We consider the topological defect lines commuting with the spectral flow and the $\CN=(4,4)$ superconformal symmetry in two dimensional non-linear sigma models on K3. By studying their fusion with boundary states, we derive a number of general results for the category of such defects. We argue that while for certain K3 models infinitely many simple defects, and even a continuum, can occur, at generic points in the moduli space the category is actually trivial, i.e. it is generated by the identity defect. Furthermore, we show that if a K3 model is at the attractor point for some BPS configuration of D-branes, then all topological defects have integral quantum dimension. We also conjecture that a continuum of topological defects arises if and only if the K3 model is a (possibly generalized) orbifold of a torus model. Finally, we test our general results in a couple of examples, where we provide a partial classification of the topological defects.

§ INTRODUCTION

Defects in a quantum field theory (QFT) or in statistical lattice models and quantum spin systems are broadly defined as  inhomogeneities  localized on submanifolds of positive codimension, and  appear in a variety of physical settings, with the codimension $1$ case playing the distinctive role of interfaces between different theories. A more specific notion is the one of topological defects, which are assumed to be invariant under continuous deformations as long as the deformations do not move the inhomogeneities past other defects or field operator insertions.  Historically, in particular in the context of $2d$ QFT,  topological defect lines (TDLs) have been studied in Conformal Field Theories (CFTs), where conformal invariance provides a very restrictive framework, even more so for Rational CFTs (RCFTs),  because of their connection to Boundary Conformal Field Theories (BCFTs), twisted boundary conditions, and orbifolds <cit.>. However most recent developments stem from the realization  that, in QFT with generic dimension $d$, topological defects provide a natural generalization of the notion of group symmetry, with the group elements corresponding to topological defects of codimension $1$ across which the value of field operators jumps by the respective symmetry actions. This picture has naturally led to considering both “higher-form symmetries”, realized by topological defects of higher codimension ( see e.g. <cit.>), and  “non-invertible symmetries”, corresponding to topological defects  which cannot be fused with another topological defect to produce the trivial defect.  In two dimensions, the group structure is in the latter case generalized by the one of fusion categories <cit.>. The classical example of fusion category symmetry is the Ising one, defined by three simple TDLs: the trivial defect, the $\ZZ_2$ Ising symmetry and the  “Kramers-Wannier” duality defect in the Ising model <cit.>, which relates spin correlators in the Ising model at some inverse temperature to disorder/twist correlators at the Kramers-Wannier dual inverse temperature.
TDLs have been widely studied in RCFTs <cit.>, gapped boundaries of $(2+1)D$ topological field theories, and anyon chains, and found to encode nontrivial topological information about a theory, such as constraints on the operator spectrum of CFTs and renormalization group flows in gauge theories <cit.>.  Non-invertible topological defects can also be considered in higher-dimensional theories where they also provide relevant information about the dynamics, so that they are by now considered a tool of paramount importance in unraveling the non-perturbative aspects of Quantum Field Theory.

One of the main hurdles in advancing our knowledge is the fairly limited number of theories where the fusion category of defects or at least a subset thereof are known. This is true even in the simple framework of 2d CFTs.
While in rational CFTs some general techniques have been developed, the much broader realm of non rational theories is largely unexplored. In fact, topological interfaces, which in general separate possibly different theories,  have been studied in a very limited number of examples, in particular for the free boson compactified on a circle and on an orbifold thereof  <cit.> and for $d$-dimensional torus models, where they are assumed to preserve a $\hat{u}(1)^{2d}$ current algebra <cit.>.

In this article, we consider topological defects in two-dimensional superconformal field theories (SCFT) arising as supersymmetric non-linear sigma models with target space a K3 surface. More precisely, we will focus on the defects that preserve the full $\CN=(4,4)$ superconformal symmetry at central charge $c=\tilde c=6$, and that are invariant under the spectral flow transformations that relate the different (NS-NS, NS-R, R-NS, R-R) sectors of the theory. Non-linear sigma models on K3 (or K3 models, for short), provide the simplest examples of Calabi-Yau compactifications in type II string theory. A generic K3 model is not a rational CFT, and it cannot be solved exactly. Nevertheless, due to the large amount of space-time and worldsheet supersymmetries, many general results about these models are known, such as the geometry of the moduli space, the elliptic genus (which is the same for every K3 model), the spectrum of short $\CN=(4,4)$ representations, and even the finite groups of symmetries at each point in the moduli space <cit.>.
For these reasons, K3 models represent the ideal framework to understand topological defects in a non-rational CFT, besides the torus models examples.

Because K3 models are not rational with respect to the $\CN=(4,4)$ superconformal algebra, one can expect infinitely many distinct simple defects. In particular, we will see some examples of K3 models where a continuum of simple non-invertible topological defects arise, a phenomenon that
has already been observed in orbifolds of torus models <cit.>. This implies that, strictly speaking, we are putting ourselves outside of the mathematical framework of fusion categories, at least in its most restrictive definitions. Nevertheless, we will assume that the basic properties of fusion categories still hold for the defects we consider. In particular, we will require that the fusion of any two defects is a superposition of finitely many simple defects. We denote by $\cTop_\calC$ the fusion category (in a broad sense) of topological defects in a K3 model $\calC$  preserving the $\CN=(4,4)$ superconformal algebra and the spectral flow. We will only focus on certain properties of this category, in particular on its fusion ring, on the behaviour of defects when moved past local operators, and on the fusion with boundary states. We will mostly ignore the detailed properties of the  fusion matrices.
The main constraints on the topological defects in $\cTop_\calC$ come from considering the fusion with boundary states representing 1/2 BPS D-branes, i.e. preserving half of the space-time supersymmetry of type II superstring compactified on $\calC$. Such D-branes are necessarily charged with respect to the $U(1)^{24}$ gauge group of R-R ground fields of the theory. One can argue that fusion with defects in $\cTop_\calC$ preserves the set of such boundary states. This means that each defect $\CL\in \cTop_\calC$ can be associated with a $\ZZ$-linear map (endomorphism) $\nL\in {\rm End}(\Gamma^{4,20}_{R-R})$ on the even unimodular lattice $\Gamma^{4,20}_{R-R}$ of D-brane R-R charges. The endomorphism $\nL$ is further constrained by the requirement that the defect cannot mix R-R ground fields belonging to different representations of the $\CN=(4,4)$ algebra. The map $\cTop_\calC \to {\rm End}(\Gamma^{4,20}_{R-R})$ gives rise to a ring homomorphism from the fusion ring of $\cTop_\calC$ the ring ${\rm End}(\Gamma^{4,20}_{R-R})$ (or rather the subring of endomorphisms satisfying suitable properties), so that properties of the former ring can be deduced by studying the latter. One can argue that the property of a defect $\CL$ to be preserved by some marginal deformation of the model depends only on $\nL$.

Using this simple idea, we are able to derive several properties of topological defects in generic K3 models (see in particular section <ref>).  We argue that $\cTop_{\calC}$ is trivial (i.e. the only simple defect is the identity) in most K3 models $\calC$, except a subset with null measure in the moduli space (Claim <ref>). While the map $\cTop_{\calC} \to {\rm End}(\Gamma^{4,20}_{R-R})$ is not injective, we can show that an endomorphism $\nL$ proportional to the identity can only be associated to a superposition of $n$ copies of the identity defect. Given a point $\calC$ in the moduli space of K3 models, we will spell out some necessary conditions for $\cTop_\calC$ to be an integral category, i.e. such that the quantum dimensions of all topological defects are integral (Claim <ref>). In particular, this condition is satisfied for the points in the moduli space  of K3 models that are attractor points for some 1/2 BPS configuration of D-branes <cit.>. We also derive some weaker constraints on the quantum dimensions that are valid everywhere in the moduli space.

While we know some examples of K3 models $\calC$ where $\cTop_\calC$ contains a continuum of topological defects, Claim <ref> implies that this cannot be the generic situation. It is natural to ask for a characterization of K3 models where such a continuum exists. In section <ref>, we conjecture that this only happens for (generalised) orbifolds of torus models.

The main limitations in our approach comes from the fact that the map $\cTop_{\calC} \to {\rm End}(\Gamma^{4,20}_{R-R})$ is, in general, neither injective, nor surjective. In particular, whenever $\cTop_{\calC}$ admits a continuum of defect $\CL_\theta$, parametrised by some real parameter(s) $\theta$, all such defects $\CL_\theta$ are necessarily mapped to the same endomorphism $\nL$. As for surjectivity, while we are able to put some constraints on the endomorphisms $\nL \in {\rm End}(\Gamma^{4,20}_{R-R})$ that arise from a defect $\CL$, we cannot determine precisely what the image of the map $\cTop_{\calC} \to {\rm End}(\Gamma^{4,20}_{R-R})$ is.

The article is structured as follows. In section <ref>, we review some basic facts about topological defects in two dimensional CFTs, and fix the notation that we will use in the rest of the paper. Section <ref> is the core of the article: after reviewing the main properties of K3 models, we describe and prove the main results of our work in sections <ref> and <ref>. We stress that the proofs are on a physics level of rigour, as they are based on various assumptions about K3 models and boundary states that are not mathematically rigorous. This is why we prefer to call such statements `Claim' rather than `Theorem'.  In section <ref> we focus on K3 models that can be described as torus orbifolds. We show that, in general, they admit a continuum of topological defects in $\cTop$. In section <ref>, we conjecture that the converse might be true: generalised torus orbifolds are actually the only K3 models for which such a continuum exists. In sections <ref> and <ref>, we describe some topological defects in $\cTop_\calC$ in a couple of interesting K3 models. While in none of these two models we were able to determine precisely the category $\cTop_\calC$, the examples are useful both to confirm some of the general arguments of section <ref>, and were used in the proof of some of the claims. Finally, in section <ref> we describe some avenues for future investigation. Various technical details of our calculations are relegated in the appendices.

§ GENERALITIES ON TOPOLOGICAL DEFECTS IN 2D CFT

In this section we give a simple and concise review about defects in two dimensional CFTs. We refer to <cit.> for more detailed information.

Usually, when we talk about a generic CFT, we characterize it by specifying the whole set of local operators and their corresponding OPEs.  However, it is well known that in many cases further extended objects associated with non-local operators can also exist in the theory. Such objects encode additional properties of the quantum field theory that are not visible at the level of the spectrum, and they are easily understood in the language of the defects.

In a generic QFT defined over a $d$-dimensional spacetime $\mathcal{M}_d$, such extended objects can be described through operators $\hat{D}_a \left( \mathcal{M}^{d-q} \right)$  supported on $(d-q)-$dimensional submanifolds of $\mathcal{M}_d$, with $q<d$. These operators are called topological in the sense that small deformations of their support manifold, which do not cross other operators of the theory, do not affect the physical observables.

In the limit where the support manifolds of two distinct defects $D_a$ and $D_b$ overlap, the generalized OPE between the corresponding operators defines a fusion algebra among defects of the form:

\begin{equation}
D_a \left( \mathcal{M}^{d-q} \right) \times D_b \left( \mathcal{M}^{d-q} \right) = \sum_c N^c_{ab} D_c \left( \mathcal{M}^{d-q} \right).
\label{fusion_alg}
\end{equation}

The simplest example of defects we can encounter in a QFT are the invertible defects, which encode information about the standard and higher-form global symmetries owned by the theory. More specifically, let $G$ be the group of $p$-form global symmetries in our QFT, then we can associate to each element $g \in G$ a $(d-p-1)-$dimensional topological defect $D_g$ such that the induced fusion algebra (<ref>) satisfies the same group multiplication law as $G$:

\begin{equation}
D_g \left( \mathcal{M}^{d-p-1} \right) \times D_{g'} \left( \mathcal{M}^{d-p-1} \right) =  D_{g''} \left( \mathcal{M}^{d-p-1} \right), \quad \quad \quad g''=gg'.
\end{equation}

The name invertible for this class of defects comes from the fact that for each of them, i.e. $D_g$, there exists a second defect $D_{g^{-1}}$ such that their fusion produces the trivial defect $D_e$ associated with the identity operator:

\begin{equation}
D_g \times D_{g^{-1}} = D_{g^{-1}} \times D_g = D_{e}, \quad \quad \quad D_e \mapsto \mathbb{I}.
\end{equation}

The above definitions apply to any QFT of generic dimension $d$, where topological defects supported on submanifolds of different codimensions may be present. In the rest of the discussion we will focus our attention on 2D QFT, where topological defects have support on oriented $1-$dimensional manifolds (lines) of the 2d spacetime. For this reason, we will refer to them as Topological Defect Lines (TDLs) and denote them with the notation $\mathcal{L}$.

In particular, if $G$ is the symmetry group of our CFT, we can associate to each element $g \in G$ an invertible TDL $\mathcal{L}_g$.

Invertible TDL in 2d associated to the element $g \in G$.

By definition of global symmetry, the elements of the group G define a non-trivial action on the bulk operators:

\begin{equation}
g: \quad \mathcal{O}_i (x_i) \quad \mapsto \quad \rho (g) \cdot \mathcal{O}_i (x_i)
\label{action_g_bulk}
\end{equation}

while they leave the correlators invariant:

\begin{equation}
\forall g \in G \quad \longrightarrow \quad \langle \prod_i \mathcal{O}_i (x_i) \rangle = \langle \prod_i \left( \rho(g) \cdot \mathcal{O}_i (x_i) \right) \rangle.
\end{equation}

We can represent the action (<ref>) in the language of topological defects through the loop contraction of the TDL $\mathcal{L}_g$ encircling the bulk operator $\mathcal{O}_{\psi} (x_i)$ as depicted on the left side of figure <ref>.

(a) Action of $\mathcal{L}_g$ on the bulk operator $\mathcal{O}_{\psi} (x_i)$.

(b) Action of $\CL_g$ on the asymptotic state $\vert \psi \rangle$.

This mechanism is also valid for general TDLs that are not associated with a global symmetry group. From now on we will use the hat notation $\hat{\mathcal{L}}$ to denote the extended operator supported on the line $\mathcal{L}$.

Invertible defects associated with the elements of a continuous global symmetry group $G$ offer the simplest explicit construction of an extended operator supported on an oriented line. In this case the Noether's theorem provides us a set of conserved currents $J_{(r)}$, such that:

\begin{equation}
\langle d \ast  J_{(r)} (x) ... \rangle =0,
\label{conservation_current}
\end{equation}

where the dots denote any operator insertion away from the point $x$. Exponentiating the integral of the Noether's currents on the support line $\CL$:

\begin{equation}
\hat{\CL}_g = e^{i \alpha^{(r)} \int_{\CL} \ast J_{(r)}}.
\end{equation}

we get the extended operator $\hat{\CL}_g$ associated with the element $g \in G$ specified by the group parameters $\left\lbrace \alpha^{(r)} \right\rbrace$.
The operator $\hat{\mathcal{L}}_g$ is topological due to the conservation law in (<ref>).

Similarly, we can define extended operators associated with elements of discrete symmetry groups satisfying the same above proprieties.

Beyond these objects, we can equip our theory with other point like operators where TDLs can terminate or join. In the first case we can associate to each TDL $\mathcal{L}$ the space $\mathcal{H}_{\mathcal{L}}$ of possible point like operators on which the line $\mathcal{L}$ can end. If $\mathcal{H}_{\mathcal{L}} \neq \emptyset$, then the line $\mathcal{L}$ is said endable, and the point like operators of $\mathcal{H}_{\mathcal{L}}$ are called defect operators.

§.§ Defining proprieties of Topological Defect Lines

Let us now focus on the case of topological defects $\CL(\gamma)$ supported on lines $\gamma$ in unitary Euclidean two dimensional CFTs.  In the same line of the Introduction, we can think about topological defects as a generalization of global symmetries. As we spell out below, the set of such defects is equipped with a composition law (<ref>) that is generally non-invertible. This means that one cannot define on this set the standard group structure, as for ordinary symmetries, but rather a fusion category.

In this subsection we will summarize the basic properties that we need in order to specify this structure.

As stated above, topological defect lines (TDLs) $\CL$ represent the fundamental objects of the structure. They are supported on oriented lines $\gamma$ on the worldsheet. On the set of distinct defects $\{\CL\}$ is defined an involution $\CL \mapsto \CL^\dual$, shown in figure <ref>(a), that corresponds to inverting the orientation of the support. A defect $\CL$ is unoriented if $\CL^\dual =\CL$.

Correlation functions with the insertions of topological defects are invariant under deformation of the support line, as long as the line is not moved past another operator insertion. We will say that a defect $\CL$ and a local operator $\phi$ are `transparent' to each other, or that $\CL$ preserves $\phi$, if the support $\CL$ can be moved past the support of $\phi$ without changing any correlation function. In general, the holomorphic and anti-holomorphic stress tensors $T(z)$ and $\tilde T(\bar z)$ are always preserved by any topological defects. In the following sections, we will consider topological defects that preserve all supercurrents generating the $\CN=(4,4)$ superconformal algebra of a K3 sigma model.

The set of TDLs in a 2d QFT always includes the Identity defect $\mathcal{I}$, typically represented by the dotted lines as depicted in figure <ref> (b). The insertion or removal of the identity defect does not change any correlation function; equivalently, $\CI$ is transparent to all local operators of the theory.

(a) Involution map $\mathcal{L} \mapsto \CL^\dual$.
(b) Identity operator in the trivial line.

Let us consider the two-dimensional CFT on the cylinder $S^1\times \RR$. A TDL $\CL$ inserted along the circle $S^1$ defines a linear operator $\hat \CL:\Hh\to \Hh$ on the Hilbert space $\Hh$ of states on $S^1$ (see figure <ref>.b). Equivalently, we can consider a closed line $\CL$ encircling the insertion point of a local operator $\phi(z,\bar z)$, corresponding to a state $\phi \in \Hh$. By shrinking the circle around the point $z$ we obtain a new local operator $(\hat \CL\phi)(z,\bar z)$.[To be precise, the definition of $\hat\CL$ on the sphere might differ from the definition of the cylinder by a phase, see section 2.4 in <cit.>. In this case, we reserve the notation $\hat \CL$ for the operator defined on the cylinder.]

