Let us discuss the possible abelian groups $H$ for which $\calC_{GTVW}/H$ is isomorphic to $\calC_{GTVW}$. One necessary condition is that the orbifold model contains bosonic chiral and antichiral algebras isomorphic to $(\widehat{su}(2)_1)^{\oplus 6}$. We can separate the subgrops $H\subset G_{GTVW}$ into two classes, depending on whether they act trivially on the chiral and anti-chiral $\CA\times \bar \CA$ or not.

In the first case, $H$ must be a subgroup of (<ref>), i.e. the intersection of $G_{GTVW}$ and the centre $(\ZZ_2^6\times\ZZ_2^6)/Z_0$. Furthermore, $\frac{\hat{N}}{\sqrt{|H|}}$ must act by an automorphism on $(\widehat{su}(2)_1)^{\oplus 6}$. All such defects $N\in \cTop$ are discussed in appendix <ref>. The results are as follows. We find three kind of duality defects (up to conjugation in $G_{GTVW}$): $\CN_{ijk}$ for all $2\le i<j<k\le 6$, with $\CN_{ijk}^2=\CI+\CL_{t_it_j}+\CL_{t_jt_k}+\CL_{t_it_k}$; $\CN_{ij,kl}$ for all pairwise distinct $i,j,k,l\in\{2,\ldots,6\}$, with $\CN_{ij,kl}^2=\CI+\CL_{t_it_j}+\CL_{t_kt_l}+\CL_{t_it_jt_kt_l}$; and $\CN_{23456}$ where $\CN_{23456}^2$ equals to the superposition of the $16$ invertible defects in $G_{GTVW}\cap (\ZZ_2^6\times\ZZ_2^6)/Z_0\cong \ZZ_2^4$.

Let us now consider the case where $H$ acts non-trivially on the chiral algebra. In order for $\calC/H$ to be isomorphic to $\calC$, there must be holomorophic currents in some twisted sector $\CH_h$, for some $h\in H$.

By a direct calculation of the $h$-twisted partition function, we found that the group $G_{GTVW}$ admits only three conjugacy classes of elements $h$ such that $\Hh_h$ contains holomorphic currents. One class is given by symmmetries of the form $t_it_jt_kt_l$, $2\le i<j<k<l\le 6$, that are contained in the group (<ref>). Each of them can be interpreted as the quantum symmetry in a description of the model as a torus orbifold $T^4/\ZZ_2$. This means that the orbifold of the CFT $\calC_{GTVW}$ by $\ZZ_2$ group generated by any such symmetries is a torus model, and in particular cannot be isomorphic to $\calC_{GTVW}$ itself. However, there could be larger abelian groups containing symmetries of the form $t_it_jt_kt_l$, under which $\calC$ is self-orbifold. In fact, we already found some of these abelian groups among the ones acting trivially on currents.

The second class are given by symmetries $Q_4$ of order four, such as, for example
(            ;       )(1)^6
in the notation of appendix <ref>; the third class is related to the second by exchanging the action on the holomorphic and anti-holomorphic sectors. The elements $Q_4$ in either the second or the third conjugacy class are quantum symmetries in a description of the model $\calC_{GTVW}$ as a $\ZZ_4$ torus orbifold $T^4/\ZZ_4$.
As in the previous case, this means that the orbifold of $\calC$ by $\langle Q_4\rangle \cong \ZZ_4$ is a torus model (more precisely, the model described in section <ref>), and therefore is not isomorphic to $\calC_{GTVW}$. On the other hand, from the results in section <ref>, we know that there are duality defects $\CN_x\in \cTop_{GTVW}$ of order $4$ for groups of  the form $\ZZ_2\times \ZZ_8$ generated by $Q_4$ and by some other symmetry $\eta_x$ of order $8$. In principle, there could be even larger abelian groups with respect to which $\calC$ is self-orbifold, but we did not attempt a full classification.

§.§ D-branes and RR charges

Let us describe the lattice $\Gamma^{4,20}$ of RR charges of the K3 model $\calC_{GTVW}$, expressed in the orthonormal basis $\{|1,i\rangle,|2,i\rangle,|3,i\rangle,|4,i\rangle\}_{i=1,\ldots,6}$ of the space $V$ of R-R ground fields that is described in appendix <ref>. In particular, for each fixed $i=1,\ldots,6$, the subset $\{|1,i\rangle,|2,i\rangle,|3,i\rangle,|4,i\rangle\}$ corresponds to a `tetrad' of states, belonging to one of the six representations of $\CA\times\bar\CA$ listed in eq. (<ref>). The first tetrad ($i=1$) is given  by the spectral flow generators and spans the subspace $\Pi\subset V$, while $\Pi^\perp$ is spanned by the remaining tetrads with $i=2,\ldots,6$.

In order to find $\Gamma^{4,20}$ the most direct way would be to find an explicit description of a suitable set of $24$ boundary states. However, we will consider a simpler method, that uses the group of symmetries $G_{GTVW}$. The outcome of this analysis is that the only even unimodular lattice $\Gamma\subset V$ with signature $(4,20)$ that is invariant under the action of the $G_{GTVW}$ on $V$, as described in appendix <ref>, is the one spanned by the columns of the following matrix:
=3pt 0.8

8     4     4     4     4     4     4     2     4     4     4     2     4     2     2     2     4     2     2     2     2     0
0     1

0     4     0     0     0     0     0     2     0     0     0     2     0     2     0     0     0     0     0     2     0     0
0     1

0     0     4     0     0     0     0     2     0     0     0     2     0     0     2     0     0     2     0     0     0     0
0     1

0     0     0     4     0     0     0     2     0     0     0     2     0     0     0     2     0     0     2     0     0     0
0     1

0     0     0     0     4     0     0     2     0     0     0     0     0     2     2     2     0     2     2     2     2     0
0     1

0     0     0     0     0     4     0     2     0     0     0     0     0     2     0     0     0     0     2     0     0     0
0     1

0     0     0     0     0     0     4     2     0     0     0     0     0     0     2     0     0     0     0     2     0     0
0     1

0     0     0     0     0     0     0     2     0     0     0     0     0     0     0     2     0     2     0     0     0     0
0     1

0     0     0     0     0     0     0     0     4     0     0     2     0     2     2     2     0     2     2     2     2     2
2     1

0     0     0     0     0     0     0     0     0     4     0     2     0     2     0     0     0     2     0     0     0     2
0     1

0     0     0     0     0     0     0     0     0     0     4     2     0     0     2     0     0     0     2     0     0     0
2     1

0     0     0     0     0     0     0     0     0     0     0     2     0     0     0     2     0     0     0     2     0     0
0     1

0     0     0     0     0     0     0     0     0     0     0     0     4     2     2     2     0     0     0     0     2     2
2     1

0     0     0     0     0     0     0     0     0     0     0     0     0     2     0     0     0     0     0     0     0     2
0     1

0     0     0     0     0     0     0     0     0     0     0     0     0     0     2     0     0     0     0     0     0     0
2     1

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     2     0     0     0     0     0     0
0     1

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     4     2     2     2     2     2
2     1

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     2     0     0     0     2
0     1

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     2     0     0     0
2     1

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     2     0     0
0     1

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     2     2
2     1

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     2
0     1

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
2     1

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
0     1

This is a slight modification (necessary because of the different signature) of the basis of vectors of the Leech lattice in <cit.>.
It is easy to verify by a direct calculation that the lattice generated by these vectors is invariant under the group $G_{GTVW}$, and that it is even and unimodular with respect to the diagonal metric $\eta={\rm diag}(1,1,1,1,-1,\ldots,-1)$ of signature $(4,20)$.

The proof that the lattice with these properties is unique (up to $O(4)\times O(20)$ transformations that do not affect the splitting $V=\Pi\oplus\Pi^\perp$) is as follows.
It is known from <cit.> that the group of symmetries $G_{\calC}$ of any K3 model $\calC$ is isomorphic to a subgroup $G_\Lambda$ of the Conway group $O(\Lambda)\cong Co_0$, the group of automorphisms of the Leech lattice $\Lambda$. Furthermore, if $\Gamma^G\subset \Gamma^{4,20}$ and $\Lambda^G\subset \Lambda$ denote the sublattices of $G$-fixed vectors and $\Gamma_G:=\Gamma^{4,20}\cap (\Gamma^G)^\perp$ and $\Lambda_G:=\Lambda\cap (\Lambda^G)^\perp$ their orthogonal complement, then there is an isomorphism $\Gamma_G\cong \Lambda_G$ (reversing the sign of the quadratic form) that is compatible with the action of $G_\calC\cong G_\Lambda$.
The relevant subgroups of $Co_0$ were classified by Höhn and Mason in <cit.>, and $G_{GTVW}$ appears as group $99$ in their table. In particular, up to conjugation in $Co_0$, <cit.> shows that there is a unique sublattice $\Lambda_G\cong \Gamma_G$ for this group; this means that $\Gamma\cap \Pi^\perp\cong \Gamma_G$ is unique up to $O(20)$ transformations of $\Pi^\perp$.
As described, for example, in <cit.> (see also <cit.>), the genus of the primitive sublattice $\Gamma_G\subset \Gamma^{4,20}$ determines the genus of its orthogonal complement $\Gamma^G$ in the even unimodular lattice $\Gamma^{4,20}$.
We checked, with some computer aid, that such a genus contains a unique isomorphism class of lattices; this means that $\Gamma\cap \Pi\cong \Gamma^G$ is also uniquely determined up to $O(4)$ transformations. Therefore, for any choice of even unimodular lattice $\Gamma\subset V$ invariant under $G_{GTVW}$, there is a primitive embedding of $\Gamma^G\oplus \Gamma_G$ into $\Gamma$. Different choice of the lattice $\Gamma$ would lead to primitive embedddings that are not related by either $O(\Gamma^G)\times O(\Gamma_G)$ or $O(\Gamma)$ automorphisms (the latter can thought of as changes of basis in a given lattice $\Gamma)$. The possible equivalence classes of such primitive embeddings, up to $O(\Gamma^G)\times O(\Gamma_G)$ and $O(\Gamma)$ transformations, are described by proposition 2.1 of <cit.>, which is a reformulation of propositions 1.4.1 and 1.6.1 of <cit.>. In particular, the fact that for group $99$ in <cit.> the index $\bar{i}_G$ equals $1$, implies that there is a unique class of such embeddings, and this concludes the proof. We refer to <cit.> for more information about embeddings of primitive sublattices into even unimodular lattices, and in particular for the meaning of the index $\bar{i}_G$.

Notice that the first four vectors in the lattice basis
8/√(8)|1,1⟩ ,    4/√(8)(|1,1⟩+|2,1⟩) ,    4/√(8)(|1,1⟩+|3,1⟩) ,    4/√(8)(|1,1⟩+|4,1⟩) ,
are contained in the subspace $\Pi\subset V$ of spectral flow operators. By Claim (<ref>), this implies that all topological defects $\CL\in \cTop_{GTVW}$ in this model have integral quantum dimension.

As described in section <ref>, with each $\CL\in \cTop_{GTVW}$ is associated a lattice endomorphism $\nL:\Gamma^{4,20}\to\Gamma^{4,20}$, which is contained in the intersection $B^{4,20}_\Pi(\ZZ)={\rm End}(\Gamma^{4,20})\cap B^{4,20}(\RR)$ of ${\rm End}(\Gamma^{4,20})$ with the $401$-dimensional real space $B^{4,20}(\RR)$ of block diagonal matrices defined in eq.(<ref>).

Using some computer aid <cit.>, we found that for the model $\calC_{GTVW}$ the $\ZZ$-module $B^{4,20}_\Pi(\ZZ)$ has maximal rank, and computed a set of $401$ generators $\{\nL_i\}_{i=1,\ldots,401}$, so that any such map $\nL$ can be written as
=∑_i k_i _i ,    k_i∈ .
We stress that not all elements $\nL\in B^{4,20}_\Pi(\ZZ)$ are expected to correspond to actual topological defects in $\cTop_{GTVW}$. The $\ZZ$-module $B_{4,20}(\ZZ)$ contains a submodule,
generated by all maps $\nL_g$ induced by invertible defects $\CL_g$, $g\in G_{GTVW}$. Notice that, since the $24$-dimensional representation of $G_{GTVW}$ is faithful, $B^{4,20}_{\Pi,inv}(\ZZ)$ is isomorphic to the group ring $\ZZ[G_{GTVW}]$. We found that $B^{4,20}_{\Pi,inv}(\ZZ)$ has also maximal rank, has index $2^{168}$ in $B^{4,20}_{\Pi}(\ZZ)$, and that for all $\nL\in B^{4,20}_{\Pi}(\ZZ)$ one has $4\cdot \nL\in  B^{4,20}_{\Pi,inv}(\ZZ)$. In particular,
B^4,20_Π()/B^4,20_Π,inv()≅_4^8⊕_2^152 .

In fact, all the examples of topological defects $\CL\in \cTop_{GTVW}$ described in this section, including the simple non-invertible ones, correspond to elements $\nL$ in $B^{4,20}_{\Pi,inv}(\ZZ)$ (recall that the map $\CL\to \nL$ is not injective).  Unfortunately, we were not able to determine which elements (if any) $\nL\in B^{4,20}_{\Pi}(\ZZ)$, $\nL\notin B^{4,20}_{\Pi,inv}(\ZZ)$, are actually induced by topological defects in $\cTop_{GTVW}$.

It is interesting to consider the action on RR charges of the duality defects $\CN_{ijk}$, described in appendix <ref>. For example, the map $\mathsf{N}_{256}$ related to the defect $\CN_{256}$ is
=3pt 0.8𝖭_256=

2     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     2     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     2     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     2     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     1     -1     -1     -1     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     1     1     -1     1     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     1     1     1     -1     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     1     -1     1     1     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     1     1     1     1     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     -1     1     1     -1     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     -1     -1     1     1     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     -1     1     -1     1     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0

While $\mathsf{N}_{256}\in B^{4,20}_{\Pi,inv}(\ZZ)$, it cannot be written as a sum of the form $\nL_g+\nL_h$,  for some $g,h\in G_{GTVW}$: one always needs an integral linear combination of (at least) four $\nL_g$, $g\in G_{GTVW}$ with (at least) one negative coefficient. This can be proved, for example, by noticing that each $\nL_g$ acts by a signed permutation in this basis, so that each row or column in $\nL_g$ has always a single non-zero entry, equal to $\pm 1$. Thus, any sum $\nL_g+\nL_h$ has at most two non-zero entries in each row or column, so it cannot be equal to $\mathsf{N}_{256}$. This example shows that the image of $\cTop_{GTVW}$ in ${\rm End}(\Gamma^{4,20})$ by  $\CL\to \nL$ is strictly larger than the image of the subcategory generated only by invertible defects.

§ ANOTHER EXAMPLE: THE GEPNER MODEL $(1)^6$

The second example of K3 model we will analyse is the Gepner model $(1)^6$, that was considered in <cit.>. In general, Gepner models are obtained by considering products of $\CN=2$ minimal models, each one having central charge $c_k=\frac{3k}{k+2}$ with $k\in \NN$, and then taking an orbifold that projects on the states carrying integral total $U(1)$ R-charge. When the total central charge is $c=6$ and the spectrum is invariant under spectral flow, then the $\CN=(2,2)$ superconformal algebra is enhanced to $\CN=(4,4)$ and the CFT is always a non-linear sigma model on K3.

