatrix}\Omega_{1}&0&0\\\ 0&\Omega_{2}&0\\\
0&0&\Omega_{3}\end{pmatrix}\,,$ (3.41)

by which only the components of the same gauge field are related on the
boundary, and of the form

$\mathbb{R}_{3}=\begin{pmatrix}\Omega_{1}&0&0\\\ 0&0&\Omega\\\
0&\Omega^{-1}&0\end{pmatrix}\,,$ (3.42)

by which components of different gauge fields are related as well. Two more
gluing matrices similar to (3.42) are also allowed by performing a cyclic
permutation of the indices $(1,2,3)$. Moreover, due to consistency reasons
explained in section 2 and the demand of a vanishing momentum flow across the
boundary we find that $\mathbb{R}_{3}^{2}=\mathbb{1}$ and
$\mathbb{R}_{3}^{T}=\mathbb{R}_{3}^{-1}$. For simplicity, we will consider
trivial acting automorphisms, i.e. $\Omega_{i}=\mathbb{1}$.

Since we want to study cases with arbitrary $N$ we found it convenient to
relate the matrices $\mathbb{R}_{N}$ to the elements of the symmetry group
$S_{N}$. In the $N=3$ case, labeling every set $A_{i\pm}$ with $i=1,2,3$ we
find that $\mathbb{R}_{3}$ corresponds to elements of $S_{3}$ of the form
$(i)(j)(k)$, i.e. a product of three distinct one-cycles, and $(i)(jk)$, i.e.
a product of a one-cycle and a two-cycle. Specifically the rule is:

$\begin{split}&(i)\leftrightarrow
A_{i+}|_{\partial\Sigma}=A_{i-}|_{\partial\Sigma}\,,\\\ &(ij)\leftrightarrow
A_{i+}|_{\partial\Sigma}=A_{j-}|_{\partial\Sigma}\,,\quad
A_{j+}|_{\partial\Sigma}=A_{i-}|_{\partial\Sigma}\,.\end{split}$

The integrable brane geometries corresponding to the boundary conditions
(3.41) and (3.42) are141414The permutation branes between $N$ group elements
$g_{i}$ are defined as  ${\mathcal{C}^{\pi}}_{f_{1},\dots,f_{i},\dots
f_{N}}=(h_{1}f_{1}h_{2}^{-1},\dots,h_{i}f_{i}h_{i+1}^{-1},\dots,h_{N}f_{N}f_{1}^{-1})$
(3.43)

$\begin{split}&(1)(2)(3):\quad{\mathcal{C}^{\pi}}_{f_{1}\,f_{2}\,f_{3}}\,,\\\
&(1)(23):\quad\left(\mathcal{C}_{f_{2}}\,,\,{\mathcal{C}^{\pi}}_{f_{1}\,f_{3}}\right)\,.\end{split}$
(3.44)

Thus we see that the boundary conditions consisting of three one-cycles
correspond to permutation branes defined on all three group elements, while
the other boundary conditions to a conjugacy class times a permutation brane.
We would like to see if this correspondence between elements of the symmetric
group and the integrable brane geometry holds for $N>3$. Towards this it is
straightforward to see that different sets of boundary conditions are
classified as $\\{n,m\\}$ where $n$ is the number of two-cycles and $m$ the
number of one-cycles with $N=2n+m$. From combinatorics each class has
$N!/(n!m!2^{n})$ elements.

Thus for $N=4$ we have the classes, $\\{0,4\\},\\{1,2\\},\\{2,0\\}$. In this
case we find that each class does not admit the same brane geometry.
Specifically we find that:

$\begin{split}&(0,4):\quad{\mathcal{C}^{\pi}}_{f\,f\,f\,f}\,,\\\
&(1,2):\quad\left(\mathcal{C}_{f},{\mathcal{C}^{\pi}}_{f\,f\,f}\right),\left({\mathcal{C}^{\pi}}_{f\,f},{\mathcal{C}^{\pi}}_{f\,f}\right),\\\
&(2,0):\quad\left(\mathcal{C}_{f},\mathcal{C}_{f},{\mathcal{C}^{\pi}}_{f\,f}\right),{\mathcal{C}^{\pi}}_{f\,f\,f\,f}\,.\end{split}$
(3.45)

Hence, to generate a general rule we need more involved combinatorics in each
class, which we have not performed. However let us mention that every two-
cycle between adjacent indices, i.e. $(i,i+1)$, results to a conjugacy class,
thus the maximum number of conjugacy classes contained in an integrable brane
configuration is $N/2$ for $N$ even and $(N-1)/2$ for $N$ odd.

##  4 The integrable branes in the CFT limits

###  4.1 Integrable branes in the conformal limits of model (II)

As has been analyzed in [31], [32] model (II) flows from a UV point for
$(\lambda_{1},\lambda_{2})=(0,0)$ towards a far IR for
$(\lambda_{1},\lambda_{2})=(\lambda_{0},\lambda_{0})$. The conformal symmetry
of the UV CFT is based on a product of two current algebra symmetries

$(G_{k_{1}}\times G_{k_{2}})_{L}\otimes(G_{k_{1}}\times G_{k_{2}})_{R}\,,$
(4.1)

while the IR CFT is a product of coset and current algebra symmetries, i.e.

$\left(\frac{G_{k_{1}}\times G_{k_{2}-k_{1}}}{G_{k_{2}}}\times
G_{k_{2}-k_{1}}\right)_{L}\otimes\left(\frac{G_{k_{1}}\times
G_{k_{2}-k_{1}}}{G_{k_{2}}}\times G_{k_{2}-k_{1}}\right)_{R}.$ (4.2)

Keeping the discussion general, conformal boundary conditions reduce the
initial symmetry group of a CFT to a large enough subgroup which guarantees
conformal invariance of the model. Here we will find the symmetries of model
(II) at its UV and IR fixed points under the presence of the branes (3.7).
Specifically we will determine the subgroups of the initial symmetry groups
(4.1) and (4.2) preserved by the corresponding branes.

The integrable branes (3.7), in the UV limit of model (II), preserve on the
boundary the diagonal subgroups of each copy of the chiral symmetry (4.1)
[10]. Put it plainly the symmetry

$(g_{1},g_{2})\mapsto(k_{1L}(\sigma_{-})g_{1}k_{1R}(\sigma_{+}),k_{2L}(\sigma_{-})g_{2}k_{2R}(\sigma_{+}))\,,$
(4.3)

of two WZW models, is reduced to

$(g_{1},g_{2})|_{\partial\Sigma}\mapsto(k_{1}(\tau)g_{1}k^{-1}_{1}(\tau),k_{2}(\tau)g_{2}k^{-1}_{2}(\tau))\,,$
(4.4)

in the presence of (3.7). This symmetry can be realized also by noticing that
in the UV limit the boundary conditions (3.12) identify the generators of the
chiral transformation (4.3) of each of the group, i.e.

$J_{1+}|_{\partial\Sigma}=-J_{1-}|_{\partial\Sigma}\,,\quad
J_{2+}|_{\partial\Sigma}=-J_{2-}|_{\partial\Sigma}\,.$ (4.5)

Turning our attention to the IR limit of the flow we see that the integrable
conditions (3.12) become

$A_{1+}(\lambda_{0})|_{\partial\Sigma}=A_{2-}(\lambda_{0})|_{\partial\Sigma}\,,\quad
A_{2+}(\lambda_{0})|_{\partial\Sigma}=A_{1-}(\lambda_{0})|_{\partial\Sigma}\,,$
(4.6)

where we remind to the reader that the presence of the branes (3.7) restrict
the deformation parameters to be equal, $\lambda_{1}=\lambda_{2}=\lambda$ and
we emphasize the fact the fields $A_{1\pm},A_{2\pm}$ are evaluated on the
fixed point $\lambda=\lambda_{0}$. According to [32] the fields
$A_{2+},A_{1-}$ satisfy a
$\hat{\mathfrak{g}}_{k_{2}-k_{1}}\oplus\hat{\mathfrak{g}}_{k_{2}-k_{1}}$
algebra corresponding to the chiral symmetry

$(G_{k_{2}-k_{1}})|_{L}\otimes(G_{k_{2}-k_{1}})|_{R}\,,$ (4.7)

of the IR CFT. From (4.6) the left moving generators $A_{1-}$ are identified
with the right moving $A_{2+}$, thus proving that the branes (3.7) preserve on
the boundary the diagonal subgroup of (4.7). What about the other set of
conformal boundary conditions in (4.6)? As has been also noted in [32] the
action (2.33) in the IR limit and under the group redefinition
$(g_{1},g_{2})\to(g^{\prime}_{1}=g_{1}g_{2},g^{\prime}_{2}=g_{2})$ is
identified as the gauge fixed version, $g^{\prime}_{3}=\mathbb{1}$, of an
action which realizes the conformal symmetry (4.2) generated by the
transformations

$(g^{\prime}_{1},g^{\prime}_{2},g^{\prime}_{3})\mapsto(Lg^{\prime}_{1}L^{-1},g^{\prime}_{2}L^{-1},Lg^{\prime}_{3}),\quad
L=L(\sigma_{+},\sigma_{-})\,,$ (4.8)

and

$(g^{\prime}_{2},g^{\prime}_{3})\mapsto(k_{L}(\sigma_{+})g^{\prime}_{2},g^{\prime}_{3}k_{R}(\sigma_{-}))\,.$
(4.9)

The branes (3.7), which under the group redefinition become
$(\mathcal{C}_{f_{1}}\mathcal{C}_{f_{2}},\mathcal{C}_{f_{2}})$, can be viewed
as the gauge fixed version of a brane geometry which when embedded in the
extended action preserves the gauge symmetry (4.8) and the diagonal subgroup
of (4.9). Thus the boundary conditions (4.6) are the gauge fixed version of
the conditions which correspond to the preservation of $\eqref{38}$ and (4.9).
We were not able to find the gauge invariant extension of the
$(\mathcal{C}_{f_{1}}\mathcal{C}_{f_{2}},\mathcal{C}_{f_{2}})$ geometry.