On the other hand, inserting the defect line $\mathcal{L}$ along the Euclidean time direction $\RR$ of $S^1\times \RR$, correspond to modifying the space of states $\Hh$. We denote by $\mathcal{H}_{\mathcal{L}}$ the new Hilbert space of states on the circle $S^1$ that are `twisted' by the defect $\CL$. For example, if $\mathcal{L}$ is an invertible defect associated with a symmetry $g \in G$, then $\mathcal{H}_{\mathcal{L}}$ is simply the g-twisted sector of the theory.

The state/operator correspondence defined by the conformal mapping from the cylinder $S^1\times \RR$ to the plane $\CC\setminus \{0\}$, allows us to identify every state $\vert \psi \rangle \in \mathcal{H}_{\mathcal{L}}$ on the cylinder with a non-local defect operator $\mathcal{O}_{\psi}$ on the plane, i.e. an operator  attached to an outgoing defect $\CL$, as shown in figure <ref>.

An endable TDL in the cylinder (figure a) produces a non-local operator in the complex plane (figure b).

Using this identification, we will refer to $\mathcal{H}_{\mathcal{L}}$ indifferently as the space of $\CL$-twisted states or as the space of $\CL$ defect operators. When $\CL=\CI$ we recover the ordinary space $\Hh_{\CI}=\Hh$ of local point-like operators of the CFT.

If the defect $\CL$ is transparent for a certain set of local (anti-)holomorphic operators, then the space $\Hh_{\CL}$ is a representation of the corresponding (anti-)chiral algebra. In particular, $\Hh_{\CL}$ is always a representations of the holomorphic and antiholomorphic Virasoro algebra.  More generally, the space $   \Hh_{\CL}$ is a $\CL$-twisted representation of the algebra of local operators on the cylinder. This means that for every $\phi\in \Hh$, $\phi(z,\bar z)$ defines a linear operator on the Hilbert space $\Hh_\CL$ that obeys the OPE relations with other local operators. By $\CL$-twisted, we mean that correlation functions on $S^1\times \RR$ with the insertion of a point operator $\phi(z,\bar z)$ and a defect line $\CL$ along $\RR$ are not quite periodic as $z$ moves around $S^1$, but have some non-trivial discontinuities at the support of the defect that depend on $\CL$. A defect $\CL$ is simple if the space $\Hh_{\CL}$ is irreducible as a twisted representation of the algebra of local operators.

When the CFT is defined on a torus $S^1\times S^1$, the modular S-transformation exchanges the insertion of a line $\CL$ along the `space' circle with the insertion along the Euclidean `time' circle. This establishes a relation between the linear operator $\hat\CL$ on $\Hh$ and the twisted space $\Hh_\CL$.

The $2$-point correlation functions on the sphere with two defect operators $\phi(z,\bar z),\psi(w,\bar w)$ connected by a defect line $\CL$ define a natural non-degenerate bilinear pairing $(\phi,\psi)$ between $\Hh_\CL$ and $\Hh_{\CL^\dual}$. The bilinear pairing is related to the hermitian product on the Hilbert space $\Hh_{\CL}$ by a anti-linear involution $\iota:\Hh_{\CL}\to \Hh_{\CL^\dual}$, such that $\langle \phi_1|\phi_2\rangle=(\iota(\phi_1),\phi_2)$.

The set of TDLs is endowed with the algebraic structure of a generally non-commutative (semi-)ring defined by the two operations of direct sum (or superposition) ($+$) and fusion ($\triangleright$).

The direct sum allows to associate to each pair of topological defects $\mathcal{L}_a$ and $\mathcal{L}_b$ a third defect $\mathcal{L}_a + \mathcal{L}_b$ such that:

\begin{equation}
\mathcal{H}_{\mathcal{L}_a + \mathcal{L}_b} = \mathcal{H}_{\mathcal{L}_a} \oplus \mathcal{H}_{\mathcal{L}_b}.
\end{equation}

This first operation is associative and commutative. Unitary CFTs are expected to be semi-simple, i.e. every defect $\CL$ can be written as a superposition of simple defects
=∑_simple _i n_i_i ,
for some non-negative multiplicities $n_i\in \ZZ_{\ge 0}$. Correspondingly, each reducible $\Hh_\CL$ can be decomposed into a direct sum of irreducible components $\Hh_\CL=\oplus_i n_i \Hh_{\CL_i}$.

If there are no additional intermediate insertions between two TDLs $\mathcal{L}_1$ and $\mathcal{L}_2$, we can define the fusion deforming one defect into the other, as shown in fig. <ref>. The resulting object is again a topological defect line denoted as:

\begin{equation}
\mathcal{L}_1 \triangleright  \mathcal{L}_2 = \mathcal{L}_1 \mathcal{L}_2.
\end{equation}

Fusion defines a notion of tensor product between defect operator spaces
__1⊗__2=_ℒ_1 ℒ_2
This second operation is associative but it is in general not commutative. The fusion of two simple defects $\CL_i$ and $\CL_j$  is not necessarily simple, so that one has a decomposition
_i_j=∑_simple _k N_ij^k _k ,
for some fusion coefficients $N_{ij}^k\in \ZZ_{\ge 0}$. Formally, this means that the set of defects has the structure of a fusion ring, with the simple defects playing the role of a distinguished basis.

Sets of defects that also satisfy commutativity for the fusion form commutative rings.

Fusion of two TDLs.

The trivial line $\mathcal{I}$ is the neutral element under fusion

\begin{equation}
\mathcal{L} \mathcal{I} = \mathcal{I} \mathcal{L} = \mathcal{L}.
\end{equation}

In analogy with the construction of the spaces $\Hh_\CL$, we can consider $k$ parallel defect lines $\CL_1,\ldots,\CL_k$ inserted in the time-like direction on the cylinder. The corresponding Hilbert space $\CH_{\CL_1,\ldots,\CL_k}$ of states on $S^1$ is identified, via the state/operator correspondence, with the space of  k-junction operators. As a convention, we choose that all the involved lines are outgoing from the junction. Once again, $\CH_{\CL_1,\ldots,\CL_k}$ are genuine representations of the chiral and anti-chiral algebras preserved by the defects, and a suitable twisted representation of the algebra of local operators.

By moving the parallel lines $\CL_1,\ldots,\CL_k$ on the cylinder $S^1\times \RR$ very close to each other, we get the identification
__1,…,_k≅__1⋯_k≅__1⊗⋯⊗__k ,
between the $k$-junction space $\Hh_{\CL_1,\ldots,\CL_k}$ and the space of defect operators of the fusion $\CL_1\cdots\CL_k$.

The subspace $V_{\CL_1,\ldots,\CL_k}\subset \Hh_{\CL_1,\ldots,\CL_k}$ of states with conformal weights $(0,0)$ correspond to junction operators that are themselves topological, i.e. such that the junction point can be moved without changing a correlation function (as long as the insertion point is not moved past the support of some other operator). We can restrict ourselves to consider the correlation functions where all the $k$-junctions with $k>1$ are topological, because all the other junction operators can be obtained by suitable OPE with local operators.

For a simple defect $\CL$, the only topological $\CL$-twisted operator is the vacuum operator for $\CL=\CI$, so that
1     for $\CL=\CI$

0     for simple $\CL\neq\CI$ .

For $k>1$, a topological operator $u\in V_{\CL_1,\ldots,\CL_{k}}$ can also be interpreted as a linear map $u:\Hh_{\CL^\dual_k }\to \Hh_{\CL_1,\ldots,\CL_{k-1}}$ that is a homomorphism of twisted representations of the algebra of local operators. In particular, $V_{\CL,\CL^\dual}={\rm Hom}(\Hh_{\CL},\Hh_{\CL})$ has always dimension at least $1$, because it contains (multiples) of the identity map $\mathbf{1}_\CL:\Hh_\CL\to \Hh_\CL$.
A defect $\CL$ is simple if and only if $\dim V_{\CL,\CL^\dual}=1$. Furthermore, for two simple defects $\CL$ and $\CL'$,
$$ \dim V_{\CL',\CL^\dual}=\begin{cases}
1 & \text{if }\CL=\CL'\\
0 & \text{otherwise}
\end{cases} .$$

As for topological $3$-junctions, one can prove that if $\CL_i$, $\CL_j$, and $\CL_k$ are simple defects, then the dimension of topological junction operators is exactly the fusion coefficient
V__i,_i,_k^=N_ij^k .
This fits with the idea that $V_{\CL_i,\CL_i,\CL_k^\dual}$ is the space ${\rm Hom}(\Hh_{\CL_k},\Hh_{\CL_i\CL_j})$ of morphisms from $\Hh_{\CL_k}$  to $\Hh_{\CL_i\CL_j}=\oplus_{l} N_{ij}^l\Hh_{\CL_l}$. Notice that for $\CL$ simple, one can think of the identity $2$-junction $1\in V_{\CL,\CL^\dual}$ as a $3$-junction with the identity defect. As a consequence, $\CL$ is simple if and only if $\dim V_{\CL,\CL^\dual,\CI}=1$, i.e. if and only if $\CI$ appears with multiplicity $1$ in the fusion of $\CL$ with its dual
^=+… .
Here, $\ldots$ denotes a sum with non-negative multiplicities over simple defects distinct from the identity.

At this point it is important to emphasize that the set of topological defect lines equipped by fusion multiplication in general does not form a group. The reason is the absence of an inverse element associated to each TDL. Only a subclass of all possible TDLs admit an inverse under fusion. They are the invertible TDLs, and they form a group with respect to this operation.

The remaining TDLs are called non-invertible and, equipped with the fusion multiplication and the direct sum, they form a more complicated algebraic structure named Fusion Ring.

A fundamental quantity that we can associate with each TDL $\CL$ is the quantum dimension $\langle \CL \rangle\equiv \langle \CL\rangle_{S^1\times \RR} $, defined as the vacuum expectation value of a defect $\CL$ wrapping the circle $S^1$ on the cylinder:

\begin{equation}
\langle \CL \rangle\equiv \langle \CL\rangle_{S^1\times \RR} := \langle 0 \vert \hat{\CL} \vert 0 \rangle.
\end{equation}

Using the modular invariance properties of the partition function with the defect $\CL$ inserted, it is easy to prove that for unitary theories with a unique vacuum the quantum dimension is bounded from below:

\begin{equation}
\langle \CL \rangle \geq 0.
\end{equation}

Such constraint is more restrictive when we consider a unitary, compact CFT, where the condition becomes:

\begin{equation}
\langle \CL \rangle \geq 1.
\end{equation}

Notice that the quantum dimension is also the absolute value of the vacuum expectation value $\langle \CL\rangle_{S^2}$ on the sphere, defined by considering an loop encircling only the vacuum on $S^2$
$$|\langle \CL\rangle_{S^2}|=\langle \CL \rangle\ ,$$
but in general the phase might be different.
The quantum dimensions provide a $1$-dimensional representation of the fusion ring, so that, in particular,
⟨_i⟩⟨_j⟩=∑_k N_ij^k ⟨_k⟩ ,
for any simple $\CL_i,\CL_j,\CL_k$. Together with the condition $\langle \CL \rangle \geq 1$, this means that for every simple $\CL_i$, $\CL_j$ of finite quantum dimension, there are only finitely many non-zero fusion coefficients $N_{ij}^k$. In this article, we only consider defects with finite quantum dimension; see <cit.> for a discussion about more general possibilities.

Equivalent configurations under the action of the fusion matrices $F^{\CL_1 , \CL_3}_{\CL_2 , \CL_4}$.

The topology of a network of defects can be modified using the fusion rules shown in fig. <ref>, where the fusion matrices $F_{\CL_2,\CL_3}^{\CL_1,\CL_4}(\CL_5,\CL_6)$ map the topological junctions $V_{\CL_1 ,\CL_2, \CL_5^\dual} \otimes V_{\CL_5 ,\CL_3 , \CL_4}$ in $V_{\CL_1 , \CL_6 , \CL_4 } \otimes V_{\CL_2 , \CL_3 , \CL_6^\dual}$. In particular,
$F^{\CL,\CL^\dual}_{{\CL^\dual}, \CL}(\CI,\CI)$ maps ${\bf 1}\otimes {\bf 1}$ to $\frac{1}{\langle \hat \CL\rangle_{S^2}}{\bf 1}\otimes {\bf 1}$, see fig.<ref>.

Equivalent configurations under the action of the fusion move. In the second term on the right: the red defect is the superposition (with multiplicity) of the simple defects $\CL_k\neq \CI$ appearing in the fusion $\CL\CL^\dual$; the topological $3$-junction operators are given by the images of ${\bf 1}\otimes {\bf 1}$ by matrices $F^{\CL , \CL^\dual}_{\CL^\dual , \CL}(\CI,\CL_k)$.

This can be used to determine how the correlation function is modified when a simple defect $\CL$ is moved past a local operator $\phi(z,\bar z)$, see figure <ref>. In particular, $\CL$ is transparent to $\phi$ if and only if $\hat\CL(\phi)=\langle \CL\rangle \phi$, because in this case one can prove that the term $\hat\CL^v(\phi)$ in figure <ref> vanishes. We will use this property repeatedly in the following.

Moving of a simple defect line through the local operator $\phi (z, \overline{z})$. The defect in the second term on the right is a superposition of all simple defect lines $\CL_i\neq \CI$ appearing in the fusion $\CL\CL^\dual=\CI+\ldots$. The factor $1/\langle \CL\rangle$ is determined by observing that the defect must be transparent to the vacuum operator and to its descendants.

Formally, the properties of TDLs described in this section can be formulated in terms of a fusion category, where the defects $\CL$ are the objects and the topological junctions $u\in V_{\CL_1,\CL_2^\dual}\equiv {\rm Hom}(\CL_2,\CL_1)$ are the morphisms. Sometimes in the definition of fusion categories one requires that the number of simple objects is finite. This condition might be in general violated when a CFT is not rational with respect to the chiral and anti-chiral algebra preserved by the topological defects, as will be the case in this article.

§ TOPOLOGICAL DEFECTS IN K3 MODELS

In this section, after a short review of non-linear sigma models on K3, we discuss the general properties of topological defects in such models.

§.§ General properties of K3 models

Let us review some of the main properties of supersymmetric non-linear sigma models on K3 (or K3 models, for short), and fix our notation for the rest of the article. See <cit.> for more information.

All K3 models contain a holomorphic and an anti-holomorphic copy of the small $\CN=4$ superconformal algebra at central charge $c=6$ <cit.>. The bosonic subalgebra of $\CN=4$ is generated by the stress-tensor and the three currents in a $\widehat{su}(2)_1$ current algebra. The $SU(2)$ group generated by the zero modes of such holomorphic currents is the R-symmetry group, and the four supercurrents belong to two R-symmetry doublets.
The unitary representations of the $\CN=4$ algebra at $c=6$ can be labeled by a pair $(h,q)$, where $h$ is the conformal weight and $q\in\{0,1/2\}$ the $SU(2)$ highest weight of the ground states in the representation; similarly, we will denote by $(h,q;\tilde h,\tilde q)$ the representations of the full (holomorphic and antiholomorphic) $\CN=(4,4)$ algebra at $(c,\tilde c)=(6,6)$. The Neveu-Schwarz (NS) version of the $\CN=4$ algebra contains two unitary short (BPS) representations $(h,q)=(0,0)$ and $(h,q)=(\frac{1}{2},\frac{1}{2})$ and infinitely many long (massive) representations $(h,0)$ with $h>0$. The spectral flow automorphism relating the NS and the Ramond (R) versions of $\CN=4$ maps the NS shorts representations $(0,0)$ and $(\frac{1}{2},\frac{1}{2})$ to the short R representations $(\frac{1}{4},\frac{1}{2})$ and $(\frac{1}{4},0)$, respectively, and the long $(h,0)$ with $h>0$ to the long R representation $(h+\frac{1}{4},\frac{1}{2})$.

Every K3 model contains a single $(0,0;0,0)$ representation of $\CN=(4,4)$ (the vacuum and its descendants), and $20$ $(\frac{1}{2},\frac{1}{2};\frac{1}{2},\frac{1}{2})$ representations. There are no fields in the $(\frac{1}{2},\frac{1}{2};0,0)$ or $(0,0;\frac{1}{2},\frac{1}{2})$; such holomorphic and anti-holomorphic free fermions are a characteristic feature of  supersymmetric sigma models on $T^4$, where they are part of a `large' $\CN=(4,4)$ superconformal algebra at $(c,\tilde c)=(6,6)$. Besides these short-short (BPS) representations, a K3 model contains infinitely many different short-long, long-short and long-long representations, with multiplicities depending on the particular model. It is believed that the $\CN=4$ superconformal algebra is the full chiral algebra of a generic K3 model, that is therefore not a rational CFT.
At special points in the moduli space of K3 models, the chiral algebra might be extended so that the CFT is rational.

K3 models are invariant under spectral flow exchanging the NS-NS sector and R-R sector. This implies that the R-R sector contains a single $(\frac{1}{4},\frac{1}{2};\frac{1}{4},\frac{1}{2})$  and $20$ $(\frac{1}{4},0;\frac{1}{4},0)$ representations, as well as infinitely many $(h,\frac{1}{2};\tilde{h},\frac{1}{2})$, with $h+\tilde h>1/2$. The four R-R ground fields of $(\frac{1}{4},\frac{1}{2};\frac{1}{4},\frac{1}{2})$ transform in a $(\bf{2},\bf{2})$ representation of the holomorphic and anti-holomorphic $SU(2)\times SU(2)$ R-symmetry group; the OPE with such fields generate the spectral flow between the NS-NS and the R-R sectors. For this reason, we will refer to such fields as the spectral flow generators.

