On symmetries of N=(4,4) sigma models on T^4

Motivated by an analogous result for K3 models, we classify all groups of symmetries of non-linear sigma models on a torus T^4 that preserve the N=(4,4) superconformal algebra. The resulting symmetry groups are isomorphic to certain subgroups of the Weyl group of E8, that plays a role similar to the Conway group for the case of K3 models. Our analysis heavily relies on the triality automorphism of the T-duality group SO(4,4,Z). As a byproduct of our results, we discover new explicit descriptions of K3 models as asymmetric orbifolds of torus CFTs.


Introduction and summary
Non-linear sigma models on tori provide the basic examples of string theory compactifications. The study of the discrete groups of symmetries and the orbifolds of supersymmetric torus models, both in heterotic or type II superstring theory, have been the subject of intensive study since the eighties [1][2][3] (see also [4] and references therein). These models represent both a fruitful arena for the formal understanding of string compactifications and are of great interest from a phenomenological viewpoint.
In this paper, we focus on type II non-linear sigma models with target space four dimensional tori T 4 and classify the groups of symmetries that commute with the ('small') world-sheet N = (4, 4) superconformal algebra. This analysis is mainly motivated by the relation between such torus models and non-linear sigma models on K3. There are few K3 models for which an explicit and complete description as conformal field theories is available; among them, torus orbifolds play a fundamental role. The interest in K3 superstring compactifications has received new impetus in the last few years with the discovery of the Mathieu moonshine phenomenon. This conjecture, originated from an observation of Eguchi, Ooguri and Tachikawa (EOT) in [5], proposes a connection between the elliptic genus of K3 and a finite sporadic simple group, the Mathieu group M 24 . After the original EOT proposal, a considerable amount of evidence in favour of this conjecture has been compiled [6][7][8][9][10] and several different incarnations of the relationship between M 24 and various string compactifications on K3 have been uncovered [11][12][13][14][15][16][17][18][19]. Despite the amount of work on the subject, however, no satisfactory explanation of this phenomenon has been provided so far.
One of the most natural approaches towards an interpretation of Mathieu Moonshine is the analysis of the groups of discrete symmetries of K3 models. Orbifolds of non-linear sigma models on T 4 provide some of the simplest examples for such an investigation [20][21][22]. In [23], all possible groups of discrete symmetries of K3 models that commute with the N = (4, 4) superconformal algebra have been classified. This result can be thought of as a stringy analogue of Mukai's theorem in algebraic geometry [24,25], where the groups of symplectic automorphisms of K3 surfaces were considered. The rather surprising outcome of this classification is that all such groups of symmetries are subgroups of the Conway group Co 0 , the group of automorphisms of the Leech lattice. Although M 24 itself is a subgroup of Co 0 , it does not appear in the list of [23]. As a consequence, no K3 sigma model admits M 24 as its group of symmetries. It was observed in [26] that, for almost all K3 models, the symmetry group is actually a subgroup of M 24 and that most of the exceptions correspond to torus orbifolds. The analysis of [26] also showed the existence of a K3 model represented by an asymmetric orbifold of a torus model by a symmetry of order 5.