###  4.2 Integrable branes in the conformal limits of model (III)

As we have mentioned model (III) is invariant under the adjoint action of $G$
(2.42). It flows from a UV CFT for $\lambda=0$ towards a CFT in the IR for
$\lambda=1/(s_{2}-3s_{1})$ [34]. That is

$\frac{G_{k_{1}}\times
G_{k_{2}}}{G_{k_{1}+k_{2}}}\Longrightarrow\frac{G_{k_{1}-k_{2}}\times
G_{k_{2}}}{G_{k_{1}}}\,.$ (4.10)

The presence of the branes (3.3) along (4.10) does not spoil gauge invariance
of the model. To see this notice that under (2.42) the action (3.28)
transforms

$\delta_{V}S=\delta_{V}S|_{\Sigma}+\delta_{V}S|_{\partial\Sigma}\,,$ (4.11)

where $\delta_{V}$ stands for the infinitesimal version of (2.42), with the
boundary term given by

$\delta_{V}S|_{\partial\Sigma}=\frac{k(\lambda^{-1}-1)}{4\pi}\int_{\partial\Sigma}{\rm
Tr}(\delta_{V}h_{1}h_{1}^{-1},\mathcal{B}_{+}-\mathcal{B}_{-})-{\rm
Tr}(\delta_{V}h_{2}h_{2}^{-1},(\mathcal{B}_{+}-\mathcal{B}_{-}))\,.$ (4.12)

To find explicitly the contribution (4.12) notice that the geometry of the
brane remains invariant, in the sense that

$(g_{1},g_{2})|_{\partial\Sigma}\mapsto(L^{-1}g_{1}L,L^{-1}g_{2}L)|_{\partial\Sigma}=(h^{\prime}_{1}f_{1}{h_{1}^{\prime}}^{-1},h^{\prime}_{2}f_{2}{h_{2}^{\prime}}^{-1})\,,$
(4.13)

with $h_{i}^{\prime}=L^{-1}h_{i}$. Thus setting
$\delta_{V}h_{1}h_{1}^{-1}=\delta_{V}h_{2}h_{2}^{-1}$ in (4.12) we find that
$\delta_{V}S|_{\partial\Sigma}=0$, while of course $\delta_{V}S|_{\Sigma}=0$.
This proves that the boundary action (3.28) remains invariant. Thus,
projecting the $2(d_{G}-r_{G})$-dimensional branes in the deformed coset space
we end up with $(d_{G}-2r_{G})$-dimensional integrable branes. In the case
$G=SU(2)$ our branes are either $0$ or $1$-dimensional.

Let us now turn our attention to the conformal limits of (4.10) for $SU(2)$.
In [40] the authors have studied the $D$-branes in a diagonal coset space CFT
of the form $SU(2)_{k}\times SU(2)_{l}/SU(2)_{m}$, with $m=k+l$. They have
argued that the conformal branes are given as a projected product of the
$SU(2)$-conjugacy classes on the coset space, i.e.

$\pi_{\text{Ad}}(\mathcal{C}_{a_{1}a_{2}a_{3}})=\pi_{\text{Ad}}((\mathcal{C}_{a_{1}}\mathcal{C}_{a_{3}},\mathcal{C}_{a_{2}}\mathcal{C}_{a_{3}}))\,,$
(4.14)

where $\pi_{Ad}$ denotes the projector and the numbers $a_{1},a_{2},a_{3}$
parametrize the elements of the Cartan torus of $SU(2)$, i.e.
$f_{i}=f_{i}(a_{i})\in SU(2)$, and correspond to the levels $k,l,m$,
respectively. Furthermore, they classified the dimensions of the branes (4.14)
in terms of the values of the $a_{i}^{\prime}s$ and found them to be $0,1$ and
$3$-dimensional.

Returning to our integrable branes (3.3) for $G=SU(2)$, corresponding to
$D0$\- and $D1$-branes, it is straightforward to see that in the UV limit of
the flow (4.10) they belong in the class (4.14) for

$a_{1}=\psi_{1}=\frac{n_{1}\pi}{k_{1}}\,,\quad
a_{2}=\psi_{2}=\frac{n_{2}\pi}{k_{2}}\,,\quad a_{3}=0\,,$ (4.15)

with $n_{i}=0\,,\dots,k_{i}\,,\,i=1,2$. The IR CFT in (4.10) is realized under
the coordinate transformation
$(g_{1},g_{2})\to(g^{\prime}_{1}=g_{1},g^{\prime}_{2}=g_{2}g_{1})$ [34]. In
these coordinates our integrable branes are written as

$({g^{\prime}}_{1},{g^{\prime}}_{2})|_{\partial\Sigma}=(\mathcal{C}_{\psi_{1}},\mathcal{C}_{\psi_{2}}\mathcal{C}_{\psi_{1}})\,.$
(4.16)

Thus in the IR they belong again in the class (4.14) for

$a_{1}=0\,,\quad a_{2}=\psi_{2}=\frac{n_{2}\pi}{k_{2}}\,,\quad
a_{3}=\psi_{1}=\frac{n_{1}\pi}{k_{1}}\,,$ (4.17)

with $n_{i}=0\,,\dots,k_{i}\,,\,i=1,2$.

The highest dimensional branes correspond to (4.14) with
$a_{1},a_{2},a_{3}\neq 0$ and as mentioned they correspond to consistent brane
geometries for the UV and IR CFTs in (4.10). However they can not be
identified as integrable brane geometries for the whole flow. To be more
specific, if we consider the boundary values
$(g_{1},g_{2})|_{\partial\Sigma}=(r_{1}r_{3},r_{2}r_{3})$ with
$r_{i}=h_{i}f_{i}h_{i}^{-1}$ we find that the boundary contribution to the
variation of (3.28) reads (see Appendix B.3)

$\begin{split}\delta
S|_{\partial\Sigma}&=\frac{k_{1}}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{1}h_{1}^{-1},\nabla_{+}g_{1}g_{1}^{-1}+\nabla_{r_{3}}g_{1}^{-1}\nabla_{-}g_{1}+(\mathbb{1}-D_{r_{3}})A_{1\tau}-\partial_{\tau}r_{3}r_{3}^{-1})\\\
&+\frac{k_{2}}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{2}h_{2}^{-1},\nabla_{+}g_{2}g_{2}^{-1}+\nabla_{r_{3}}g_{2}^{-1}\nabla_{-}g_{2}+(\mathbb{1}-D_{r_{3}})A_{2\tau}-\partial_{\tau}r_{3}r_{3}^{-1})\\\
&+\frac{1}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{3}h_{3}^{-1},(D_{r_{3}}-\mathbb{1})(k_{1}g_{1}^{-1}\nabla_{-}g_{1}+k_{2}g_{2}^{-1}\nabla_{-}g_{2})\\\
&+(D_{r_{3}}-1)(k_{1}A_{1\tau}+k_{2}A_{2\tau})+(k_{1}+k_{2})\partial_{\tau}r_{3}r_{3}^{-1})\,.\end{split}$
(4.18)

Using now the constraints (3.31), this can be rewritten as

$\begin{split}\delta
S|_{\partial\Sigma}&=\frac{k}{4\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{1}h_{1}^{-1},(\lambda^{-1}-1)(\mathcal{B}_{+}-D_{r_{3}}\mathcal{B}_{-})+2s_{1}(\mathbb{1}-D_{r_{3}})A_{1\tau}-2s_{1}\partial_{\tau}r_{3}r_{3}^{-1})\\\
&+\frac{k}{4\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{2}h_{2}^{-1},(1-\lambda^{-1})(\mathcal{B}_{+}-D_{r_{3}}\mathcal{B}_{-})+2s_{2}(\mathbb{1}-D_{r_{3}})A_{2\tau}-2s_{2}\partial_{\tau}r_{3}r_{3}^{-1})\\\
&+\frac{1}{2\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{3}h_{3}^{-1},(D_{r_{3}}-\mathbb{1})(k_{1}A_{1\tau}+k_{2}A_{2\tau})+(k_{1}+k_{2})\partial_{\tau}r_{3}r_{3}^{-1})\,.\end{split}$
(4.19)

The vanishing of the boundary terms lead to the boundary conditions

$\begin{split}((\lambda^{-1}+4s_{1}s_{2})\mathbb{1}-&4s_{1}s_{2}D_{r_{3}})\mathcal{B}_{+}|_{\partial\Sigma}=((\lambda^{-1}+4s_{1}s_{2})D_{L_{3}}-4s_{1}s_{2}\mathbb{1})\mathcal{B}_{-}|_{\partial\Sigma}\,,\\\
&\partial_{\tau}r_{3}r_{3}^{-1}|_{\partial\Sigma}=(\mathbb{1}-D_{r_{3}})(s_{1}A_{1\tau}+s_{2}A_{2\tau})|_{\partial\Sigma}\,.\end{split}$
(4.20)

The field dependent gluing of the gauge fields $\mathcal{B}_{\pm}$ on the
boundary, suggests that (4.20) could be derived with the method presented in
section 2 by allowing the possibility to glue $T(\pi,0;z)$ to a gauge
transformed reflected transport matrix. In this case the integrable boundary
conditions take the form [27]

$L_{\tau}(z)|_{\partial\Sigma}=D_{r_{3}}L_{\tau}(z^{-1})+\partial_{\tau}r_{3}r_{3}^{-1}|_{\partial\Sigma}\,,\quad
r_{3}\in G\,.$ (4.21)

Using however the form of the Lax (2.44) it is straightforward to see that due
to consistency $D^{2}_{r_{3}}=\mathbb{1}$, leading to $r_{3}=\mathbb{1}$ which
is the case already studied.

##  5 Generalized permutation branes in the integrable flows

###  5.1 Preliminaries on the generalized permutation branes

In [17] the authors defined a consistent brane configuration, involving a
permutation automorphism, on a product of WZW models $G_{k_{1}}\times
G_{k_{2}}$. Geometrically such branes, called generalized permutation branes,
wrap the submanifold

${\mathcal{D}^{\pi}}_{f}=\left\\{\left((h_{1}fh_{2}^{-1})^{k^{\prime}_{2}},(h_{2}fh_{1}^{-1})^{k^{\prime}_{1}}\right)=\left({v_{1}}^{k^{\prime}_{2}},{v_{2}}^{k^{\prime}_{1}}\right)|\forall\,h_{1}\,,\,h_{2}\in
G\right\\}\,,$ (5.1)

where $k^{\prime}_{i}=k_{i}/k$ and $k=\text{gcd}(k_{1},k_{2})$. In general the
dimensions of these branes are $2d_{G}-r_{G}$, but for $f=e$, where $e$ is the
identity element of the group, is $d_{G}$ and the expression (5.1) simplifies
to

$\mathcal{D}^{\pi}(e)=\left\\{\left(v^{k^{\prime}_{2}},v^{-k^{\prime}_{1}}\right)|\,\forall\,v=h_{1}h_{2}^{-1}\in
G\right\\}\,.$ (5.2)

Notice that in the case of equal levels the brane geometry (5.1) reduce to
that of the permutation branes (3.13).