It is often useful to think of the K3 model as the internal CFT in a six dimensional compactification of type IIA superstring on $\RR^{1,5}\times K3$. In this case, one should also consider the NS-R and R-NS sectors of the K3 sigma model, tensored with the corresponding sectors of the space-time and superghost CFTs, and with the correct GSO projections. The NS-R and R-NS sectors are also related to the NS-NS and R-R sectors by spectral flow, so that they contain a single $(\frac{1}{4},\frac{1}{2};0,0)$ and $(0,0;\frac{1}{4},\frac{1}{2})$ representations. The corresponding physical string states are the space-time gravitinos, whose zero modes is associated with the space-time $\CN=(1,1)$ supersymmetry in six dimensions.

In general, sigma models on K3 and on $T^4$ are the only known (and, conjecturally, the only consistent) unitary SCFTs  with $\CN=(4,4)$ superconformal algebra at $(c,\tilde c)=(6,6)$ and whose spectrum is invariant under spectral flow.

Every K3 model admits deformations that preserve the full $\CN(4,4)$ superconformal symmetry, corresponding to an $80$-dimensional space of exactly marginal operators contained in the $20$ NS-NS $(\frac{1}{2},\frac{1}{2};\frac{1}{2},\frac{1}{2})$  representations of $\CN = (4,4)$. There are compelling arguments suggesting that conformal perturbation by any such operator converges in a neighborhood of each model; we will assume that this is true.
The corresponding $80$-dimensional moduli space $\CM_{K3}$ of K3 models is given by a quotient
where $O(4,20,\ZZ)$ is the integral orthogonal group. Here, the Teichmüller space $\CT_{K3}$ is an open subset in the  Grassmannian
parametrising positive definite four-dimensional subspaces $\Pi$ within the real space $\RR^{4,20}$ with signature $(4,20)$. The moduli space $\CM_{K3}$ admits the following physical interpretation: $O(4,20,\ZZ)$ is the T-duality group, and can be identified with the group of automorphisms of the lattice $\Gamma^{4,20}$ of D-brane charges, which is an even unimodular lattice of signature $(4,20)$. One can think of this lattice (or rather its dual) as being embedded in the $24$-dimensional real space $V$ of (CPT self-conjugate) Ramond-Ramond ground states with conformal weights $h=\tilde h=\frac{1}{4}$:
Γ^4,20⊂V:={R-R ground states}≅Γ^4,20⊗≅^4,20 .
The space $V$ contains a positive definite subspace V⊃Π:={spectral flow generators}≅^4,0 , spanned by the four spectral flow generators, i.e. the R-R ground states in a $(\frac{1}{4},\frac{1}{2};\frac{1}{4},\frac{1}{2})$ representation of $\CN=(4,4)$.  The orthogonal complement $\Pi^\perp\subset V$ is the space of R-R ground states in the $20$ $\CN=(4,4)$ representations $(\frac{1}{4},0;\frac{1}{4},0)$ V⊃Π^⊥:={states in $(\frac{1}{4},0;\frac{1}{4},0)$ representations}≅^0,20 .Then, $\CM_{K3}$ is essentially the Grassmannian of the four-dimensional subspaces $\Pi\cong\RR^{4,0}$ within $V\cong \Gamma^{4,20}\otimes \RR$, modulo lattice automorphisms (T-dualities) $O(4,20,\ZZ)\cong O(\Gamma^{4,20})$. To be precise, one needs to exclude some points in this Grassmannian, where the CFT is believed to be inconsistent <cit.>. From a string theoretical point of view, these are the points in the moduli space of type IIA superstrings where some D-brane becomes exactly massless, so that the perturbative description breaks down even at small string coupling.

Henceforth, we will denote by
_Π:=K3 model corresponding to Π⊂V≅Γ^4,20⊗the K3 model corresponding to a choice of $\Pi\subset V$, i.e. to a point in the Grassmannian $\CT_{K3}$.

§.§ Symmetries and topological defects

The main focus of this article are the topological defects in non-linear sigma models of K3. A generic topological defect commutes only with the chiral and anti-chiral Virasoro algebra, i.e. it is `transparent' only to the holomorphic and antiholomorphic stress-energy tensor $T(z)$ and $\tilde T(\bar z)$. In this article, we will focus on a subcategory of topological defects $\CL$ that satisfy some further constraints, namely:

* They commute with the full $\CN=(4,4)$ superconformal algebra, i.e. they are transparent to all supercurrents and to the $\widehat{su}(2)_1$ R-symmetry currents, besides the stress-energy tensor;

* They commute with spectral flow generators. This implies that they are transparent to the four R-R ground fields in the $(\frac{1}{4},\frac{1}{2};\frac{1}{4},\frac{1}{2})$ representation of $\CN=(4,4)$. When the K3 model is the internal SCFT in a full type IIA compactification, we also require the defect to be transparent with respect to the NS-R and R-NS ground fields corresponding to the space-time gravitini. These fields generate the purely holomorphic or purely anti-holomorphic spectral flows.

For a K3 model $\calC\equiv \calC_\Pi$, corresponding to a choice of a four dimensional positive definite subspace $\Pi\subset \Gamma^{4,20}\otimes \RR$, we denote by
_≡_Πthe category of topological defects of $\calC_\Pi$ satisfying the properties 1 and 2.

Properties 1 and 2 lead to a number of important consequences:

* As explained in section <ref>, each topological defect $\CL$ is associated with a linear operator
:→on the Hilbert space of states on the circle $S^1$ (or, equivalently, the space of local point-like operators) of the K3 model $\calC$. The action of $\hat\CL$ is defined  by inserting a defect $\CL$ along the circle $S^1$ on a cylinder  $S^1\times \RR$. Because of the properties 1 and 2, this operator commutes with the $\CN=(4,4)$ algebra and the spectral flow.  Therefore, it maps $\CN=4$ primaries into $\CN=4$ primaries in the same representation. Furthermore, because the defect $\CL$ is transparent with respect to the spectral flow generators, once the action of $\hat\CL$  is known on one of the sectors ($\Hh_{NS-NS}$, $\Hh_{R-R}$, $\Hh_{R-NS}$ or $\Hh_{NS-R}$), then it is uniquely determined in all the other sectors as well.

* In general, the properties of topological defects in fermionic CFTs are more complicated than the ones in purely bosonic ones. In particular, fermionic CFTs can contain topological defects $\CL$ of `q-type', that admit topological $2$-junctions $\psi\in V_{\CL,\CL^\dual}$ with odd fermion number $(-1)^{F_L+F_R}=-1$, see <cit.>. However, none of the topological defects in the category $\cTop$ is of q-type. Indeed, in the K3 models we consider, the holomorphic and anti-holomorphic fermion numbers $(-1)^{F_L}$ and $(-1)^{F_R}$ can be identified with the central $\ZZ_2$ elements in the holomorphic and anti-holomorphic $SU(2)$ R-symmetry group. The topological defects $\CL\in \cTop$ are transparent to the $\widehat{su}(2)_1$ currents in the $\CN=(4,4)$ algebra, and, as a consequence, they commute with the $SU(2)$ R-symmmetry groups generated by their zero modes, and therefore with the fermion numbers $(-1)^{F_L}$ and $(-1)^{F_R}$. Furthermore, all spaces of defect and junction operators are representations of the $\CN=(4,4)$ superconformal algebra, and the fermion number of each state is determined by their $SU(2)$ R-symmetry charges. In particular, topological operators with $h=0=\tilde h$ have zero R-symmetry charges, and therefore they are always bosons.

In fact, each $\CL\in \cTop_\calC$ induces a topological defect in the purely bosonic CFT $\calC^{bos}$ obtained by a type $0$ GSO projection, i.e. by including only the NS-NS and R-R fields of the K3 model $\calC$ with positive fermion number. Furthermore, all properties of the topological defect $\CL$ in the original supersymmetric model $\calC$ are completely determined in terms of the action of the operator $\hat \CL$ in the bosonic model $\calC^{bos}$.
Note, however, that the definition of the category $\cTop_\calC$ is much more natural in the supersymmetric setup. In particular, we do not know whether the condition of preserving the $\CN=(4,4)$ superconformal algebra admits an equivalent formulation in the bosonic model $\calC^{bos}$.

* With each defect $\CL$ is associated a $\CL$-twisted space of states $\Hh_{\CL}$, with different sectors $\Hh_{\CL}^{NS-NS}$, $\Hh_{\CL}^{R-R}$, $\Hh_{\CL}^{R-NS}$, $\Hh_{\CL}^{NS-R}$. When $\CL\in \cTop_\calC$, each of these sectors decomposes into representations of the $\CN=(4,4)$ superconformal algebra. Furthermore, the sectors are related to one each other by spectral flow.

Topological defects that are invertible form  the group of symmetries of the K3 model. In <cit.>, all group of symmetries commuting with the $\CN=(4,4)$ algebra and the spectral flow generators have been classified. In particular, consider a K3 model $\calC_\Pi$ in the moduli space $\CM_{K3}$, corresponding to the choice of the positive definite four-dimensional subspace $\Pi$ of spectral flow generators in the space of RR ground fields $V\cong \Gamma^{4,20}\otimes  \RR$. Then,  the group $G_\Pi$ of symmetries of $\calC_\Pi$ satisfying 1 and 2 is isomorphic to the subgroup ${\rm Stab}(\Pi)$ of $O(\Gamma^{4,20})\cong O(4,20,\ZZ)$ fixing $\Pi$ pointwise <cit.>.

Let us revisit the argument that led to this result, and then discuss to what extent such argument can be generalized to the case of topological defects.
Every symmetry $g\in G_\Pi$ maps 1/2 BPS boundary states to 1/2 BPS boundary states, and therefore maps the lattice $\Gamma^{4,20}$ of RR charge vectors into itself. The map must be linear and preserve the bilinear form of the lattice – indeed, the bilinear form is a Witten index counting the Ramond ground states for open strings suspended between two D-branes, and is invariant under the action of $G_\Pi$. Therefore, every $g\in G_\Pi$ induces an automorphism of the lattice $\Gamma^{4,20}$. Furthermore, by property 2, the induced action on RR ground states must act trivially on the spectral flow generators in $\Pi$. We conclude that there is a homomorphism $\rho:G_\Pi \to {\rm Stab}(\Pi)\subset O(\Gamma^{4,20})$. Then one proves that such homomorphism is both injective and surjective. To show surjectivity, one notices that $O(\Gamma^{4,20})\cong O(4,20,\ZZ)$ is the T-duality group, and ${\rm Stab}(\Pi)$ is a subgroup of dualities mapping the model $\calC_\Pi$ into itself, i.e. self-dualities. But all self-dualities are symmetries of the model, so they must correspond to some $g\in G_\Pi$. As for injectivity, let $K_\Pi\subseteq G_\Pi$ be the kernel of $\rho$. Then $K_\Pi$  acts trivially on $\Gamma^{4,20}$, and, by linearity, on all RR ground states. But the RR ground states in the twenty $(\frac{1}{4},0;\frac{1}{4},0)$ representations of $\CN=4$ are related by spectral flow to the $80$-dimensional space of exactly marginal operators in the NS-NS sector. Thus, the symmetries in $K_\Pi$ act trivially on all such exactly marginal operators, and as a consequence they are not broken by any deformation of the model. Given that the moduli space $\CM_{K3}$ is connected, we conclude that the kernel of $\rho$ is the same group $K_\Pi\equiv K$ for all K3 models. At this point, one just needs to consider a simple example of non-linear sigma model on K3, where the group $K$ can be explicitly described – for example the model considered in section <ref>. It turns out that $K$ is trivial in that model, and therefore is trivial everywhere in the moduli space $\CM_{K3}$. We conclude that $\rho:G_\Pi \to {\rm Stab}(\Pi)$ is an isomorphism. In <cit.>, it was then proved that every group of the form ${\rm Stab}(\Pi)$ is isomorphic to a subgroup of the Conway group $Co_0$, the group of automorphisms of the Leech lattice $\Lambda$, fixing a sublattice of $\Lambda$ of rank at least $4$. All subgroups of $Co_0$ that are lattice stabilizers were classified in <cit.>.

Let us now discuss how a similar argument could be generalized to a classification of topological defects $\CL\in \cTop$. As described in section <ref>, the fusion of a boundary state $||\alpha\rrangle$ and a defect $\CL$ yields a new boundary state $||\CL\alpha\rrangle$. In particular, the defects $\CL\in\cTop$  preserve the space-time supersymmetry, so they map 1/2 BPS D-branes into 1/2 BPS D-branes, and RR charge vectors to RR charge vectors. Therefore, we have a map

\begin{align}\label{homtoend}
\cTop_\Pi &\to {\rm End}(\Gamma^{4,20})\\
\CL&\mapsto \nL\notag
\end{align}

that assigns a $\ZZ$-linear function $\nL:\Gamma^{4,20}\to\Gamma^{4,20}$ to each defect $\CL\in\cTop_\Pi$. This map is compatible with fusion product, i.e. it gives rise to a ring homomorphism from the fusion ring of $\cTop_\Pi$ to ${\rm End}(\Gamma^{4,20})$. The extension of $\nL$  by linearity to the real space of R-R ground fields $V\cong \Gamma^{4,20}\otimes \RR$ coincides with the restriction $\hat \CL_{|V}\in {\rm End}_{\RR}(V)$ of the linear operator $\hat \CL$ to $V$, _|V:V→V .To summarize:

(a) The restriction $\hat\CL_{|V}:V\to V$ maps $\Gamma^{4,20}\subset V$ into $\Gamma^{4,20}$, i.e. it is the extension by $\RR$-linearity of some lattice endomorphism $\nL:\Gamma^{4,20}\to \Gamma^{4,20}$.

Henceforth, we use the symbol $\nL$ to denote both maps $\nL:\Gamma^{4,20}\to\Gamma^{4,20}$ and $\hat \CL_{|V}:V\to V$. Because $\hat\CL$ commutes with the $\CN=(4,4)$ algebra, it cannot mix RR ground fields in different representations. Furthermore, the condition that the spectral flow generators are transparent with respect to $\CL$ implies that the map $\hat\CL$ acts on them in the same way as on the vacuum, i.e. by multiplication by the quantum dimension $\langle \CL\rangle\ge 1$. The following property then follows:

(b) $\nL:V\to V$ is block-diagonal with respect to the orthogonal decomposition $V=\Pi\oplus \Pi^\perp$, i.e. $\nL(\Pi)\subseteq \Pi$ and $\nL(\Pi^\perp)\subseteq\Pi^\perp$. Furthermore, the restriction $\nL_{\rvert \Pi}$ is proportional to the identity $\nL_{\rvert\Pi}=\langle \CL\rangle{\rm id}_\Pi$, where $\langle \CL\rangle\ge 1$ is the quantum dimension of the defect.

It is useful to introduce the real vector space
d·1_4×4     0

0     b_20×20
)|d∈, b_20,20∈Mat_20×20()}of block diagonal real $24\times 24$ matrices, with a $4\times 4$ upper left-corner proportional to the identity and an unconstrained $20\times 20$ lower-right block, and its subset

d·1_4×4     0

0     b_20×20
)|d≥1, b_20,20∈Mat_20×20()}⊂B^4,20() .Upon choosing a suitable orthonormal basis of $V$, compatible with the splitting $V\cong \Pi\oplus\Pi^\perp$, the space of linear maps $V\to V$ that satisfy property (b) can be identified with $B^{4,20}_+(\RR)$. Furthermore, if we define
B^4,20_Π():=End(Γ^4,20)∩B^4,20() ,        B^4,20_Π,+():=End(Γ^4,20)∩B^4,20_+() ,
then the maps $V\to V$ satisfying both properties (a) and (b) can be identified with $B^{4,20}_{\Pi,+}(\ZZ)$. Then the image of the map (<ref>) is actually contained in $B^{4,20}_{\Pi,+}(\ZZ)$, so that we can restrict the target and consider the map

\begin{align}\label{homtoB}
\cTop_\Pi &\to B^{4,20}_{\Pi,+}(\ZZ)\subset {\rm End}(\Gamma^{4,20})\\
\CL&\mapsto \nL\notag
\end{align}
which gives rise to a homomorphism of semirings. As the notation suggests, the intersections $B^{4,20}_{\Pi}(\ZZ)$ and $B^{4,20}_{\Pi,+}(\ZZ)$ depend on the way the four-dimensional space $\Pi$ is embedded in $\Gamma^{4,20}\otimes \RR$, i.e. on the point on the moduli space $\CM_{K3}$.