In particular, as the name suggests, the $(1)^6$ model $\calC_{(1)^6}$ is obtained by taking six copies of the $k=1$ minimal model with $c_{k=1}=1$. We refer to <cit.> for a description of this CFT as a Gepner model.
Here, we notice that this particular Gepner model admits different descriptions that might be more useful to study symmetries, defects and D-branes.

§.§ The $(1)^6$ Gepner model as a free scalar CFT

Let us describe the K3 model $\calC_{(1)^6}$ as a rational CFT. Both chiral and anti-chiral superalgebras $\CA$ and $\bar\CA$ contain[Notice that this is not  the full chiral algebra of this CFT.] the product of six copies of the $\CN=2$ superconformal algebra at $c=1$. It is known (see appendix <ref>) that the bosonic subalgebra of $\CN=2$ at $c=1$ can be completely described in terms of a chiral free boson on a circle of suitable radius. This means that the K3 model $\calC_{(1)^6}$ is generated by $6$ holomorphic and six antiholomorphic chiral free bosons

$$ i\partial X^k(z)\ ,\qquad\qquad i\bar \partial X^k(z)\ ,\qquad\qquad k=1,\ldots,6\ ,
together with the vertex operators $V_\lambda(z,\bar z)\sim :e^{i(\vec{\lambda}_L\cdot \vec{X}_L(z)+\vec{\lambda}_R\cdot \vec{X}_R(\bar z))}:$ of conformal weights $(h_L,h_R)=(\frac{\vec\lambda_L^2}{2},\frac{\vec\lambda_L^2}{2})$, where $\lambda\equiv (\vec{\lambda_L},\vec{\lambda}_R)$ takes values in a suitable integral (odd) lattice $\Upsilon\in \RR^{6,6}$. Let $\{e_1,\ldots,e_6,\tilde e_1,\ldots,\tilde e_6\}$ denote an orthonormal basis of $\RR^{6,6}$, so that $\lambda_L=\sum_i^6\lambda_ie_i$ and $\lambda_R=\sum_k^6 \tilde \lambda_k\tilde e_k$. The lattice $\Upsilon$ is given by the union $\Upsilon=\Upsilon^{NS-NS}\cup \Upsilon^{R-R}$ of a NS-NS and a R-R component.

The chiral and anti-chiral algebras $\CA$ and $\bar\CA$ are generated by the currents $i\partial X^k$ and $i\bar\partial X^k$, and by purely holomorphic (respectively, anti-holomorphic) vertex operators $V_{(\vec \lambda_L,0)}(z)$ (respectively, $V_{(0,\vec \lambda_R)}(\bar z)$)  with $\vec \lambda_L$ takes values in a suitable six dimensional positive definite lattice $\Lambda_\CA\subset\RR^6$, such that
(Υ^NS-NS∩^6,0)≅Λ_ ,    (Υ^NS-NS∩^0,6)≅Λ_(-1)  ,where $\Lambda_\CA(-1)$ denotes the lattices with opposite quadratic form. In particular, the $(\CN=2)^6\subset \CA$ algebra can be described as the lattice super vertex operator algebra (SVOA) associated with the odd lattice $(\sqrt{3}\ZZ)^6\equiv (\sqrt{3}\ZZ)\oplus \ldots \oplus (\sqrt{3}\ZZ)$ (six copies), so that

$$ (\sqrt{3}\ZZ)^6\subset \Lambda_\CA\ .

The $(\CN=2)^6$ algebra is not the full chiral algebra $\CA$ of the model $\calC_{(1)^6}$: one needs to extend this SVOA by two additional holomorphic currents $V_{\vec\pm (\lambda_L,0)}(z)$, corresponding to the  lattice vectors

$$ \pm \vec\lambda_L=\pm (\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})\subset \RR^6\ .$$

The full chiral and antichiral algebras $\CA$ and $\bar\CA$  of the theory are therefore both isomorphic to the lattice SVOA associated
with the integral odd lattice
Λ_ ={ n/√(3)(1,1,1,1,1,1)+√(3)(x_1,…,x_6)|n,x_1,…,x_6∈}⊂^6 .
The bosonic chiral algebra is the lattice VOA associated with the even lattice
Λ^bos_ ={ n/√(3)(1,1,1,1,1,1)+√(3)(x_1,…,x_6)|n,x_1,…,x_6∈, ∑_ix_i∈2}⊂Λ_ .
Therefore, the NS-NS lattice $\Upsilon^{NS-NS}$ contains $\Lambda\oplus \Lambda(-1)$ as a sublattice, corresponding to the chiral and anti-chiral algebra $\CA\times \bar\CA$.

The representations of the algebra $\CA\times \bar\CA$ are labeled by cosets $[v]:v+(\Lambda_{\CA}\oplus \Lambda_{\CA}(-1))$, for suitable $v\in \RR^{6,6}$ so that $\Upsilon^{NS-NS}$ and $\Upsilon^{R-R}$ decompose as unions of cosets
Υ^NS-NS=⋃_[v]∈A_(NS-NS)  v+(Λ_⊕Λ_(-1))
Υ^R-R=⋃_[v]∈A_(R-R)  v+(Λ_⊕Λ_(-))
In particular, the NS-NS sector contains the representations

$$ A_{(NS-NS)} =\Bigl\{\bigl[\tfrac{1}{\sqrt{3}}(a_1,\ldots,a_6;a_1,\ldots,a_6)\bigr]\ \mid\ a_i\in \ZZ/3\ZZ,\ ,\ \sum_{i=1}^6 a_i\in 3\ZZ\Bigr\}
while the R-R sector includes

$$ A_{(R-R)} =\Bigl\{\bigl[\tfrac{1}{\sqrt{3}}(a_1,\ldots,a_6;a_1,\ldots,a_6)\bigr]\ \mid\ a_i\in \frac{1}{2}+\ZZ/3\ZZ,\ ,\ \sum_{i=1}^6 a_i\in 3\ZZ\Bigr\}\ .
Notice that we have the identifications

$$ [v]\sim [v+\frac{1}{\sqrt{3}}(1,1,1,1,1,1;1,1,1,1,1,1)]
and the fusion rules

$$ [v]\times [v']=[v+v']

In this description, the $\CN=4$ superconformal algebra at $c=6$ is given as follows. The $\hat{su}(2)_1$ R-symmetry algebra is generated by
J^3(z)=1/2√(3)∑_k=1^6 i∂X^k(z) ,    J^±(z)=

while the supercurrents are given by
G^±(z)=√(2/3)∑_k=1^6 V_±√(3)e_k(z) ,    G̃^±(z)=√(2/3)∑_k=1^6 V_∓√(3)e_k±(1/√(3),…,1/√(3);0,…,0)(z)
The $24$ R-R ground states are given by $V_\lambda$ with the following vectors $\lambda\in \Upsilon^{R-R}$:

\begin{align*}
&\pm\frac{1}{2\sqrt{3}}(1,\ldots,1;1,\ldots,1)\qquad \pm\frac{1}{2\sqrt{3}}(1,\ldots,1;-1,\ldots,-1)\qquad 4\text{ spectral flow generators}\\
&\frac{1}{2\sqrt{3}}(1,1,1,-1,-1,-1;1,1,1,-1,-1,-1)\text{ and permutations}\qquad 20\text{ operators}\ .
\end{align*}

The symmetries of the model preserving the $\CN=(4,4)$ algebra form a group G_(1)^6≅_3^4⋊A_6 ,i.e. the extension by $\ZZ_3^4$ of the alternating group $A_6$ of even permutations of six objects <cit.>. Here, the normal subgroup $\ZZ_3^4$ is the group of CFT symmetries acting trivially on the chiral and anti-chiral algebras $\CA\times \bar\CA$. It is generated by elements $t_1^{m_1}\cdots t_6^{m_6}$ acting by
∏_k=1^6t_k^m_k(V_λ) =
e^2πi/3 ∑_k=1^6m_ka_kV_λ ,    λ∈1/√(3)(a_1,…,a_6;a_1,…,a_6)+(Λ_⊕Λ_(-1)) ,
where $m_1,\ldots,m_6\in \ZZ/3\ZZ$ satisfy the condition $\sum_{k=1}^6 m_k\equiv 0\mod 3$. Notice that, because of the condition $\sum_{k=1}^6 a_k\equiv 0\mod 3$, we have that the action is trivial whenever $m_1=\ldots=m_6$
t_1t_2t_3t_4t_5t_6=t_1^2t_2^2t_3^2t_4^2t_5^2t_6^2=1 .

The alternating group $A_6$ acts by even permutations $\sigma$ on the left- and right-moving $\hat{u}(1)$ currents, and transforms the vertex operators accordingly:

$$ i\partial X^k\mapsto i\partial X^{\sigma(k)}\ ,\qquad i\bar\partial X^k\mapsto i\bar\partial X^{\sigma(k)}\ ,\qquad V_{\sum_k(\lambda_ke_k+\tilde\lambda_k\tilde e_k)}\mapsto \pm V_{\sum_k(\lambda_ke_{\sigma(k)}+\tilde\lambda_k\tilde e_{\sigma(k)})}\ ,
where the sign in front of each $V_\lambda$ must be chosen in such a way that the OPE between vertex operators is preserved.

The full group of symmetries of the CFT is $[(SU(2)\times U(1)^5)\times (SU(2)\times U(1)^5)]\rtimes S_6$, where the two $SU(2)\times U(1)^5$ factors are generated by the zero modes of the holomorphic and anti-holomorphic currents. The permutation group $S_6$ acts simultaneously on each of the $U(1)^5$ factors as in the $5$-dimensional irreducible representation.

§.§ The $(1)^6$ model as a torus orbifold $T^4/\ZZ_3$

The $\calC_{(1)^6}$ Gepner model can also be described in terms of a torus orbifold $T^4/\ZZ_3$. In fact, there are actually many different ways to obtain this model from an orbifold.
One simple way to prove that $\calC_{(1)^6}$ is a torus orbifold is to notice that the symmetry group $\ZZ_3^4:A_6$ contains some symmetry $Q$ of order $3$ (e.g. the element of $\ZZ_3^4$ with $m_1=m_2=m_3=1$ and $m_4=m_5=m_6=0$) whose trace on the $24$ dimensional representation of RR ground states $V$ is $-3$. The orbifold of $\calC_{(1)^6}$ by $Q$ gives a sigma model $\CT:=\calC_{(1)^6}/\langle Q\rangle$ on $T^4$, as can be checked by verifying that the elliptic genus of $\calC_{(1)^6}/\langle Q\rangle$ is $0$ <cit.>. By reversibility of the orbifold procedure, because $\CT=\calC_{(1)^6}/\langle Q\rangle$ with $\langle Q\rangle\cong\ZZ_3$, we must conclude that the K3 model $\calC_{(1)^6}$ is an orbifold $\calC_{(1)^6}=\CT/\langle g\rangle$ of the $T^4$ model $\CT$ by a cyclic group $\langle g\rangle\cong \ZZ_3$, and $Q$ is the `quantum symmetry' acting trivially on the untwisted sector and multiplying by $e^{\frac{2\pi i k}{3}}$ the $g^k$-twisted sector.

Knowing the action of $Q$ on the states of $\calC_{(1)^6}$ allows us to identify the untwisted sector and the $g^k$ twisted sector in the bosonic description of the previous section:

$$ \text{untwisted sector: } x_1+x_2+x_3\equiv 0\mod 3\ ,$$

$$ g^k\text{twisted sector: } x_1+x_2+x_3\equiv k\mod 3\ .$$

It is easy to construct the sigma model $\CT$ on $T^4$ corresponding to the orbifold $\CT=\calC_{(1)^6}/\langle Q\rangle$.  We know that the symmetry $Q$ can be written as
V_λ↦e^2πi(δ_L·λ_L-δ_R·λ_R) V_λ=V_λwhere
is a vector such that $3\delta\in \Gamma$ and $\delta_L^2-\delta_R^2\in 2\ZZ$. The untwisted sector of $\calC_{(1)^6}/\langle Q\rangle$ contains the currents $i\partial X^i(z),i\bar\partial X^i(\bar z)$, as well as the vertex operators $V_\lambda$, $\lambda\in \Upsilon^{Q}$, where
Υ^Q={λ∈Υ|δ_L·λ_L-δ_R·λ_R∈} .
The full orbifold $\calC_{(1)^6}/\langle Q\rangle$ is generated by the $i\partial X^i(z),i\bar\partial X^i(\bar z)$ currents together with the vertex operators $V_\lambda$ with $\lambda$ taking values in the extended lattice
Υ'=Υ^Q∪(δ+Υ^Q)∪(2δ+Υ^Q) .
Let us consider the holomorphic fields of this orbifold. As expected for a supersymmetric sigma model on $T^4$, there are four holomorphic fields of weight $1/2$, namely

$$ \chi_1^{\pm}(z)\sim V_{\pm (\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},0,0,0;0,\ldots,0)}\qquad\qquad \chi_2^{\pm}(z)\sim V_{\pm (0,0,0,\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}};0,\ldots,0)}\ .
By taking normal ordered products of pairs of these four fermions, one obtains the $6$ currents of a `fermionic' $so(4)_1=su(2)_1\oplus su(2)_1$ algebra, where one of the $su(2)_1$ is the R-symmetry of the $\CN=4$ superconformal algebra with currents (<ref>), and the second commuting $su(2)_1$ algebra is generated by

$$ i\sum_{k=1}^3(\partial X^k-\partial X^{k+3})\ ,\qquad V_{\pm (\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}};0,\ldots,0)}(z)\ .
Besides the $so(4)_1$ currents that can be obtained from the OPE of the spin $1/2$ fields, the orbifold $\calC_{(1)^6}/\langle Q\rangle$ contains $16$ additional spin $1$ fields, that can be arranged into two commuting algebras

$$ \frac{i}{\sqrt{2}}(\partial X^1-\partial X^2)\ ,\qquad \frac{i}{\sqrt{2}}(\partial X^2-\partial X^3)

$$ V_{\pm (\frac{-2}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},0,0,0;0,\ldots,0)}(z)\,\quad V_{\pm (\frac{1}{\sqrt{3}},\frac{-2}{\sqrt{3}},\frac{1}{\sqrt{3}},0,0,0;0,\ldots,0)}(z)\ ,\quad
V_{\pm (\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{-2}{\sqrt{3}},0,0,0;0,\ldots,0)}(z)

$$ \frac{i}{\sqrt{2}}(\partial X^4-\partial X^5)\ ,\qquad \frac{i}{\sqrt{2}}(\partial X^5-\partial X^6)

$$ V_{\pm (0,0,0,\frac{-2}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}};0,\ldots,0)}(z)\,\quad V_{\pm (0,0,0,\frac{1}{\sqrt{3}},\frac{-2}{\sqrt{3}},\frac{1}{\sqrt{3}};0,\ldots,0)}(z)\ ,\quad
V_{\pm (0,0,0,\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{-2}{\sqrt{3}};0,\ldots,0)}(z)\ .
These two commuting algebras have dimension $8$ and rank $2$, and this is enough to conclude that they are both isomorphic to $su(3)_1$. Thus, the orbifold model $\calC_{(1)^6}/\langle Q\rangle$ has both a chiral and anti-chiral algebras isomorphic to $(su(3)_1)^2$, besides the $so(4)_1$ from the free fermions. For this reason, we denote this torus model as $\CT_{A_2^2}$, so that
_(1)^6/⟨Q⟩=_A_2^2 .
The $(su(3)_1)^2$ currents include the superconformal descendants $i\partial Z_i^\pm$ of the free fermions $\chi^\pm_i$. The scalar fields $Z_1^+,Z_2^+$ can be identified with the complex coordinates on the target torus $T^4$, with $Z_1^-,Z_2^-$ their complex conjugate. Notice that the currents $i\partial Z_i^\pm$ are linear combinations of the $i\partial X^k$ and of suitable vertex operators $V_\lambda$; this means that one cannot give the $X^k$ any geometric interpretation as the coordinates on the target torus.