Furthermore, it was found that the restriction of the WZ three-form on (5.1)
satisfies the condition (3.6) with $\omega_{\text{WZ}}$ given by

$\begin{split}\omega_{\text{WZ}}(h_{1},h_{2})&=\frac{k^{\prime}_{1}k^{\prime}_{2}}{k}\left({\rm
Tr}(h_{1}^{-1}dh_{1}\wedge fh_{2}^{-1}dh_{2}f^{-1})+{\rm
Tr}(h_{2}^{-1}dh_{2}\wedge fh_{1}^{-1}dh_{1}f^{-1})\right)\\\
&+k_{1}\sum_{i=1}^{k^{\prime}_{2}-1}(k^{\prime}_{2}-i){\rm
Tr}(v_{1}^{i}v_{1}^{-1}dv_{1}v_{1}^{-i}\wedge v_{1}^{-1}dv_{1})\\\
&+k_{2}\sum_{i=1}^{k^{\prime}_{1}-1}(k^{\prime}_{1}-i){\rm
Tr}(v_{2}^{i}v_{2}^{-1}dv_{2}v_{2}^{-i}\wedge v_{2}^{-1}dv_{2})\,.\end{split}$
(5.3)

For topological reasons, discussed in 3.4, it is deduced in [17] that
$f=\exp(\pi i\lambda/\kappa)$ where $\lambda$ is an integral weight of
$\mathfrak{g}$ and $\kappa=\text{lcm}(k_{1},k_{2})$. Thus, the number of
stable generalized permutation branes coincides with the number of the
ordinary branes in $G$ with level $\kappa$.

Let us finally comment on the symmetries of the model in the presence of
(5.1). It is invariant under the chiral transformation (4.3) in the bulk,
which on the boundary reduces to

$\begin{split}&(g_{1},g_{2})|_{\partial\Sigma}\mapsto(kg_{1}k^{-1},kg_{2}k^{-1})\,,\end{split}$
(5.4)

in agreement with the boundary equations of motion found in [17]

$k_{1}J_{1+}+k_{2}J_{2-}|_{\partial\Sigma}=-k_{1}J_{1-}-k_{2}J_{2-}|_{\partial\Sigma}\,,$
(5.5)

Thus, the generalized permutation branes preserve the
$\hat{\mathfrak{g}}_{k_{1}+k_{2}}$ Kac-Moody algebra. Notice that the boundary
conditions (5.5) do not preserve the Virassoro algebra associated with
$\hat{\mathfrak{g}}_{k_{1}}\oplus\hat{\mathfrak{g}}_{k_{2}}$. However in [17]
and [18] the authors have proven that (5.1) are consistent solutions of the
DBI action, thus proving their conformal invariance. In the following sections
we will investigate the possibility for such a brane geometry to belong in the
family of integrable branes for the RG flows under study, specifically those
with IR fixed points.

###  5.2 Generalized permutation branes in the deformed models

####  5.2.1 GPB in the deformed coset space

Let us now embed (5.1) in (3.28). In order to determine the boundary
contribution in the variation of (3.28) first we have to specify the induced
two-form on the brane worldvolume (5.1). It turns out that

$\displaystyle\omega_{k_{1},k_{2};\lambda}=$
$\displaystyle\frac{1}{4\pi}\Big{(}\omega_{WZ}(h_{1},h_{2})+\sum_{i=1}^{2}2k_{i}(1-\lambda){\rm
Tr}(dg_{i}g_{i}^{-1},\Lambda^{-T}_{i,i+1}(s_{i}g_{i}^{-1}dg_{i}+s_{i+1}g_{i+1}^{-1}dg_{i+1}))|_{{\mathcal{D}^{\pi}}_{f}}$
$\displaystyle-2\lambda s_{i}s_{i+1}k_{i}{\rm
Tr}(dg_{i}g_{i}^{-1},\Lambda_{i,i+1}^{-T}(D_{i+1}^{T}-\mathbb{1})g_{i}^{-1}dg_{i})|_{{\mathcal{D}^{\pi}}_{f}}\Big{)}\,,$
(5.6)

where $\omega_{\text{WZ}}(h_{1},h_{2})$ is given in (5.3). Then we find that

$\begin{split}\delta
S|_{\partial\Sigma}&=\frac{k(\lambda^{-1}-1)}{4\pi}\int_{\partial\Sigma}{\rm
Tr}(\delta
h_{1}h_{1}^{-1},((1-D^{T}_{v_{1}})^{-1}-(1-D_{v_{2}})^{-1})(\mathcal{B}_{+}-\mathcal{B}_{-}))\\\
&-\frac{k(\lambda^{-1}-1)}{4\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
h_{2}h_{2}^{-1},((1-D^{T}_{v_{2}})^{-1}-(1-D_{v_{1}})^{-1})(\mathcal{B}_{+}-\mathcal{B}_{-}))\,.\end{split}$
(5.7)

It is straightforward to see that its vanishing leads to the conclusion that
the brane geometry (5.1) with the two-form (5.3) solve the boundary conditions
(2.52), thus consisting an integrable brane configuration.

The brane geometry (5.1) preserves gauge invariance of model (III). To see
this notice that the boundary conditions (5.1) remain fixed under the gauge
transformation (2.42)

$(g_{1},g_{2})|_{\partial\Sigma}\mapsto(Lg_{1}L^{-1},Lg_{2}L^{-1})|_{\partial\Sigma}=((h^{\prime}_{1}f{h^{\prime}_{2}}^{-1})^{k^{\prime}_{2}},(h^{\prime}_{2}f{h^{\prime}_{1}}^{-1})^{k^{\prime}_{1}})\,,$
(5.8)

where $h^{\prime}_{1}=Lh_{1}$ and $h^{\prime}_{2}=Lh_{2}$. Setting thus,
$\delta_{V}h_{1}h_{1}^{-1}=\delta_{V}h_{2}h_{2}^{-1}$ in (5.7) we find that
the boundary term in (4.11) vanishes, proving that model (III) preserves its
gauge invariance under the presence of (5.1). Projecting the
$(2d_{G}-r_{G})$-dimensional branes in the deformed coset space we end up with
$(d_{G}-r_{G})$-dimensional integrable branes. For $G=SU(2)$ such branes are
$D2$-branes.

According to the analysis of section 3.4, which may equally well be performed
here, we conclude that the model (III) admits $\kappa=\text{lcm}(k_{1},k_{2})$
integrable branes, additionally to the $(k_{1}+1)(k_{2}+1)$ given in (3.3),
wrapped around ${\mathcal{D}^{\pi}}_{f}$.

####  5.2.2 GPB in the deformed current algebra models

For the sake of simplicity we will consider the lowest dimensional brane (5.2)
embedded in the target space of the sigma-model (3.20). The boundary two-form,
trivializing $H_{k_{1},k_{2};\lambda_{1},\lambda_{2}}$ on (5.2) is

$\begin{split}\omega_{k_{1},k_{2};\lambda_{1},\lambda_{2}}=\frac{1}{4\pi}&\Big{(}\omega_{WZ}(h_{1},h_{2})+\sum_{i=1}^{2}\lambda_{i}k^{(i+1)}{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}^{T}_{i+1,i}g_{i+1}^{-1}dg_{i+1})|_{\mathcal{D}^{\pi}(e)}\\\
&+\lambda_{i}\lambda_{i+1}k_{i}{\rm
Tr}(dg_{i}g_{i}^{-1}\wedge\mathcal{O}^{T}_{i+1,i}D_{i+1}^{T}g_{i}^{-1}dg_{i})|_{\mathcal{D}^{\pi}(e)}\Big{)}\,,\end{split}$
(5.9)

Varying the action and restricting our attention to the contribution from the
boundary we find that (see Appendix B.2)

$\displaystyle\delta
S_{k_{1},k_{2};\lambda_{1},\lambda_{2}}|_{\partial\Sigma}$
$\displaystyle=\frac{\sqrt{k_{1}k_{2}}}{2\pi}\int_{\partial\Sigma}{\rm
Tr}(\delta
h_{1}h_{1}^{-1},(1-D^{T}_{v})^{-1}((\lambda_{1}^{-1}-\lambda_{0}^{-1})A_{1+}+(\lambda_{2}^{-1}-\lambda_{0})A_{2+}$
$\displaystyle-(\lambda_{1}^{-1}-\lambda_{0})A_{1-}-(\lambda_{2}^{-1}-\lambda_{0}^{-1})A_{2-})$
$\displaystyle+\frac{\sqrt{k_{1}k_{2}}}{2\pi}\int_{\partial\Sigma}{\rm
Tr}(\delta
h_{2}h_{2}^{-1},(1-D_{v})^{-1}((\lambda_{1}^{-1}-\lambda_{0}^{-1})A_{1+}+(\lambda_{2}^{-1}-\lambda_{0})A_{2+}$
$\displaystyle-(\lambda_{1}^{-1}-\lambda_{0})A_{1-}-(\lambda_{2}^{-1}-\lambda_{0}^{-1})A_{2-})\,.$
(5.10)

This indicates that the brane (5.2) with the boundary two-form (5.9) is a
solution of the boundary conditions

$(\lambda_{1}^{-1}-\lambda_{0}^{-1})A_{1+}+(\lambda_{2}^{-1}-\lambda_{0})A_{2+}|_{\partial\Sigma}=(\lambda_{1}^{-1}-\lambda_{0})A_{1-}+(\lambda_{2}^{-1}-\lambda_{0}^{-1})A_{2-}|_{\partial\Sigma}\,.$
(5.11)

Unfortunately, we were not able to derive (5.11) with the boundary monodromy
method presented in the first section, thus we can not be sure if model (II)
continues to be integrable in the presence of the brane (5.2). However, we
will make a few observations regarding its presence in the RG flow under study
and especially at its fixed points.

Consider the case where one of the deformation parameters freezes out, e.g.
$(\lambda_{1},\lambda_{2})=(\lambda,0)$.151515Notice that this RG flow can not
be considered for the branes (3.7) studied in 4.1, since they require
$\lambda_{1}=\lambda_{2}$ This theory smoothly flows from a UV CFT, for
$\lambda=0$, towards a CFT in the IR for $\lambda=\lambda_{0}$. That is

$G_{k_{1}}\times G_{k_{2}}\Longrightarrow G_{k_{1}}\times G_{k_{2}-k_{1}}\,.$
(5.12)

The IR CFT can be realized under the group redefinition
$(g_{1},g_{2})\to(g^{\prime}_{1}=g_{2}g_{1},g^{\prime}_{2}=g_{2})$. It is
straightforward to see that the boundary conditions (5.11) in the flow (5.12),
expressed in terms of the group elements are

$(\lambda_{0}-\lambda)J_{1+}+\lambda_{0}^{-1}(J_{2+}+\lambda
D_{2}J_{1+})|_{\partial\Sigma}=(\lambda-\lambda_{0}^{-1})J_{2-}-\lambda_{0}(J_{1-}+\lambda
D_{1}^{T}J_{2-})|_{\partial\Sigma}\,.$ (5.13)

In the UV limit, $\lambda=0$, (5.13) reduce to (5.5) as expected, in agreement
with the symmetry (5.4) preserved by the brane (5.2). Turning our attention to
the IR limit, $\lambda=\lambda_{0}$, (5.13) becomes