It is plausible that the maps $\nL$ satisfy some further constraints associated with unitarity.
Let $\CL$ be a simple defect, so that the Hilbert space $\Hh_{\CL\CL^\dual}$ admits an orthogonal decomposition as
$$\Hh_{\CL\CL^\dual} \cong \Hh\oplus \Hh_{\CL\CL^\dual-\CI}\ ,$$
where $\Hh_{\CL\CL^\dual-\CI}$ is a sum (with suitable multiplicities) of simple defect spaces $\Hh_{\CL_i}$ with $\CL_i\neq \CI$. Suppose that, in a correlation function, we move the support of the defect line $\CL$ past the insertion point $z$ of a point-like operator $\phi(z)$, with $\phi\in \CH$. Then, $\phi(z)$ gets replaced with the sum of an operator $\frac{1}{\langle \CL\rangle}\hat\CL(\phi)\in \Hh$ plus (possibly vanishing) contributions from each of the components of $\Hh_{\CL\CL^\dual-\CI}$ (see figure <ref> in section <ref>). This move defines a linear map $\Hh\to \Hh_{\CL\CL^\dual}$. In a unitary CFT, it is natural to expect such a map to be an isometry; we call this assumption a strong unitarity hypothesis. Because for a simple defect, the contribution  $\frac{1}{\langle \CL\rangle}\hat\CL(\phi)\in \Hh$ is orthogonal to the contributions from $\Hh_{\CL\CL^\dual-\CI}$, we get as a consequence

(ϕ)^2/|⟨⟩|^2≤ϕ^2 ,    ∀ϕ∈ ,
where the equality holds if and only if all the other contributions vanish. We call the condition (<ref>) the weak unitarity hypothesis. Notice that if (<ref>) holds for all simple defects $\CL$, then it must hold for all superpositions as well. While we do not know any counterexample to these hypotheses[There are well-known counterexamples in non-unitary theories though. For example, the Lee-Yang model with central charge $-\frac{22}{5}$ admits a simple defect $\CL$ of quantum dimension $\langle \CL\rangle=\frac{\sqrt{5}-1}{2}$ where one of the eigenvalues of $\hat\CL$ is $\frac{\sqrt{5}+1}{2}>\langle \CL\rangle$ <cit.>.], we are not aware of any general proof either. A proof of (<ref>) was given in <cit.> (see proposition 8), under certain conditions on $\CL$ and $\phi$. In particular, eq.(<ref>) holds for all Verlinde lines in unitary rational CFT. Unfortunately, K3 models are not rational with respect to the $\CN=(4,4)$ algebra and we were not able to prove that such conditions are satisfied for all $\CL\in \cTop$. If (<ref>) holds, an immediate consequence is the following:

(c) (Assuming (<ref>) holds.) The operatorial norm $\|\nL\|:=\sup_{0\neq v\in V}\frac{\|\nL v\|}{\|v\|}$ equals $\langle \CL\rangle$. Here, $\|v\|$ denotes the Euclidean norm on $V$.

Let us define the `bounded' sets

d·1_4×4     0

0     b_20×20
)|d≥1, b_20,20∈Mat_20×20(), b_20,20≤d}⊂B^4,20_+() ,and
B^4,20_Π,+,b()=End(Γ^4,20)∩B^4,20_+,b() .If property (c) holds we can further restrict the target of the map (<ref>) to

\begin{align}\label{homtoBbound}
\cTop_\Pi &\to B^{4,20}_{\Pi,+,b}(\ZZ)\subset {\rm End}(\Gamma^{4,20})\\
\CL&\mapsto \nL\ .\notag
\end{align}
The map (<ref>) still gives rise to a homomorphism of semirings. Notice that, for any given real number $d>0$, there are finitely many maps $\nL\in B^{4,20}_{\Pi,+,b}(\ZZ)$ with quantum dimension $\langle \CL\rangle\le d$.
We will not use property (c) to prove any of the claims in the rest of the paper.

The set $B^{4,20}_{\Pi,+}(\ZZ)$ contains the group ${\rm Stab}(\Pi)\subset O(\Gamma^{4,20})$ of lattice automorphisms fixing $\Pi$. In fact,  ${\rm Stab}(\Pi)$ can be characterized as the subset of invertible elements in $B^{4,20}_{\Pi,+}(\ZZ)$, this follows immediately by noticing that if $\nL\in {\rm End}(\Gamma^{4,20})$ admits a multiplicative inverse $\nL^{-1}\in {\rm End}(\Gamma^{4,20})$, then both $\nL$ and $\nL^{-1}$ are in $O(\Gamma^{4,20})$.

Therefore, the semiring homomorphism (<ref>) (or (<ref>), if (<ref>) holds) can be understood as an extension of the group  isomorphism $\rho:G_\Pi\to {\rm Stab}(\Pi)$.

Unfortunately, in general we expect the homomorphism (<ref>) to be neither injective nor surjective. As for surjectivity, recall that a topological defect $\CL$ is invertible if and only if its quantum dimension is $\langle \CL\rangle=1$. On the other hand, it is quite easy to construct elements of $B^{4,20}_{\Pi,+}(\ZZ)$ that have dimension $d=1$ but are not invertible, and therefore are not in the image of $\rho$. Of course, one could put further restrictions on $B^{4,20}_{\Pi,+}(\ZZ)$ by simply excluding such elements. However, we have no guarantee that all the elements of $B^{4,20}_{\Pi,+}(\ZZ)$ with dimension larger than $1$ are associated with topological defects.

As for injectivity, we can try to run an argument analogous to the one used in <cit.> to classify the symmetry groups $G_\Pi$. Let $\mathsf{K}_\Pi$ denote the subcategory of topological defects $\CL$ of the model $\calC_\Pi$ preserving $\CN=(4,4)$ and spectral flow, and such that the corresponding maps $\nL$ are proportional to the identity on $V$, i.e. such that
ρ(𝖪_Π)⊆{=d·id_V, d≥1}⊂B^4,20_Π,+() .
We can prove the following:

For all K3 models $\calC_\Pi$ the category $\mathsf{K}_\Pi$  of defects preserving $\CN=(4,4)$ and spectral flow, and acting by multiplication by some $d\in \RR$ on the space of RR ground fields $V$ is generated by the trivial defect
𝖪_Π={d, d∈} .

Let $\CL$ be a defect in $\mathsf{K}_\Pi$. Because $\CL$ is transparent to the spectral flow generators, the real number $d$ must be the quantum dimension $d=\langle \CL\rangle$. As a consequence, all R-R ground fields are transparent to $\CL$. This implies that $\CL$ is transparent to all exactly marginal operators of $\calC_\Pi$, and it cannot be lifted by any deformation of the model. Because the moduli space of K3 is connected, this means that the category $\mathsf{K}_\Pi$  is the same for all K3 models. Therefore, it is sufficient to determine $\mathsf{K}_\Pi\equiv \mathsf{K}$ in a specific model. In section <ref>, we will show that in a certain torus orbifold the defects of $\mathsf{K}_\Pi$ are necessarily superpositions of $d$ copies of the identity defect (see Claim <ref>); in particular, $d$ is a natural number.

This result generalizes the analogous statement for symmetry groups that the kernel of $\rho$ is trivial. However, in the case of defects, this is not sufficient to conclude that $\rho$ is injective. If $g,h$ are elements of the group $G_\Pi$, then $\rho(g)=\rho(h)$ implies that $\rho(gh^{-1})= \rho(g)\rho(h)^{-1}=1$, and therefore $gh^{-1}\in \ker\rho$ must be the identity and $g=h$. But if $\CL$ and $\CL'$ are non-invertible defects, the fact that $\rho(\CL)=\rho(\CL')$ does not imply that $\CL$ and $\CL'$ can be obtained from each other by fusion with a defect in $\mathsf{K}$. Indeed, in sections <ref>, <ref> and <ref> we will see examples of continuous families of distinct defects $\CL_\theta$, all with the same image $\nL$.

The possible quantum dimensions of defect $\CL\in \cTop_\Pi$ are strongly constrained by properties (a) and (b). In section <ref> we will prove the following:

The quantum dimension $\langle \CL\rangle$ of a defect $\CL\in \cTop_\Pi$ is an algebraic integer of degree at most $6$. Furthermore, if $\Pi\cap \Gamma^{4,20}\neq 0$, then $\langle \CL\rangle$ is integral for all $\CL\in \cTop_{\Pi}$.

We recall that an algebraic integer is the root of a monic polynomial $p(x)$ with integral coefficients, and its degree is $d$ if any such $p(x)$ has degree at least $d$. We do not know whether the upper bound on the degree is sharp.
A slightly weaker necessary condition for the quantum dimension to be integral is given in proposition <ref> in section <ref>.

The condition $\Pi\cap \Gamma^{4,20}\neq 0$ has a nice physical interpretation. Let $v\neq 0$  be a primitive vector in $\Pi\cap \Gamma^{4,20}$. Because the subspace $\Pi$ is positive definite, the vector $v$ has positive norm $v^2>0$. From the viewpoint of type IIA superstring, a primitive vector $v\in \Gamma^{4,20}$ with $v^2>0$ represents the charge of a BPS D0-D2-D4-brane configuration.[While the charges of BPS D-branes span the lattice $\Gamma^{4,20}$, a generic vector $v\in \Gamma^{4,20}$ is the charge of a system of branes and anti-branes that is not by itself BPS.] The mass of such a BPS configuration depends on the moduli, and is proportional to $v_\Pi^2$, where $v_\Pi$ and $v_\perp$ are the orthogonal projections of $v$ along $\Pi$ and $\Pi^\perp$, so that $v^2=v_\Pi^2-v_\perp^2$. An attractor point in the moduli space for the BPS state with charge $v$ is a point where the BPS mass $v_\Pi^2=v^2+v_\perp^2$ is minimized, and  this happens if and only if $v\in \Pi$ <cit.>. Thus, the points in the moduli space where $\Pi \cap  \Gamma^{4,20}\neq 0$ are exactly the attractor points for some BPS brane configurations. Claim <ref> then implies that whenever the K3 model $\calC$ is `attractive', all topological defects $\CL\in \cTop_\calC$ have integral quantum dimension.

By combining Claims <ref> and <ref>, we get:

For a generic K3 sigma model $\calC_\Pi$, the only topological defects in $\cTop_\Pi$ are integral multiples of the identity.

See section <ref> for the proof. As a consequence, for any non-trivial topological defect $\CL\in \cTop_\Pi$ there is a deformation of the K3 model $\calC_\Pi$ that lifts it.

§.§ Action on D-branes

Let us consider the set of BPS boundary states in a K3 model $\calC_\Pi$, corresponding to D-branes in the full string theory that are $1/2$-BPS, i.e. that preserve $8$ out of $16$ space-time supersymmetries.

As in the previous section, we denote by $V$ the $24$-dimensional real space of RR ground states that are CPT self-conjugate. The hermitian form on $\CH$ induces a positive definite bilinear form on $V$.

With each 1/2-BPS boundary state $||\alpha\rrangle$ is associated a $24$-dimensional charge vector $q_\alpha\in V^*$. The pairing of $q_\alpha$ with a R-R ground state $\psi\in V$ is given by the
amplitude q_α(ψ):=α||ψ⟩∈ ,    on a half-cylinder S^1×_≤0={(x,t)|x∈/2π, t≤0}where $|\psi\rangle$ is  the asymptotic state at $t\to-\infty$ and $\alpha$ the boundary condition at $t=0$, see figure <ref>.

Amplitude $\llangle  \alpha \vert\vert \psi \rangle$ in the half-cylinder $S^1 \times \mathbb{R}_{\leq 0}$ with boundary $\vert\vert \alpha \rrangle$ at $t=0$.

Geometrically, if $S$ is the target K3 surface, we can identify $V$ with the even real cohomology $H^{even}(S,\RR)$ (in fact, RR ground fields correspond to harmonic forms on $S$), and the lattice of D-brane charges with $H_{even}(S,\ZZ)$, the integral even homology. We can define a bilinear form $(q_\alpha,q_\beta)$ on the lattice of D-brane charges, corresponding to the Mukai pairing on $H_{even}(S,\ZZ)$:
Physically, the bilinear form is given by a string amplitude on a bounded cylinder $S^1\times [0,1]$ with boundaries $\alpha$ and $\beta$, describing a loop of Ramond open strings with periodic conditions for the fermions.
This amplitude reduces to an index counting Ramond ground states in the space $\CH^{open}_{R,\alpha,\beta}$ of  open strings with Chan-Paton factors $\alpha$–$\beta$. In particular, $(q_\alpha,q_\beta)$  is just an integral (in fact, even) number independent of the size of the cylinder. With respect to this pairing, the RR charge lattice is isomorphic to the (unique) even unimodular lattice with signature $(4,20)$.

In the closed string channel, the amplitude gets contributions only from the Ramond-Ramond ground fields propagating between the corresponding boundary states, with the insertion of a left-moving fermion number $(-1)^{F_L+1}$. The latter acts by $+1$ on $\Pi$ (i.e. on the spectral flow generators in the $(\frac{1}{4},\frac{1}{2};\frac{1}{4},\frac{1}{2})$ representation of $\CN=(4,4)$) and by $-1$ on $\Pi^\perp$ (i.e. on states in $(\frac{1}{4},0;\frac{1}{4},0)$), and can be used to define a non-degenerate bilinear form with signature $(4,20)$ on $V$
(ψ,ψ'):=-⟨ψ|(-1)^F|ψ'⟩ .
Let us focus on the dual of the charge lattice,  i.e. the lattice $\Gamma^{4,20}\subset V$ of states with respect to which the charge of any boundary state is integral
Γ^4,20:={Ψ∈V|α|Ψ⟩∈ , ∀|| α} .
Since the charge lattice is self-dual, $\Gamma^{4,20}$ is again an even unimodular lattice of signature $(4,20)$ with respect to the bilinear form (<ref>).

Let us now consider, as above, a half-cylinder amplitude $\llangle\alpha|\hat \CL|\psi\rangle$ for a boundary $\alpha$ and a state $\psi\in V$, with the insertion of a topological defect line $\CL\in\cTop$ wrapping once along the circle $S^1$.

Amplitude $\llangle \alpha \vert \hat{\mathcal{L}} \vert \psi \rangle$ in the half-cylinder $S^1 \times \mathbb{R}_{\leq 0}$ with insertion of a TDL $\mathcal{L} \in Top$.

By moving the line $\CL$ along the cylinder to $t\to -\infty$  or $t\to 0$,  $\llangle\alpha|\hat \CL|\phi\rangle$ can be interpreted as either the RR charge of the D-brane $\alpha$ with respect to $\hat\CL|\psi\rangle$, or as the charge of $\CL^\dual ||\alpha\rrangle$ with respect to $|\psi\rangle$. Therefore,
q_α(ψ)=α||ψ⟩=q_^α(ψ) .
As in the previous section, we denote by
$ \nL:=\hat\CL_{\rvert V}\ ,
$ the restriction of $\hat\CL$ to $V$.
In particular, if $\Psi\in \Gamma^{4,20}\subset V$, then
q_α(Ψ)=q_^α(Ψ)∈ ,
for all $||\alpha\rrangle$. It follows that $\nL(\Psi)\in \Gamma^{4,20}$, i.e. $\hat\CL$ gives rise to an endomorphism $\nL\in {\rm End}_\ZZ(\Gamma^{4,20})$ of the unimodular lattice $\Gamma^{4,20}$

\begin{align*}
\nL:\Gamma^{4,20}&\to\Gamma^{4,20}\\
\Psi&\mapsto \nL(\Psi)\ .
\end{align*}

The operator $\hat\CL$ commutes with the left-moving fermion number $(-1)^{F_L}$
[(-1)^F_L,]=0 ,
so that the linear map $\nL$ preserves the orthogonal decomposition $V=\Pi\oplus \Pi^\perp$. In particular, it is block diagonal with respect to an orthonormal basis $\{\psi_1,\ldots,\psi_{24}\}\subset V$ satisfying  ψ_1,…,ψ_4∈Π ,    ψ_5,…,ψ_24∈Π^⊥ .
Finally, we notice that on a spectral flow generators $\psi\in \Pi$, $\nL$ acts by
(ψ)=⟨⟩ψ ,    ∀ψ∈Π⊂V ,
i.e. the $4\times 4$ upper-left block of $\nL$ is just $\langle \CL\rangle$ times the identity. This argument leads to the conditions (a) and (b) described in section <ref>.

An easy but powerful consequence of this construction is the following:

Let $\CL\in\cTop_{\calC_\Pi}$, and suppose there is some $\Psi\in \Gamma^{4,20}$, $\Psi\neq 0$, such that $\nL(\Psi)=\langle \CL\rangle\Psi$, where $\langle\CL\rangle$ is the quantum dimension of $\CL$. Then, $\langle \CL\rangle$ is integral. In particular, if for a certain model $\calC_\Pi$, one has $\Pi\cap \Gamma^{4,20}\neq 0$, then all $\CL\in \cTop_{\calC_\Pi}$ have integral quantum dimension.

Indeed, without loss of generality, we can assume the vector $\Psi\in \Gamma^{4,20}$ is primitive (i.e., not an integral multiple of a shorter vector in the lattice). Then, because $\langle\CL\rangle\Psi\in \Gamma^{4,20}$, it must necessarily be an integral multiple of $\Psi$.

More generally, because $\hat\CL$ acting on the space of RR ground fields is represented by an integral $24\times 24$ matrix, it satisfies $p(\hat\CL)=0$ where $p(x)$, the characteristic polynomial, is a monic polynomial of degree $24$ with integral coefficients. This implies that the quantum dimension $\langle \CL\rangle$ is also a root of the same monic polynomial, i.e. it is an algebraic integer. Let $d=\langle\CL\rangle$ be an algebraic integer, and let $r(x)$ be the minimal polynomial for $d$, i.e. the least degree monic polynomial with integral coefficient having $d$ as a root. Since $d$ has multiplicity at least $4$ as an eigenvalue, it follows that $r(x)^4$ divides $p(x)$, so that $r(x)$ must have degree at most $6$. We conclude that:

The quantum dimension $\langle \CL\rangle$ of any defect $\CL\in\cTop$ is an algebraic integer of degree at most $6$.

This result, together with Claim <ref>, leads us to a proof of Claim <ref>.