This amount of work on non-linear sigma models on K3 led to a rather paradoxical situation: in fact, although non-linear sigma models on T 4 are much better understood than K3 models, the classification of the groups of symmetries preserving the N = (4, 4) superconformal algebra was known only for the latter. The goal of this paper is to fill in this gap. This completes the classification for all (known) models with N = (4,4) superconformal symmetry at central charge c =c = 6. Our main result is the following: The group G of symmetries of a supersymmetric non-linear sigma model on T 4 that commutes with the (small) N = (4, 4) superconformal algebra is where G 0 is one of the following finite subgroups of SU(2) × SU(2): -Geometric groups: C 2 , C 4 , C 6 , D 8 (2), D 12 (3), T 24 -Non-geometric groups: This rather abstract description of the symmetry groups deserves some further clarification (see section 2 for details). The continuous normal subgroup U(1) 4 × U(1) 4 is generated by the zero modes of four holomorphic and four antiholomorphic u(1) currents ∂X i and∂X i , and contains in particular the group U(1) 4 of translations along the four directions on the torus. The group G 0 = G/(U(1) 4 × U(1) 4 ) includes the rotations (fixed point automorphism) of the target space. If G 0 is generated only by automorphisms of the target space, we say that the group is (purely) geometric. In this case, the requirement that the superconformal algebra is preserved implies that G 0 is a subgroup of SU (2). The properties of the purely geometric groups are well-known [27] and their classification follows from a theorem by Fujiki [28]. In general, the group G 0 might contain elements that act asymmetrically on the left-and the right-moving fields (non-geometric group). In this case, G 0 is a subgroup of the product of two SU (2), acting separately on the left-and right-moving fields. The SU(2) subgroup of geometric symmetries is embedded diagonally in this general SU(2) × SU (2). Both in the geometric and non-geometric case, G 0 must act by automorphisms on the Narain lattice Γ 4,4 . This condition constrains G 0 to be a discrete subgroup of SU(2) × SU(2) that depends on the specific torus model. In every torus model, the group G 0 contains a central involution that flips the sign of all target space coordinates. For a generic model, this is the only non-trivial element of G 0 . The precise classification of all possible discrete subgroups G 0 , or rather of the quotients G 1 = G 0 /Z 2 by this central involution, is better understood by considering the action on the lattices of D-brane charges. In particular, as discussed in section 3, the lattices Γ 4,4 even and Γ 4,4 odd of even and odd dimensional D-brane charges, are both isomorphic to the winding-momentum (Narain) lattice. This is a peculiarity of non-linear sigma models on four dimensional tori and is closely related to the triality automorphism of the group Spin (4,4). The role of triality for non-linear sigma models on T 4 was emphasized in [29,30].
In section 4, using this triality, we obtain a useful and suggestive characterization of the groups G 1 as subgroups of W + (E 8 ), the group of even Weyl transformations of the E 8 root lattice. More precisely, the possible groups G 1 are precisely the subgroups W + (E 8 ) that fix pointwise a sublattice of E 8 of rank at least four. This result is the closest analogue of the classification theorem for K3 models: in this case, the possible groups of symmetries are exactly those subgroups of the Conway group Co 0 that fix pointwise a sublattice of the Leech lattice of rank at least four [23]. Whether the appearance of the Leech and of the E 8 lattice and their group of automorphisms is just an accident or if it has some physical meaning is still an open question.
Finally, in section 5, we discuss the orbifolds of torus models by cyclic groups of symmetries. By construction, these orbifolds, if consistent, must enjoy a N = (4, 4) superconformal symmetry at central charge c =c = 6, so that they are necessarily non-linear sigma models on T 4 or K3. It is easy to see that the orbifolds by cyclic subgroups of U(1) 4 × U(1) 4 always give rise to torus models. For each symmetry g ∈ G with non-trivial image in G 0 , we classify the nature (i.e., the topology of the target space) of the corresponding orbifold model. In this sense, it is useful to think of g as an element in a Z 2 central extension Z 2 .W + (E 8 ) of the Weyl group of E 8 . Indeed, the nature of the orbifold only depends on the conjugacy class of g within Z 2 .W + (E 8 ). As a consistency check of our analysis, we verify that the classification of torus orbifolds matches perfectly with the results of [26], where a similar investigation was considered for orbifolds of K3 models.
Possible generalizations and applications of our results are discussed in section 6. Some technical details about finite subgroups of SU(2)×SU(2) and integral lattices are relegated to the appendices.

Generalities on torus models
In this section we review the main features of supersymmetric non-linear sigma models on T 4 and fix our notation and conventions.
The anti-holomorphic fields a (z),ψ a (z), a = 1, . . . , 4, and their modesα a n ,ψ a r satisfy analogous relations. The chiral algebra contains, in addition to the u(1) 4 'bosonic' algebra generated by j a , a = 1, . . . , 4, also a 'fermionic' so(4) 1 Kac-Moody algebra generated by : ψ a (z)ψ b (z):, 1 ≤ a < b ≤ 4, and 16 fields of weight (3/2, 0) of the form : ψ i (z)j k (z):. In particular, any torus model contains a (small) N = (4, 4) superconformal algebra with central charge c =c = 6, that will be described in detail in section 2.1. The zero modes of the currents generate a continuous group of symmetries of the conformal field theory. In particular, we denote by the groups of symmetries generated by the zero modes j a 0 andj a 0 of the left-and rightmoving currents and by the groups generated by the zero modes of : ψ a ψ b : and :ψ aψb :, 1 ≤ a < b ≤ 4.