$k_{1}J^{\prime}_{1+}+(k_{2}-k_{1})J^{\prime}_{2+}|_{\partial\Sigma}=-k_{1}J^{\prime}_{1-}-(k_{2}-k_{1})J^{\prime}_{2-}|_{\partial\Sigma}\,,$
(5.14)

where we expressed the fields in terms of
$(g^{\prime}_{1},g^{\prime}_{2})$.161616We used the identities
$J^{\prime}_{1+}=J_{2+}+D_{2}J_{1+}\,,\quad
J^{\prime}_{1-}=J_{1-}+D_{1}^{T}J_{2-}\,.$ (5.15) Notice that the currents
appearing in (5.14) are the diagonal combination of the generators of the Kac-
Moody algebra of the IR CFT. This agrees with the fact that the brane (5.2)
under the redefinition $(g_{1},g_{2})\to(g^{\prime}_{1},g^{\prime}_{2})$,
transforms as

${\mathcal{D}^{\pi}}_{e}=\left(v^{k^{\prime}_{2}},v^{-k^{\prime}_{1}}\right)\to\left(v^{k^{\prime}_{2}-k^{\prime}_{1}},v^{-k^{\prime}_{1}}\right)\,,$
(5.16)

which corresponds to the lowest dimensional GPB embedded in the IR CFT
$G_{k_{1}}\times G_{k_{2}-k_{1}}$, since
$k=\text{gcd}(k_{1},k_{2})=\text{gcd}(k_{1},k_{2}-k_{1})$ for $k_{2}>k_{1}$.
Let us point out that the above result holds only for the brane (5.2) as the
higher dimensional ones (5.1), under the previous field redefinition,
transform as

${\mathcal{D}^{\pi}}_{f}\to\left\\{\left({v_{2}}^{k^{\prime}_{2}}{v_{1}}^{k^{\prime}_{1}},{v_{2}}^{k^{\prime}_{2}}\right)|\forall\,h_{1}\,,\,h_{2}\in
G\right\\}=\mathcal{M}_{f}\,,$ (5.17)

which do not belong in the class of the GPBs for the product $G_{k_{1}}\times
G_{k_{2}-k_{1}}$.171717Based on the arguments of [17], one can show that the
number of stable $\mathcal{M}_{f}$ branes embedded in the IR CFT
$G_{k_{1}}\times G_{k_{2}-k_{1}}$ is $\kappa=\text{lcm}(k_{1},k_{2})$. Thus,
as the previously defined geometries the number of stable branes (5.1) remains
invariant along the whole deformation line (5.12).

Thus, we conclude that the brane configuration

$\begin{split}&\mathcal{D}^{\pi}(e)=\left\\{\left(v^{k^{\prime}_{2}},v^{-k^{\prime}_{1}}\right)|\,\forall\,v=h_{1}h_{2}^{-1}\in
G\right\\}\,,\\\
\omega_{k_{1},k_{2};\lambda}=&\omega_{WZ}(h_{1},h_{2})+\sqrt{k_{1}k_{2}}\lambda{\rm
Tr}(dg_{1}g_{1}^{-1}\wedge g_{2}^{-1}dg_{2})|_{\mathcal{D}^{\pi}(e)}\,,\\\
\end{split}$ (5.18)

with $\omega_{\text{WZ}}(h_{1},h_{2})$ given in (5.3), embedded in the RG flow
(5.12) smoothly flows from a conformal brane in the UV, for $\lambda=0$, to a
conformal brane in the IR for $\lambda=\lambda_{0}$. However its integrability
preserving nature between the two CFT points remains open. We expect that the
same results hold for the higher dimensional GPB branes given in (5.1).

In the case where none of the deformation parameters is set to zero the model
flows to the IR CFT given in (4.2) for
$(\lambda_{1},\lambda_{2})=(\lambda_{0},\lambda_{0})$. The boundary conditions
(5.11), evaluated at the IR fixed point, become

$A_{2+}(\lambda_{0},\lambda_{0})|_{\partial\Sigma}=A_{1-}(\lambda_{0},\lambda_{0})|_{\partial\Sigma}\,,$
(5.19)

which, as was mentioned before, relate the left-right generators of the
current algebra symmetry of the IR CFT (4.2). Thus the presence of the brane
(5.2) in the IR limit, breaks down the symmetry group (4.2) to the diagonal
subgroup of (4.7). Finally let us mention that in the case of equal levels
$k_{1}=k_{2}$ the whole set up is reduced to the integrability preserving
permutation branes (3.19) embedded in model (I) (3.3).

###  5.3 The lowest dimensional GPB for $G=SU(2)$

The action describing the entire flow (5.12), for a worldsheet with no
boundaries, is

$S_{k_{1},k_{2};\lambda}=S_{k_{1}}(g_{1})+S_{k_{2}}(g_{2})-\frac{\sqrt{k_{1}k_{2}}\lambda}{\pi}\int{\rm
Tr}(J_{1+},J_{2-})\,.$ (5.20)

The induced metric and boundary two-form on the worldvolume of (5.2) will be
derived from the background fields of (5.20) for $g_{i}=SU(2)$, $i=1,2$ and in
the parametrization (3.35). Doing so, we extract the background metric and
$H$-field of (5.20) but we do not present them here.

As has been noted in [17] the parametrization (3.35) proves to be convenient
for the description of the brane (5.2). Being specific every point on the
brane satisfies the relation

$\psi_{1}=k^{\prime}_{2}\psi\,,\quad\psi_{2}=-k^{\prime}_{1}\psi\,,\quad\theta_{1}=\theta_{2}=\theta\,,\quad\phi_{1}=\phi_{2}=\phi.\,$
(5.21)

where $g_{i}=g_{i}(\psi_{i},\theta_{i},\phi_{i})$, $i=1,2$ and
$v=v(\psi,\theta,\phi)$. Thus, we find that the induced metric reads

$\begin{split}d\hat{s}^{2}&=f(k_{1},k_{2},\lambda)d\psi^{2}+g(k_{1},k_{2};\lambda)(d\theta^{2}+\sin^{2}(\theta)d\phi^{2})\,,\end{split}$
(5.22)

where

$\begin{split}&f(k_{1},k_{2},\lambda)=k^{\prime}_{1}k^{\prime}_{2}(k_{1}+k_{2}-2k_{2}\lambda\lambda_{0})\,,\\\
&g(k_{1},k_{2};\lambda)=(k_{1}-k_{2}\lambda\lambda_{0})\sin^{2}(k^{\prime}_{2}\psi)+k_{2}(1-\lambda\lambda_{0})\sin^{2}(k^{\prime}_{1}\psi)\\\
&\hskip
56.9055pt+k_{2}\lambda\lambda_{0}\sin^{2}((k^{\prime}_{1}-k^{\prime}_{2})\psi)\,.\end{split}$
(5.23)

From (5.18) and using the form of $\eqref{413}$ found in [17] we find that the
boundary two- form is

$\omega_{k_{1},k_{2};\lambda}=h(k_{1},k_{2},\lambda)\sin(\theta)d\theta\wedge
d\phi\,,$ (5.24)

where

$\begin{split}h(k_{1},k_{2},\lambda)&=k_{2}\sin(2k^{\prime}_{1}\psi)-k_{1}\sin(2k^{\prime}_{2}\psi)\\\
&-4k_{2}\lambda_{0}\lambda\sin(k^{\prime}_{2}\psi)\sin(k^{\prime}_{1}\psi)\sin((k^{\prime}_{2}-k^{\prime}_{1})\psi)\,.\end{split}$
(5.25)

It is straightforward to see that for $\lambda=0$ the induced fields
$\eqref{A4}$ and $\eqref{A6}$ become those found in [17]. Turning our
attention to the IR limit $\lambda=\lambda_{0}=\sqrt{k_{1}/k_{2}}$ we find
what we expected from the analysis in the previous section, i.e. the fields on
the brane are the ones found in the UV limit under the redefinition $k_{2}\to
k_{2}-k_{1}$

$\displaystyle
d\hat{s}^{2}=k^{\prime}_{1}(k^{\prime}_{2}-k^{\prime}_{1})k_{2}d\psi^{2}+(k_{1}\sin^{2}((k^{\prime}_{2}-k^{\prime}_{1})\psi)-(k_{2}-k_{1})\sin^{2}k^{\prime}_{1}\psi)(d\theta^{2}+\sin(\theta)^{2}d\phi^{2})\,,$
$\displaystyle\omega_{k_{1},k_{2},\lambda_{0}}=(k_{1}\sin(2(k^{\prime}_{2}-k^{\prime}_{1})\psi)-(k_{2}-k_{1})\sin(2k^{\prime}_{1}\psi))\sin(\theta)d\theta\wedge
d\phi\,.$ (5.26)

Notice that in the case of equal levels $k_{1}=k_{2}$ the whole set up is
reduced to the lowest dimensional integrability preserving permutation brane,
${\mathcal{C}^{\pi}}_{f,f^{-1}}$, embedded in the RG flow described by (2.1)
for $(\lambda,\lambda_{2})=(\lambda,0)$. It is straightforward to see that the
metric (5.22) reduces to the metric of an $S^{3}$-sphere with a deformed
radius embedded in the diagonal and symmetric way in the target space of the
corresponding sigma-model

$ds^{2}=k(1-\lambda)(d\psi^{2}+\sin^{2}\psi(d\theta^{2}+\sin^{2}\theta
d\phi^{2}))\,.$ (5.27)

In this case the two-form (5.24) vanishes, in agreement with [15] [16].

##  6 Conclusions

In this paper we studied integrable brane configurations embedded in three
types of generalized $\lambda$-deformed models, denoted as model (I), (II) and
(III). The model (I) and (II) correspond to the deformation of two WZW models
based on the product group, $G_{k}\times G_{k}$ and $G_{k_{1}}\times
G_{k_{2}}$ respectively. The model (III) corresponds to the deformation of a
diagonal coset space CFT of the form $G_{k_{1}}\times
G_{k_{2}}/G_{k_{1}+k_{2}}$. Model (I) flows to a strongly coupled theory,
while (II) and (III) to exact CFTs in the far IR.

To obtain the integrable boundary conditions along their RG flows, we
constructed the simplest forms of boundary monodromy matrices and demanded, in
each case, that they generate conserved charges in the presence of boundaries.
Using two different definitions of the reflection transformation we found that
the models (I) and (II) admit two kinds of integrable conditions, relating the
components of the same and of different gauge fields. In the case of (I), both
types are consistent with the vanishing of the momentum flow from the
boundary, and have a nice geometric interpretation as a product of G-conjugacy
classes, $\mathcal{C}_{f_{1}f_{2}}$, and twisted conjugacy classes,
${\mathcal{C}^{\pi}}_{f_{1}f_{2}}$. We saw that the conjugacy classes require
the couplings to be equal, $\lambda_{1}=\lambda_{2}$, while for the twisted
ones this is not a requirement. Model (II) admits one set of integrable
boundary conditions consistent with the vanishing of the momentum flow, which
correspond to branes wrapping around $\mathcal{C}_{f_{1}f_{2}}$. The other
set, even though it preserves integrability of the model, does not have a
geometric interpretation in terms of Dirichlet and Neumann boundary
conditions. Finally, model (III) admits integrable brane configurations that
are also described by a product of $G$-conjugacy classes.