Consider a K3 model $\calC_\Pi$. Let $\Psi_1,\ldots,\Psi_{24}$ be generators of the lattice $\Gamma^{4,20}\subset V$, let $d:=\langle \CL\rangle$ be the quantum dimension of a defect $\CL\in \cTop_\Pi$, and let $\QQ[d]$ be the extension of the field $\QQ$ by the algebraic integer $d$.    Then, for a generic model $\calC_\Pi$, we expect the scalar products $\langle \psi|\Psi_1\rangle,\ldots,\langle \psi|\Psi_{24}\rangle$, with some spectral flow generator $\psi\in \Pi$, to generate a space of maximal rank $24$ over the field extension $\QQ[d]$. Indeed, because the set of algebraic numbers has measure zero in $\RR$, for a generic model $\calC_\Pi$ every non-trivial linear relation among $\langle \psi|\Psi_1\rangle,\ldots,\langle \psi|\Psi_{24}\rangle$ will contain some transcendental coefficient.
Let us define the matrix elements $\nL_{ij}$ by $\nL|\Psi_i\rangle=\sum_j \nL_{ij}|\Psi_j\rangle$, so that $\nL_{ij}\in \ZZ$. Using
d⟨ψ|Ψ_i⟩= ⟨ψ||Ψ_i⟩=∑_j=1^24 _ij ⟨ψ|Ψ_j⟩ ,
we get
∑_j (dδ_ij-_ij)⟨ψ|Ψ_j⟩=0    ∀i=1,…,24 .
Because $d\delta_{ij}-\nL_{ij}\in \QQ[d]$ and $\langle\psi|\Psi_j\rangle$ are linearly independent over $\QQ[d]$, it follows that
dδ_ij-_ij=0    ∀i,j=1,…,24 .
This means that $\nL$ is $d$ times the identity, and by proposition <ref> $d$ is integral. By Claim <ref>, this means that $\CL=d\CI$, and we conclude.

§.§ Defects as boundaries in the doubled theory and other approaches

In this section we discuss other approaches to topological defects that we are not used elsewhere in this article.

First of all, a topological defect in a CFT $\calC$ can be described as a boundary state in the doubled theory $\calC\times\calC^*$, where $\calC^*$ is obtained from $\calC$ by worldsheet parity. In particular, when $\calC$ is a sigma model on K3, the doubled theory $\calC\times\calC^*$ is a sigma model on the Cartesian product of two K3 surfaces. This CFT contains a $\CN=(4,4)$ superconformal algebra at central charge $c=\tilde c=12$. Notice that the $\CN=4$ algebra at $c=12$ contains a copy of $\widehat{su}(2)_2$ at level $2$, rather than $1$.

The space of R-R ground fields of weight $h=\bar h=\frac{1}{2}$ of $\calC\times \calC^*$ has dimension $24^2=576$, and is isomorphic to $V\otimes V^*\cong {\rm End}_\RR(V)$. Similarly, the lattice of RR charges is the tensor product[Actually, because $\Gamma^{4,20}$ is unimodular, the non-degenerate bilinear form defines isomorphisms $\Gamma^{4,20}\cong(\Gamma^{4,20})^*$ and $V\cong V^*$. ] $\Gamma^{4,20}\otimes (\Gamma^{4,20})^*\cong  {\rm End}_\ZZ(\Gamma^{4,20})$. Therefore, for each given $\CL\in \cTop_\calC$ one can naturally identify $\nL\in {\rm End}_\ZZ(\Gamma^{4,20})$ with the RR charge of the corresponding boundary state in $\calC\times \calC^*$.
It follows that property (a) is just quantization of RR charges in the doubled theory.

In order to understand the analogue of property (b), let us consider the decomposition of the space V⊗V^*≅V⊗V=(Π⊕Π^⊥)⊗(Π⊕Π^⊥) into representations of the $\CN=(4,4)$ algebra. There are three kind of (short) Ramond representations of $\CN=4$ at $c=12$ with $h=\frac{1}{2}$, that are labeled by the highest weight (charge) $q\in \{0,1/2,1\}$ with respect to the $\widehat{su}(2)_2$ algebra. In particular, the subspace $\Pi^\perp\otimes \Pi^\perp$ is contained in $20^2=400$ representations of $\CN=(4,4)$ with holomorphic and antiholomorphic charges $q=\tilde q=0$; the $80$-dimensional subspaces $\Pi\otimes \Pi^\perp$ and $\Pi^\perp\otimes \Pi$ are contained in $40$ $\CN=(4,4)$-representations with $(q,\tilde q)=(\frac{1}{2},0)$ and $(q,\tilde q)=(0,\frac{1}{2})$, respectively; finally, the $16$-dimensional subspace $\Pi\otimes \Pi$ decomposes into one $\CN=(4,4)$-representation with $(q,\tilde q)=(1,1)$ ($9$ RR ground states), one representation with $(q,\tilde q)=(1,0)$ ($3$ states), one with $(q,\tilde q)=(0,1)$ ($3$ states) and one with $(q,\tilde q)=(0,0)$ (one state). This gives an orthogonal decomposition
V⊗V^*=Σ_0,0⊕Σ_1/2,0⊕Σ_0,1/2⊕Σ_1,0⊕Σ_0,1⊕Σ_1,1 ,
where $\Sigma_{q,\tilde q}$ is the subspace of RR ground states of $\calC\times\calC$ contained in the $(q,\tilde q)$  representation of $\CN=(4,4)$. Here,

$$ \dim\Sigma_{0,0}=401\ ,\quad  \dim\Sigma_{\frac{1}{2},0}=\dim\Sigma_{0,\frac{1}{2}}=80\ ,\quad \dim\Sigma_{1,0}=\dim\Sigma_{0,1}=3\ ,\quad \dim\Sigma_{1,1}=9\ .

It is clear that property (b) for $\CL\in\cTop_\calC$ is equivalent to the condition the corresponding boundary state in $\calC\times\calC^*$ is charged only under the $401$ RR ground states in $\Sigma_{0,0}$, and neutral with respect to the other ones. In particular, the condition that the restriction of $\nL$ to $\Pi$ is proportional to the identity implies that the map $\nL$ commutes with the left- and right-moving $SU(2)$ R-symmetry groups; in turn, this means that the corresponding boundary state in $\calC\times \calC$ can only be charged under the only singlet in $\Pi\otimes \Pi$. In other words, the space $B^{4,20}(\RR)$ can be identified with the $401$-dimensional subspace $\Sigma_{0,0}$ of R-R ground states of $\calC\times\calC^*$.

This construction also provides some intuition as for why, for a generic K3 sigma model $\calC$, the only simple defect in $\cTop_\calC$ is the identity.
Indeed, while there are infinitely many boundary states in $\calC\times\calC^*$, whose RR charges span the whole lattice $\Gamma^{4,20}\otimes (\Gamma^{4,20})^*$, it is not obvious at all that any of these lattice vectors is contained in the subspace $\Sigma_{0,0}\equiv B^{4,20}(\RR)$. Let us elaborate on this point in more detail. We know that in any K3 model $\calC$ there is the identity defect $\CI$, and the charge of the corresponding boundary state in the doubled theory is an element $\mathsf{1}\in \Gamma^{4,20}\otimes (\Gamma^{4,20})^*$ lying in the $401$-dimensional subspace $\Sigma_{0,0}$ with $(q,\tilde q)=(0,0)$. Under the identification $\Gamma^{4,20}\otimes (\Gamma^{4,20})^*\cong {\rm End}(\Gamma^{4,20})$, $\mathsf{1}$ simply corresponds to the identity map. Therefore, the intersection
(Γ^4,20⊗(Γ^4,20)^*)∩Σ_0,0=End(Γ^4,20)∩B^4,20() = B^4,20_Π()
is always at least one-dimensional.
Now, deformations of the K3 model correspond to $O(4,20,\RR)$ transformations of $V\cong \Pi\oplus \Pi^\perp$ within $\Gamma^{4,20}\otimes \RR$. While $V$ is an irreducible representation of the orthogonal group $O(4,20,\RR)$ (the vector representation), the tensor product $V\otimes V^*\cong V^{\otimes 2}$ decomposes as
$$V^{\otimes 2}=\mathbf{1}\oplus \wedge^2 V\oplus {\rm Sym^2_0}V\ ,$$
where $\mathbf{1}$, $\wedge^2 V$, and ${\rm Sym^2_0}V$ are, respectively, the trivial, the anti-symmetric and the traceless symmetric representations of $O(4,20,\RR)$.
Under the identification $V\otimes V^*\cong {\rm End}_\RR(V)$, clearly the trivial representation $\mathbf{1}$ corresponds to the identity map, i.e. to the charge $\mathsf{1}$ of the identity defect $\CI$. Because it is invariant under $O(4,20,\RR)$ transformations, the charge $\mathsf{1}\in \Gamma^{4,20}\otimes (\Gamma^{4,20})^*$ is always contained in $\Sigma_{0,0}$. This fits with the obvious fact that the identity defect is not lifted by any deformation of the K3 model. On the other hand, both irreducible representations $\wedge^2 V$, and ${\rm Sym^2_0}V$ have non-trivial intersection with $\Sigma_{0,0}$ and with other $\Sigma_{q,\tilde q}$. Thus, a generic $O(4,20,\RR)$ transformation will mix any vector in the $400$-dimensional space
$$(\Sigma_{0,0}\cap \mathsf{1}^\perp) \subset \wedge^2 V\oplus {\rm Sym^2_0}V$$
with vectors in the other components $\Sigma_{q,\tilde q}$. Therefore, generically, we do not expect any other lattice vector in $\Gamma^{4,20}\otimes (\Gamma^{4,20})^*$ to lie along $\Sigma_{0,0}\cap \mathsf{1}^\perp$. This means that the intersection (<ref>) is generically one-dimensional, and contains only the maps $\nL$ that are multiple of the identity.

Let us now comment on other common approaches to determine topological defects in two dimensional conformal field theories. One simple and powerful method pioneered by Petkova and Zuber in <cit.> consists in imposing the analogue of the Cardy condition for boundary states. In practice, a generic defect $\CL$ preserving the chiral and antichiral algebra $\CA\times\bar \CA$ is parametrized in terms of the (unknown) eigenvalues of $\hat\CL$ on the primary fields. One can then compute the torus $\CL$-twined partition function, obtained by inserting the $\CL$-line along the `space' direction of the worldsheet torus, as a function of these parameters. A modular S-transformation relates this $\CL$-twined partition function to the $\CL$-twisted one, where the defect runs along the `time' direction. Because the $\CL$-twisted partition function is just a trace over the  $\CL$-twisted space $\Hh_\CL$, it must decompose into characters of the $\CA\times\bar\CA$ algebra with non-negative integral coefficients.
Imposing such property on the latter coefficients gives a set of quantization conditions on the unknown parameters of the defect $\CL$, whose solutions form a lattice. In rational CFTs, where there is a finite number of primary fields and of unknown parameters, this method is very effective and very often allows one to determine all possible topological defects in the theory.[More generally, in order to determine all simple defects in a rational CFT, one needs to impose a Cardy-like condition to all possible products $\CL_1\CL_2^\dual$, for all pairs of simple defects $\CL_1$, $\CL_2$. In the doubled theory, the Petkova-Zuber method just corresponds to imposing the Cardy condition for open strings stretched between any pair of boundary states.]

The main difficulty in applying this method to study $\cTop_\calC$ in a K3 model $\calC$ is that the theory is not rational with respect to the $\CN=(4,4)$ algebra, so that there infinitely many primary operators and unknown parameters. It is known that the character of the $\CN=4$ algebra (in particular, for the BPS representations) are mock modular forms <cit.>, and their S-tranformations give rise to a continuum of massive characters.
As is well-known from the study of boundary states in non-rational theories, imposing Cardy-like conditions to these cases is technically very difficult.

Despite these obstacles, modular properties of torus amplitudes have been successfully used in the past to get information about the action of finite symmetry groups on the space $\Hh$ (or, at least, on some subspace) of states  of a K3 model. In particular, when $\CL$ is an invertible defect preserving the superconformal algebra, one can consider the $\CL$-twined elliptic genus, which is a (weakly) holomorphic Jacobi function for a certain subgroup of the modular group $SL(2,\ZZ)$. Because the space of such Jacobi functions is finite dimensional, the problem is once again reduced to determining a finite number of unknown coefficients. In  particular, these methods were applied in the context of Moonshine conjectures for K3 models <cit.>.

One crucial step in these methods is the fact that, for an invertible symmetry of order $n$, the $\CL$-twined genus is modular with respect to a level $n$ congruence subgroup of $SL(2,\ZZ)$. This is a consequence of the fusion relation $\CL^n=\CI$. Furthermore, all possible orders $n$ of these symmetries can be determined by the general classification theorem of <cit.>.  Unfortunately, when $\CL$ is an unknown non-invertible defect in some K3 model, we do not have any information about the relations in the fusion ring generated by $\CL$. In principle, the decomposition of $\CL^k$ might involve new simple defects at every power $k$. For this reason, one cannot predict what the modular properties of the $\CL$-twined genus are. In fact, we hope that the methods developed in the present article will provide enough information about the fusion ring of $\cTop_\calC$ so as to make these methods effective.

Finally, when the CFT $\calC$ can be obtained as a IR fixed of a RG flow from a Landau-Ginzburg model, there are well-developed techniques to study its topological defects and their fusion with supersymmetric boundary conditions, relying in particular on matrix factorization <cit.>. It would be interesting to apply these methods to K3 models and use them to verify and extend the general results of our article.

§ TOPOLOGICAL DEFECTS IN TORUS ORBIFOLDS

Many interesting examples of K3 models can be described in terms of orbifolds of supersymmetric sigma models on $T^4$ by some finite group of symmetries. In this section, we will show that for all such K3 models $\calC$ the category $\cTop_\calC$ contains some family of simple defects $\CL_\theta$ parametrized by continuous parameters $\theta$. This result is an immediate generalization of known properties of the orbifold $S^1/\ZZ_2$ of a single free boson on a circle <cit.>.

§.§ Supersymmetric sigma models on $T^4$

A supersymmetric sigma model $\CT$ on $T^4$ contains four holomorphic and four anti-holomorphic $u(1)$ currents $i\partial X^k$, $i\bar\partial X^k$, $k=1,\ldots, 4$, as well as their superpartners, the four holomorphic and anti-holomorphic free fermions $\psi^k$, $\tilde \psi^k$ with weights $(h,\bar h)=(\frac{1}{2},0)$ and $(0,\frac{1}{2})$, respectively. The full current algebra is $\widehat{so}(4)_1\oplus u(1)^4$, where the six currents of $\widehat{so}(4)_1\cong \widehat{su}(2)_1\oplus \widehat{su}(2)_1$ are given by normal-ordered products $:\!\!\psi\psi\!\!:$ of pairs of free fermions.  These fields generate the full chiral and anti-chiral algebra of the $T^4$ model at a generic point in the moduli space – the algebra can be enhanced at subloci of positive codimension.

The NS-NS primary operators with respect to this algebra  are the vertex operators $V_\lambda(z,\bar z)$ labeled by vectors λ≡(λ_L,λ_R)=(λ^1_L,…,λ^4_L;λ^1_R,…,λ^4_R)∈^8 ,with conformal weights $h=\frac{\lambda_L\cdot \lambda_L}{2}$ and $\bar h=\frac{\lambda_R\cdot \lambda_R}{2}$. For each given $T^4$ model, the allowed vectors $\lambda\in \RR^8$ are the points of a $8$-dimensional lattice $\Gamma^{4,4}$ (the Narain lattice) that is even and unimodular with respect to the bilinear form
(λ,μ):=λ_L·μ_L-λ_R·μ_R ,    λ,μ∈Γ^4,4 ,
with signature $(4,4)$. Notice that all even unimodular lattices $\Gamma^{4,4}$ with signature $(4,4)$ are isomorphic to each other. There is a (non-unique) choice of basis $\{n_1,\ldots,n_4,w_1,\ldots,w_4\}$ for $\Gamma^{4,4}$ with respect to which the bilinear form is
( n_i,n_j) =0=( w_i,w_j) ,    ( n_i,w_j)=δ_ij .The vector $\lambda\in \RR^8$  represents the weights of the corresponding primary state $|\lambda\rangle$ with respect to the $U(1)^4\times U(1)^4$ group generated by the four bosonic currents $i\partial X^k$, $i\bar\partial X^k$. In other words, the invertible defects W_θ ,        θ∈(Γ^4,4⊗)/Γ^4,4≅(/)^8  ,corresponding to the $U(1)^8$ symmetries, commute with the full chiral algebra and act by
Ŵ_θ|λ⟩= e^2πi (θ,λ)|λ⟩ ,
on the primary state $\lambda$. Geometrically, $U(1)^8\cong U(1)^4\times U(1)^4$ is the product of the $U(1)^4$ group of translations along the four direction of the torus $T^4$, times the $U(1)^4$ group of translations along the T-dual torus.

The RR sector representations of the chiral algebra are labeled by the same vectors $\lambda\in \Gamma^{4,4}$. For each such $\lambda\in \Gamma^{4,4}$, there are $16$ degenerate ground states with conformal weights $h=\frac{\lambda_L\cdot \lambda_L}{2}$ and $\bar h=\frac{\lambda_R\cdot \lambda_R}{2}$, which form an irreducible representation of the Clifford algebra of zero modes of the $8$ fermions $\psi^k\,\tilde \psi^k$, $k=1,\ldots,4$. With respect to the holomorphic and antiholomorphic $\widehat{so}(4)_1\cong \widehat{su}(2)_1 \oplus \widehat{su}(2)_1 $ algebras and the corresponding $Spin(4)\cong SU(2)\times SU(2)$ groups, the $16$ ground states decompose into four irreducible representations with weights $(s_1,s_2;\tilde s_1,\tilde s_2)$
(1/2,0;1/2,0) ,     (1/2,0;0,1/2) ,     (0,1/2;1/2,0) ,     (0,1/2;0,1/2) .