The real vector space space Π = Γ 4,4 w-m ⊗ R ∼ = R 4,4 splits into an orthogonal sum Π = Π L ⊕ Π R of a positive-definite subspace Π L , spanned by vectors of the form ( v L , 0), and a negative-definite Π R spanned by (0, v R ). Each torus model is uniquely determined by the relative position of Π L and Π R with respect to the lattice Γ 4,4 w-m and we obtain the usual Narain moduli space A concrete realisation of this CFT is given in terms of a supersymmetric non-linear sigma model on a torus T 4 = R 4 /L with a constant antisymmetric B-field B. We fix once and for all a set of orthonormal coordinates for the euclidean space R 4 , so that the geometry of the torus is encoded in the shape of the lattice L ⊂ R 4 . The four scalar fields X a (z,z) = φ a (z) +φ a (z), a = 1, . . . , 4, describing the target space coordinates, are related to the CFT currents by j a = i∂X a ,j a = i∂X a . The corresponding lattice of (α 0 ,α 0 )-eigenvalues is given by where L * ⊂ R 4 denotes the dual lattice. The fermionic fields ψ a ,ψ a are, up to normalization, the supersymmetric partners of the scalars X a .
Each of the abstract conformal field theories, defined in terms of the relative position of Π L and Π R with respect to the lattice Γ 4,4 w-m , is reproduced by several different geometric descriptions in terms of non-linear sigma models. Technically, the choice of any such description amounts to choosing two sublattices Λ m and Λ w of Γ 4,4 w-m , such that Λ ⊥ m = Λ m , Λ ⊥ w = Λ w (i.e., they are maximal isotropic sublattices) and Λ m ⊕ Λ w ∼ = Γ 4,4 w-m . Different choices are related to one another by automorphisms of Γ 4,4 w-m (interpreted as T-dualities). In particular, the choice of Λ m determines an isometry γ : Π L → Π R (−1), characterised by the property that γ( λ L ) = λ R for each λ = ( λ L , λ R ) ∈ Λ m ⊂ Π L ⊕ Π R (here, Π R (−1) is obtained from Π R by flipping the sign of the metric). Then one can define a positive definite lattice L * ⊂ Π L ∼ = Π R (−1) ∼ = R 4 of rank four by The dual lattice L = (L * ) * ⊂ R 4 is interpreted as the lattice of winding vectors, so that the CFT can be described as a non-linear sigma model on the torus T 4 := R 4 /L. 1 Finally, the choice of a constant background B-field is encoded in the choice of the lattice Λ w . Indeed, for a given Λ m , the most general isotropic sublattice Λ w ⊂ Γ 4,4 w-m is of the form Here, the linear map B is given by a real antisymmetric matrix B ij = −B ji as in terms of a basis l 1 , . . . , l 4 of L and of the dual basis l * 1 , . . . , l * 4 of L * . The matrix B ij , and consequently the lattice Λ w , is determined by the decomposition Γ 4,4 w-m ⊗ R = Π L ⊕ Π R and by the choice of Λ m ∼ = Γ 4,4 w-m , up to a shift by an integral antisymmetric matrix.

A first classification of symmetries
Our goal is to classify all possible groups G of symmetries of the OPE that fix the small N = (4, 4) superconformal algebra. We focus only on symmetries that act non-trivially on the NS-NS sector; the induced action on the R-R sector is discussed in section 3.1.
Let us consider first the symmetries that act trivially on the fermionic fields ψ a ,ψ a as well as on the bosonic currents j a ,j a , a = 1, . . . , 4, so that the N = (4, 4) algebra is automatically invariant. These symmetries act non-trivially only on the fields V λ and the only linear transformations preserving the OPE (2.7)-(2.9) are for some ( v L , v R ) ∈ (Γ 4,4 ⊗ R)/Γ 4,4 . Therefore, the normal subgroup of G fixing the fields ψ a ,ψ a , j a ,j a , a = 1, . . . , 4, is the group U(1) 4 L × U(1) 4 R generated by j a 0 andj a 0 as in (2.2). Thus, G can always be written as a product where G 0 := G/(U(1) 4 L × U(1) 4 R ). We have reduced our problem to the classification of the possible groups G 0 that act non-trivially on the fundamental u(1) currents and fermions. In order to preserve the OPEs (2.1) and to fix the supercurrent : ψ a (z)j a (z): , and its right-moving analogue, a symmetry g must act by a simultaneous orthogonal transformation on the fermions and on the u(1) currents, i.e.