After the identification of the integrable boundary conditions and their
geometric interpretation as D-branes, we proceeded with the study of their
stability points and their geometry. As in [26] we found that all occurrences
of the deformation in the quantization condition (3.33) cancel out, enforcing
the D-branes to sit at localized positions. This result, in the case of models
(II) and (III), indicates that the number of stable branes in the UV and IR
fixed points of their RG flows remains invariant, despite the loss of bulk
degrees of freedom dictated by Zamolodchikov’s C-theorem. Subsequently, we
proceeded with the identification of the surviving symmetries of the latter
CFTs. Being specific, the model (II) flows from the CFT (4.1) to the CFT
(4.2). The presence of the brane $\mathcal{C}_{f_{1}f_{2}}$ along its
deformation line preserves in the UV the diagonal subgroup of each copy of the
chiral symmetry (4.3). In the IR it can be realized as the gauge fixed version
of a brane geometry that respects its gauge symmetry (4.8) and the diagonal
subgroup of (4.9). For the model (III) we found that the integrable branes
preserve its gauge invariance along the whole deformation line. Additionally,
for $G=SU(2)$ and in the two CFT limits, we saw that they can be identified
with the $0,1-$dimensional Cardy branes for the diagonal coset spaces
$SU(2)_{k}\times SU(2)_{l}/SU(2)_{k+l}$.

Finally, we considered the newest class, at least to our knowledge, of non
factorizing D-branes known as generalized permutation branes,
${\mathcal{D}^{\pi}}_{f}$. We saw that they can be thought of as additional
integrable brane geometries for model (III), which also preserve its gauge
symmetry. On the contrary we were not able to identify them as integrable
brane geometries for model (II), however we saw that they interpolate between
two conformal brane configurations in the UV and IR fixed points of its RG
flow. As the previously defined geometries, their quantization condition
remains invariant when we turn on the deformation leading to an invariant
brane spectrum between the two CFT limits of (II) and (III). Finally, we
considered the lowest dimensional GPB embedded in the deformed
$SU(2)_{k_{1}}\times SU(2)_{k_{2}}$ CFT and we extracted its induced fields.

There are several directions for future study: One, for example, could try to
construct a more exotic generalized transport matrix (2.9) in order to derive
the boundary conditions (5.11) as conditions that preserve the bulk
integrability of model (II) and subsequently derive the infinite tower of
conserved local and non local higher spin charges. Furthermore, it would be
interesting to find integrable boundary conditions for the models recently
constructed in [41, 42, 43, 44] which admit Lax pairs that do not have a
simple spectral dependence, and apply the sigma-model approach in order to
give a geometric picture as D-branes. Lastly, one might want to consider the
boundary conditions we studied on the upper half plane as was done in [45] in
the case of the single $\lambda$-deformed model.

## Acknowledgement

I thank G. Georgiou and K. Siampos for usefull discussions on the topic of
D-branes. I also appreciate the fruitful and very interesting comments of S.
Driezen and D. Thompson on this work. Finally I am grateful to K. Sfetsos for
suggesting this topic and for a careful reading of the manuscript. Part of
this work was funded by the State Scholarships Foundation (I.K.Y).

##  Appendix A Integrable branes in the $\lambda$-model

Here we will show in detail, using the approach of section 2, that the brane
geometry

${\mathcal{C}^{\omega}}_{f}=\\{\omega(h)fh^{-1}|\forall h\in G\\}\,.$ (A.1)

corresponds to the integrable boundary conditions found in [26][27].

The effective $\lambda$-deformed group $G$ and symmetric space $G/H$ action
can be put in the following compact form

$S_{k;\lambda}=S_{k}(g)-\frac{k}{\pi}\int_{\Sigma}{\rm
Tr}(\partial_{+}gg^{-1},(\mathcal{P}-D^{T})^{-1}g^{-1}\partial_{-}g)\,,$ (A.2)

where the operator $\mathcal{P}$ is defined as

$\mathcal{P}=\begin{cases}\lambda^{-1},\quad\text{group space}\,,\\\
\mathcal{P}^{(0)}+\lambda^{-1}\mathcal{P}^{(1)},\quad\text{symmetric
space}\,.\end{cases}$ (A.3)

In the latter case $\mathcal{P}^{(i)}$ are the projectors along the
$\mathbb{Z}_{2}$-decomposition of the Lie algebra
$\mathfrak{g}=\mathfrak{g}^{(0)}\oplus\mathfrak{g}^{(1)}$.

In the presence of boundaries the action (A.2) is modified as

$S_{k;\lambda}=\int_{\Sigma}L_{k;\lambda}+\int_{M^{\prime}}H_{k;\lambda}-\int_{D}\omega_{k;\lambda}\,,$
(A.4)

where181818Our worldsheet conventions are
$\eta_{\tau\tau}=-\eta_{\sigma\sigma}=\eta^{\tau\tau}=-\eta^{\sigma\sigma}=1\,,\quad\epsilon_{\tau\sigma}=-\epsilon^{\tau\sigma}=-\epsilon_{\sigma\tau}=\epsilon^{\sigma\tau}=-1\,.$
(A.5) Lightcone coordinates are defined as $\sigma_{\pm}=\tau\pm\sigma$.

$\begin{split}&L_{k,\lambda}=-\frac{k}{8\pi}{\rm
Tr}(g^{-1}\partial_{\mu}g,g^{-1}\partial^{\mu}g)-\frac{k}{4\pi}{\rm
Tr}(\partial^{\mu}gg^{-1},(\mathcal{P}-D^{T})^{-1}g^{-1}\partial_{\mu}g)\,,\\\
&H_{k,\lambda}(g)=\frac{k}{4\pi}(H_{WZ}(g)+d{\rm
Tr}(dgg^{-1}\wedge(\mathcal{P}-D^{T})^{-1}g^{-1}dg))\,.\end{split}$ (A.6)

The three-dimensional space $M^{\prime}$ is defined such that $\partial
M^{\prime}=\Sigma\cup D$ and

$H_{k,\lambda}|_{\text{brane}}=d\omega_{k,\lambda}\,.$ (A.7)

Using the brane geometry (A.1) and the equation (A.7) we find that

$\omega_{k,\lambda}(h)=\frac{k}{4\pi}(\omega_{WZ}(h)+{\rm
Tr}(dgg^{-1}\wedge(\mathcal{P}-D^{T})^{-1}g^{-1}dg)|_{{\mathcal{C}^{\omega}}_{f}})\,,$
(A.8)

where [8]

$\omega_{\text{WZ}}(f)={\rm tr}(h^{-1}dh\wedge\Omega^{T}fh^{-1}dhf^{-1})\,,$
(A.9)

with $\Omega$ satisfying the condition $\Omega^{T}=\Omega^{-1}$.191919To see
this notice that in order for $H_{WZ}(g)|_{{\mathcal{C}^{\omega}}_{f}}$ to
equal $\omega_{WZ}(f)$ the following relation must hold  ${\rm
Tr}(t^{A},[t^{B},t^{C}])={\rm
Tr}(\Omega(t^{A}),[\Omega(t^{B}),\Omega(t^{C})])={\rm
Tr}(\Omega(t^{A}),\Omega[t^{B},t^{C}])\,.$ (A.10) Now we are in position to
derive the boundary conditions which correspond to the brane (A.1). We begin
with the variation of the first term in (A.4) where we restrict ourselves in
the boundary terms while the bulk terms will be omitted. Doing so we find that

$\begin{split}\int\delta
L_{k;\lambda}|_{\partial\Sigma}&=\frac{k}{4\pi}\int_{\partial\Sigma}({\rm
Tr}(g^{-1}\delta g,g^{-1}\partial_{\sigma}g)+{\rm Tr}(\delta
gg^{-1},(\mathcal{P}-D^{T})^{-1}g^{-1}\partial_{\sigma}g)\\\ &+{\rm
Tr}(g^{-1}\delta g,(\mathcal{P}-D)^{-1}\partial_{\sigma}gg^{-1})\\\
&=\frac{k}{4\pi}\int_{\partial\Sigma}({\rm Tr}(g^{-1}\delta
g,g^{-1}\partial_{\sigma}g)-{\rm Tr}(\delta gg^{-1},A^{L}_{\sigma})+{\rm
Tr}(g^{-1}\delta g,A^{R}_{\sigma}))\,,\end{split}$ (A.11)

where for convenience we have defined

$\begin{split}A^{L}_{\mu}=-(\mathcal{P}-D^{T})^{-1}g^{-1}\partial_{\mu}g,\quad
A^{R}_{\mu}=(\mathcal{P}-D)^{-1}\partial_{\mu}gg^{-1}\,,\quad\mu=\tau,\sigma\,.\end{split}$
(A.12)

Using (A.1) one can see that on the boundary, arbitrary variations of $g$ can
be written as

$\begin{split}&\delta gg^{-1}|_{\partial\Sigma}=(\Omega-D)\delta hh^{-1}\,,\\\
&g^{-1}\delta g|_{\partial\Sigma}=(D^{T}\Omega-\mathbb{1})\delta
hh^{-1}\,.\end{split}$ (A.13)

Plugging (A.13) in (A.11) we find that

$\begin{split}\int\delta
L_{k;\lambda}|_{\partial\Sigma}=\frac{k}{4\pi}\int_{\partial\Sigma}{\rm
Tr}(\delta
hh^{-1},\Omega^{T}\partial_{\sigma}gg^{-1}-g^{-1}\partial_{\sigma}g-(\Omega^{T}-D^{T})A^{L}_{\sigma}+(\Omega^{T}D-\mathbb{1})A^{R}_{\sigma})\,.\end{split}$
(A.14)

Similarly, the variation of the boundary two-form (A.8) reads 202020We used
the fact that $\partial\Sigma=-\partial D$ and the relation
$(D^{T}\Omega-\Omega^{T}D)\partial_{\tau}hh^{-1}=\Omega^{T}g^{-1}\partial_{\tau}g+\partial_{\tau}gg^{-1}\,.$
(A.15)

$\begin{split}\int\delta\omega_{k;\lambda}|_{\partial
D}=-\frac{k}{4\pi}\int_{\partial\Sigma}{\rm Tr}(\delta
hh^{-1},\Omega^{T}\partial_{\tau}gg^{-1}+(\Omega^{T}D-\mathbb{1})A^{R}_{\tau}+g^{-1}\partial_{\tau}g+(\Omega^{T}-D^{T})A^{L}_{\tau})\,.\end{split}$
(A.16)