The chiral algebra of the $T^4$ model contains several copies of the small $\CN=4$ superconformal algebra, corresponding to a choice of $\widehat{su(2)}_1$ current algebra (the R-symmetry algebra) inside $\widehat{so(4)}_1$; we choose a small $\CN=(4,4)$ once and for all. The four R-R ground states with $\lambda=0$ (i.e. conformal weights $h=\bar h=\frac{1}{4}$) that are charged with respect to both the holomorphic and antiholomorphic $\widehat{su}(2)_1$ algebra are the spectral flow generators, and belong to a single $(\frac{1}{4},\frac{1}{2};\frac{1}{4},\frac{1}{2})$ representation of $\CN=(4,4)$. The remaining $12$ RR ground states belong to two $(\frac{1}{4},\frac{1}{2};\frac{1}{4},0)$, two $(\frac{1}{4},0;\frac{1}{4},\frac{1}{2})$, and four $(\frac{1}{4},0;\frac{1}{4},0)$ representations of $\CN=(4,4)$.

For a generic $T^4$ model $\CT$, the group $G_\CT$ of symmetries preserving the small $\CN=(4,4)$ algebra and the spectral flow generators is
G_≅U(1)^8⋊_2 ,
where the $\ZZ_2$ symmetry $\calR$ is the coordinate reflection
∂X^k→-∂X^k ,    ∂̅X^k→-∂̅X^k    ψ^k→-ψ^k ,    ψ̃^k→-ψ̃^k .
The fusion of $\calR$ with $W_\theta$ is
W_θ=W_-θ .
At special loci in the moduli space, $G_\CT$ can be enhanced to a larger group. In general, the group $G_{\CT}$ is always a group extension
1⟶U(1)^8 ⟶G_⟶G^0_⟶1 ,
where $G^0_\CT\subset O(\Gamma^{4,4})\cong O(4,4,\ZZ)$ is a finite group of lattice automorphisms, that always contains  the reflection (<ref>) as a central element. Physically, $O(\Gamma^{4,4})\cong O(4,4,\ZZ)$ is the group of torus T-dualities, and $G^0_\CT$ is the group self-dualities of the model $\CT$. See <cit.> for a complete classification of the groups $G^0_\CT$.

§.§ Continuous defects in $T^4/\ZZ_2$.

Given any torus model $\CT$, the orbifold $\calC=\CT/\ZZ_2$ by the $\ZZ_2$ symmetry (<ref>) is a non-linear sigma model on K3.

A torus orbifold $\calC=\CT/\ZZ_2$ always admits an invertible defect $Q$ (the quantum symmetry) acting by $-1$ on the twisted sector and trivially on the untwisted one. Besides $Q$, it also contains some invertible simple defects that are induced by the simple defects $W_\theta$ of the torus model $\CT$ that commute with $\calR$, i.e such that $\theta\equiv -\theta\mod \Gamma^{4,4}$. More precisely, with each defect $W_{\frac{\lambda}{2}}$, $\lambda\in \Gamma^{4,4}/2\Gamma^{4,4}$ of $\CT$, are associated two defects
$\eta_\frac{\lambda}{2},\eta'_{\frac{\lambda}{2}}\equiv Q\eta_\frac{\lambda}{2}$ of $\calC$. In particular, for $\lambda=0$, one has $\eta_0=I$ (the identity defect) and $\eta'_0=Q$.

While the defects $W_{\frac{\lambda}{2}}$, $\lambda\in \Gamma^{4,4}$, generate an abelian group of symmetries $\ZZ_2^8\subset U(1)^8$ of the torus model $\CT$, the group generated by the $\eta_{\lambda/2}$ is a non-abelian extension of $\ZZ_2^8$ by a central $\ZZ_2$ generated by the quantum symmetry $Q$. Consider a basis $\{n_1,\ldots,n_4,w_1,\ldots,w_4\}$ for $\Gamma^{4,4}$ as in (<ref>). The defects $\eta_{n_i}$, $\eta_{w_i}$ and $Q$ obey the following relations
η_n_i/2^2=I=η_w_i/2^2 ,    Q^2=1 ,    η_n_i/2Q=Qη_n_i/2 ,    η_w_i/2Q=Qη_w_i/2
If $\lambda=\sum_i (a_in_i+b_iw_i)$, $a_i,b_i\in \ZZ/2\ZZ$, is an element of $\Gamma^{4,4}/2\Gamma^{4,4}$, then we define
η_λ/2=η_1/2∑_i (a_in_i+b_iw_i):=η_a_1n_1/2⋯η_a_4n_4/2η_b_1w_1/2⋯η_b_4w_4/2 ,     η'_λ/2:=Qη_λ/2 .
We have the fusion rules
Q^2=I ,    η_λ/2 Q=Qη_λ/2  ,    (η_λ/2)^2=Q^(λ,λ)/2
which imply
η_n_i/2^2=I=η_w_i/2^2 ,    η_λ/2η_μ/2=Q^(λ,μ)η_μ/2η_λ/2
Different choices of the basis $\{n_1,\ldots,n_4,w_1,\ldots,w_4\}$ just lead to exchanging $\eta_{\frac{\lambda}{2}}\leftrightarrow \eta'_{\frac{\lambda}{2}}$ for some values of $\theta$ – this is just a relabeling of the defects. Altogether, these invertible elements have group-like fusion rules, corresponding to an extraspecial group $2^{1+8}$ <cit.>
1⟶⟨Q⟩≅_2 ⟶2^1+8⟶_2^8⟶1 .

Besides invertible defects, torus orbifolds $\CT/\ZZ_2$ always contain a continuum of defects $T_\theta$, parametrized by $\theta \in ((\RR/ \ZZ)^8)/{\pm 1}$, that preserve the $\CN=(4,4)$ algebra and the spectral flow generators, and that are induced by the $\calR$-invariant superposition $W_\theta+W_{-\theta}$ of topological defects  of the torus model $\CT$. The defects $T_\theta$ have dimension $2$ and satisfy the fusion rules

$$ T_\theta T_{\theta'}=T_{\theta+\theta'}+T_{\theta-\theta'}\ ,
While $W_\theta+W_{-\theta}$ are clearly a superposition, the $T_\theta$ are actually simple for generic values of $\theta$. The only exceptions are when $\theta$ is one of the $\calR$-fixed points $\theta=\frac{\lambda}{2} \in \frac{1}{2}\Gamma^{4,4}/\Gamma^{4,4}$, and in this case they are superpositions
T_λ/2=η_λ/2+Qη_λ/2 .
Notice that $T_0=\CI+Q$, so that
(T_θ)^2=T_0+T_2θ=+Q+T_2θ ,
that implies that $T_\theta$ is unoriented, $(T_\theta)^\dual=T_{\theta}$. The fusion with the invertible defects is
QT_θ=T_θQ=T_θ ,    η_λ/2T_θ=T_θη_λ/2=T_θ+λ/2 ,
where in the last identity, one uses $\frac{\lambda}{2}\equiv -\frac{\lambda}{2} \mod \Gamma^{4,4}$, so that $T_{\theta+\frac{\lambda}{2}}=T_{\theta-\frac{\lambda}{2}}$.

According to <cit.>, the operators $\hat T_{\theta}$ associated with these defects act on the untwisted sector as the operators $\hat W_\theta+\hat W_{-\theta}$ in the original theory, while they annihilate the twisted sector. On the RR ground states, all $\hat T_\theta$ act by twice the identity on the states in the untwisted sector, and annihilate all the states in the twisted sector.

If $2\theta\in (\Gamma^{4,4}\otimes\RR)/\Gamma^{4,4}$ is a  $\calR$-fixed point, i.e. if $2\theta\equiv -2\theta\mod \Gamma^{4,4}$, then $\theta=\frac{\lambda}{4}$ for some $\lambda\in \Gamma^{4,4}$, and the fusion product (<ref>) becomes
(T_λ/4)^2=+Q+η_λ/2+Qη_λ/2 .
The right-hand side is just the sum over all invertible defects in the order $4$ group generated by $Q$ and $\eta_{\frac{\lambda}{2}}$ (this is either $\ZZ_2\times \ZZ_2$ or $\ZZ_4$, depending on whether $(\eta_\frac{\lambda}{2})^2$ equals $\CI$ or $Q$, i.e. if $(\lambda,\lambda)$ equals $0$ or $2$ mod $4$). Therefore, $T_{\frac{\lambda}{4}}$ is a duality defect, providing an equivalence of our theory to the orbifold by this subgroup.

§.§ Topological defects in $T^4/\ZZ_N$ models

Let us generalize the construction of the previous sections to the case of a K3 model $\calC$ obtained as the orbifold $\CT/\ZZ_N$ of a torus model $\CT$ by a cyclic group $\langle g\rangle\cong\ZZ_N$.

In order for $\CT/\ZZ_N$ to be a K3 model, all holomorphic and anti-holomorphic fields of spin $1/2$ must be projected out. Because the $U(1)^4\times U(1)^4$ group generated by $W_\theta$ acts trivially on such fields, this implies that $g$ must be a lift of an automorphism of the Narain lattice $\Gamma^{4,4}$. Furthermore, because the eight fermions transform in the same representation as $\Gamma^{4,4}\otimes \RR$, we require $g$ to have no fixed vectors in such representation. Notice that, for $N>2$, the group $G_\CT$ preserving the small $\CN=(4,4)$  algebra and spectral flow contains such lifts only at special loci in the moduli space of torus models. All such models and symmetries were classified in <cit.>.

The quantum symmetry $Q$ has order $N$ and acts by multiplication by a phase $e^{\frac{2\pi ik }{N}}$ on the $g^k$-twisted sector. The orbifold procedure can be described as the gauging of the finite group $\langle g\rangle$, and defect line $Q^k$ is interpreted as a Wilson line  associated with the $1$-dimensional representation $\rho_k$ of $\langle g\rangle$, where $\rho_k(g)=e^{\frac{2\pi ik}{N}}$. In particular, operators that were local in the original torus model $\CT$ and transforming in the $\rho_k$ representation of $\langle g\rangle$ become operators in the defect space $\CH_{Q^k}$ in the orbifold $\calC$.

For each $\theta\in \Gamma^{4,4}\otimes \RR/\Gamma^{4,4}$, there  is a continuum of topological defects $T_\theta\in \cTop_\calC$ of dimension $N$ induced by a superposition of defects $W_{g^k(\theta)}$ of the torus model $\CT$
W_θ+W_g(θ)+…+W_g^N-1(θ)⟶T_θ .
The defect $T_\theta$ only depends on the orbit of $\theta$ with respect to the action of $\langle g\rangle$, and its dual is $(T_\theta)^\dual=T_{-\theta}$. For generic $\theta$, when the $\ZZ_N$-orbit has $N$ distinct elements $g^k(\theta)$, $T_\theta$ is simple, while it decomposes
into a sum of $N/d$ simple defects when $\theta$ has a non-trivial stabiliser $\ZZ_{N/d}\subset \ZZ_N$. More precisely, when the stabiliser subgroup $Stab(\theta)$ is  $\langle g^{d}\rangle\cong\ZZ^{N/d} $ for some $d|N$, there are $N/d$ simple defects $\eta_{\theta},Q\eta_\theta,Q^{2}\eta_\theta,\ldots,Q^{N/d-1}\eta_\theta$  of dimension $d$ induced by
∑_k=0^d-1 W_g^k(θ)⟶η_θand we have a decomposition
T_θ= η_θ+ Qη_θ+…+Q^N/d-1η_θ .
This decomposition can be understood as follows: when $g^d(\theta)=\theta$, each defect space $\CH_{W_{g^k(\theta)}}$ in the torus model, with $k=0,\ldots,d-1$,  carries a non-trivial representation of $\langle g^d\rangle$, and operators in $\CH_{W_{g^k(\theta)}}$ with different $g^d$-charge must belong to different (non-isomorphic) irreducible modules  of the orbifold theory $\calC$. The OPE with defect operators in $\CH_Q$ modifies the $g^d$-charge, and therefore maps each of these $N/d$ modules into one another. Therefore, the defect lines for these modules can be written as $Q^k\eta_\theta\equiv \eta_\theta Q^k$, $k=0,\ldots,N/d-1$. We notice that operators in $Q^{N/d}$ have trivial $g^d$ charge, so that we have the fusion rule
Q^N/dη_θ=η_θQ^N/d=η_θ .
This fusion rule also implies that the linear operator $\hat\eta_\theta:\CH\to\CH$ associated with $\eta_\theta$ must annihilate all $g^k$-twisted states, unless $k$ is a multiple of $d$. Similarly, the fusion rule
QT_θ=T_θQ=T_θ ,
implies that $\hat T_\theta$ annihilates all $g^k$-twisted sectors, for all $k\neq 0\mod N$.

The fusion rules are
T_θT_θ'=∑_k=0^N-1 T_θ+g^k(θ') ,
and, in particular, for generic $\theta$,
T_θT_-θ=T_0+∑_k=1^N-1 T_θ-g^k(θ)=1+Q+…+Q^N-1+∑_k=1^N-1 T_(1-g^k)(θ) .
Finally, because  the reflection (<ref>) commutes with every automorphism of the lattice $\Gamma^{4,4}$, we have that $\calR$ is also an invertible topological defect of the orbifold model $\calC$.

The elements $\theta$ that are stabilised by $g$ are all the solutions $x\in (\Gamma^{4,4}\otimes\RR)/\Gamma^{4,4}$ of the equation
(1-g)(x)≡0Γ^4,4 .
Because by hypothesis $g$ has no fixed vectors on $\Gamma^{4,4}\otimes\RR$ then
1+g+…g^N-1=0 ,
and $(1-g)$ is invertible, with inverse
(1-g)^-1=-1/N(g+2g^2+…+(N-1)g^N-1) ,
as can be easily verified. Thus, the fixed vectors $x$ are all elements of
((1-g)^-1Γ^4,4)/Γ^4,4⊆(1/NΓ^4,4)/Γ^4,4 .
The number of distinct points in this quotient is given by $\det(1-g)$ and can be easily computed once the eigenvalues of $g$ are known.

Consider for example the case when $N$ is prime. The analysis in <cit.> shows that the possible values are $N=2,3,5$ (for $N=5$, the symmetry $g$ has no geometric interpretation as an automorphism of the target $T^4$ torus.). In this case, for any $\theta\in (\Gamma^{4,4}\otimes\RR)/\Gamma^{4,4}$, either the orbit of $\theta$ contains $N$ distinct elements,  or $\theta$ is fixed by the full group $\langle g\rangle$. For each $N$, the eigenvalues of $g$ are all primitive $N$-th roots of unity with the same multiplicity <cit.>.  Therefore, the number $\det(1-g)$ of distinct points in the quotient $((1-g)^{-1}\Gamma^{4,4})/\Gamma^{4,4}$ equals $2^8$ for $N=2$, $3^4$ for $N=3$ and $5^2$ for $N=5$.
The corresponding simple defects $\eta_{x}$ are invertible, and together with $Q$ they generate some non-abelian central extension of $\ZZ_2^8$, $\ZZ_3^4$ and $\ZZ_5^2$ called extraspecial groups $2^{1+8}$, $3^{1+4}$ and $5^{1+2}$ for $N=2$, $N=3$ and $N=5$, respectively:
1⟶⟨Q⟩≅_N⟶N^1+k⟶_N^k→1 .
In particular, two lines $\eta_{x}$ and $\eta_{x'}$ do not necessarily commute
η_xη_x'=Q^c_g(x,x')η_x'η_x .
The $2$-cocycle $c_g(x,x')$ (<ref>) characterizing the central extension $N^{1+k}$ are determined by the 't Hooft anomaly in the $U(1)^8$ symmetry of the torus model, see <cit.>.

Similar arguments hold when $N$ is not prime. The possible values of $N$, in this case, are $4,6,8,10,12$ (they are the values $o(\pm g_0)$ in table 2 of <cit.>; the eigenvalues of $g$ when acting on the space $\Gamma^{4,4}\otimes \RR$ are denoted by $\pm\zeta_L,\pm\zeta_R,\pm\zeta_L^{-1},\pm\zeta_R^{-1}$ in the same table). For each $d|N$, consider the element $g^d$. If $1$ is an eigenvalue of $g^d$, then the points $x\in (\Gamma^{4,4}\otimes\RR)/\Gamma^{4,4}$ stabilized by $g^d$ form a continuum, corresponding to the $g^d$-fixed subspace of $\Gamma^{4,4}\otimes \RR$. On the other hand, suppose that none of the eigenvalues of $g^d$ equals $1$, so that $(1-g^d)$ is invertible. In this case, the $g^d$ fixed vectors $x\in \Gamma^{4,4}\otimes\RR/\Gamma^{4,4}$ are the $\det(1-g^d)$ elements in the quotient $((1-g^d)^{-1}\Gamma^{4,4})/\Gamma^{4,4}$. When $d=1$, the corresponding defects $\eta_x$ are invertible, and form a group that is a central extension of some $\ZZ_N^k$ by the quantum symmetry $Q$.

Consider the case where $g$ has  prime order $N$, and $(1-g)$ is invertible.
Let $x\in (\Gamma^{4,4}\otimes \RR)/\Gamma^{4,4}$, $x\neq 0$, be a non-trivial fixed point of $g$, i.e. such that $x\equiv g(x)\mod \Gamma^{4,4}$; this is the image $x=(1-g)^{-1}\lambda$ of some vector $\lambda\in\Gamma^{4,4}$. If we apply the operator $(1-g)^{-1}$ once again to $x$, we obtain a vector
v:=(1-g)^-1x ,
with some special properties. In particular, we get
(1-g^k)v=(1+g+…+g^k-1)(1-g)(1-g)^-1x=kxΓ^4,4 .
Let us define the simple defect
_x:=T_v=∑_k=0^N-1 W_g^k(v)=∑_k=0^N-1 W_v-kx .
We get

\begin{align} \CN_x^2&=\calR T_{v}\calR T_{v}=T_{-v}T_v=\sum_{k=0}^{N-1}T_{v-g^k(v)}=\sum_{k=0}^{N-1}T_{kx}\notag\\&=(1+\eta_x+\ldots+\eta_{(N-1)x})(1+Q+\ldots+Q^{N-1})\ ,
\end{align}
_xη_x=T_vη_x=T_v+x=∑_k=0^N-1 W_v-kx+x=∑_k=0^N-1 W_v-(k-1)x=_x
Therefore, $\CN_x$ is a duality defect for the abelian group of order $N^2$ generated by $\eta_x$ and $Q$. This group could be isomorphic to either $\ZZ_N\times\ZZ_N$ or $\ZZ_{N^2}$, depending on the norm of $x$. This argument shows that the torus orbifold $\CT/\langle g\rangle$ is self-orbifold with respect to any such group of symmetries.