As a CFT, any supersymmetric non-linear sigma model $\CT_{A_2^2}$ on $T^4$ can be described as the product $\CT_{A_2^2}=F(4)\otimes \bar{F}(4)\times \CT^{bos}_{A_2^2}$ of a theory $F(4)\otimes \bar{F}(4)$ of four chiral and four antichiral free fermions with central charges $(2,2)$, times the bosonic sigma model $\CT^{bos}_{A_2^2}$ on the same torus $T^4$, with central charges $(4,4)$. Let us focus on the bosonic sigma model factor, whose fields are characterised by having non-singular OPE with all the chiral and anti-chiral free fermions (i.e. they commute with all the free fermion modes). This is an honest bosonic CFT, with modular invariant partition function.  It contains the chiral algebra $(su(3)_1)^2$ – the free fermions have zero charge with respect to this algebra, so their OPE with the currents is non-singular. More generally, every $V_\lambda$ with $\lambda$ orthogonal to the vectors

$$ \pm (\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},0,0,0;0,\ldots,0), \qquad \pm (0,0,0,\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}};0,\ldots,0)
and their anti-holomorphic counterparts, belong to the bosonic sigma model.

As we show below, the bosonic sigma model $\CT^{bos}_{A_2^2}$ on $T^4$ is just the product $\CT^{bos}_{A_2^2}=\CT^{bos}_{A_2}\otimes \CT^{bos}_{A_2}$ of two isomorphic bosonic sigma models $\CT^{bos}_{A_2}$ on $T^2$, each one corresponding to the diagonal modular invariant for the algebra $su(3)_1$.
Recall that $su(3)_1$ admits three different modules, which we denote by $M_{\bf 1}$ (the vacuum), $M_{\bf 3}$ and $M_{\bar{\bf 3}}$; the latter modules are charge conjugate of each other, and their highest weight vector has weight $2/3$. Therefore, the spectrum of the diagonal modular invariant is

$$ (M_{\bf 1}\otimes \overline{M}_{\bf 1})\oplus (M_{\bf 3}\otimes \overline{M}_{\bf 3})\oplus (M_{\bar{\bf 3}}\otimes \overline{M}_{\bar{\bf 3}})\ .
More precisely, the first $\CT^{bos}_{A_2}$ is generated by the first (holomorphic and anti-holomorphic) $su(3)_1$ factor, by the $9$ non-holomorphic vertex operators of conformal weights $(2/3,2/3)$

$$ V_{(\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0,0;\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0,0)}\quad V_{(0,\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0;0,\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0)}\quad V_{(\frac{-1}{\sqrt{3}},0,\frac{1}{\sqrt{3}},0,0,0;\frac{-1}{\sqrt{3}},0,\frac{1}{\sqrt{3}},0,0,0)}

$$ V_{(\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0,0;\frac{-1}{\sqrt{3}},0,\frac{1}{\sqrt{3}},0,0,0)}\quad V_{(0,\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0;\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0,0)}\quad V_{(\frac{-1}{\sqrt{3}},0,\frac{1}{\sqrt{3}},0,0,0;0,\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0)}

$$ V_{(\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0,0;0,\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0)}\quad V_{(0,\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0;\frac{-1}{\sqrt{3}},0,\frac{1}{\sqrt{3}},0,0,0)}\quad V_{(\frac{-1}{\sqrt{3}},0,\frac{1}{\sqrt{3}},0,0,0;\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0,0)}
that are the ground states of the $M_{\bf 3}\otimes \overline{M}_{\bf 3}$ modules, and by the $9$ vertex operators $V_\lambda$ with the opposite signs for $\lambda$, namely

$$ V_{-(\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0,0;\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0,0)}\quad  V_{-(0,\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0;0,\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0)}\quad V_{-(\frac{-1}{\sqrt{3}},0,\frac{1}{\sqrt{3}},0,0,0;\frac{-1}{\sqrt{3}},0,\frac{1}{\sqrt{3}},0,0,0)}

$$ V_{-(\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0,0;\frac{-1}{\sqrt{3}},0,\frac{1}{\sqrt{3}},0,0,0)}\quad V_{-(0,\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0;\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0,0)}\quad V_{-(\frac{-1}{\sqrt{3}},0,\frac{1}{\sqrt{3}},0,0,0;0,\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0)}

$$ V_{-(\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0,0;0,\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0)}\quad V_{-(0,\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0;\frac{-1}{\sqrt{3}},0,\frac{1}{\sqrt{3}},0,0,0)}\quad V_{-(\frac{-1}{\sqrt{3}},0,\frac{1}{\sqrt{3}},0,0,0;\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},0,0,0,0)}
that are the ground states of the charge conjugate module $M_{\bar{\bf 3}}\otimes \overline{M}_{\bar{\bf 3}}$. The second $\CT^{bos}_{A_2}$ torus model is generated in a similar way, by exchanging the first free set of bosonic coordinates $X^1,X^2,X^3$ with the second set $X^4,X^5,X^6$.

The analysis in this section is sufficient to identify $\CT_{A_2^2}$ with the supersymmetric $T^4$ sigma model denoted by $A_2^2$ in section 4.4.4 of <cit.>. Geometrically, the target space is the product $T^2\times T^2$ of two tori $T^2=\CC/(\ZZ+e^{\frac{2\pi i}{3}}\ZZ)$ with a suitable B-field.

The K3 model $\calC_{(1)^6}$ can be obtained by the orbifold of $\CT_{A_2^2}$ by a symmetry $g$ of order $3$. More precisely, $g$ belongs the class denoted by 3A in <cit.>, and corresponds to a geometric rotation obtained by multiplying the complex coordinates $(z_1,z_2)$ parametrizing the first and second torus $T^2=\CC/(\ZZ+e^{\frac{2\pi i}{3}}\ZZ)$ by $(z_1,z_2)\mapsto (e^{\frac{2\pi i}{3}}z_1,e^{-\frac{2\pi i}{3}}z_2)$. Indeed, only for this class of symmetries the orbifold $\CT_{A_2^2}/\langle g\rangle$ contains $6$ R-R ground states in the untwisted sector and $9+9$ in the two twisted sectors, as expected for the model we are considering. The $g$-invariant untwisted sector contains a subalgebra $\widehat{su}(2)_1\oplus \widehat{u}(1)$ from the `fermionic' $\widehat{so}(4)_1$ algebra of $\CT_{A_2^2}$, as well as subalgebras $\widehat{u}(1)^2\subset \widehat{su}(3)_1$ for each of the factors in the bosonic $\widehat{su}(3)_1$. Altogether, we get $\widehat{su}(2)_1\oplus \widehat{u}(1)^5$, which is the current algebra of $\calC_{(1)^6}=\CT_{A_2^2}/\langle g\rangle$ – no further currents come from the twisted sectors.

The full symmetry group of the $\CT_{A_2^2}$ torus model preserving the small $\CN=(4,4)$ algebra and the spectral flow generators is $U(1)^8\rtimes G_0$ where $G_0$ is a group of order $36$ acting by automorphisms on the winding-momentum lattice.
The subgroup commuting with $g$ is $\langle g\rangle\times (\ZZ_3^4\rtimes \ZZ_6)$, where $\ZZ_3^4\subset U(1)^8$ and $\ZZ_6$ is generated $\calR:(z_1,z_2)\mapsto (-z_1,-z_2)$ and by a non-geometric symmetry of order $3$ that rotates only the holomorphic currents $(\partial Z_1,\partial Z_2)\mapsto (e^{\frac{2\pi i}{3}}\partial Z_1,e^{-\frac{2\pi i}{3}}\partial Z_2)$ while keeping the anti-holomorphic currents $(\bar\partial Z_1,\bar\partial Z_2)$ fixed. These torus symmetries induce a group of symmetries of the orbifold $\calC_{(1)^6}=\CT_{A_2^2}/\langle g\rangle$ isomorphic to $3^{1+4}\rtimes \ZZ_6$, obtained from $\langle g\rangle\times (\ZZ_3^4\rtimes \ZZ_6)$ by first quotienting out $\langle g\rangle$, and then taking a non-trivial central extension by the quantum symmetry $\langle Q\rangle\cong\ZZ_3$, as described in section <ref>.

§.§ Symmetries and topological defects

After describing in some detail the model $\calC_{(1)^6}$, we are now ready to discuss the topological defects in $\cTop_{(1)^6}$ that preserve the $\CN=(4,4)$ algebra and the spectral flow. Such a category includes:

* Invertible defects. There is one defect $\CL_g\in \cTop_{(1)^6}$ for each $g\in G_{(1)^6}\cong \ZZ_3^4\rtimes A_6$.

* Defects acting by automorphisms on the chiral algebra. The Verlinde lines preserving the full chiral and anti-chiral algebras $\CA\times \bar\CA$ must be in one-to-one correspondence with the representations of $\CA\times \bar\CA$ and with the same fusion rules. This implies that the Verlinde lines are all invertible, and generate the subgroup $\ZZ_2^4\subset G_{(1)^6}$.

It is more interesting to consider topological lines acting on $\CA\times \bar\CA$ by algebra automorphisms that fix the $\CN=(4,4)$ subalgebra, and do not lift to CFT symmetries. The analysis is very similar to the one in appendix <ref>, so we just summarize the main points. Recall that the full group of symmetries of the CFT is $((SU(2)\times U(1)^5)^2)\rtimes S_6$). Up to conjugation by such CFT symmetries, we can focus on defects that act on the chiral algebra $\CA$ by an outer automorphism $\rho_L$, while acting trivially on $\bar\CA$.
Outer automorphisms preserving the $\CN=4$ subalgebra correspond to even permutations in $A_6$; it is sufficient to consider a representative $\rho_L$ for each even conjugacy class in $S_6$, i.e. for each even cycle shape.
There are five possible cycle shapes, corresponding to the partitions $1+1+2+2$, $1+1+1+3$, $3+3$, $2+4$, $1+5$. Let $\CL_{(\rho_L,1)}$ denote one of these defects.
Then, the  $\CA\times \bar \CA$ representation labeled by $a_1,\ldots,a_6\in\ZZ/3\ZZ$ is annihilated by the operator $\hat\CL_{(\rho_L,1)}$ unless it satisfies  (a_ρ_L(1),…,a_ρ_L(6)) = (a_1,…,a_6)+n(1,…,1) ,for some $n\in \ZZ/3\ZZ$. For a given $(\rho_L,1)$, the number of simple defects $\CL_{(\rho_L,1)}$ equals the number of representations satisfying (<ref>), and can be obtained from each other by conjugation $\CL_g \CL_{(\rho_L,1)}\CL_g^\dual$ by invertible defects $\CL_g$ with $g\in \ZZ_3^4\subset G_{(1)^6}$.
Let us also notice that for every invertible defect $\CL_g$ with $g\in \ZZ_3^4\subset G_{(1)^6}$ and acting trivially on all representations satisfying (<ref>), one has $\CL_g\CL_{(\rho_L,1)}=\CL_{(\rho_L,1)}\CL_g=\CL_{(\rho_L,1)}$.

For all $\CL_{(\rho_L,1)}$ we can find an invertible $\CL_g$ such that $\CN=\CL_g\CL_\rho$ is unoriented – this implies that $\CN$ acts on the chiral and antichiral algebras by  involutions $(\rho'_L,\rho'_R)$,
ρ'_L^2=1=ρ'_R^2 ,    ρ'_Lρ'_R=ρ_L .Furthermore, because $\CN^2$ acts trivially on the chiral and antichiral algebra, it must be a superposition of Verlinde lines, i.e. invertible defects in $\ZZ_3^4\subset G_{(1)^6}$. This implies that all the $\CN$ obtained in this way are duality defects for some subgroup $H\subseteq \ZZ_3^4$
^2=∑_h∈H _h ,
of order $|H|=\langle \CN\rangle ^2$. In the following table, for each partition of $\{1,\ldots,6\}$, we report the cycle shape of a generic automorphism $\rho_L\in A_6$, the dimension $\langle \CN\rangle$ of the duality defect $\CN$ acting on $\CA\times\bar\CA$ by automorphisms $(\rho'_L,\rho'_R)$ as in (<ref>), the group $H\subseteq \ZZ_3^4$ appearing in (<ref>), and the generators of $H$ in the form (<ref>):

Partition     $\rho_L$       $\langle \CN\rangle$     $H$      Generators of $H$

$1+1+2+2$       $(ij)(kl)$     $3$     $\ZZ_3\times \ZZ_3$     $t_it_j^{-1},\quad  t_kt_l^{-1}$

$1+1+1+3$       $(ijk)$     $3$     $\ZZ_3\times \ZZ_3$     $t_it_j^{-1},\quad  t_it_jt_k$

$3+3$       $(ijk)(lmn)$     $3$     $\ZZ_3\times \ZZ_3$     $t_it_j^{-1}t_l^{-1}t_m,\quad  t_it_jt_k$

$2+4$       $(ij)(klmn)$     $9$     $\ZZ_3^4$     All $\prod_k t_k^{m_k}$

$1+5$       $(ijklm)$     $9$     $\ZZ_3^4$     All $\prod_k t_k^{m_k}$

This analysis implies that the K3 model $\calC_{(1)^6}$ is self-orbifold with respect to the groups $H\subseteq \ZZ_3^4$ in this table.

* Defects induced by the description as a torus orbifold $T^4/\ZZ_3$. We have seen in section <ref> that $\calC_{(1)^6}=\CT_{A_2^2}/\langle g\rangle$ for a suitable torus symmetry $g$ of order $3$ preserving the small $\CN=(4,4)$ superconformal algebra. From the general analysis of section <ref>, this implies $\cTop_{(1)^6}$ contains a continuum of topological defects $T_\theta$ of quantum dimension $3$, that are induced by superpositions $W_\theta+W_{g(\theta)}+W_{g^2(\theta)}$ of torus model defects, and that are simple for generic $\theta\in \Gamma^{4,4}/(\Gamma^{4,4}\otimes \RR)$. When $x\in \Gamma^{4,4}/(\Gamma^{4,4}\otimes \RR)$ is a $g$-fixed point, i.e. $g(x)=x\mod \Gamma^{4,4}$, the defect $T_x$ decomposes as a superposition $\eta_x+Q\eta_x+Q^2\eta_x$, where $Q$ is a quantum symmetry of order $3$ and $\eta_x$ are invertible defects. The invertible defects $\eta_x$ obtained in this way, together with $Q$, generate an extraspecial group $3^{1+4}$, that must be a subgroup of $G_{(1)^6}$. Indeed, the quantum symmetry $Q$ can be identified, for example, with a symmetry $t_1t_2t_3=t_4t_5t_6$ acting as in (<ref>), and  the subgroup of $G_{(1)^6}$ commuting with $Q$ is isomorphic to $3^{1+4}\rtimes \ZZ_6$. The generators of the latter group are induced by the symmetries of the torus model $\CT$ that commute with $g$. The group $3^{1+4}\rtimes \ZZ_6$ contains the subgroup $\ZZ_3^4$ fixing the chiral and anti-chiral algebra $\CA\times \bar\CA$ – indeed, the index of $3^{1+4}\rtimes \ZZ_6$ in $G_{(1)^6}$ is $20$, which is not divisible by $3$. However, none of the elements $\eta_x$ are contained in this $\ZZ_3^4$ group, because they all act non-trivially on some holomorphic or antiholomorphic operator.