Combining (A.14),(A.16) we find that the total boundary contribution is

$\delta S|_{\partial\Sigma}=\frac{k}{2\pi}\int_{\partial\Sigma}{\rm tr}(\delta
hh^{-1},\Omega^{T}\nabla_{+}gg^{-1}+g^{-1}\nabla_{-}g+(\Omega^{T}-\mathbb{1})(A_{+}+A_{-}))\,.$
(A.17)

Specializing to the case of the $\lambda$-deformed $G$-model this becomes
212121We used the constraint equations for the $\lambda$ $G$-model
$\nabla_{+}gg^{-1}=(\lambda^{-1}-1)A_{+},\quad
g^{-1}\nabla_{-}g=-(\lambda^{-1}-1)A_{-}$ (A.18)

$\delta S|_{\partial\Sigma}=\frac{k}{2\pi}\int_{\partial\Sigma}{\rm tr}(\delta
hh^{-1},(\lambda^{-1}\Omega^{T}-\mathbb{1})A_{+}-(\lambda^{-1}\mathbb{1}-\Omega^{T})A_{-})\,.$
(A.19)

Thus the brane configuration $\eqref{A2''''}$ and $\eqref{269}$ is a solution
of the boundary conditions

$(\lambda^{-1}\mathbb{1}-\Omega)A_{+}|_{\partial\Sigma}=(\lambda^{-1}\Omega-\mathbb{1})A_{-}|_{\partial\Sigma}\,,$
(A.20)

where we used also the fact that $\Omega^{T}=\Omega^{-1}$. We know from [26]
that in order for (A.1) to be an integrable brane geometry $\Omega$ should be
an involutive automorphism. In this case (A.20) reduces to the integrable
boundary conditions, found in the same work,

$A_{+}|_{\partial\Sigma}=\Omega
A_{-}|_{\partial\Sigma}\,,\quad\Omega^{2}=\mathbb{1}\,.$ (A.21)

Turning our attention now to the $\lambda$-deformed $G/H$-model we decompose
$\delta hh^{-1}$ in its subgroup and coset components denoted with the indices
$(0)$ and $(1)$ respectively. We also assume that $\Omega$ respects the
$\mathbb{Z}_{2}$-decomposition of $\mathfrak{g}$. Doing so we find that (A.17)
becomes 222222We used the constraint equations for the $\lambda$ $G/H$-model
$\begin{split}&(\nabla_{+}gg^{-1})^{(0)}=0,\quad(g^{-1}\nabla g)^{(0)}=0\,,\\\
&(\nabla_{+}gg^{-1})^{(1)}=(\lambda^{-1}-1)A^{(1)}_{+},\quad(g^{-1}\nabla_{-}g)^{(1)}=-(\lambda^{-1}-1)A^{(1)}_{-}\,.\end{split}$
(A.22)

$\begin{split}\delta
S|_{\partial\Sigma}&=\frac{k}{2\pi}\int_{\partial\Sigma}{\rm tr}({\delta
hh^{-1}},(\Omega^{T}-1)A_{\tau})_{(0)}\\\ &+{\rm tr}(\delta
hh^{-1},(\lambda^{-1}\Omega^{T}-\mathbb{1})A_{+}-(\lambda^{-1}\mathbb{1}-\Omega^{T})A_{-})_{(1)}\,.\end{split}$
(A.23)

The vanishing of (A.23) leads to the boundary conditions
$\Omega(\mathfrak{g}^{(0)})=\mathbb{1}$ unless
$A^{(0)}_{\tau}|_{\partial\Sigma}=0$ and $A^{(1)}_{+}|_{\partial\Sigma}=\Omega
A^{(1)}_{-}|_{\partial\Sigma}$ with
$\Omega^{2}(\mathfrak{g}^{(1)})=\mathbb{1}$. These are exactly the integrable
boundary conditions found in [27] for $\mathcal{W}=\mathbb{1}$.

##  Appendix B Derivation of the boundary terms for model (I)-(III)

In this section we will present the technical details for the derivation of
the boundary conditions for the integrable branes of models (I), (II) and
(III).

###  B.1 Model (I)

We start with the variation of the first term in (3.5). As before we will
concentrate ourselves only on boundary terms. For further simplification we
will omit the label $(k;\lambda_{1},\lambda_{2})$ and as before we will
define,

$\begin{split}&A^{R}_{1\mu}=\lambda_{1}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{1}D_{2})^{-1}(\partial_{\mu}g_{1}g_{1}^{-1}+\lambda_{2}D_{1}\partial_{\mu}g_{2}g_{2}^{-1})\,,\\\
&A^{L}_{1\mu}=-\lambda_{1}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{2}^{T}D_{1}^{T})^{-1}(g^{-1}_{2}\partial_{\mu}g_{2}+\lambda_{2}D_{2}^{T}g^{-1}_{1}\partial_{\mu}g_{1})\,,\\\
&A^{R}_{2\mu}=\lambda_{2}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{2}D_{1})^{-1}(\partial_{\mu}g_{2}g_{2}^{-1}+\lambda_{1}D_{2}\partial_{\mu}g_{1}g_{1}^{-1})\,,\\\
&A^{L}_{2\mu}=-\lambda_{2}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{1}^{T}D_{2}^{T})^{-1}(g^{-1}_{1}\partial_{\mu}g_{1}+\lambda_{1}D_{1}^{T}g^{-1}_{2}\partial_{\mu}g_{2})\,,\end{split}$
(B.1)

with $\mu=\tau,\sigma$. Doing so we find

$\begin{split}\int\delta
L|_{\partial\Sigma}&=\frac{k}{4\pi}\int_{\partial\Sigma}\sum_{i=1}^{2}{\rm
Tr}(g_{i}^{-1}\delta g_{i},g_{i}^{-1}\partial_{\sigma}g_{i})-{\rm Tr}(\delta
g_{i}g_{i}^{-1},A^{L}_{i\sigma})+{\rm Tr}(g_{i}\delta
g_{i},A^{R}_{i+1\sigma})\,.\end{split}$ (B.2)

Using (3.7) we find that arbitrary variations of the group elements on the
boundary can be written as

$\begin{split}&g_{i}^{-1}\delta g_{i}=(D_{i}^{T}-1)\delta
h_{i}h_{i}^{-1}\,,\\\ &\delta g_{i}g_{i}^{-1}=(1-D_{i})\delta
h_{i}h_{i}^{-1}\,,\quad i=1,2\,.\end{split}$ (B.3)

Substituting (B.3) in (B.2) we find

$\begin{split}\int\delta
L|_{\partial\Sigma}&=\frac{k}{4\pi}\int_{\partial\Sigma}\sum_{i=1}^{2}{\rm
Tr}(\delta
h_{i}h_{i}^{-1},\partial_{\sigma}g_{i}g_{i}^{-1}-g^{-1}_{i}\partial_{\sigma}g_{i}-(1-D_{i}^{T})A^{L}_{i\sigma}+(D_{i}-1)A^{R}_{i+1\sigma})\,.\end{split}$
(B.4)

Similarly, the variation of the boundary two-form (3.8) gives

$\begin{split}\int\delta\omega|_{\partial
D}&=-\frac{k}{4\pi}\int_{\partial\Sigma}\sum_{i=1}^{2}{\rm Tr}(\delta
h_{i}h_{i}^{-1},\partial_{\tau}g_{i}g_{i}^{-1}+g_{i}^{-1}\partial_{\tau}g_{i}+(D_{i}-1)A^{R}_{i+1\tau}+(1-D_{i}^{T})A^{L}_{i\tau})\,.\end{split}$
(B.5)

Combining (B.4) and (B.5) we find that the total boundary contribution reads

$\begin{split}\delta
S|_{\partial\Sigma}&=\frac{k}{2\pi}\int_{\partial\Sigma}\sum_{i=1}^{2}{\rm
Tr}(\delta
h_{i}h_{i}^{-1},\partial_{+}g_{i}g_{i}^{-1}+g^{-1}_{i}\partial_{-}g_{i}+(1-D_{i}^{T})A_{i-}+(D_{i}-1)A_{i+1+})\\\
&=\frac{k}{2\pi}\int_{\partial\Sigma}\sum_{i=1}^{2}{\rm Tr}(\delta
h_{i}h_{i}^{-1},\nabla_{+}g_{i}g_{i}^{-1}+g_{i}^{-1}\nabla_{-}g_{i}+(A_{i+}-A_{i+1+})+(A_{i-}-A_{i+1-}))\,.\end{split}$
(B.6)

To pass from the first to the second equality we used the definition of the
covariant derivative
$\nabla_{\mu}g_{i}=\partial_{\mu}g_{i}-A_{i\mu}g_{i}+g_{i}A_{i+1\mu}$ [31].

Turning our attention to the brane (3.13) we find that arbitrary variations of
the group elements on the boundary can be written as

$\begin{split}&g_{i}^{-1}\delta g_{i}=D_{i}^{T}\delta h_{i}h_{i}^{-1}-\delta
h_{i+1}h_{i+1}^{-1}\,,\\\ &\delta g_{i}g_{i}^{-1}=\delta
h_{i}h_{i}^{-1}-D_{i}\delta h_{i+1}h_{i+1}^{-1}\,,\quad i=1,2\,.\end{split}$
(B.7)

Following the same steps as before we find

$\begin{split}\int_{\Sigma}\delta
L&=\frac{k}{4\pi}\int_{\partial\Sigma}\sum_{i=1}^{2}{\rm Tr}(\delta
h_{i}h_{i}^{-1},\partial_{\sigma}g_{i}g_{i}^{-1}-A^{R}_{i\sigma}-D_{i}A^{R}_{i+1\sigma}-g_{i+1}^{-1}\partial_{\sigma}g_{i+1}+D_{i+1}^{T}A^{L}_{i+1\sigma}-A^{L}_{i\sigma})\,,\end{split}$
(B.8)

and

$\begin{split}\int_{D}\delta\omega&=-\frac{k}{4\pi}\int_{\partial\Sigma}\sum_{i=1}^{2}{\rm
Tr}(\delta
h_{i}h_{i}^{-1},\partial_{\tau}g_{i}g_{i}^{-1}-A^{R}_{i\tau}-D_{i}A^{R}_{i+1\tau}+g_{i+1}^{-1}\partial_{\tau}g_{i+1}-D_{i+1}^{T}A^{L}_{i+1\tau}+A^{L}_{i\tau})\,,\end{split}$
(B.9)

where the two-form is given in (3.19). Combining (B.8) and (B.9) we find that
the total boundary contribution in the variation of (3.5) is

$\displaystyle\delta S_{\partial\Sigma}$
$\displaystyle=\frac{k}{2\pi}\int_{\partial\Sigma}\sum_{i=1}^{2}{\rm
Tr}(\delta
h_{i}h_{i}^{-1},\partial_{+}g_{i}g_{i}^{-1}-A_{i+}+D_{i}A_{i+1+}+g_{i+1}^{-1}\partial_{-}g_{i+1}-D_{i+1}^{T}A_{i+1-}+A_{i-})$
$\displaystyle=\frac{k}{2\pi}\int_{\partial\Sigma}\sum_{i=1}^{2}{\rm
Tr}(\delta
h_{i}h_{i}^{-1},\nabla_{+}g_{i}g_{i}^{-1}+g_{i+1}^{-1}\nabla_{-}g_{i+1})\,.$
(B.10)

###  B.2 Model (II)

Since the boundary conditions (3.2) correspond to the brane (3.7), we skip the
technical details as they are similar to (B.2)-(B.6). We will show however
that the brane geometry (5.2) with the boundary two form (5.9) corresponds to
the boundary conditions (5.11).