More generally, one expects a continuum of defects in $\cTop_\calC$ whenever $\calC$ can be described as an orbifold of a torus model $\CT$ by some (possibly non-abelian) symmetry group $G$ of $\CT$. Such defects are induced by superpositions $\sum_{g\in G} W_{g(\theta)}$ of topological defects of torus models, and are simple for generic values of $\theta$.

§.§ An example of torus orbifold $T^4/\ZZ_4$

Let us consider an example of the general theory described in the previous section. We consider a torus model $\CT$ with a large group $U(1)^8\rtimes G^0_\CT$ of symmetries commuting with a small $\CN=(4,4)$ superconformal algebra, where $G^0_\CT$ has order $192$. This is the torus model described in section 4.4.1 of <cit.>, and contains a $\widehat{so}(8)_1$ chiral and anti-chiral algebra. The $\ZZ_2$ orbifold by $\calR$ (the centre of $G^0_\CT$) gives the K3 model with the `largest symmetry group' <cit.> that we will consider in section <ref>.

The winding-momentum lattice $\Gamma^{4,4}$ of this model is given by the columns of the following matrix

2     0     0     -1     1     1     1     1

0     2     0     -1     1     0     0     1

0     0     2     -1     0     1     0     0

0     0     0     -1     0     0     1     0

0     0     0     0     1     1     1     1

0     0     0     0     1     0     0     -1

0     0     0     0     0     1     0     0

0     0     0     0     0     0     1     0

In each column, the first four coordinates are the eigenvalues $\lambda_L^1,\ldots,\lambda_L^4$ of the zero modes of $\partial X^1,\ldots, \partial X^4$, while the last four are the eigenvalues $\lambda_R^1,\ldots,\lambda_R^4$ of the zero modes of $\bar\partial X^1,\ldots, \bar\partial X^4$.

Notice that the vertex operators $V_\lambda(z)$ corresponding to the first four columns are holomorphic currents ($h=\frac{\lambda_L\cdot\lambda_L}{2}=1$, $\bar h=\frac{\lambda_R\cdot\lambda_R}{2}=0$), which together with $\partial X^k$ generate the $\widehat{so}(8)_1$ current algebra.

Let us focus on a symmetry $g\in G^0_\CT$ of order $N=4$, acting by

-1     0     0     0     0     0     0     0

0     -1     0     0     0     0     0     0

0     0     -1     0     0     0     0     0

0     0     0     -1     0     0     0     0

0     0     0     0     0     1     0     0

0     0     0     0     -1     0     0     0

0     0     0     0     0     0     0     1

0     0     0     0     0     0     -1     0

on $\Gamma^{4,4}\otimes \RR$. One can easily verify that this is a lattice automorphism, $g\in O(\Gamma^{4,4})$, and that $\det(1-g)=64\neq 0$. This is a symmetry in the class $-4A$, in the notation of <cit.>, and the orbifold $\CT/\langle g\rangle$ is again the K3 model with largest symmetry group that we will describe in section <ref> (see section $6$ of <cit.>).

Let us consider the $g$-fixed points in $\Gamma^{4,4}\otimes\RR/\Gamma^{4,4}$. As explained in the previous section they correspond to the points
with generators

where $y_1$ and $y_2$ have order $2$, while $u_1$ and $u_2$ have order $4$ (modulo $\Gamma^{4,4}$).

The invertible topological defects $W_x$ of the torus model $\CT$, where $x\in ((1-g)^{-1}\Gamma^{4,4})/\Gamma^{4,4}$ are $g$-fixed points, induce invertible defects $\eta_x,Q\eta_x,Q^2\eta_x,Q^3\eta_x$ in the K3 model $\CT/\langle g\rangle$, where $Q$ is the quantum symmetry of order $4$. The group they generate is a central extension of $((1-g)^{-1}\Gamma^{4,4})/\Gamma^{4,4}\cong \ZZ_2\times \ZZ_2\times \ZZ_4\times \ZZ_4$ by $\langle Q\rangle\cong \ZZ_4$. In order to determine the central extension in detail, one needs to know the 't Hooft anomaly for the abelian group
H=⟨g⟩×((1-g)^-1Γ^4,4)/Γ^4,4≅_4×_2×_2×_4×_4 .
The anomaly is encoded in a cohomology class $[\omega]\in H^3(H,U(1))$, with representative a $3$-cocycle $\omega:H\times H\times H\to U(1)$. It is known that
H^3(_2^2×_4^3,U(1))≅_2^18×_4^7 ,and a basis of generators can be found, for example, in <cit.>. In order to determine which class in $H^3(H,U(1))$ is relevant in this case, one can just consider, for each $k\in H$, the failure of the level matching condition for the $k$-twisted sector of $\CT$. In particular, if $k\in H$ has order $o(k)$, the level matching condition is satisfied if the spin ${spin}(k)$ (i.e. the difference $h-\bar h$ of conformal weights) of the $k$-twisted states take values in $\frac{1}{o(k)}\ZZ$; in this case, the restriction of the anomaly class $[\omega]$ to the cyclic group $\langle k\rangle$ is trivial. More generally, if the restriction of $[\omega]$ to $\langle k\rangle$ is non-trivial, one has the following relation <cit.>
e^2πi o(k)spin(k)=∏_i=1^o(k)-1ω(k,k^i,k) ,
between the spin and the cocycle $\omega$.
If we know the spin of the $k$-twisted sectors for all $k\in H$, we can use (<ref>) to determine the class $[\omega]$. In fact, for elements $x\in ((1-g)^{-1}\Gamma^{4,4})/\Gamma^{4,4}\subset H$ the spin of the $x$-twisted ground state is
spin(x)=(x,x)/2 .
On the other hand, the cohomology class $[\omega]$ is trivial when restricted to the cyclic subgroup $\langle g\rangle\subset H$ (see <cit.>). More generally, the restriction of $[\omega]$ to any cyclic group of the form $\langle gx\rangle\subset H$ is trivial for all $x\in ((1-g)^{-1}\Gamma^{4,4})/\Gamma^{4,4}$, because $g$ and $gx$ are conjugate within the group $G$. These data are sufficient to determine the class $[\omega]$ uniquely.

Once the cocycle $\omega$ representing the 't Hooft anomaly of $H$ is known, one can determine the central extension of $((1-g)^{-1}\Gamma^{4,4})/\Gamma^{4,4}$  by the quantum symmetry $Q$ that is induced on the orbifold theory $\CT^4/\langle g\rangle$. In particular, the commutation relations are

\eta_{y_1}\eta_{y_2}=Q^2\eta_{y_2}\eta_{y_1}\ , \qquad\qquad \eta_{u_1}\eta_{u_2}=Q\eta_{u_2}\eta_{u_1}\ ,\qquad\qquad
\eta_{y_j}\eta_{u_k}=Q^2\eta_{u_k}\eta_{y_j}\ ,\qquad j,k=1,2\ .
Furthermore, while $\eta_{y_1}$ and $\eta_{y_2}$ can be chosen of order $2$
η_y_1^2=1=η_y_2^2 ,
the symmetries $\eta_{u_1}$ and $\eta_{u_2}$ have order $8$
η_u_1^4=Q^2=η_u_2^4 ,        Q^4=1 .

The orbifold theory $\CT/\langle g\rangle$ contains a continuum of topological defects $T_\theta$ of dimension $4$, with $\theta\in (\Gamma^{4,4}\otimes\RR)/\Gamma^{4,4}$, that are simple for generic $\theta$. These defects, as well as the defects of the form $Q^kT_\theta$, $k=0,1,2,3$, are induced by the superpositions $\sum_{j=0}^3 W_{g^j(\theta)}$ in the torus model
W_θ+W_g(θ)+W_g^2(θ)+W_g^3(θ) ⟶T_θ, QT_θ, Q^2T_θ, Q^3T_θ .
When $\theta\equiv x\in  ((1-g)^{-1}\Gamma^{4,4})/\Gamma^{4,4}$ is one of the $g$-fixed points, $T_\theta\equiv T_x$ is not simple, but becomes a superposition of four invertible defects
T_x=η_x+Qη_x+Q^2η_x+Q^3η_x ,
and satisfies $QT_x=T_x$.

An intermediate case occurs when $\theta$ is fixed by $g^2$, but not by $g$. This happens when $\theta$ is of the form $\theta=(\theta_L,0)$, so that $g(\theta)=-\theta$. In this case, $T_\theta$ decomposes as a sum
T_θ=ξ_θ+Qξ_θ ,
where $\xi_{\theta}$ and $Q\xi_\theta$ are dimension two defects induced by $W_\theta+W_{-\theta}$ on $\CT$
W_θ+W_-θ ⟶ξ_θ,Qξ_θ .The defects $\xi_\theta$, where $\theta=(\theta_L,0)$, are simple for generic $\theta_L$, while they decompose as $\xi_x=\eta_x+Q\eta_x$ when $x\equiv (\theta_L,0)$ is fixed by $g$. Furthermore, the $\xi_\theta$ are always unoriented and satisfy
Q^2ξ_θ=ξ_θQ^2=ξ_θ ,        ξ_θξ_θ'=ξ_θ+θ'+ξ_θ-θ' .

As described in the previous section, for each $g$-fixed vector $x\in ((1-g)^{-1}\Gamma^{4,4})/\Gamma^{4,4}$, we can consider $v=(1-g)^{-1}x$. There are two cases to be considered, depending on whether $x$ is fixed by $g^2$ or not. In the first case, $x$ is a linear combination
x=a_1y_1+a_2y_2+2b_1u_1+2b_2u_2 ,
with $a_1,a_2,b_1,b_2\in \{0,1\}$. Notice the $x$ has always order $2$ in this case, $2x\in \Gamma^{4,4}$, and $g(x)=-x\mod \Gamma^{4,4}$. In this case, we have
+ b_1

so that $g(v)=-v$ and $g^2(v)=v$. Thus, in the orbifold theory $\CT/\langle g\rangle$ there are two dimension $2$ unoriented simple topological defects, $\xi_v$ and $Q\xi_v$, induced by the defect $W_{v}+W_{-v}$ of $\CT$, as in (<ref>).
The fusion of $\xi_v$ and $Q\xi_v$ with themselves gives
ξ_v^2=(Qξ_v)^2=ξ_0+ξ_x=1+Q^2+η_x+Q^2η_x .
Furthermore, because $v+x\equiv -v\mod \Gamma^{4,4}$, one gets
ξ_vη_x=η_xξ_v=ξ_v .
Therefore, both $\xi_v$ and $Q\xi_v$ are duality defects for the $\ZZ_2\times \ZZ_2$ group generated by $\eta_x$ and $Q^2$.

In the second case, where $x$ is not fixed by $g^2$, we have
x=a_1y_1+a_2y_2+c_1u_1+c_2u_2 ,
where $a_1,a_2,c_1,c_2\in \{0,1\}$, with at least one among $c_1$ and $c_2$ being odd. In this case, $x$ has order $4$, i.e. $4x\in \Gamma^{4,4}$, and $v:=(1-g)^{-1}x$ is not fixed by either $g$ or $g^2$. Thus, the orbifold theory $\CT/\langle g\rangle$ contains four dimension $4$ defects $Q^kT_v$, $k=0,1,2,3$, induced by $\sum_{k=0}^3W_{g^k(v)}$, as in (<ref>). The treatment is analogous to the case described in the previous section, where the order $N$ of $g$ was a prime number. One defines
_x:=T_v ,
and the resulting topological defect $\CN_x$ is unoriented and satisfies
_x^2= (1+Q+Q^2+Q^3)(1+η_x+η_2x+η_3x)
Thus, it is a duality defect for the $\ZZ_2\times \ZZ_8$ group with generators $\eta_x$ of order $8$ and $Q\eta_x^2$ of order $2$.

§.§ K3 models with continuous defects: a conjecture

In the previous sections, we have seen that all K3 models $\calC$ that can be described as torus orbifolds contain a continuous family of topological defects $\CL_\lambda\in \cTop_\calC$, that preserve the $\CN=(4,4)$ algebra and spectral flow, and that are simple for generic values of $\lambda$.
On the other hand, in claim <ref> we argued that for a generic K3 model $\cTop_\calC$ is trivial, and in particular has no such family of defects. Thus, it is natural to look for a characterisation of the K3 models admitting such continuous families.
We propose the following conjecture:

Let $\calC$ be a K3 model that admits a continuous family of topological defects $\CL_\lambda\in \cTop_\calC$, preserving the $\CN=(4,4)$ superconformal algebra and spectral flow, and simple for all $\lambda$ except possibly a zero measure set. Then, $\calC$ is the (generalised) orbifold of a torus model $\CT$.

Let us recall the notion of generalised orbifold <cit.>. In a `standard' orbifold of a CFT $\calC$ by a finite group of symmetries $G$, one can consider the superposition $A=\sum_{g\in G} \CL_g$ of all invertible defects in the group.
Then, the Hilbert space of local operators in the orbifold theory $\calC/G$ is just the subspace of $G$-invariant operators in $\CH_{A}$, and their correlation functions can be obtained as correlation functions in the original theory with an insertion of a fine enough network of defects of type $A$.
This construction can be generalised to any topological defect $A$, for which there exist some `multiplication' and `co-multiplication' maps, i.e. topological junction operators $\mu:\CH_A\otimes \CH_A\to \CH_A$ and  $\tilde \mu:\CH_A\to\CH_A\otimes \CH_A$ satisfying some suitable conditions – essentially, that the associator of the multiplication map is trivial, and that $\tilde \mu\circ\mu$ is the identity.
When such properties are satisfied, one can define a new consistent CFT, the generalised orbifold $\calC/A$, whose space of local operators is a suitable subspace of $\CH_A$, and whose correlation functions are obtained by inserting networks of defects $A$ in the $\calC$ correlators, with the operators $\mu$ and $\tilde\mu$ inserted at trivalent junctions.
Furthermore, the generalised orbifold procedure is always reversible: if $\calC'=\calC/A$ is obtained as a generalised orbifold of some CFT $\calC$, then $\calC=\calC'/A'$ is a generalised orbifold of $\calC'$.

Let us now sketch an argument for a possible proof of conjecture <ref>. According to <cit.>, the presence of a continuum of operators $\CL_\lambda$ is related to the presence of a conserved current $j\in \CH_{\CL_\lambda\CL_\lambda^\dual}$:
j:=J(z)dz+J̃(z̅)dz̅ ,
where $\bar \partial J(z)=0=\partial \tilde J(\bar z)$. This current is such that an infinitesimal deformation $\CL_{\lambda+\delta\lambda}$ of the defect $\CL_\lambda$ can be obtained as
_λ+δλ=_λe^iδλ∫j ,
where $j\in \CH_{\CL_\lambda\CL_\lambda^\dual}\cong \CH_{\CL_\lambda}\otimes\CH^*_{\CL_\lambda}$ is interpreted as a linear operator from $\CH_{\CL_\lambda}$ to itself, and the integration $\int j$ is over the support of the defect $\CL_\lambda$.

In the case we are interested in, since $\CL_\lambda$ and $\CL_{\lambda+\delta\lambda}$ are defects in $\cTop_\calC$, then the current $j$ must preserve the $\CN=(4,4)$ superconformal algebra and the spectral flow. In particular, $J(z)$ and $\tilde J(\bar z)$ must be neutral with respect to the holomorphic and anti-holomorphic $\widehat{su}(2)_1$ R-symmetry, and must be the supersymmetric descendants of holomorphic and anti-holomorphic spin $1/2$ fields. This implies that the NS-NS sector of $\CH_{\CL_\lambda\CL_\lambda^\dual}$ contains two $\CN=(4,4)$ BPS representation $(\frac{1}{2},\frac{1}{2};0,0)$ and $(0,0;\frac{1}{2},\frac{1}{2})$ containing the currents $J(z)$ and $\tilde J(\bar z)$. As a matter of fact, because the ground states of these representations are a $\widehat{su}(2)_1$ doublet on complex spin $1/2$ fields, and because $(\CL_\lambda\CL_\lambda^\dual)^\dual=\CL_\lambda\CL_\lambda^\dual$, there must be $4$ holomorphic and $4$ anti-holomorphic Majorana spin $1/2$ fields (in other words, one needs to include also the CPT conjugates of these BPS representations).
The OPE of the four holomorphic spin $1/2$ fields is very constrained, and only the vacuum operator can appear in a singular term.
We can take $\CL_{orb}\subseteq \CL_\lambda\CL_\lambda^\dual$ to be the smallest unoriented ($\CL_{orb}^\dual=\CL_{orb}$) defect containing $I$, with $\CL_{orb}\subset (\CL_{orb})^2$, and such that $\CH_{\CL_{orb}}$ contains the four holomorphic and antiholomorphic spin $1/2$ fields and is closed under OPE.
Then $\CH_{\CL_{orb}}$ contains a holomorphic and a antiholomorphic copy of the algebra generated by $\CN=4$ and the free fermions.
This is just the chiral algebra of a generic $T^4$ model, i.e. the algebra of four free bosons and four free fermions. Furthermore, $\CH_{\CL_{orb}}$ decomposes into (possibly twisted) representations of this chiral and anti-chiral algebra.
What remains to prove is that $A:=\CL_{orb}$ (or possibly some extension of $\CL_{orb}$), satisfies all properties such that the generalised orbifold $\calC/A$ is well-defined. It might be possible to prove this fact using properties of the category of (twisted) representations of the chiral algebra of $T^4$. If the orbifold $\calC/A$ is consistent, then it contains four free fermions and four free bosons and has central charges $c=6=\tilde c$, so that it is necessarily a sigma model $\CT$ on $T^4$.
By reversibility of the orbifold procedure, we conclude that $\calC$ is a generalised orbifold of the torus model $\CT$.