* Duality defects. We do not attempt a classification of all abelian groups $H\subset G_{(1)^6}$ with respect to which $\calC_{(1)^6}$ is self-orbifold, i.e $\calC/\langle H\rangle\cong \calC_{(1)^6}$, but just mention a few examples. When $H$ leaves invariant the full chiral and anti-chiral algebra, then the corresponding duality defect $\CN$ must act on $\CA\times\bar\CA$ by automorphisms. All cases where $\CN$ preserves the $\CN=(4,4)$ algebra were considered in point 2 above. As discussed in <ref>, the continuum of defects induced by the torus orbifold construction contains also the duality defects $\CN$ for the order $9$ groups $H$ generated by $\eta_x$ and $Q$. Notice that the element $\eta_x$, and therefore the group $H$, does not act trivially on $\CA\times \bar\CA$, so that $\CN$ is not one of the defects considered in point 2 above.

Let us describe the lattice $\Gamma^{4,20}$ of R-R charges in this model. A set of boundary states generating the full lattice was determined in <cit.>, using the Gepner model description. The same lattice can be also obtained in a way similar as section <ref>, using arguments based purely on the symmetry group $G_{(1)^6}$ of the model and on lattice theory. In particular, $G_{(1)^6}$ correspond to group $101$ in the list of <cit.>, and their results imply that there is a unique point in the moduli space of K3 models with this symmetry group. It is easier to describe $\Gamma^{4,20}$ as a complex lattice in $\CC^{2,10}$, the complex vector space with indefinite sesquilinear form
$$\langle z,w\rangle=\bar z_1w_1+\bar z_2w_2-\bar z_3w_3-\ldots-\bar z_{12}w_{12}\ ,$$
of signature $(2,10)$, where $z\equiv (z_1,\ldots z_{12})\in \CC^{2,10}$ and $w\equiv (w_1,\ldots w_{12})\in \CC^{2,10}$.  Using the notation $\omega:=e^{\frac{2\pi i}{3}}$ and $\theta=\omega-\bar\omega=i\sqrt{3}$, we find that $\Gamma^{4,20}\subset \CC^{2,10}$ is generated by the rows of the following matrix
=3pt 0.8

3     0     0     0     0     0     0     0     0     0     0     0

3 ω    0     0     0     0     0     0     0     0     0     0     0

0     3 θ    0     0     0     0     0     0     0     0     0     0

0     3 θω    0     0     0     0     0     0     0     0     0     0

0     -3     -3     0     0     0     0     0     0     0     0     0

0     3 ω̅    3 ω̅    0     0     0     0     0     0     0     0     0

0     -3     0     -3     0     0     0     0     0     0     0     0

0     3 ω̅    0     3 ω̅    0     0     0     0     0     0     0     0

0     -3     0     0     -3     0     0     0     0     0     0     0

0     3 ω̅    0     0     3 ω̅    0     0     0     0     0     0     0

0     -3     0     0     0     -3     0     0     0     0     0     0

0     3 ω̅    0     0     0     3 ω̅    0     0     0     0     0     0

θω    -θω    -θω    -θω    θω    -θω    θω    0
0     0     0     0

-θω    -θω    θω    -θω    -θω    -θω    0     θω
0     0     0     0

-θω    0     0     θω    -θω
-θω    0     0     θω    0     0     0

0     -θω    θω    0     θω
-θω    0     0     0     θω    0     0

θω    0     -θω    θω    0
-θω    0     0     0     0     θω    0

0     θω    θω    θω    θω    0     0     0     0     0     0     θω

θ    -θ    -θ    -θ    θ    -θ
θ    0     0     0     0     0

-θ    -θ    θ    -θ    -θ    -θ    0
θ    0     0     0     0

-θ    0     0     θ    -θ    -θ    0     0     θ    0
0     0

0     -θ    θ    0     θ    -θ    0     0     0     θ    0

0     θ    θ    θ    θ    0     0     0     0     0     0

0     2 ω-ω̅    1     1     1     1     1     1     1     1     1     4

The image of this complex lattice standard map

$$ \CC^{2,10}\in (z_1,\ldots,z_{12}) \mapsto (\Re(z_1),\Im(z_1),\ldots, \Re(z_{12}),\Im(z_{12}))\in \RR^{4,20},
defines a real even unimodular lattice of signature $(4,20)$, as can be checked by a direct computation. The construction is analogous to the definition of the complex Leech lattice (see for example chapter 7 section 8 in <cit.>), with some modifications due to the different signature. The first two columns in (<ref>) correspond to complex coordinates $z_1,z_2$ for the space $\Pi\subset V$ of spectral flow generators, while the remaining $10$ columns are complex coordinates $z_3,\ldots,z_{12}$ for its orthogonal complement. Notice that first four vectors (rows) in the lattice basis, namely

\begin{array}{cccccccccccc}
(3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0) \\
(3 \omega  & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0) \\
(0 & 3 \theta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0) \\
(0 & 3 \theta \omega  & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0)\\
\end{array}

are contained in the subspace $\Pi$. By claim <ref>, the existence of such charges imply that all defects $\CL\in \cTop_{(1)^6}$ have integral quantum dimension.

The coordinates $z_3,\ldots,z_{12}$ are a complex basis for the space $V\cap \Pi^\perp$ of R-R ground states in the $(\frac{1}{4},0;\frac{1}{4},0)$ representation on $\CN=(4,4)$. In particular, the basis consists of the $20$ highest weight vectors for the chiral and anti-chiral algebra $\CA\times \bar\CA$ labeled by

$$ [a_1,\ldots, a_{6}]=[+\frac{1}{2},+\frac{1}{2},+\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}]\qquad \text{and permutations.}
Let us denote by $|i,j,k\rangle$, $1\le i<j<k\le 6$, the highest weight vector with the $+$ signs in positions $i,j,k$, i.e. in the $\CA\times\bar\CA$ representation $[a_1,\ldots,a_6]$ with

$$ a_n=\begin{cases}
+1/2 & \text{if } n\in \{i,j,k\}\\
-1/2 & \text{if } n\notin \{i,j,k\}\ .
\end{cases}
This is a complex basis, since the CPT conjugate of $[a_1,\ldots, a_6]$ is the opposite $[-a_1,\ldots,-a_6]$. We identify the states $|i,j,k\rangle$ with the complex coordinates $z_3,z_3^*,\ldots,z_{12},z_{12}^*$ as follows:

z_3→|1,2,3⟩    z_3^*→|4,5,6⟩           z_8→|1,3,4⟩    z_8^*→|2,5,6⟩

z_4→|1,2,4⟩    z_4^*→|3,5,6⟩           z_9→|1,5,6⟩    z_9^*→|2,3,4⟩

z_5→|1,2,5⟩    z_5^*→|3,4,6⟩           z_10→|1,3,5⟩    z_10^*→|2,4,6⟩

z_6→|1,4,6⟩    z_6^*→|2,3,5⟩           z_11→|1,3,6⟩    z_11^*→|2,4,5⟩

z_7→|1,4,5⟩    z_7^*→|2,3,6⟩           z_12→|1,2,6⟩    z_12^*→|3,4,5⟩

The group of invertible defect in $\cTop_{(1)^6}$ is $G_{(1)^6}\cong \ZZ_3^4\rtimes A_6$. All such symmetries act trivially on the spectral flow operators, and therefore on the complex coordinates $z_1,z_2$. A permutation $\sigma\in A_6$ acts in the obvious way on the states $|i,j,k\rangle$
|i,j,k⟩↦|σ(i),σ(j),σ(k)⟩ .
Using the correspondence (<ref>), it is immediate to derive the action of $\sigma$ on the coordinates $z_3,z_3^*,\ldots,z_{12},z_{12}^*$.

The elements of the subgroup $\ZZ_3^4$ are labeled by $(m_1,\ldots,m_6)\in (\ZZ/3\ZZ)^6$, with the condition $\sum_i m_i\equiv 0\mod 3$ and modulo $(1,1,1,1,1,1)$. They multiply the states $|i,j,k\rangle$, and therefore complex coordinates $z_3,\ldots, z_{12}$, by some cubic roots of unity, determined by
|i,j,k⟩↦exp(2πi/3 [∑_n∈{i,j,k} m_n - ∑_n∉{i,j,k} m_n])|i,j,k⟩ .
It is easy to check that these transformations of the space $\CC^{2,10}$ correspond to automorphisms of the lattice spanned by the rows of <ref>.

As for the model $\calC_{GTVW}$, we computed a set of generators for the intersection

$$ B^{4,20}_{(1)^6}(\ZZ):= {\rm End}(\Gamma^{4,20})\cap B^{4,20}(\RR)\ ,
between the $\ZZ$-linear endomorphisms of the lattice $\Gamma^{4,20}$ and the $401$-dimensional real space $B^{4,20}(\RR)$ of block diagonal $24\times 24$ matrices, with a upper left $4\times 4$  block proportional to the identity, and an unconstrained lower right $20\times 20$ block, see (<ref>). As in the previous example, the $\ZZ$-module $B^{4,20}_{(1)^6}(\ZZ)$ has maximal rank $401$, i.e. equal to the dimension of $B^{4,20}(\RR)$.

We also computed the submodule

$$ B^{4,20}_{(1)^6,inv}(\ZZ)\subset B^{4,20}_{(1)^6}(\ZZ)
generated by the $\nL$ where $\CL$ is a superposition of invertible defects in $\cTop_{(1)^6}$. As in the previous example, we find that the submodule $B^{4,20}_{(1)^6,inv}(\ZZ)$ has full rank $401$, with quotient
B^4,20_(1)^6()/B^4,20_(1)^6,inv()≅_3^104 .

For all the defects $\CL\in \cTop_{(1)^6}$ that are discussed above, the corresponding map $\nL$ is contained in $B^{4,20}_{(1)^6}(\ZZ)$, i.e. it can be written as an integral linear combination $\nL=\sum_g n_g \nL_g$ of maps $\nL_g$ related to invertible defects $\CL_g$, $g\in G_{(1)^6}$, for some $n_g\in \ZZ$. However, some $\nL$ cannot be written as linear combinations $\sum_g n_g \nL_g$ with non-negative integral coefficients $n_g\in \ZZ_{\ge 0}$. An example is given by the duality defect $\CN_{(12)(34)}$ considered in point $2$ above, that acts on the chiral algebra $\CA$ by a permutation with cycle shape $(12)(34)$, while preserving the anti-chiral algebra $\bar\CA$. The square of $\CN_{(12)(34)}$ is a superposition of $9$ invertible defects in the $\ZZ_3\times \ZZ_3$ group generated by $t_1t_2^2$ and $t_3t_4^2$
_(12)(34)^2=(+_t_1t_2^2+_t_1^2t_2)(+_t_3t_4^2+_t_3^2t_4) .
This determines (up to a sign) the action of $\CN_{(12)(34)}$ on the basis (<ref>) of RR ground states states in the $(\frac{1}{4},0;\frac{1}{4},0)$ representation of $\CN=(4,4)$, and therefore on the space $\CC^{2,10}$ with coordinates $z_1,\ldots,z_{12}$. In particular, the action is via the diagonal matrix
=3pt 0.8

3     0     0     0     0     0     0     0     0     0     0     0

0     3     0     0     0     0     0     0     0     0     0     0

0     0     -3     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0

0     0     0     0     0     0     0     0     0     0     0     0
where the minus sign is fixed by the requirement that this linear map induces an endomorphisms of the lattice of D-brane charges. On the other hand, all invertible defects $\CL_g$ act on the basis (<ref>) by multiplication by some cubic root of unity followed by a permutation. This means that in the corresponding matrix $\nL_g$ the sum of the entries in each row or column is a cubic root of unity. Because it is impossible to obtain $-3$ as a sum over three cubic roots of unity, it follows that $\mathsf{N}_{(12)(34)}$ cannot be written as a sum of three matrices $\nL_{g_1}+\nL_{g_2}+\nL_{g_3}$ for any $g_1,g_2,g_3\in G_{(1)^6}$.

§ CONCLUSIONS

In this article, we discuss some general properties of the categories $\cTop_{\calC}$ of topological defects preserving the $\CN=(4,4)$ superconformal algebra and the spectral flow in a supersymmetric non-linear sigma model on K3 $\calC$.  In particular, we focus on the fusion of such topological defects with boundary states corresponding to BPS D-branes, and define a homomorphism from the fusion ring of $\cTop$ to the ring of $\ZZ$-linear endomorphisms on the lattice of D-brane charges. This construction, together with some standard assumptions about the moduli space of K3 models, allows us to derive a number of general properties of the topological defects in $\cTop_\calC$. For example, we show that the set of K3 models where $\cTop$ is not trivial (i.e., where there are simple defects distinct from the identity) has zero measure in the moduli space. Furthermore, we provide some restrictions on the possible quantum dimensions of the defects in $\cTop$ and a sufficient condition on the model $\calC$ (or rather on the corresponding point in the moduli space) for all quantum dimensions to be integral. Finally, we apply our methods in a couple of well understood examples of K3 models.

There are many directions of investigation that would be interesting to pursue:

* One of the main tools in our analysis is the map $\cTop \to {\rm End}(\Gamma^{4,20})$ that associates with each topological defect $\CL$ a $\ZZ$-linear endomorphism $\nL$ of the lattice $\Gamma^{4,20}$ of RR charges of the model. In section <ref> we provided some conditions that must be satisfied by the endomorphisms $\nL$ that are associated with some defect. It is clear, though, that in general these conditions are not sharp enough to identify the image of the $\cTop \to {\rm End}(\Gamma^{4,20})$. Can we refine them?

* Similarly, we know that the map $\cTop \to {\rm End}(\Gamma^{4,20})$ is in general not injective. In fact, in the two K3 models $\calC$ that we studied, we found some families of defects $\CL_\theta\in \cTop_{\calC}$ depending on real parameters $\theta$, corresponding to the same $\nL\in {\rm End}(\Gamma^{4,20})$. On the other hand, there are some cases (for example, when $\nL$ is a multiple of the identity) where we were able to prove that $\nL$ is the image of a single $\CL$. Can we determine for which $\nL$ the corresponding defect $\CL$ is unique?

Let us give some more insight into this question. It is known that each topological defect $\CL$ in the CFT $\calC$ can be described as a boundary state in the double theory $\calC\times \bar \calC$, see section <ref>. In this description, it is natural to identify $\nL\in {\rm End}(\Gamma^{4,20})$ with the R-R charge $\nL\in \Gamma^{4,20}\otimes (\Gamma^{4,20})^*$ of the boundary state in the double theory. Thus, a continuum of defects for $\nL$ in the model $\calC$ corresponds to a moduli space of D-branes with a given charge $\nL$ in the double theory. In particular, the twisted conserved currents implementing the deformations of the topological defect (see the discussion in section <ref> and references therein) should correspond to suitable massless modes for open strings with both ends on the D-brane in the double theory. It is natural to wonder whether there can be examples where there is a discrete set of defects $\CL\in \cTop$ with the same image $\nL$ via the map $\cTop\to {\rm End}(\Gamma^{4,20})$. They would correspond to a discrete set of different BPS D-branes carrying the same R-R charge in some non-linear sigma model on $K3\times K3$.