Here, ρ(g L ) andρ(g R ) are SU(2) matrices and ρ(g L ) * andρ(g R ) * are their complex conjugate. It is straightforward to verify that these transformations leave all generators of the N = (4, 4) superconformal algebra invariant.
It is also useful to give a description of the group SU(2) L × SU(2) R in terms of quaternions. Let i, j, k be imaginary units satisfying the usual quaternionic multiplication rule We have the standard identification of the space of quaternions H with R 4 so that we can think of Γ 4,4 as a lattice within H ⊕ H by Furthermore, the group unit quaternions forms a copy of SU(2) Under this identification, the pair (g L , g R ) ∈ SU(2) L × SU(2) R can be regarded as a pair of unit quaternions and the action on the fields is simply given by (left) multiplication We stress that the action (2.38) of (g L , g R ) ∈ G 0 on the fields V λ is only defined modulo To summarize, we have found the following characterization of the group G: The group G of symmetries of a supersymmetric non-linear sigma model on T 4 preserving a small N = (4, 4) superconformal algebra generated by (2.15)-(2.20), is given by a product R is generated by the zero modes of the bosonic currents j a ,j a , a = 1, . . . , 4, and the quotient is an automorphism of the windingmomentum lattice Γ 4,4 . In particular, (g L , g R ) ∈ G 0 acts as in (2.34)-(2.38) on the fields.
In other words, G 0 is the intersection of the compact group SU(2) L × SU(2) R and the discrete group SO(Γ 4,4 ), when both groups are embedded in the orthogonal group SO(4, 4, R) acting on Π L ⊕ Π R ∼ = R 4,4 . Clearly, these embeddings and therefore the intersection G 0 depend on the moduli, i.e. on the relative position of the subspaces Π L and Π R with respect to the lattice Γ 4,4 .
The group G 0 always contains a central Z 2 subgroup generated by the involution (−1, −1) ∈ SU(2) L × SU(2) R that flips the signs of all bosonic currents and all fermions; this is indeed a symmetry of all torus models. On the other hand, a generic deformation of the model will break any symmetry g ∈ G 0 not contained in this central Z 2 subgroup. We conclude that the generic symmetry group G is In the following sections, we will give a complete classification of the possible discrete subgroups G 0 ⊂ SU(2) L × SU(2) R and of the corresponding torus models.

Ramond-Ramond sector, D-branes and triality
In this section, we consider the action of the symmetry group G on the Ramond-Ramond fields and on the lattice of D-brane charges. This will lead to a useful characterization of the possible symmetry groups G in section 4.1.

Representation of the symmetries on the Ramond-Ramond sector
The Ramond-Ramond sector of the sigma model forms a representation of the algebra of fermionic zero modes ψ a 0 ,ψ a 0 . In particular, the ground states have conformal weight ( 1 4 , 1 4 ) and span a 16 dimensional space H. A convenient basis is given by simultaneous eigenvectors  The left-and right-moving world-sheet fermion number operators are defined by and are preserved by the 'fermionic' SO(4) f,L × SO(4) f,R transformations. We denote by Π even and Π odd the subspaces of R-R ground states with (−1) F L +F R = 1 and (−1) F L +F R = −1, respectively.
Let us consider the action of the group G on the R-R ground states. Clearly, the normal subgroup U(1) 4 L ×U(1) 4 R acts trivially on this space, so we can focus on the quotient G 0 . An element g = (g L , g R ) ∈ G 0 acts by an SU(2) A × SU(2)Ã transformation on the fermionic fields ψ a ,ψ a and the action is represented by some SU(2) matrices ρ(g L ),ρ(g R ) as in (2.30)-(2.33). In order to preserve the OPE, g has to act by the same SU(2) A × SU(2)Ã transformation also on the R-R ground fields. Assume, for the sake of simplicity, that this transformation is contained in the Cartan torus generated by A 3 0 andÃ 3 0 . If the eigenvalues of ρ(g L ) andρ(g R ) are ζ L , ζ −1 L , ζ R , ζ −1 R , then the action of g in the R-R sector is given by Thus, the eigenvalues of g on the eigenspaces of (−1) Notice that the eigenspace with (−1) F L = (−1) F R = −1 is fixed by g. Furthermore, the central involution (−1, −1) ∈ SU(2) L × SU(2) R acts trivially on the whole space Π even with positive total fermion number (−1) F L +F R = +1. Thus, the group acting faithfully on Π even is G 1 = G 0 /(−1, −1) which is a subgroup of (SU(2) L × SU(2) R )/(−1, −1). These conclusions are valid even when the action of g on the fermions is not generated by A 3 0 andÃ 3 0 , since by a suitable conjugation within SU(2) A × SU(2)Ã one can always reduce g to this Cartan torus while preserving the eigenspaces of (−1) F L and (−1) F R . As will be described in the following sections, it is easier to characterize the possible groups G of symmetries by considering their action on the eigenspace Π even with (−1) F L +F R = +1.