Varying the first term in (3.20) we find

$\begin{split}\int\delta
L|_{\partial\Sigma}&=\sum_{i=1}^{2}\frac{k_{i}}{4\pi}\int_{\partial\Sigma}{\rm
Tr}(g_{i}^{-1}\delta g_{i},g_{i}^{-1}\partial_{\sigma}g_{i})-{\rm Tr}(\delta
g_{i}g_{i}^{-1},A^{L}_{i\sigma})+{\rm Tr}(g_{i}\delta
g_{i},A^{R}_{i+1\sigma})\,,\end{split}$ (B.11)

where

$\begin{split}&A^{R}_{1\mu}=\lambda_{1}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{1}D_{2})^{-1}(\lambda_{0}\partial_{\mu}g_{1}g_{1}^{-1}+\lambda_{2}D_{1}\partial_{\mu}g_{2}g_{2}^{-1})\,,\\\
&A^{L}_{1\mu}=-\lambda_{1}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{2}^{T}D_{1}^{T})^{-1}(\lambda_{0}^{-1}g^{-1}_{2}\partial_{\mu}g_{2}+\lambda_{2}D_{2}^{T}g^{-1}_{1}\partial_{\mu}g_{1})\,,\\\
&A^{R}_{2\mu}=\lambda_{2}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{2}D_{1})^{-1}(\lambda_{0}^{-1}\partial_{\mu}g_{2}g_{2}^{-1}+\lambda_{1}D_{2}\partial_{\mu}g_{1}g_{1}^{-1})\,,\\\
&A^{L}_{2\mu}=-\lambda_{2}(\mathbb{1}-\lambda_{1}\lambda_{2}D_{1}^{T}D_{2}^{T})^{-1}(\lambda_{0}g^{-1}_{1}\partial_{\mu}g_{1}+\lambda_{1}D_{1}^{T}g^{-1}_{2}\partial_{\mu}g_{2})\,.\end{split}$
(B.12)

Arbitrary variations of the group elements on the boundary can be written as
[17]

$\begin{split}&g_{1}^{-1}\delta
g_{1}|_{\partial\Sigma}=\frac{1-D_{1}^{T}}{1-D_{v}^{T}}(D^{T}_{v}\delta
h_{1}h_{1}^{-1}-\delta h_{2}h_{2}^{-1})\,,\\\ &g_{2}^{-1}\delta
g_{2}|_{\partial\Sigma}=\frac{1-D_{2}^{T}}{1-D_{v}}(D_{v}\delta
h_{2}h_{2}^{-1}-\delta h_{1}h_{1}^{-1})\,.\end{split}$ (B.13)

Substituting (B.13) in (B.11) we find

$\displaystyle\int\delta L|_{\partial\Sigma}$
$\displaystyle=\frac{1}{4\pi}\int_{\partial\Sigma}\sum_{i=1}^{2}{\rm
Tr}((D_{r_{i}}-1)^{-1}\delta
h_{i}h_{i}^{-1},k_{i}(\partial_{\sigma}g_{i}g_{i}^{-1}-g^{-1}_{i}\partial_{\sigma}g_{i}-(1-D_{i}^{T})A^{L}_{i\sigma}$
$\displaystyle+(D_{i}-1)A^{R}_{i+1\sigma})+k_{i+1}(\partial_{\sigma}g_{i+1}g_{i+1}^{-1}+g^{-1}_{i+1}\partial_{\sigma}g_{i+1}-(1-D_{i+1}^{T})A^{L}_{i+1\sigma}$
$\displaystyle+(D_{i+1}-1)A^{R}_{i\sigma}))\,.$ (B.14)

where we defined $r_{1}=r_{2}^{-1}=v$. The variation of the WZ boundary two-
form (5.3) has been computed in [17]. Thus the variation of (5.9) reads

$\displaystyle\int\delta\omega|_{\partial D}$
$\displaystyle=-\frac{1}{4\pi}\int_{\partial\Sigma}\sum_{i=1}^{2}{\rm
Tr}((D_{r_{i}}-1)^{-1}\delta
h_{i}h_{i}^{-1},k_{i}(\partial_{\tau}g_{i}g_{i}^{-1}+g^{-1}_{i}\partial_{\tau}g_{i}+(1-D_{i}^{T})A^{L}_{i\tau}$
$\displaystyle+(D_{i}-1)A^{R}_{i+1\tau})+k_{i+1}(\partial_{\tau}g_{i+1}g_{i+1}^{-1}+g^{-1}_{i+1}\partial_{\tau}g_{i+1}-(1-D_{i+1}^{T})A^{L}_{i+1\tau}$
$\displaystyle+(D_{i+1}-1)A^{R}_{i\tau}))\,.$ (B.15)

Combining (B.2) and (B.2) we find that the total contribution from the
boundary in the variation of (3.20) is

$\displaystyle\delta S_{\partial\Sigma}=\frac{1}{2\pi}\int_{\partial\Sigma}$
$\displaystyle\sum_{i=1}^{2}{\rm Tr}((D_{r_{i}}-1)^{-1}\delta
h_{i}h_{i}^{-1},k_{i}(\nabla_{+}g_{i}g_{i}^{-1}+g_{i}^{-1}\nabla_{-}g_{i}+A_{i+}-A_{i+1+}+$
$\displaystyle
A_{i-}-A_{i+1-})+k_{i+1}(\nabla_{+}g_{i+1}g_{i+1}^{-1}+g_{i+1}^{-1}\nabla_{-}g_{i+1}+A_{i+1+}-A_{i+}$
$\displaystyle+A_{i+1-}-A_{i-}))\,.$ (B.16)

###  B.3 Model (III)

Here we will skip the intermediate details and we will just present the
necessary results for the derivation of the boundary conditions (4.18), (5.7)

####  B.3.1 Boundary conditions (4.18)

$\begin{split}&\delta
g_{i}g_{i}^{-1}|_{\partial\Sigma}=(1-D_{i}D_{r_{3}}^{T})\delta
h_{i}h_{i}^{-1}+(D_{i}D_{r_{3}}^{T}-D_{i})\delta h_{3}h_{3}^{-1}\\\
&g_{i}^{-1}\delta g_{i}|_{\partial\Sigma}=(D^{T}_{i}--D_{r_{3}}^{T})\delta
h_{i}h_{i}^{-1}+(D_{r_{3}}^{T}-1)\delta h_{3}h_{3}^{-1}\,,\quad
i=1,2\,,\end{split}$ (B.17) $\displaystyle\int\delta L|_{\partial\Sigma}$
$\displaystyle=\sum_{i=1}^{2}\frac{k_{i}}{4\pi}\int_{\partial\Sigma}{\rm
Tr}(\delta
h_{i}h_{i}^{-1},\partial_{\sigma}g_{i}g_{i}^{-1}-D_{r_{3}}g_{i}\partial_{\sigma}g_{i}-(1-D_{r_{3}}D^{T}_{i})A^{L}_{i\sigma}+(D_{i}-D_{r_{3}})A^{R}_{i\sigma})$
$\displaystyle+\sum_{i=1}^{2}\frac{k_{i}}{4\pi}\int_{\partial\Sigma}{\rm
Tr}(\delta
h_{3}h_{3}^{-1},(D_{3}-1)g_{i}^{-1}\partial_{\sigma}g_{i}-(D_{r_{3}}D^{T}_{i}-1)A^{L}_{i\sigma}+(D_{r_{3}}-1)A^{R}_{i\sigma})\,,$
(B.18) $\displaystyle\int\delta\omega|_{\partial D}$
$\displaystyle=-\sum_{i=1}^{2}\frac{k_{i}}{4\pi}\int_{\partial\Sigma}{\rm
Tr}(\delta
h_{i}h_{i}^{-1},\partial_{\tau}g_{i}g_{i}^{-1}+D_{r_{3}}g_{i}\partial_{\tau}g_{i}+(1-D_{r_{3}}D^{T}_{i})A^{L}_{i\tau}+(D_{i}-D_{r_{3}})A^{R}_{i\tau}$
$\displaystyle-2\partial_{\tau}r_{3}r_{3}^{-1})-\sum_{i=1}^{2}\frac{k_{i}}{4\pi}\int_{\partial\Sigma}{\rm
Tr}(\delta
h_{3}h_{3}^{-1},(D_{3}-1)g_{i}^{-1}\partial_{\tau}g_{i}+(D_{r_{3}}D^{T}_{i}-1)A^{L}_{i\tau}$
$\displaystyle+(D_{r_{3}}-1)A^{R}_{i\tau}+2\partial_{\tau}r_{3}r_{3}^{-1})\,.$
(B.19)