As a second observation, we notice that a given topological defect $\CL$ is preserved by a marginal deformation of the K3 model if and only if the defect is `transparent' for the corresponding exactly marginal operator. This property is completely encoded in the form of $\nL$. As a consequence, all defects $\CL$ with the same $\nL$ are either all lifted or all preserved by a certain deformation. Thus, for each $\nL$, one can identify some connected families of K3 models $\calC$ in the moduli space, with the property that $\cTop_{\calC}$ contains a set of defects $\CL$ with action $\nL$ on the D-brane charges. These arguments suggest that the cardinality of this set of defects might only depend on the connected family, and not on the particular K3 model.

* In section <ref> we conjectured that $\cTop_{\calC}$ contains a continuum of defects only if $\calC$ is a (possibly generalised) orbifold of a torus model. It would be very interesting to either prove this conjecture or to find a counterexample. Notice that both models studied in our article are actually torus models. In this respect, it would be very useful to study an example of K3 model that is not a torus model. Unfortunately, while torus orbifolds are necessarily a zero measure set in the moduli space of K3 models (they are a discrete union of subloci of dimension at most $16$), finding such an example of K3 model that is reasonably under control (e.g. it is rational) is not an easy task.  In particular, we do not know of any simple criterion to determine whether a given K3 model admits or not a description as a generalised orbifold of a torus model.

* Given a symmetry $g$ of a K3 model $\calC$, one can define a twining genus $\phi_g(\tau,z)$ by computing the elliptic genus of the model $\calC$ with the insertion of the topological defect $\CL_g$ along one of the cycles of the torus. In the same spirit, one can define a twining genus $\phi_\CL(\tau,z)$ for any topological defect $\CL\in \cTop_{\calC}$. On general grounds, one expects such functions to be holomorphic and modular, in a suitable sense.[More precisely, we expect them to be weak Jacobi forms of weight $0$ and index $1$ with respect to a suitable subgroup of $SL(2,\ZZ)$.] Twining genera play a prominent role in the calculation of the microstates degeneracy for 1/4-BPS black hole in string compactifications on K3 and on orbifolds thereof (CHL models). See for example
They are also the main characters in the `moonshine' conjectures for string theory on K3

The study of twining genera related to topological defects, rather than symmetries, opens a number of new paths for the research in these subjects.

* A mysterious correspondence has been observed <cit.> between symmetries of K3 sigma models and automorphisms of a certain $\CN=1$ supersymmetric vertex operator algebra  $V^{s\natural}$ (a holomorphic superconformal field theory, in physics parlance) with central charge $12$. In particular, most (though, probably, not all) of the twining genera $\phi_g$ of K3 models can be exactly reproduced by certain $g$-twisted supertraces in $V^{s\natural}$. It is reasonable to expect the SCFT $V^{s\natural}$ to admit suitable topological defects that preserve the $\CN=1$ superconformal symmetry – to the best of our knowledge, such defects have not been studied yet. While one cannot define boundary states in a purely holomorphic theory, we do expect some modification of our methods to apply to this case as well. We are planning to describe such methods in a forthcoming paper <cit.>. It is natural to wonder whether the K3-VOA correspondence extends to the case of defects.

Acknowledgements. The research of S.G and R.V. has been supported by a BIRD-2021 project (PRD-2021) and by the  PRIN Project n. 2022ABPBEY, “Understanding quantum field theory through its deformations”. The research of R.A. has been supported by the grants CEX2020001007-S and PID2021-123017NB-I00, funded by MCIN/AEI/10.13039/501100011033 and by ERDF A way of making Europe. S.G. and R.V. would like to thank the Pollica Physics Center for support and hospitality during the programme “New connections between Physics and Number Theory” where work on this paper was undertaken, and to thank the participants to the programme for stimulating discussions.

§ GENERATORS OF $\ZZ_2^8:M_{20}$

In this section, we provide an explicit description of the group of symmetries $G_{GTVW}\cong \ZZ_2^8:M_{20}$ of the K3 model $\calC_{GTVW}$ considered in section <ref>; see <cit.> and <cit.> for a derivation. Recall that the full symmetry group of the CFT is $(SU(2)^6\times SU(2)^6):S_6$, where the $SU(2)^6$ factors are generated by the zero modes of the holomorphic and antiholomorphic currents, while $G_{GTVW}\cong\ZZ_2^8:M_{20}$ is the subgroup that preserves the $\CN=(4,4)$ superconformal algebra and the spectral flow. Every element of $(SU(2)^6\times SU(2)^6):S_6$ can be written as $(A_1,\ldots,A_6\ B_1,\ldots,B_6)\pi$ where $A_i$ and $B_i$ are $SU(2)$ matrices, and $\pi\in S_6$ is a permutation. As discussed in section <ref>, $SU(2)^6\times SU(2)^6$ does not act faithfully on the operators of the CFT. There is a subgroup $Z_0\cong \ZZ_2^5$ of the center $\ZZ_2^6\times \ZZ_2^6$ of $SU(2)^6\times SU(2)^6$ that acts trivially on all states of the theory (see eq.(<ref>)). This means that there is an ambiguity in writing the symmetries  as $(A_1,\ldots,A_6\ B_1,\ldots,B_6)\pi$, as one can multiply by any element in $Z_0\cong \ZZ_2^5$.

The $SU(2)$ matrices that one needs to write all elements of $\ZZ_2^8:M_{20}$ are
1     0
0     1
,    =
0     1
-1     0
,    =
0     i
i     0
,    =
i     0
0     -i

and their opposite matrices
-1     0
0     -1
,    =
0     -1
1     0
,    =
0     -i
-i     0
,    =
-i     0
0     i
Notice that $\ii,\jj,\kk$ obey the quaternionic relations $\ii^2=\jj^2=\kk^2=-1$, $\ii\jj=\kk=-\jj\ii$. Finally we need the matrices
1-i/2     1+i/2

-1-i/2     1+i/2
,    Ω^†=
1+i/2     1+i/2

1-i/2     1-i/2

that satisfy $\Omega^3=-\one$, so that $\Omega^6=\one$ and $\Omega^\dag=-\Omega^2$. Furthermore, $\Omega \ii\Omega^\dag=\kk$, $\Omega \jj\Omega^\dag=\ii$, $\Omega \kk\Omega^\dag=\jj$. The permutations $\pi$ will be denoted by their non-trivial cycles.

In this notation, the generators of $\ZZ_2^8:M_{20}$ are as follows:

* Generators in the $\ZZ_2^6\times \ZZ_2^6$ subgroup of $SU(2)^6\times SU(2)^6$:

\begin{align}
&t_{23}=( \one\mone\mone\one\one\one\  \one\one\one\one\one\one)(1)^6\\
&t_{24}=(\one\mone\one\mone\one\one\  \one\one\one\one\one\one)(1)^6\\
&t_{25}=(\one\mone\one\one\mone\one\  \one\one\one\one\one\one)(1)^6\\
&t_{26}=(\one\mone\one\one\one\mone\  \one\one\one\one\one\one)(1)^6
\end{align}
These generators form a $\ZZ_2^4$ group, with $\ZZ_2^4\subset \ZZ_2^8\subset \ZZ_2^8:M_{20}$, that contains $10$ elements $t_{ij}$, $2\le i<j\le 6$ with non-trivial $SU(2)$ factors $A_i=A_j=\mone$, and $5$ elements $t_{ijkl}$, $2\le i<j<k<l\le 6$, with non-trivial $SU(2)$ factors $A_i=A_j=A_k=A_l=\mone$.

* Further generators of the $\ZZ_2^8$ normal subgroup of $\ZZ_2^8:M_{20}$

\begin{align}
&s_1=(\one \, \one \, \mi \, \mi \, \mi\, \mi\, ;\  \one\,\one\,\ii\,\ii\,\ii\,\ii)(1)^6\\
&s_2=(\one\,\one\,\mj\,\mj\,\mj\,\mj\,;\  \one\,\one\,\jj\,\jj\,\jj\,\jj)(1)^6\\
&s_3=(\one\,\ii\,\kk\,\mj\,\mi\,\one\,;\  \one\,\ii\,\kk\,\jj\,\ii\,\one)(1)^6\\
&s_4=(\one\,\kk\,\jj\,\mi\,\mk\,\one\,;\  \one\,\kk\,\jj\,\ii\,\kk\,\one)(1)^6
\end{align}

* Generators of the $\ZZ_2^4$ subgroup of $M_{20}\cong \ZZ_2^4.A_5$ (modulo elements of the form $t_{ijkl}$)

\begin{align}
&v_1=(\one\,\one\,\ii\,\ii\,\ii\,\ii\,;\ \one \, \one \, \kk \, \kk \, \kk\, \kk )(1)^6\\
&v_2=(\one\,\one\,\jj\,\jj\,\jj\,\jj\,;\  \one\,\one\,\ii\,\ii\,\ii\,\ii)(1)^6\\
&v_3=(\one\,\jj\,\jj\,\one\,\ii\,\kk\,;\  \one\,\ii\,\ii\,\one\,\kk\,\jj)(1)^6\\
&v_4=(\one\,\ii\,\jj\,\kk\,\one\,\ii\,;\  \one\,\kk\,\ii\,\jj\,\one\,\kk)(1)^6
\end{align}

Altogether, the $t_{ij}$, $s_1,\ldots,s_4$, and $v_1,\ldots,v_4$ generate the subgroup of $\ZZ_2^8:M_{20}$ contained in $SU(2)^6\times SU(2)^6$.

* Generators of the quotient $A_5$ of $M_{20}\cong \ZZ_2^4.A_5$

\begin{align}
&p_1=(\one\,\one\,\one\,\one\,\one\,\one\, ;\  \one\,\one\,\one\,\one\,\one\,\one)(34)(56)\\
&p_2=(\one\,\one\,\one\,\one\,\one\,\one\, ;\  \one\,\one\,\one\,\one\,\one\,\one)(35)(46)\\
&p_3=(\one\,\one\,\Omega^{\dag}\,\Omega\,\one\,\one\, ;\  \one\,\one\,\Omega^{\dag}\,\Omega\,\one\,\one)(25)(34)
\end{align}

Action on the currents. The $18$ left-moving currents of $su(2)^6$ can be denoted by a pair $(\vec\sigma\cdot \vec n,i)_L$, where $i=1,\ldots, 6$ denotes the $su(2)_1$ factors, and $\vec\sigma\cdot \vec n$ is a traceless hermitian matrix, written a linear combination of Pauli matrices with coefficients $\vec n=(n_1,n_2,n_3)\in \RR^3$, $\vec n\cdot \vec n=1$. The action of $g=(A_,\ldots,A_6; B_1,\ldots,B_6)\pi$ on $(\vec\sigma\cdot \vec n,i)_L$ is
where we make the permutation act before the $SU(2)^6\times SU(2)^6$ adjoint action. Similarly, the action on the right-moving currents is
g·(σ⃗·n⃗,i)_R=(B_π(i)σ⃗·n⃗B_π(i)^†,π(i))_R .

Action on the RR ground states. An orthonormal basis of the $24$-dimensional space of RR ground states is given by six `tetrads' of states $|1,i\rangle$, $|2,i\rangle$, $|3,i\rangle$, $|4,i\rangle$, $i=1,\ldots,6$. One can assign with each element $|a,i\rangle$ in a tetrad a $2\times 2$ matrix $U_a$, $a=1,2,3,4$, as follows

\begin{align}
|1,i\rangle &\longrightarrow U_1=\frac{1}{\sqrt{2}} \begin{pmatrix}
0 & 1\\ -1 & 0
\end{pmatrix}=\frac{\ii}{\sqrt{2}}\\
|2,i\rangle &\longrightarrow U_2=\frac{1}{\sqrt{2}} \begin{pmatrix}
1 & 0\\ 0 & 1
\end{pmatrix}=\frac{\one}{\sqrt{2}}\\
|3,i\rangle &\longrightarrow U_3=\frac{1}{\sqrt{2}} \begin{pmatrix}
i & 0
\\0 & -i
\end{pmatrix}=-\frac{\kk}{\sqrt{2}}\\
|4,i\rangle &\longrightarrow U_4=\frac{1}{\sqrt{2}} \begin{pmatrix}
0 & i\\ i & 0
\end{pmatrix}=\frac{\jj}{\sqrt{2}}\ ,
\end{align}
in such a way that $\langle a,i|b,j\rangle =\delta_{ij}\Tr(U_a^\dag U_b)$. The action of $g=(A_1,\ldots,A_6;B_1,\ldots,B_6)\pi$ on $|a,i\rangle\equiv (U_a,i)$, $a=1,\ldots,4$, $i=1,\ldots,6$ is
g·(U_a,i)=(A_π(i)U_aB_π(i)^t,π(i)) .

§ A DUALITY DEFECT NOT IN $\CTOP_{GTVW}$

Consider the K3 model $\calC_{GTVW}$ described in section <ref> and let $H\subset G_{GTVW}$ the abelian group $H\cong \ZZ_2$ generated by the symmetry $g=t_5t_6$.
By computing its partition function, one can show that the orbifold $\calC/H$ is a consistent K3 model with the same bosonic chiral and anti-chiral algebras and representations as $\calC$; therefore, the two CFTs must be isomorphic $\calC\cong \calC/H$. In particular, all holomorphic and anti-holomorphic currents are $g$-invariant; the $g$-twisted sector contains no further currents.
Thus, we expect a topological duality defect $\CN$ such that
^2=+_g ,    _g=_g= .
In this section, we describe the duality defect $\CN$ and show that $\CN\not\in \cTop_{GTVW}$, and in particular that $\CN$ does not preserve the $\CN=(4,4)$  superconformal algebra.

The fusion rules for $\CN\equiv \CN_{56}$ imply that $\hat \CN$ must act with eigenvalues $\pm \sqrt{2}$ on the states that are fixed by $g$, while eigenstates of $g$ with eigenvalue $-1$ must be in the kernel. On the other hand, these topological defects $\CN$ cannot commute with the whole $(\widehat{su}(2)_1)^{\oplus 6}$ algebra – we have already identified all such defects, and they are all invertible.

What is the possible action on the chiral algebra?

* Consider moving a defect $\CN$ through local holomorphic operators in $\CA$, we obtain a map from  $\CA$ to itself in a way compatible with OPE. This means that the defect should act on $\CA$ by an automorphism of order $2$.

* The action of $\CN$ on $\CA$ should not lift to a symmetry of the whole CFT. Indeed, if this was the case, then for a suitable CFT symmetry $h$, the fusion $\CL_h\CN$ would act trivially on $\CA$. But the only simple defects with this property are the Verlinde lines for the algebra $\CA$ and all such defects are invertible. It would follow that $\CN$ is a superposition of invertible defects, which cannot be true for a duality defect.

* Because $\hat \CN^2$ acts trivially on the $g$-invariant space of states $\CH^g$, then $\CN$ acts by permutations on the set of $\CA\times\bar\CA$ representations contained in $\CH^g$. In particular, it cannot map any such representation into one that is not contained in $\CH^g$.

The group of outer automorphisms of the chiral and antichiral algebra is the $S_6\times S_6$ permutation group acting separately on the holomophic and anti-holomorphic  currents. Only the diagonal $S_6\subset S_6\times S_6$ lifts to a symmetry of the whole theory; the other elements of do not preserve the set of $\CA\times\bar\CA$ representations in the space of states $\CH$. If instead we consider the subgroup of $S_6\times S_6$ that preserve only the set of representations contained in the subspace $\CH^g\subset \CH$ of $g$-invariant states, then there is one non-trivial choice (modulo the diagonal $S_6$ symmetry): the involution exchanging the $5$-th and the $6$-th $\widehat{su}(2)_1$ components on the (say) holomorphic side, while keeping the anti-holomorphic side fixed.