In the examples of continuous defects discussed in the previous sections, $\CL_{orb}$ was given by $I+Q+\ldots+Q^{N-1}$, and indeed the orbifold of $\calC$ by $\CL_{orb}$ gives back $\CT$.

§ AN EXAMPLE: A K3 MODEL WITH $\ZZ_2^8:M_{20}$ SYMMETRY GROUP

In this section, we discuss the topological defects $\CL\in \cTop_\calC$ preserving the $\CN=(4,4)$ superconformal algebra (SCA) in a particular K3 sigma model $\calC_{GTVW}$, the `most symmetric model' of <cit.> (see also <cit.>). The bosonic parts of the holomorphic and anti-holomorphic chiral algebras of this model are isomorphic to $\CA:=(\widehat{su}(2)_1)^{6}$ (six copies of the $su(2)$ current algebra at level $1$). Recall that $\widehat{su}(2)_1$ has two irreducible representations with conformal weights $0$ and $1/4$, and the fusion ring of  $\widehat{su}(2)_1$ is the group ring of the cyclic group $\ZZ_2$ of order two, which we represent as the additive group with elements $0,1$. Each irreducible $\CA$-module is labeled by an element in $\ZZ_2^{6}$, as $M_{a_1,\ldots, a_6}$, $a_i\in\{0,1\}$.

The NS-NS sector of $\calC_{GTVW}$ is given by the sum
⊕_[a_1,…,a_6;b_1,…,b_6]∈A_NS-NS M_a_1,…, a_6⊗M̅_b_1,…, b_6 ,
where the set $A_{NS-NS}\subset \ZZ_2^6\times \ZZ_2^6$ is given by the disjoint union
of a `bosonic' subset (corresponding to the subsector with positive fermion number)
A_(NS-NS)^+={[a_1,…,a_6;b_1,…,b_6]∈_2^6×_2^6|a_i=b_i,  ∑_i a_i≡02}
and a `fermionic' one
A_(NS-NS)^-={[a_1,…,a_6;b_1,…,b_6]∈_2^6×_2^6|a_i=b_i+1,  ∑_i a_i≡02} .
Similarly, the R-R sector is a sum over the modules whose labels take values in the set $A_{R-R}=(A_{R-R})^+\sqcup (A_{R-R})^-$ with
A_(R-R)^+={[a_1,…,a_6;b_1,…,b_6]∈_2^6×_2^6|a_i=b_i,  ∑_i a_i≡12}
A_(R-R)^-={[a_1,…,a_6;b_1,…,b_6]∈_2^6×_2^6|a_i=b_i+1,  ∑_i a_i≡12} .
The bosonic CFT $\calC^{bos}_{GTVW}$ that contains only the states in the $(NS-NS)^+$ and $(R-R)^+$ sectors of the SCFT $\calC_{GTVW}$ is just the diagonal modular invariant of the $(\widehat{su}(2)_1)^{\oplus 6}$ algebra.

The K3  model $\calC_{GTVW}$ can also be defined as the orbifold $\CT/\ZZ_2$ of a particular torus model $\CT$ by the $\ZZ_2$ symmetry $\calR$ reflecting all torus coordinates, see section <ref>. Alternatively, it can be described as a $\CT/\ZZ_4$ orbifold of the same torus model $\CT$ by a symmetry of order $4$, as described in section <ref>. See also <cit.> for more details on this construction.

This model contains several different copies of the $\CN=(4,4)$ superconformal algebra, related to one each other by symmetries of the CFT. We focus on one of them, such that the $\widehat{su}(2)_1$ subalgebra of the (anti-)holomorphic $\CN=4$ is identified with the first factor in the (anti-)chiral algebra $(\widehat{su}(2)_1)^{6}$. The four supercurrents of the holomorphic $\CN=(4,4)$ are suitably chosen ground states in the $M_{1,1,1,1,1,1}\otimes \bar{M}_{0,0,0,0,0,0}$ module of $\CA\times\bar\CA$. The space of ground states of this module is isomorphic to a tensor product $(\CC^2)^{\otimes 6}$, and the $\CN=4$ supercurrents can be nicely described in terms of a quantum error correcting code <cit.>. The full (bosonic and fermionic) chiral algebra of the theory is generated by the bosonic $(\widehat{su}(2)_1)^{6}$ and one of the supercurrents.

The $24$ (real) Ramond-Ramond states with conformal weight $h=\bar h=\frac{1}{4}$ generating the space $V$ are the ground states in the six $\CA\times\bar\CA$ representations

\begin{align}
&[1,0,0,0,0,0;1,0,0,0,0,0]\, ,\ [0,1,0,0,0,0;0,1,0,0,0,0]\, ,\ [0,0,1,0,0,0;0,0,1,0,0,0]\notag\\
&[0,0,0,1,0,0;0,0,0,1,0,0]\, ,\ [0,0,0,0,1,0;0,0,0,0,1,0]\, ,\ [0,0,0,0,0,1;0,0,0,0,0,1]\label{RRreps}
\end{align}
each one containing $4$ ground states; we will call any such a set of four states a tetrad. In particular, the first tetrad, i.e. the ground states in $[1,0,0,0,0,0;1,0,0,0,0,0]$, are the spectral flow generators, spanning $\Pi\subset V$.

The full group of symmetry of $\calC_{GTVW}$ is $(SU(2)^6\times SU(2)^6)\rtimes S^6$, where the two $SU(2)^6$ subgroups are generated by the zero modes of the holomorphic and anti-holomorphic currents, while $S^6$ is the group of permutations of the six $\widehat{su}(2)_1$ factors in $(\widehat{su}(2)_1)^{6}$ acting diagonally on the holomorphic and anti-holomorphic currents. To be precise, such a group does not act faithfully on $\CH$, because a certain subgroup $Z_0\subset SU(2)^6\times SU(2)^6$ acts trivially on all the states of the theory. Indeed, let t_i,t̃_i ,     i=1,…,6,be the generators of the centre $\ZZ_2^6\times \ZZ_2^6$ of $SU(2)^6\times SU(2)^6$, where $t_i$ and $\tilde t_i$  act on the representation $M_{a_1,\ldots, a_6}\otimes \bar M_{b_1,\ldots, b_6}$ by multiplication by $(-1)^{a_i}$ and $(-1)^{b_i}$, respectively. Then, the subgroup
Z_0={ ∏_i=1^6 (t_it̃_i)^r_i |∑_i=1^6 r_i∈2}≅_2^5
acts trivially on all states of the theory. Thus, the group acting faithfully is $((SU(2)^6\times SU(2)^6)/Z_0)\rtimes S^6$. In particular, the quotient $(\ZZ_2^6\times \ZZ_2^6)/Z_0 \cong \ZZ_2^7$ is the group of symmetries commuting with the full bosonic chiral algebra $\CA$, and we can take
t_1,…, t_6, t̃_1 ,
as representatives of the generators (modulo $Z_0$).

In <cit.> it was shown that the subgroup $G_{GTVW}\subset ((SU(2)^6\times SU(2)^6)/Z_0)\rtimes S^6$ fixing the $\CN=(4,4)$ superconformal algebra and the spectral flow generators is finite and isomorphic to a split extension $\ZZ_2^8:M_{20}$ of the Mathieu group $M_{20}$ by $\ZZ_2^8$. In turn, $M_{20}$ is a split extension $\ZZ_2^4:A_5$ of the alternating group $A_5$ by $\ZZ_2^4$. See appendix <ref> for a description of the generators. The intersection
G_GTVW∩(_2^6×_2^6)/Z_0={ ∏_i=2^6 t_i^r_i |∑_i=2^6 r_i∈2}≅_2^4 ,
is the subgroup of symmetries commuting with both $\CN=(4,4)$ superconformal algebra and with the bosonic current algebra $(\widehat{su}(2)_1)^{6}\times (\widehat{su}(2)_1)^{6}$.

§.§ Topological defects

Let us now consider the possible topological defects $\CL\in \cTop_{GTVW}\equiv \cTop_{\calC_{GTVW}}$ preserving the $\CN=(4,4)$ algebra and spectral flow. The following result was used in section <ref> to prove that an analogous statement holds for the category of defects $\cTop_\Pi$ of any K3 models $\calC_\Pi$ (see claim <ref>).

The only defects $\CL\in \cTop_{GTVW}$ that are transparent to all $24$ R-R operators in $V$ are integral multiples of the identity defects.

To prove this statement, we notice that if $\CL$ is transparent to the $\CN=(4,4)$ algebra and to all R-R operators in $V$, then it is also transparent to all the operators that can be obtained from their OPE. Thus, we just need to prove that one can obtain all operators in $\CH$ by OPE of R-R operators in $V$ and their $\CN=(4,4)$ descendants. In fact, by taking the OPE of the four operators the $i$-th tetrad, $i=1,\ldots, 6$, one can obtain the currents in the holomorphic and anti-holomorphic $i$-th $\widehat{su}(2)_1$ factor. Thus, in this way we can generate the full chiral and antichiral algebra $\CA\times\bar\CA$. Furthermore, the fusion rules between $\CA$ modules imply that the OPE of the $\CA\times\bar\CA$ primary fields in the six modules (<ref>), together with the ones in the $[1,1,1,1,1,1;0,0,0,0,0,0]$ and $[0,0,0,0,0,0; 1,1,1,1,1,1]$ modules, where the supercurrents live, generate every other $\CA\times\bar\CA$ primary operator in the spectrum, and we conclude.

Some examples of defects in $\cTop_{GTVW}$ are as follows:

* Invertible defects. We have one simple defect $\CL_g$ for each $g\in G_{GTVW}\cong \ZZ_2^8:M_{20}\cong \ZZ_2^8:(\ZZ_2^4:A_5)$.

* Defects induced by the torus orbifold description. Since the model $\calC_{GTVW}$ can be defined as a torus orbifold $\CT/\ZZ_2$, it inherits all defects from the torus model $\CT$. In particular, as described in section <ref>, $\cTop_{GTVW}$ contains a continuum of non-invertible defects $T_\theta$, $\theta\in (\Gamma^{4,4}\otimes\RR)/\Gamma^{4,4}$ of dimension $2$ that are simple for generic values of $\theta$. One can identify the first two tetrads in (<ref>) as the RR ground states in the untwisted sector of the orbifold, and the last four tetrads as the twisted sector. Then, all operators $\hat T_\theta$ act on the first two tetrads by multiplication by $2$, while they annihilate the last four tetrads. Notice that symmetries in $G_{GTVW}$ permute the second tetrad with the last four, while keeping the first tetrad of spectral flow generators fixed; therefore, they change the identification of the tetrads with the twisted or untwisted sector of the orbifold. This means that fusion of $T_\theta$ with invertible defects in $\cTop_{GTVW}$ provide further sets of continuous defects of dimension $2$. Any such defect acts by multiplication by $2$ on the spectral flow generators (the first tetrad) and on one more tetrad (the untwisted sector RR ground states), while annihilating the remaining four tetrads (the twisted sector).

The model $\calC_{GTVW}$ can also be described as a $T^4/\ZZ_4$ orbifold. In fact, this is exactly the model described in section <ref>. We denote by $Q_4$ the quantum symmetry of order $4$ in this torus orbifold description. Among the elements of the group $G_{GTVW}$, the quantum symmetries $Q_4$ can be characterised by their eigenvalues on the $24$-dimensional space $V$ of RR ground states ($\pm 1$ with multiplicity $4$ each, and $\pm i$ with multiplicity $8$ each).  From the $T^4/\ZZ_4$ description, one can deduce that $\cTop_{GTVW}$ must contain a continuum of topological defects $T^{(4)}_\theta$ of order $4$ and a continuum of topological defects $\xi_\theta$ of order $2$. The former act on the first tetrad of RR ground states by multiplication by $4$, while they annihilate all other $5$ tetrads. The latter act by multiplication by $2$ on two tetrads (including the first), while they annihilate the remaining four tetrads.

* Verlinde lines for $(\widehat{su}(2)_1)^{\oplus 6}$. Since this SCFT is rational with respect to the chiral algebra $\CA$, it is useful to consider the topological defects that preserve the whole algebra $\CA$. For rational CFTs, there is a finite number of such defects. In particular, for a bosonic CFT corresponding to the diagonal invariant of $\CA$, simple topological defects preserving $\CA$ are completely classified, and are given by the Verlinde line defects. In general, Verlinde lines are in one-to-one correspondence with representations of the algebra $\CA$, and their fusion ring is the same as the fusion ring of $\CA$ representations. Let us consider the simple topological defects of the bosonic CFT $\calC_{GTVW}^{bos}$ that preserve the $(\widehat{su}(2)_1)^{\oplus 6}$ chiral and anti-chiral algebras. They are $2^6$ Verlinde lines, one for each $(\widehat{su(2)}_1)^{\oplus 6}$ representation, and obey group-like $\ZZ_2^6$ fusion rules; this means that they are all invertible, so that they form a $\ZZ_2^6$ group of symmetries. The lifts of such symmetries from the bosonic CFT $\calC_{GTVW}^{bos}$ to the SCFT $\calC_{GTVW}$, together with the fermion number, generate the full group $(\ZZ_2^6\times \ZZ_2^6)/Z_0\cong \ZZ_2^7$ of symmetries fixing all $(\widehat{su}(2)_1)^{\oplus 6}$. Notice that not all Verlinde defects preserve the $\CN=(4,4)$ superconformal algebra and the spectral flow generators – $\CA$ only contains the even subalgebra of $\CN=4$, but not the supercurrent. In fact, only  the subgroup (<ref>)
of $(\ZZ_2^6\times \ZZ_2^6)/Z_0$ is contained in $G_{GTVW}$ and therefore gives rise to defects in $\cTop_{GTVW}$. In particular, $t_1$, $ \tilde t_1$, and $t_1\cdots t_6$ can be identified, respectively, with the left-moving and right-moving fermion number, and with the $\ZZ_2$ symmetry acting by $-1$ on all R-R states. In conclusion, Verlinde lines only provide topological defects that are generated by ordinary symmetries.

* Lines preserving some `large' chiral algebra. There are two ways we can generalize the construction of Verlinde lines in the previous point. One is to consider topological defects that act on the $(\widehat{su}(2)_1)^{\oplus 6}$ chiral and anti-chiral algebra by some non-trivial automorphism. The full analysis is performed in appendix <ref>. The final outcome is that, besides the invertible defects, there are a few more duality defects (see the next point), related to the fact that the SCFT $\cTop_{GTVW}$ is self-orbifold $\cTop_{GTVW}=\cTop_{GTVW}/H$ with respect to certain subgroups $H$ of the subgroup $G_{GTVW}\cap (\ZZ_2^6\times \ZZ_2^6)/Z_0$ of symmetries fixing all the $(\widehat{su}(2)_1)^{\oplus 6}$ currents. Actually, it turns out that the duality defects obtained in this way are elements in the family of continuous defects $T_\theta$, at special values of $\theta$. The possibility that some $T_\theta$ are duality defects was discussed in section <ref>.

The other possibility to discover new objects in $\cTop_{GTVW}$ is to consider the defects that preserve some chiral algebra $\CB$ that is smaller than the full chiral algebra of the theory, but such that the theory is still rational with respect to $\CB$.
There are many simple subalgebras of the bosonic $(\widehat{su}(2)_1)^{\oplus 6}$ whose representation theory are well known, such as, for example, products of affine subalgebras $\hat h\subset (\widehat{su}(2)_1)^{\oplus 6}$ and cosets $(\widehat{su}(2)_1)^{\oplus 6}/h$. The main difficulty with this approach is to find the defects that preserve the $\CN=4$ supercurrents. We were not able to find any new defects in $\cTop_{GTVW}$ using this technique.

* Duality defects. By definition, a duality defect $N$ is such that the fusion with the reversed orientation defect $N^\dual$ is a superposition of invertible defects
N^N=∑_h∈H _h ,    N_h=N ,
for some group $H$ of symmetries. Duality defects occur when a model $\calC$ is self-orbifold with respect to the group $H$, i.e. if the orbifold theory $\calC/H$ is a consistent CFT isomorphic to $\calC$. This means that there is an isomorphism between the space $\CH$ of local operators of $\calC$ and the one of  $\calC/H$ such that correlation functions are the same. In this case, moving a local operator of $\calC$ through $N$ gives the corresponding operator in $\calC/H$.
In particular, when $N =N^\dual$ is unoriented, and $H$ is an abelian group, the category generated by $N$ and $\CL_h$, $h\in H$, is called a Tambara-Yamagami (TY) category, with fusions
N^2=∑_h∈H _h ,    _h N=N=N_h .
If $N\in \cTop_{GTVW}$ is a duality defect, then the corresponding $H$ must be an subgroup of $G_{GTVW}$, the group of symmetries acting trivially on the $\CN=(4,4)$ superconformal algebra and spectral flow. Notice, however, that the converse is not true in general: even if a certain subgroup $H\subset G_{GTVW}$ is such that $\calC_{GTVW}/H\cong \calC_{GTVW}$, the corresponding duality defect $N$ is not necessarily transparent to the $\CN=(4,4)$ algebra and spectral flow, and in this case it is not in $\cTop_{GTVW}$ (see appendix <ref> for an example).