####  B.3.2 Boundary conditions (5.7)

$\begin{split}&g_{1}^{-1}\delta
g_{1}|_{\partial\Sigma}=\frac{1-D_{1}^{T}}{1-D_{v_{1}}^{T}}(D^{T}_{v_{1}}\delta
h_{1}h_{1}^{-1}-\delta h_{2}h_{2}^{-1})\,,\\\ &g_{2}^{-1}\delta
g_{2}|_{\partial\Sigma}=\frac{1-D_{2}^{T}}{1-D_{v_{2}}^{T}}(D^{T}_{v_{2}}\delta
h_{2}h_{2}^{-1}-\delta h_{1}h_{1}^{-1})\,.\end{split}$ (B.20)
$\displaystyle\int\delta
L|_{\partial\Sigma}=\frac{1}{4\pi}\sum_{i=1}^{2}\int_{\partial\Sigma}$
$\displaystyle{\rm Tr}(\delta
h_{i}h_{i}^{-1},k_{i}(1-D^{T}_{v_{i}})^{-1}(\partial_{\sigma}g_{i}g_{i}^{-1}-g_{i}^{-1}\partial_{\sigma}g_{i}-(1-D_{i}^{T})A^{L}_{i\sigma}$
$\displaystyle+(D_{i}-1)A^{R}_{i\sigma})-k_{i+1}(D_{v_{i+1}}-1)^{-1}(\partial_{\sigma}g_{i+1}g_{i+1}^{-1}+g_{i+1}^{-1}\partial_{\sigma}g_{i+1}$
$\displaystyle-(1-D_{i+1}^{T})A^{L}_{i+1\sigma}+(D_{i+1}-1)A^{R}_{i+1\sigma}))\,.$
(B.21) $\displaystyle\int\delta\omega|_{\partial
D}=-\frac{1}{4\pi}\sum_{i=1}^{2}\int_{\partial\Sigma}$ $\displaystyle{\rm
Tr}(\delta
h_{i}h_{i}^{-1},k_{i}(1-D^{T}_{v_{i}})^{-1}(\partial_{\tau}g_{i}g_{i}^{-1}+g_{i}^{-1}\partial_{\tau}g_{i}+(1-D_{i}^{T})A^{L}_{i\tau}$
$\displaystyle+(D_{i}-1)A^{R}_{i\tau})-k_{i+1}(D_{v_{i+1}}-1)^{-1}(\partial_{\tau}g_{i+1}g_{i+1}^{-1}+g_{i+1}^{-1}\partial_{\tau}g_{i+1}$
$\displaystyle+(1-D_{i+1}^{T})A^{L}_{i+1\tau}+(D_{i+1}-1)A^{R}_{i+1\tau}))\,.$
(B.22) $\displaystyle\delta
S|_{\partial\Sigma}=-\frac{1}{4\pi}\sum_{i=1}^{2}\int_{\partial\Sigma}$
$\displaystyle{\rm Tr}(\delta
h_{i}h_{i}^{-1},k_{i}(1-D^{T}_{v_{i}})^{-1}(\partial_{+}g_{i}g_{i}^{-1}+g_{i}^{-1}\partial_{-}g_{i}+(1-D_{i}^{T})A_{i-}$
$\displaystyle+(D_{i}-1)A_{i+})-k_{i+1}(D_{v_{i+1}}-1)^{-1}(\partial_{+}g_{i+1}g_{i+1}^{-1}+g_{i+1}^{-1}\partial_{-}g_{i+1}$
$\displaystyle+(1-D_{i+1}^{T})A_{i+1-}+(D_{i+1}-1)A_{i+1+}))\,.$ (B.23)

## References

  * [1] H. Ishikawa, Boundary states in coset conformal field theories, Nucl. Phys. B629 (2002), 209-232  [arXiv:hep-th/0111230 [hep-th]].
  * [2] M. R. Gaberdiel and T. Gannon, Boundary states for WZW models, Nucl. Phys. B639 (2002), 471-501  [arXiv:hep-th/0202067 [hep-th]].
  * [3] G. Felder, J. Frohlich, J. Fuchs and C. Schweigert, The Geometry of WZW branes, J. Geom. Phys. 34 (2000), 162-190  [arXiv:hep-th/9909030 [hep-th]].
  * [4] S. Fredenhagen and V. Schomerus, D-branes in coset models, JHEP 02 (2002), 005  [arXiv:hep-th/0111189 [hep-th]].
  * [5] A. Y. Alekseev and V. Schomerus, D-branes in the WZW model, Phys. Rev. D60 (1999), 061901  [arXiv:hep-th/9812193 [hep-th]].
  * [6] T. Kubota, J. Rasmussen, M. A. Walton and J. G. Zhou, Maximally symmetric D-branes in gauged WZW models, Phys. Lett. B544 (2002), 192-198  [arXiv:hep-th/0112078 [hep-th]].
  * [7] S. Stanciu, D-branes in group manifolds, JHEP 01 (2000), 025  [arXiv:hep-th/9909163 [hep-th]].
  * [8] S. Stanciu, A Note on D-branes in group manifolds: Flux quantization and D0-charge, JHEP 10 (2000), 015  [arXiv:hep-th/0006145 [hep-th]].
  * [9] C. Bachas, M. R. Douglas and C. Schweigert, Flux stabilization of D-branes, JHEP 05 (2000), 048  [arXiv:hep-th/0003037 [hep-th]].
  * [10] S. Elitzur and G. Sarkissian, D branes on a gauged WZW model, Nucl. Phys. B625 (2002), 166-178  [arXiv:hep-th/0108142 [hep-th]].
  * [11] K. Gawedzki, Conformal field theory: A Case study,  [arXiv:hep-th/9904145 [hep-th]].
  * [12] S. Stanciu, An illustrated guide to D-branes in SU(3), [arXiv:hep-th/0111221 [hep-th]].
  * [13] K. Gawedzki, Boundary WZW, G / H, G / G and CS theories, Annales Henri Poincare 3 (2002), 847-881  [arXiv:hep-th/0108044 [hep-th]].
  * [14] A. Recknagel, Permutation branes, JHEP 04 (2003), 041  [arXiv:hep-th/0208119 [hep-th]].
  * [15] J. M. Figueroa-O’Farrill and S. Stanciu, D-branes in $AdS(3)\times S^{3}\times S^{3}\times S^{1}$, JHEP 04 (2000), 005  [arXiv:hep-th/0001199 [hep-th]].
  * [16] G. Sarkissian and M. Zamaklar, Symmetry breaking, permutation D-branes on group manifolds: Boundary states and geometric description, Nucl. Phys. B696 (2004), 66-106  [arXiv:hep-th/0312215 [hep-th]].
  * [17] S. Fredenhagen and T. Quella, Generalised permutation branes, JHEP 11 (2005), 004  [arXiv:hep-th/0509153 [hep-th]].
  * [18] S. Fredenhagen and C. Restuccia, DBI analysis of generalised permutation branes, JHEP 01 (2010), 065  [arXiv:0908.1049 [hep-th]].
  * [19] S. Fredenhagen and M. R. Gaberdiel, Generalised N=2 permutation branes, JHEP 11 (2006), 041  [arXiv:hep-th/0607095 [hep-th]].
  * [20] E. K. Sklyanin, Boundary Conditions for Integrable Quantum Systems, J. Phys. A21 (1988), 2375-2389
  * [21] G. W. Delius, Soliton preserving boundary condition in affine Toda field theories, Phys. Lett. B444 (1998), 217-223  [arXiv:hep-th/9809140 [hep-th]].
  * [22] A. Dekel and Y. Oz, Integrability of Green-Schwarz Sigma-Models with Boundaries, JHEP 08 (2011), [arXiv:1106.3446 [hep-th]].
  * [23] N. Mann and S. E. Vazquez, Classical Open String Integrability, JHEP 04 (2007), 065  [arXiv:hep-th/0612038 [hep-th]].
  * [24] K. Sfetsos, Integrable interpolations: From exact CFTs to non-Abelian T-duals, Nucl. Phys. B880 (2014), 225-246  [arXiv:1312.4560 [hep-th]].
  * [25] T. J. Hollowood, J. L. Miramontes and D. M. Schmidtt, Integrable Deformations of Strings on Symmetric Spaces, JHEP 11 (2014), 009  [arXiv:1407.2840 [hep-th]].
  * [26] S. Driezen, A. Sevrin and D. C. Thompson, D-branes in $\lambda$-deformations, JHEP 09 (2018), 015  [arXiv:1806.10712 [hep-th]].
  * [27] S. Driezen, A. Sevrin and D. C. Thompson, Integrable asymmetric $\lambda$-deformations, JHEP 04 (2019), 094  [arXiv:1902.04142 [hep-th]].
  * [28] G. Georgiou and K. Sfetsos, A new class of integrable deformations of CFTs, JHEP 03 (2017), 083  [arXiv:1612.05012 [hep-th]].
  * [29] G. Georgiou, E. Sagkrioti, K. Sfetsos and K. Siampos, Quantum aspects of doubly deformed CFTs, Nucl. Phys. B919 (2017), 504-522  [arXiv:1703.00462 [hep-th]].
  * [30] G. Georgiou, K. Sfetsos and K. Siampos, Double and cyclic $\lambda$-deformations and their canonical equivalents, Phys. Lett. B771 (2017), 576-582  [arXiv:1704.07834 [hep-th]].
  * [31] G. Georgiou and K. Sfetsos, Integrable flows between exact CFTs, JHEP 11 (2017), 078  [arXiv:1707.05149 [hep-th]].
  * [32] G. Georgiou, G. P. D. Pappas and K. Sfetsos, Asymmetric CFTs arising at the IR fixed points of RG flows, Nucl. Phys. B958 (2020), 115138  [arXiv:2005.02414 [hep-th]].
  * [33] K. Sfetsos and D. C. Thompson, “Spacetimes for $\lambda$-deformations,” JHEP 12 (2014), 164  [arXiv:1410.1886 [hep-th]].
  * [34] K. Sfetsos and K. Siampos, Integrable deformations of the $G_{k_{1}}\times G_{k_{2}}/G_{k_{1}+k_{2}}$ coset CFTs, Nucl. Phys. B927 (2018), 124-139  [arXiv:1710.02515 [hep-th]].
  * [35] S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici and G. Sarkissian, D-branes in the background of NS five-branes, JHEP 08 (2000), 046  [arXiv:hep-th/0005052 [hep-th]].
  * [36] K. Gawedzki, I. Todorov and P. Tran-Ngoc-Bich, Canonical quantization of the boundary Wess-Zumino-Witten model, Commun. Math. Phys. 248 (2004), 217-254  [arXiv:hep-th/0101170 [hep-th]].
  * [37] J. M. Figueroa-O’Farrill and S. Stanciu, More D-branes in the Nappi-Witten background, JHEP 01 (2000), 024  [arXiv:hep-th/9909164 [hep-th]].
  * [38] C. Klimcik and P. Severa, Poisson Lie T duality: Open strings and D-branes, Phys. Lett. B376 (1996), 82-89  [arXiv:hep-th/9512124 [hep-th]].
  * [39] B. Fraser, D-branes (or not) in the non-Abelian T-dual of the $SU(2)$ WZW model, Phys. Lett. B784 (2018), 307-311  [arXiv:1806.00713 [hep-th]].
  * [40] A. Wurtz, D-branes in the diagonal SU(2) coset, JHEP 01 (2006), 154  [arXiv:hep-th/0512126 [hep-th]].
  * [41] G. Georgiou and K. Sfetsos, Novel all loop actions of interacting CFTs: Construction, integrability and RG flows, Nucl. Phys. B937 (2018), 371-393  [arXiv:1809.03522 [hep-th]].
  * [42] G. Georgiou and K. Sfetsos, The most general $\lambda$-deformation of CFTs and integrability, JHEP 03 (2019), 094  [arXiv:1812.04033 [hep-th]].
  * [43] G. Georgiou, Webs of integrable theories, Nucl. Phys. B965 (2021), 115340  [arXiv:2006.12525 [hep-th]].
  * [44] G. Georgiou, Hamiltonian integrability of the webs of integrable theories, Nucl. Phys. B969 (2021), 115474  [arXiv:2104.04355 [hep-th]].
  * [45] K. Sfetsos and K. Siampos, $\lambda$-Deformations in the upper-half plane, Nucl. Phys. B968 (2021), 115451  [arXiv:2103.08650 [hep-th]].