Therefore, modulo automorphisms that lift to CFT symmetries, this is the only possible action of $\CN$ on the algebra $(\widehat{su}(2)_1)^{ 6}$.

We can calculate the $\CN$-twined partition functions
Z^NSNS_1,=_NS-NS(q^L_0-c/24q̅^L̅_0-c̅/24) ,    Z̃^NSNS_1,=_NS-NS((-1)^F+F̃q^L_0-c/24q̅^L̅_0-c̅/24)
Z^RR_1,=_R-R(q^L_0-c/24q̅^L̅_0-c̅/24) ,    Z̃^RR_1,=_R-R((-1)^F+F̃q^L_0-c/24q̅^L̅_0-c̅/24)
using the characters of $\widehat{su}(2)_1$, namely
ch_1,0(τ,z)= θ_3(2τ,2z)/η(τ) ,    ch_1,1/2(τ,z)= θ_2(2τ,2z)/η(τ) ,
from which
ch_1,0(τ,0)= θ_3(2τ,0)/η(τ)=η(2τ)^5/η(τ)^3η(4τ)^2 ,    ch_1,1/2(τ,0)= θ_2(2τ,0)/η(τ)=2η(4τ)^2/η(2τ)η(τ) .
The S-transformations are
ch_1,0(-1τ,0)=1/√(2)(ch_1,0(τ,0)+ch_1,1/2(τ,0)) ,  ch_1,1/2(-1τ,0)=1/√(2)(ch_1,0(τ,0)-ch_1,1/2(τ,0))
Recall $\hat \CN$ acts on the $t_5t_6$ invariant states by $\sqrt{2}$ times the permutation of the fifth and sixth holomorphic $\widehat{su}(2)_1$ components.
It follows that the $\CN$-twined partition function are

\begin{align*} Z^{NSNS}_{1,\CN}=\sqrt{2}&\sum_{\substack{a_1,\ldots, a_4\in\{0,\frac{1}{2}\}\\ \sum a_i\in \ZZ}}\Bigl[\prod_{i=1}^4\left|{\rm ch}_{1,a_i}(\tau,0)\right|^2\bigl({\rm ch}_{1,0}(2\tau,0)\overline{{\rm ch}_{1,0}(\tau,0)}^2+{\rm ch}_{1,\frac{1}{2}}(2\tau,0)\overline{{\rm ch}_{1,\frac{1}{2}}(\tau,0)}^2\bigr)\\
&+\prod_{i=1}^4{\rm ch}_{1,a_i}(\tau,0)\overline{{\rm ch}_{1,\frac{1}{2}-a_i}(\tau,0)}\bigl({\rm ch}_{1,0}(2\tau,0)\overline{{\rm ch}_{1,\frac{1}{2}}(\tau,0)}^2+{\rm ch}_{1,\frac{1}{2}}(2\tau,0)\overline{{\rm ch}_{1,0}(\tau,0)}^2\bigr)\Bigr]
\end{align*}

\begin{align*} \tilde{Z}^{NSNS}_{1,\CN}=\sqrt{2}&\sum_{\substack{a_1,\ldots, a_4\in\{0,\frac{1}{2}\}\\ \sum a_i\in \ZZ}}\Bigl[\prod_{i=1}^4\left|{\rm ch}_{1,a_i}(\tau,0)\right|^2\bigl({\rm ch}_{1,0}(2\tau,0)\overline{{\rm ch}_{1,0}(\tau,0)}^2+{\rm ch}_{1,\frac{1}{2}}(2\tau,0)\overline{{\rm ch}_{1,\frac{1}{2}}(\tau,0)}^2\bigr)\\
&-\prod_{i=1}^4{\rm ch}_{1,a_i}(\tau,0)\overline{{\rm ch}_{1,\frac{1}{2}-a_i}(\tau,0)}\bigl({\rm ch}_{1,0}(2\tau,0)\overline{{\rm ch}_{1,\frac{1}{2}}(\tau,0)}^2+{\rm ch}_{1,\frac{1}{2}}(2\tau,0)\overline{{\rm ch}_{1,0}(\tau,0)}^2\bigr)\Bigr]
\end{align*}

\begin{align*} Z^{RR}_{1,\CN}=\sqrt{2}&\sum_{\substack{a_1,\ldots, a_4\in\{0,\frac{1}{2}\}\\ \sum a_i\in \frac{1}{2}+\ZZ}}\Bigl[\prod_{i=1}^4\left|{\rm ch}_{1,a_i}(\tau,0)\right|^2\bigl({\rm ch}_{1,0}(2\tau,0)\overline{{\rm ch}_{1,0}(\tau,0)}^2+{\rm ch}_{1,\frac{1}{2}}(2\tau,0)\overline{{\rm ch}_{1,\frac{1}{2}}(\tau,0)}^2\bigr)\\
&+\prod_{i=1}^4{\rm ch}_{1,a_i}(\tau,0)\overline{{\rm ch}_{1,\frac{1}{2}-a_i}(\tau,0)}\bigl({\rm ch}_{1,0}(2\tau,0)\overline{{\rm ch}_{1,\frac{1}{2}}(\tau,0)}^2+{\rm ch}_{1,\frac{1}{2}}(2\tau,0)\overline{{\rm ch}_{1,0}(\tau,0)}^2\bigr)\Bigr]
\end{align*}

\begin{align*} \tilde{Z}^{RR}_{1,\CN}=\sqrt{2}&\sum_{\substack{a_1,\ldots, a_4\in\{0,\frac{1}{2}\}\\ \sum a_i\in \frac{1}{2}+\ZZ}}\Bigl[\prod_{i=1}^4\left|{\rm ch}_{1,a_i}(\tau,0)\right|^2\bigl({\rm ch}_{1,0}(2\tau,0)\overline{{\rm ch}_{1,0}(\tau,0)}^2+{\rm ch}_{1,\frac{1}{2}}(2\tau,0)\overline{{\rm ch}_{1,\frac{1}{2}}(\tau,0)}^2\bigr)\\
&-\prod_{i=1}^4{\rm ch}_{1,a_i}(\tau,0)\overline{{\rm ch}_{1,\frac{1}{2}-a_i}(\tau,0)}\bigl({\rm ch}_{1,0}(2\tau,0)\overline{{\rm ch}_{1,\frac{1}{2}}(\tau,0)}^2+{\rm ch}_{1,\frac{1}{2}}(2\tau,0)\overline{{\rm ch}_{1,0}(\tau,0)}^2\bigr)\Bigr]
\end{align*}

By a direct calculation we obtain
Z̃^RR_1,=(16+36q+96q^2+…)+O(q̅^1) .
This shows that  $\CN$ does not preserve any holomorphic supercurrent. Indeed, if a holomorphic supercurrent were preserved by $\CN$, then $\tilde{Z}^{RR}_{1,\CN}$ would receive contributions only from the RR ground states, and it would be a constant in $q$.

§ DEFECTS ACTING BY AUTOMORPHISMS OF THE CHIRAL ALGEBRA

In this section we classify the topological defects  $\CL\in \cTop_{GTVW}$ of the model $\calC_{GTVW}$ of section <ref>, such that $\hat \CL$ acts on all holomorphic fields generating the bosonic chiral algebra $\CA\cong (\widehat {su}(2)_1)^6$ by an algebra automorphism $\rho_L$ times the quantum dimension $\langle \CL\rangle$. This implies that when a defect line $\CL$ is moved past the insertion point of one of the holomorphic currents $j(z)$, the latter gets simply transformed into the current $\rho_L(j(z))$.  Similarly, we require the defect to act on the anti-holomorphic fields by a (possibly different) automorphism $\rho_R$. The group of automorphisms of $(\widehat {su}(2)_1)^6$ is $SO(3)^6\rtimes S_6$, where $SO(3)^6$ are the inner automorphisms generated by the zero modes of the currents; note that the center $\ZZ_2^6$ of $SU(2)^6$ acts trivially on the algebra itself.
We still require $\rho_L$ and $\rho_R$ to act trivially on the $\CN=(4,4)$ superconformal algebra; this condition constrains $\rho_L$ and $\rho_R$ to be in a finite subgroup of $SO(3)^6\rtimes S_6$. If the automorphisms $(\rho_L,\rho_R)$ extends to a symmetry of the CFT, then there is an invertible defect $\CL_g$, for some  $g\in G_{GTVW}\cong \ZZ_2^8:M_{20}$, such that the fusion $\CL_g\CL$ acts trivially on the whole $\CA\otimes\bar\CA$. The only simple defects with this property are the Verlinde lines discussed in section <ref>, where it is shown that they are all invertible. This means that, in this case, $\CL$ is just a superposition of invertible defects.

Therefore, in order to get some new defects, we have to require that $(\rho_L,\rho_R)$ does not lift to a symmetry of the CFT. Now, inner  automorphisms $SU(2)^6\times SU(2)^6$ are generated by the current zero modes, and therefore always define CFT symmetries. The group of outer automorphisms of $\CA\times \bar\CA$ is $S_6\times S_6$, and in this case, only the diagonal $S_6^{diag}\subset S_6\times S_6$, permuting the holomorphic and antiholomorphic $\widehat{su}(2)_1$ factors in the same way, lifts to a CFT symmetry.
Thus, the defects acting by algebra automorphisms, modulo fusion with invertible defects from the left or from the right, correspond to non-trivial double cosets in ((SU(2)^6×SU(2)^6)⋊S_6^diag)\((SU(2)^6×SU(2)^6)⋊(S_6×S_6))/((SU(2)^6×SU(2)^6)⋊S_6^diag) .

For each such coset, we can always choose a representative $(\rho_L,\rho_R)$ with $\rho_R=1$. Furthermore, we can choose one $\rho_L$ for each $S_6$-conjugacy class, i.e. for each possible cycle shape.

Finally, we require $\rho_L$ to preserve the $\CN=4$ superconformal algebra and the spectral flow generators. This implies that the induced permutation must be contained in the $A_5\subset S_6$ subgroup of even permutations fixing the first $\widehat{su}(2)_1$ factor. There are only three possible non-trivial cycle shapes, corresponding to the partitions $1+1+1+3$, $1+1+2+2$ and $1+5$. Let us consider each of these three cases in detail.

Partition $1+1+1+3$. It is known (see appendix <ref>), that $G_{GTVW}$ contains a symmetry
g_3=p_1∘p_3=(  Ω Ω^†    ;    Ω Ω^†  )(265) ,
where we denote the elements of $(SU(2)^6\times SU(2)^6)\rtimes S_6^{diag}$ by $(A_1,\ldots,A_6\, ; B_1,\ldots,B_6)\pi$, with $A_i,B_i\in SU(2)$ and $\pi\in S_6^{diag}$, and where
1-i/2     1+i/2

-1-i/2     1+i/2
∈SU(2) .
This means that if we define $(\rho_L,\rho_R)$ as

$$ \rho_L=(\one\,\one\,\Omega\,\Omega^{\dag}\,\one\,\one)(265)\ ,\qquad \rho_R=1\ .
the left automorphism $\rho_L$ acts on the holomorphic algebra in the same way as the symmetry element $g_3$. This ensures that the holomorphic $\CN=4$ supercurrents are invariant under $\rho_L$. However, $\rho_R$ is not the same as for $g_3$, and in particular the left- and right-moving $\widehat{su}(2)_1$ factors are permuted in different ways. This means that the pair $(\rho_L,\rho_R)$ cannot be extended to a symmetry of the whole CFT.

More explicitly, any such symmetry would have to map any field in the representation $[110000;110000]$, for example, to a field in a representation $[100010;110000]$; however, while the former field is in the spectrum of the theory, the latter is not. This implies that there cannot be any invertible defect acting by $(\rho_L,\rho_R)$ on the chiral and antichiral algebras. However, there is no obstruction to having a non-invertible defect with such an action. Indeed, a non-invertible defect $\CL$, when circling a field in the $[110000;110000]$ representation, can simply annihilate it. More generally, the operator $\hat\CL$ associated with any such defect needs to annihilate any field in a representation $[a_1\ldots,a_6;b_1\ldots b_6]$ such that $[\rho_L(a_1\ldots,a_6);\rho_R(b_1\ldots b_6)]$ is not in the spectrum. The NS-NS representations $[a_1\ldots,a_6;b_1\ldots b_6]$ that are not necessarily annihilated by $\CL$ are the ones satisfying  a_2=a_5=a_6 ,    b_2=b_5=b_6 ,and can be grouped into four sets:

\begin{align*}
\Omega_1&=\{[000000;000000],\ [111111;000000],\ [000000;111111],\ [111111;111111] \}\ ,\\
\Omega_2&=\{[101000;101000],\ [010111;101000],\ [101000;010111],\ [010111;010111] \}\ ,\\
\Omega_3&=\{[100100;100100],\ [011011;100100],\ [100100;011011],\ [011011;011011] \}\ ,\\
\Omega_4&=\{[001100;001100],\ [110011;001100],\ [001100;110011],\ [110011;110011] \}\ .
\end{align*}
The four members in each set are related to each other by the action of the $\CN=(4,4)$ supercurrents. Therefore, the requirement that the $\CN=(4,4)$ algebra is invariant under $\hat\CL$ implies that the action of $\hat\CL$ is the same on all representations in the same set $\Omega_i$. We conclude that the action of $\hat\CL$ on the NS-NS sector depends only on four parameters $\alpha_1,\ldots,\alpha_4$ as
=∑_i=1^4 α_iP^i_ρ_L,ρ_R .
Here, for each $i=1,\ldots,4$, $P^i_{\rho_L,\rho_R}$ acts by the automorphism $(\rho_L,\rho_R)$ on the representations in the set $\Omega_i$, while it annihilates all fields in the representations in $\Omega_j$ for $j\neq i$.

Requiring $\CL$ to commute with the spectral flow operators fixes the action of $\hat\CL$ on the R-R sector to be of the same form (<ref>) with the same parameters $\alpha_i$, where now each $P^i_{\rho_L,\rho_R}$ is non-zero only on the set of representations $\tilde\Omega_i$ with

\begin{align*}
\tilde\Omega_1&=\{[100000;100000],\ [011111;100000],\ [100000;011111],\ [011111;011111] \}\ ,\\
\tilde\Omega_2&=\{[001000;001000],\ [110111;001000],\ [001000;110111],\ [110111;110111] \}\ ,\\
\tilde\Omega_3&=\{[000100;000100],\ [111011;000100],\ [000100;111011],\ [111011;111011] \}\ ,\\
\tilde\Omega_4&=\{[101100;101100],\ [010011;101100],\ [101100;010011],\ [010011;010011] \}\ .
\end{align*}

The four parameters $\alpha_1,\ldots,\alpha_4$ are constrained by the Cardy-like conditions that are obtained by considering the torus partition function with the defect line $\CL$ wrapping one of the cycles. Notice that if $(\alpha_1,\alpha_2,\alpha_3,\alpha_4)$ correspond to a consistent defect $\CL$ acting by $(\rho_L,\rho_R)$ on the chiral and antichiral algebras, then also $(\alpha_1,-\alpha_2,-\alpha_3,\alpha_4)$, $(\alpha_1,\alpha_2,-\alpha_3,-\alpha_4)$, and $(\alpha_1,-\alpha_2,\alpha_3,-\alpha_4)$ correspond to consistent defects, since they can be obtained by fusion of $\CL$ with the invertible defects acting trivially on all the currents and all the $\CN=(4,4)$ supercurrents (in particular, fusion with the symmetries $t_2t_3$, $t_2t_4$ and $t_3t_4$ gives all such defects; fusion with the symmetries $t_2t_5$, $t_2t_6$, and $t_5t_6$ leaves each of these defects invariant).

Let us consider the possible fusion products of the simple defect $\CL$ with $\alpha_1=\ldots=\alpha_4$. The dual defect $\CL^\dual$ acts on the space of states by the adjoint operator
^= ∑_i=1^4 α_i^*P^i_ρ_L^-1,ρ_R^-1
so that
the product $\CL\CL^\dual$ acts by
^=∑_i=1^4 |α_i|^2 P^i ,
where $P^i$ is the projector on the representations in the set $\Omega_i$. On the other hand, we know that $\CL\CL^\dual=\CI+\ldots $ acts trivially on the whole algebra $\widehat{su}(2)_1^6\oplus \widehat{su}(2)_1^6$ as well as on the $\CN=(4,4)$ algebra. This means that it must be a superposition of the invertible defects $\CL_{t_it_j}$ generating the $\ZZ_2^4$ subgroup of $\ZZ_2^8:M_{20}$ that is contained in the centre of $SU(2)^6\times SU(2)^6$. Furthermore, $\CL\CL^\dual$ must annihilate the representations $[a_1,\ldots,b_6]$ of the chiral algebra that do not satisfy (<ref>), and must be a sum with positive coefficients of the projectors $P_i$. The only possibility is
^=+_t_2t_5+_t_2t_6+_t_5t_6 ,
so that

$$ |\alpha_i|=\langle \CL \rangle =\langle \CL^\dual \rangle=2\ .

The torus partition function with the insertion of $\CL$ is
Z_(τ)=(q^L_0-c/24q̅^L̅_0-c̅/24)=∑_i=1^4 α_i (P_ρ_L,ρ_R^i q^L_0-c/24q̅^L̅_0-c̅/24) ,

\begin{align*}\Tr(P_{\rho_L,\rho_R}^i q^{L_0-\frac{c}{24}}\bar q^{\bar L_0-\frac{\bar c}{24}})=\sum_{[a_1,\ldots,b_6]\in\Omega_i} \Bigl[&\ch_{a_1}(\tau,0)\ch_{a_2}(3\tau,0)\ch_{a_3}(\tau,\frac{1}{6})\ch_{a_4}(\tau,\frac{1}{6})\\
\end{align*}

Let us calculate $Z_\CL$ with the ansatz
α_1=α_2=α_3=α_4= ⟨⟩ Using the $su(2)$ characters, $Z_{\mathcal{L}} (\tau)$ takes the form:

\begin{equation*}
\begin{split}
Z_{\mathcal{L}} (\tau) &  =  \frac{\langle \CL \rangle }{\overline{ \eta (\tau)^6} \eta (\tau)^3 \eta(3 \tau)} \left\lbrace \left( \overline{\theta_3 (2 \tau)^6} + \overline{\theta_2 (2 \tau)^6} \right)\left[ \theta_3 (2 \tau) \theta_3(6 \tau) \theta_3 \left( 2 \tau, \frac{1}{3} \right)^2 + \theta_2 (2 \tau) \theta_2(6 \tau) \theta_2 \left( 2 \tau, \frac{1}{3} \right)^2  \right] +\right. \\
& + \left. \left( \overline{\theta_3 (2 \tau)^2} \overline{\theta_2 (2 \tau)^4}+ \overline{\theta_2 (2 \tau)^2} \overline{\theta_3 (2 \tau)^4} \right) \left[ 2 \left( \theta_3 \left( 2 \tau, \frac{1}{3} \right) \theta_2 \left( 2 \tau, \frac{1}{3} \right) \right) \left( \theta_3 (2 \tau) \theta_2 (6 \tau)  + \theta_2 (2 \tau) \theta_3 (6 \tau)  \right) +   \right. \right. \\
& \left. \left.   +  \left( \theta_3 (2 \tau) \theta_3 (6 \tau) \theta_2 \left( 2 \tau, \frac{1}{3} \right)^2 + \theta_2 (2 \tau) \theta_2 (6 \tau) \theta_3 \left( 2 \tau, \frac{1}{3} \right)^2 \right) \right] \right\rbrace. \\
\end{split}
\end{equation*}

Applying the transformation $\tau \mapsto - 1 / \tau$ we get:

\begin{equation*}
\begin{split}
& Z^{\mathcal{L}} (\tau)  =  \frac{ \langle \CL \rangle q^{1/18}}{32 \overline{\eta (\tau)^6} \eta(\tau)^3 \eta (\tau / 3)} \left\lbrace \left[ \left( \theta_3 \left( \frac{\tau}{2} \right) \theta_3 \left( \frac{\tau}{6} \right) \theta_4 \left( \frac{\tau}{2} ; \frac{\tau}{6} \right)^2 +  \theta_4 \left( \frac{\tau}{2} \right) \theta_4 \left( \frac{\tau}{6} \right) \theta_3 \left( \frac{\tau}{2} ; \frac{\tau}{6} \right)^2\right) \right. + \right. \\
& \left. \left. + 2 \left( \theta_3 \left( \frac{\tau}{2} ; \frac{\tau}{6} \right) \theta_4 \left( \frac{\tau}{2} ; \frac{\tau}{6} \right) \right)\left( \theta_3 \left( \frac{\tau}{2} \right) \theta_4 \left( \frac{\tau}{6} \right) + \theta_4 \left( \frac{\tau}{2} \right) \theta_3 \left( \frac{\tau}{6} \right)\right) \right] \left( \overline{\theta_3 \left( \frac{\tau}{2}\right)^4} \overline{\theta_4 \left( \frac{\tau}{2}\right)^2} + \overline{\theta_4 \left( \frac{\tau}{2}\right)^4} \overline{\theta_3 \left( \frac{\tau}{2}\right)^2} \right) + \right. \\
& \left. +  \left[ \theta_3 \left( \frac{\tau}{2} \right) \theta_3 \left( \frac{\tau}{6} \right) \theta_3 \left( \frac{\tau}{2} ; \frac{\tau}{6} \right)^2 + \theta_4 \left( \frac{\tau}{2} \right) \theta_4 \left( \frac{\tau}{6} \right) \theta_4 \left( \frac{\tau}{2} ; \frac{\tau}{6} \right)^2 \right] \left( \overline{\theta_3 \left(\frac{\tau}{2} \right)^6} + \overline{\theta_4 \left(\frac{\tau}{2} \right)^6} \right) \right\rbrace, \\
\end{split}
\end{equation*}

which series expansion in $q$ and $\bar{q}$ is:

\begin{equation*}
\begin{split}
Z^{\mathcal{L}} = & \frac{\langle \CL \rangle }{ q^{1/4} \overline{q}^{1/4} } \left[ \left(\frac{ q^{1/6}}{2}+\frac{3 }{2}q^{1/2}+2  q^{2/3}+3  q^{5/6}+6  q+9  q^{7/6}+12  q^{4/3}+\frac{39}{2}  q^{3/2}+28  q^{5/3}+36  q^{11/6}+ \right. \right. \\
& \left. +54  q^2+  \frac{149}{2}  q^{13/6} +96  q^{7/3}+\frac{273}{2}  q^{5/2}+182  q^{8/3}+234  q^{17/6}+O\left(q^{3}\right) \right) + \bar{q}^{1/2} \left(6  q^{1/6}+40  q^{1/3} +\right. \\
& \left. +82  q^{1/2}+128  q^{2/3}+  196  q^{5/6}+320  q+524  q^{7/6}+776  q^{4/3}+1098  q^{3/2}+1616  q^{5/3}+2320  q^{11/6}+ \right. \\
& \left. \left. + O\left(q^{2}\right)\right) + \bar{q} \left(33  q^{1/6} +160  q^{1/3} +355  q^{1/2}+ 548  q^{2/3} +838  q^{5/6}+O \left( q \right) \right) +O\left(\overline{q}^{3/2} \right) \right] . \\
\end{split}
\end{equation*}

As predicted above, we find that such expansion produces only integer non-negative coefficients coefficients if $\langle \CL \rangle =2$.

Notice that the automorphism $(\rho_L,\rho_R)$ has order $3$, so that $\CL^2$ acts on the chiral algebra by $(\rho_L^2,\rho_R^2)=(\rho_L^{-1},\rho_R^{-1})$ (the same as $\CL^\dual$), while $\CL^3$ acts trivially on the chiral algebra. This suggests
^2=2^ ,
^3=2^=2(+_t_2t_5+_t_2t_6+_t_5t_6) ,
which fits with the quantum dimensions.

We notice that if $\CL_{p_1}$ is the invertible defect corresponding to the symmetry
$$ p_1=(\one\,\one\,\one\,\one\,\one\,\one\, ;\  \one\,\one\,\one\,\one\,\one\,\one)(34)(56)$$
of order $2$, then $\CL_{p_1}\CL\CL_{p_1}$ acts by $(\rho_L^{-1},\rho_R^{-1})$ on the chiral algebra. This suggests that
_p_1_p_1=^ .
As a consequence, if we define
_256:=_p_1 ,
then $\CN_{256}$ is unoriented $\CN_{256}=\CN_{256}^\dual$, and
_256^2= +_t_2t_5+_t_2t_6+_t_5t_6 .
This means that $\CN_{256}$ is the duality defect related to the fact that the theory is self-orbifold with respect to the $\ZZ_2\times \ZZ_2$ group of symmetries with generators $t_2t_5$ and $t_2t_6$. Similarly, by conjugating by invertible defects, for every $1<i<j<k\le 6$ one can find duality defects $\CN_{ijk}$ of order $2$ such that
_ijk^2= +_t_it_j+_t_jt_k+_t_it_k .

Partition $1+1+2+2$. Let us now consider the class of defects acting by
ρ_L=(     )(34)(56) ,    ρ_R=1 ,
on the chiral and anti-chiral algebra.  We denote such defects by $\CL^{(34)(56)}_a$, $a=1,2,\ldots$. Because $\rho_L$ act in the same way as $p_1\in G_{GTVW}$ on the chiral algebra, $\hat{\CL}^{(34)(56)}_a$ must preserve the $\CN=(4,4)$ SCA. The only representations where $\hat \CL^{(34)(56)}_a$ can be non-zero are the ones satisfying
a_3=a_4 ,    a_5=a_6 ,    b_3=b_4 ,    b_5=b_6 .
Once again, we arrange such representations (in the NS-NS sector) in sets

\begin{align*}
\Omega_1&=\{[000000;000000],\ [111111;000000],\ [000000;111111],\ [111111;111111] \}\ ,\\
\Omega_2&=\{[110000;110000],\ [001111;110000],\ [110000;001111],\ [001111;001111] \}\ ,\\
\Omega_3&=\{[001100;001100],\ [110011;001100],\ [001100;110011],\ [110011;110011] \}\ ,\\
\Omega_4&=\{[000011;000011],\ [111100;000011],\ [000011;111100],\ [111100;111100] \}\ ,
\end{align*}
so that

$$ \hat\CL^{(34)(56)}_a=\sum_{i=1}^4 \alpha_{i}P^i_{(34)(56)}\ .
Because there are $4$ parameters $\alpha_i$, we expect (at most) $4$ simple defects of this kind. Let us denote by $\CL:=\CL^{(34)(56)}_1$ one of these defects. Then, by fusion of $\CL^{(34)(56)}$ with the invertible defects $\CL_{t_2t_3}$, $\CL_{t_2t_5}$, $\CL_{t_3t_5}$, we find three more simple defects $\CL^{(34)(56)}_2$, $\CL^{(34)(56)}_3$, $\CL^{(34)(56)}_4$ whose parameters  $\alpha_i$ differ only by signs. The product $\CL\CL^\dual=\CL^\dual\CL=\CI+\ldots$ acts trivially on the whole chiral and antichiral algebra, as well as on the $\CN=(4,4)$ superconformal algebra, and annihilate the representations that do not satisfy (<ref>). This leads to
^=+_t_3t_4+_t_5t_6+_t_3t_4t_5t_6 .
It follows that
|α_1|=|α_2|=|α_3|=|α_4|=||=2 .

The torus partition function is

$$Z_{\CL}(\tau)=\Tr(\hat\CL q^{L_0-\frac{c}{24}}\bar q^{\bar L_0-\frac{\bar c}{24}})=\sum_{i=1}^4 \alpha_i \Tr(P_{\rho_L,\rho_R}^i q^{L_0-\frac{c}{24}}\bar q^{\bar L_0-\frac{\bar c}{24}})$$

\begin{align*}\Tr(P_{(34)(56)}^i q^{L_0-\frac{c}{24}}\bar q^{\bar L_0-\frac{\bar c}{24}})=\sum_{[a_1,\ldots,b_6]\in\Omega_i} \Bigl[&\ch_{a_1}(\tau,0)\ch_{a_2}(\tau,0)\ch_{a_3}(2\tau,0)\ch_{a_5}(2\tau,0)\\
\end{align*}

Using the ansatz $\alpha_i = \langle \mathcal{L}^{(34)(56)} \rangle$, the partition function with a $\CL^{(34)(56)}$ insertion can be written in terms of the Theta functions as:

\begin{equation*}
\begin{split}
Z_{\mathcal{L}^{(34)(56)}} (\tau) & =  \frac{\langle \CL^{(34)(56)} \rangle}{\overline{\eta (\tau)^6} \eta(\tau)^2 \eta (2 \tau)^2} \left\lbrace  \left( \overline{\theta_3 (2 \tau)^6} + \overline{\theta_2 (2 \tau)^6}\right) \left( \theta_3 (2 \tau)^2 \theta_3 (4 \tau)^2 + \theta_2 (2 \tau)^2 \theta_2 (4 \tau)^2 \right) +  \right. \\
& \left. +  \left( \overline{\theta_2 (2 \tau)^2} \overline{\theta_3 (2 \tau)^4} + \overline{\theta_3(2 \tau)^2} \overline{\theta_2 (2 \tau)^4} \right) \left( \theta_2 (2 \tau)^2 \theta_3 (4 \tau)^2 + \theta_3 (2 \tau)^2 \theta_2 (4 \tau)^2 \right) + \right. \\
& \left. + 2 \left( \overline{\theta_3 (2 \tau)^4} \overline{\theta_2 (2 \tau)^2} + \overline{\theta_3 (2 \tau)^2} \overline{\theta_2 (2 \tau)^4}\right) \left( \theta_3 (2 \tau)^2 \theta_3 (4 \tau) \theta_2 (4 \tau) + \theta_2 (2 \tau)^2 \theta_3 (4 \tau) \theta_2 (4 \tau) \right) \right\rbrace.
\end{split}
\end{equation*}

Upon an $S$-transformation, we obtain the twisted function $Z^{\mathcal{L}^{(34)(56)}} (\tau) = Z_{\mathcal{L}^{(34)(56)}} \left( - 1 / \tau \right)$ in the form:

\begin{equation*}
\begin{split}
Z^{\mathcal{L}^{(34)(56)}} (\tau) & =  \frac{\langle \CL^{(34)(56)} \rangle }{32 \overline{\eta (\tau)^6}\eta (\tau)^2 \eta (\tau/2)} \left\lbrace  \left(  \overline{\theta_3 \left( \frac{\tau}{2} \right)}^6  +  \overline{\theta_4 \left( \frac{\tau}{2}
\right)}^6 \right) \left( \theta_3 \left( \frac{\tau}{2} \right)^2  \theta_3 \left( \frac{\tau}{4} \right)^2 +  \theta_4 \left( \frac{\tau}{2} \right)^2  \theta_4 \left( \frac{\tau}{4} \right)^2 \right)  \right. \\
& \left. + \left( \overline{\theta_4 \left( \frac{\tau}{2} \right)}^2 \overline{\theta_3 \left( \frac{\tau}{2} \right)}^4 + \overline{\theta_3 \left( \frac{\tau}{2} \right)}^2 \overline{\theta_4 \left( \frac{\tau}{2} \right)}^4  \right) \left[ \left( \theta_4 \left( \frac{\tau}{2} \right)^2 \theta_3 \left( \frac{\tau}{4}\right)^2 + \theta_3 \left( \frac{\tau}{2} \right)^2 \theta_4 \left( \frac{\tau}{4}\right)^2  \right) + \right. \right. \\
& \left. \left. + 2 \left( \theta_3 \left( \frac{\tau}{4} \right) \theta_4 \left( \frac{\tau}{4} \right) \right) \left( \theta_3 \left( \frac{\tau}{2} \right)^2 + \theta_4 \left( \frac{\tau}{2} \right)^2 \right) \right] \right\rbrace, \\
\end{split}
\end{equation*}

whose expansion in $q$ and $\bar{q}$ and is:

\begin{equation*}
\begin{split}
Z^{\mathcal{L}^{(34)(56)}} (\tau) & =  \frac{\langle\mathcal{L}^{(34)(56)} \rangle}{q^{1/8} \overline{q}^{1/4}} \left[  \left(\frac{1}{2}+5 q^{1/2}+\frac{47 q}{2}+75 q^{3/2}+\frac{403 q^2}{2}+501 q^{5/2}+1158 q^3+2502 q^{7/2}+\frac{10309}{2} q^4+ \right.\right. \\
& \left. +10228 q^{9/2}+ O\left(q^{5}\right)\right)+\overline{q}^{1/2} \left(6+32 q^{1/4}+128 q^{3/8}+60 q^{1/2} +192 q^{3/4}+512 q^{7/8}+282 q+ \right. \\
& \left. + 672 q^{5/4}+1792 q^{11/8}+900 q^{3/2}+1984 q^{7/4}+5120 q^{15/8}+ O \left(q^2 \right) \right) + \overline{q} \left(33+128 q^{1/4}+512 q^{3/8}+ \right.  \\
& \left. \left. +330 q^{1/2}+768 q^{3/4}+2048 q^{7/8}+1551 q+2688 q^{5/4}+7168 q^{11/8}+O \left( q^{3/2} \right) \right) + O \left( \bar{q}^{3/2} \right) \right] \\
\end{split}
\end{equation*}

The minimal integer value required for the quantum dimension of $\mathcal{L}^{(34)(56)}$ in order to have just positive integer coefficients in this expansion is $\langle\mathcal{L}^{(34)(56)} \rangle=2$. The three additional simple defects, obtained by fusion with $\CL_{t_2 t_3}, \CL_{t_2 t_5}$ and $\CL_{t_3t_5}$, act on the representations $\Omega_i$ through the operators $ \pm 2 P^i_{(34)(56)}  $, where the minus sign appears for $i=2,3$, or $i=2,5$, or $i=3,5$, respectively. In all such cases, the $q$-expansion of the resulting $\CL$-twisted partition functions have positive integer coefficients, as expected.

Because the coefficients $\alpha_i$ are real and $(\rho_L,\rho_R)$ has order two, we find that $\CL^\dual=\CL$ is unoriented, and
^2=+_t_3t_4+_t_5t_6+_t_3t_4t_5t_6 .
This implies that $\CL^{(34)(56)}$ coincides with the duality defect $\CN_{34,56}\equiv \CL^{(34)(56)}$ related to the fact that the theory is self-orbifold with respect to the $\ZZ_2\times \ZZ_2$ subgroup of $\ZZ_2^8:M_{20}$ generated by $t_3t_4$ and $t_5t_6$.    Similarly, for every choice of pairwise distinct $i,j,k,l\in \{2,\ldots,6\}$, we have duality defects $\CN_{ij,kl}\equiv \CN_{kl,ij}$ of dimension $2$ such that
_ij,kl^2=+_t_it_j+_t_kt_l+_t_it_jt_kt_l .
Notice that, in a suitable description of the model $\calC_{GTVW}$ as a torus orbifold $T^4/\ZZ_2$, the symmetries $t_it_j$, $t_kt_l$ and $t_it_jt_kt_l$ can be identified, respectively, with $\eta_\frac{\lambda}{2}$, $Q\eta_\frac{\lambda}{2}$, and $Q$, where $\eta_\frac{\lambda}{2}$ is induced by the half-periods along one of the directions of $T^4$, and $Q$ is the quantum symmetry of the orbifold, see sections <ref>. This suggests that $\CN_{ij,kl}$ can be identified with the topological defect $T_{\frac{\lambda}{4}}$ in eq.(<ref>).

Partition $1+5$. Finally, let us consider the class of defects that act by the automorphism
ρ_L=(  Ω Ω^†  )(23645) ,    ρ_R=1
of order $5$, preserving the $\CN=(4,4)$ supercurrents (in our conventions, the permutation acts after the $SU(2)^6$ transformation). The only possible for a simple defect $\CL^{(23645)}$ in this class is as follows (we set $\CL\equiv \CL^{(23645)}$ in this part)
=⟨⟩P_(23645) ,
where $P_{(23645)}$ acts by the automorphism $(\rho_L,\rho_R)$ on the four NS-NS representations

$$[000000;000000],\ [111111;000000],\ [000000;111111],\ [111111;111111] $$

that satisfy
a_2=a_3=a_4=a_5=a_6 ,    b_2=b_3=b_4=b_5=b_6
and annihilates any other representation in the NS-NS sector. The characters contributing to torus NS-NS partition function are:
Z_^(23645)(τ)=⟨⟩∑_a,b=0^1 _a(τ,0)_a(5τ,0)_b(τ,0)^6 ,
which expressions in terms of the $su(2)$ characters is:

\begin{equation*}
\begin{split}
Z_{\CL^{(23645)}}(\tau)= & \langle \CL\rangle \frac{1}{\overline{\eta (\tau)^6}} \frac{1}{\eta (\tau) \eta (5 \tau)} \left( \overline{\theta_3 (2 \tau)}^6 + \overline{\theta_2 (2 \tau)}^6\right) \left( \theta_3 (2 \tau) \theta_3 (10 \tau) + \theta_2 (2 \tau) \theta_2 (10 \tau) \right)
\end{split}
\end{equation*}

The twined partition function, obtained from the previous one  by modular transformation, takes the form:

\begin{equation*}
Z^{\CL^{(23645)}}(\tau)= \frac{\langle \CL\rangle}{16 \overline{\eta (\tau)}^6 \eta (\tau) \eta (\tau /5)} \left( \overline{\theta_3 \left( \frac{\tau}{2} \right)}^6 + \overline{\theta_4 \left( \frac{\tau}{2} \right)}^6 \right) \left( \theta_3 \left( \frac{\tau}{2} \right) \theta_3 \left(  \frac{\tau}{10}\right) + \theta_4 \left( \frac{\tau}{2} \right) \theta_4 \left(  \frac{\tau}{10}\right) \right),
\end{equation*}

with corresponding series expansion given by:

\begin{equation*}
\begin{split}
& Z^{\CL^{(23645)}} (\tau) =  \frac{\langle \CL\rangle}{q^{1/20} \overline{q}^{1/4}}  \left[ \left(\frac{1}{4}+\frac{3 }{4} q^{1/5}+q^{3/10}+q^{2/5}+q^{1/2}+\frac{7 }{4} q^{3/5}+3 q^{7/10}+\frac{13 }{4} q^{4/5}+4 q^{9/10}+\frac{11 }{2}q+ \right. \right. \\
& \left. \left.+7 q^{11/10} + \frac{19 }{2} q^{6/5}+11 q^{13/10}+\frac{55 q^{7/5}}{4} q^{7/5}+18 q^{3/2}+\frac{83 }{4} q^{8/5}+26 q^{17/10}+\frac{129 }{4} q^{9/5}+39 q^{19/10}+O \left( q^2 \right) \right) + \right. \\
&  \left. +\overline{q}^{1/2} \left(15+45 q^{1/5}+60 q^{3/10}+60 q^{2/5}+60 q^{1/2}+105 q^{3/5}+180 q^{7/10}+195 q^{4/5}+240 q^{9/10}+O \left( q \right) \right) + \right. \\
& \left. + \overline{q} \left(\frac{129}{2}+\frac{387 }{2} q^{1/5}+258 q^{3/10}+258 q^{2/5}+ 258 q^{1/2}+\frac{903 }{2} q^{3/5}+774 q^{7/10}+\frac{1677 }{2} q^{4/5}+O \left( q^{9/10} \right) \right) + O(\overline{q}^{3/2}) \right] \\
\end{split}
\end{equation*}

From such expansion we can extract the condition for the minimal value of the quantum dimension at $\langle \CL\rangle =4$.

The product $\CL\CL^\dual$ must act trivially on the whole chiral and antichiral algebra and annihilate any representation that does not satisfy (<ref>). Because the defect with smallest quantum dimension satisfying these properties has dimension $16$, this implies $\langle \CL\rangle=4$ and
^=+∑_2≤i<j≤6 _t_it_j+∑_i=2^6 _t_it_2t_3t_4t_5t_6 .

The defects $\CL^{(26534)}$, $\CL^{(24356)}$, $\CL^{(25463)}\equiv \CL^\dual$ can be obtained by conjugation $\CL_g \CL^{(23645)}\CL_g^\dual$ with suitable invertible defects $\CL_g$, with $g\in \ZZ_2^8:M_{20}$, so that all such defects have dimension $4$, and are the unique simple defects acting with the given automorphism on the chiral algebra.

Let $p_2\in \ZZ_2^8:M_{20}$ be the order $2$ symmetry acting by $p_2=(\one\,\one\,\one\,\one\,\one\,\one\, ;\  \one\,\one\,\one\,\one\,\one\,\one)(35)(46)$, so that $\CL^\dual=\CL_{p_2} \CL\CL_{p_2}$. Then $(\CL_{p_2}\CL)^\dual=\CL_{p_2}\CL$ is unoriented, has dimension $4$, and
(_p_2)^2=+∑_2≤i<j≤6 _t_it_j+∑_i=2^6 _t_it_2t_3t_4t_5t_6 .
This means that $\CN_{23456}:=\CL_{p_2}\CL$ is the duality defect related to the fact that the theory is self-orbifold under the $\ZZ_2^4$ group of symmetries generated by $t_it_j$, $2\le i<j\le 6$.

§ THE $\CN=2$ $C=1$ SUPERCONFORMAL ALGEBRA AS A FREE BOSON.

The $\CN=2$ superconformal algebra at central charge $c_k=\frac{3k}{k+2}$, $k\in \NN$, can be described in terms of a coset $\frac{\widehat{su}(2)_k\oplus \widehat{u}(1)_{4}}{\widehat{u}(1)_{2k+4}}$, which provides the bosonic subalgebra of $\CN=2$. In the particular case of $k=1$, the bosonic subalgebra of the $\CN=2$ algebra, simplifies and becomes essentially the $\hat{u}(1)_{12}$ algebra <cit.>.
Here, $\hat{u}(1)_{2l}$ denotes[Our normalization differs by a factor $2$ from the conventions in <cit.>: the algebra $\hat{u}(1)_{2l}$ in this paper is denoted as $\hat{u}(1)_{l}$ in <cit.>. ] the $c=1$ chiral algebra generated by a single chiral free boson $i\partial X_L(z)=\sum_n \alpha_nz^{-n-1}$ (whose modes generate the Heisenberg algebra), together with the holomorphic vertex operators $V_{n\sqrt{2l}}(z)\sim :e^{in\sqrt{2l}X_L(z)}:$, $n\in \ZZ$, of $\alpha_0$-eigenvalue $n\sqrt{2l}$ and conformal weight $n^2l$. In other words, this is the lattice VOA associated with the lattice $\sqrt{2l}\ZZ$. The irreducible representations $M^l_{[x]}$ of $\hat{u}(1)_{2l}$ are labeled by $[x]\in \ZZ/2l\ZZ$, and the characters are given by
K^2l_[x](τ,z)=∑_Q∈x/2l+ q^lQ^2y^Q/η(τ)
,     y=e^2πi z ,
where $Q$ is the $\alpha_0$-eigenvalue divided by $\sqrt{2l}$ (this strange normalization is justified below). In particular, the ground state of $M^l_{[x]}$ has conformal weight $h= \frac{x^2}{4l}$  and charge $Q=\frac{x}{2l}$, where $x\in\ZZ$ is a representative of $[x]\in \ZZ/2l\ZZ$ in the range $-l+1\le x\le l$.

The $\CN=2$ superconformal algebra at $c=1$ is obtained by adjoining the bosonic algebra $\hat{u}(1)_{12}$ (i.e., $l=6$) with its module $M^6_{[6]}$, which contains the two supercurrents and the other fermionic fields. The charge $Q$ is the $U(1)$ R-charge with the standard normalization such that the supercurrents have charge $\pm 1/2$. In general, the $\CN=2$ modules are given by sums $M^6_{[x]}\oplus M^6_{[x+6]}$ of $\hat{u}(1)_{12}$ modules, and the representations are NS or Ramond depending on whether $[x]$ is even or odd, respectively.
In other words, the $\CN=2$ algebra at $c=1$ can be identified with the lattice SVOA related with the odd lattice $\sqrt{3}\ZZ$. The representations of this superalgebra are given by $R_{[x]}=M^6_{[x]}\oplus M^6_{[x+6]}$ where now $x\in \ZZ/6\ZZ$, and correspond to the lattice cosets  $\frac{x}{2\sqrt{3}}+\sqrt{3}\ZZ\subset \frac{1}{2\sqrt{3}}\ZZ$.

Gepner models are usually described in terms of the coset algebra $(\hat{su}(2)_1\oplus \hat{u}(1)_4)/\hat{u}(1)_{6}$, whose representations are labeled as $[l,m,s]\equiv [1-l,m+3,s+2]$, $l=0,1$, $m\in\ZZ/6\ZZ$, $s\in \ZZ/4\ZZ$ with $l+m+s\equiv 0\mod 2$. As mentioned above, the algebra $(\hat{su}(2)_1\oplus \hat{u}(1)_4)/\hat{u}(1)_{6}$ is isomorphic to $\hat{u}(1)_{12}$, and the respective representations can be identified as follows:

\begin{align*}
&[0,m,0] \quad \equiv \quad M^6_{[m]}\qquad m\in\{0,\pm 2\}\ ,\\
&[0,m,2] \quad \equiv \quad M^6_{[m+6]}\qquad m\in\{0,\pm 2\}\\
&[0,m,1] \quad \equiv \quad M^6_{[m]}\qquad m\in\{3,\pm 1\}\ ,\\
&[0,m,-1] \quad \equiv \quad M^6_{[m+6]}\qquad m\in\{3,\pm 1\}\ .
\end{align*}

In terms of representations of the $\CN=2$ superconformal algebra, one has the following identifications

\begin{align*}
&[0,m,0]\oplus [0,m,2]\quad \equiv \quad R_{[m]}\qquad m\in\{0,\pm 2\}\ ,\\
&[0,m,1]\oplus [0,m,-1]\quad \equiv \quad R_{[m]}\qquad m\in\{3,\pm 1\}\ .
\end{align*}

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