In the above derivation we have been slightly naive in the identification of the action of g on the R-R sector. In general, a SO(4) f,L × SO(4) f,R transformation on the fundamental fermions in the NS-NS sector determines the transformation on the R-R sector only up to a sign -the symmetries (−1) J 3 0 +A 3 0 and (−1)J 3 0 +Ã 3 0 act trivially on the NS-NS sector and by a minus sign on the R-R sector. Therefore, for a generic subgroup G of SO(4) f,L × SO(4) f,R acting on the NS-NS sector, the induced group acting on the R-R sector and preserving the OPE is a Z 2 -central extension of G.
However, the groups G 0 we are considering are actually contained in the subgroup SU(2) A ×SU(2)Ã of SO(4) f,L ×SO(4) f,R generated by A 3 0 , A ± 0 ,Ã 3 0 ,Ã ± 0 , so that their central extension in the R-R sector is always the trivial one Z 2 × G 0 . Therefore, we do not lose any information if we simply focus on the component of Z 2 × G 0 that fixes the four R-R states with (−1) F L = (−1) F R = −1. An analogous condition was considered in the classification of the groups of symmetries of K3 models in [23].

D-branes and triality
Fundamental D-branes are defined, in a given geometric interpretation of the torus model, by imposing either Neumann or Dirichlet boundary conditions in each direction. Each of these boundary conditions preserve a particular bosonic u(1) 4 subalgebra of the original For example, consider a geometric realization of the model, as described in section 2, associated with a maximal isotropic sublattice Λ m ≡ span Z {λ 1 , . . . , λ 4 } ⊂ Γ 4,4 , which determines the lattice L * of space-time momenta on a torus R 4 /L as in eq.(2.10). Then, the Dirichlet boundary conditions describing a D0-brane in the Ramond-Ramond sector are enforced by Here, η ∈ {±1} and Λ m determine which N = 4 and which u(1) 4 subalgebras are preserved by the boundary conditions, µ ≡ 1 √ 2 ( l * , l * ) ∈ Λ m , with l * ∈ L * , labels the distinct representations of this preserved u(1) 4 subalgebra and |µ; Λ m , η denotes the Ishibashi state in the corresponding sector. The boundary state ||a; Λ m , η is then obtained by a superposition of the Ishibashi states for the different values of µ ∈ Λ m ||a; Λ m , η = N µ∈Λm e 2πi(a•µ) |µ; Λ m , η .
The modulus a ∈ (Λ w ⊗ R)/Λ w represents the position of the D0-brane on the fourdimensional torus and N is a suitable normalization. The treatment of the NS-NS sector is analogous, the only difference being in the boundary conditions (3.4), where the fermion modes ψ r ,ψ −r are defined for r ∈ 1 2 + Z. In a full ten dimensional superstring theory, space-time supersymmetric D-branes are obtained by tensoring these boundary states with the analogous D-brane states in the remaining 6 space-time directions and considering a suitable GSO projected combination of NS-NS and R-R contributions.
The Ramond-Ramond charge of a D-brane corresponds to the ground state component of the boundary state ||a; Λ, η and is completely determined by eq.(3.4) for r = 0 up to normalization, which in turn is fixed by the Cardy and factorisation conditions. Let us focus on the case η = 1 and use a simplified notation ||Λ ≡ ||a; Λ, η = 1 , where we drop the dependence on the modulus a. We denote by Ψ Λ the R-R ground state component of where the operators generate the real Clifford algebra associated with the space Γ 4,4 ⊗R, with anticommutation relations The overlap between two D-branes is given by which defines a non-degenerate bilinear form of signature (8,8) on the lattice of D-brane charges. Since all c(v) anticommute with the chirality operator Ψ Λ must have definite chirality. D-brane charges with opposite chiralities are orthogonal with respect to (3.8), so that the lattice of D-brane charges splits as an orthogonal sum Γ 4,4 even ⊕ ⊥ Γ 4,4 odd of lattices isomorphic to Γ 4,4 . In a geometric description of the model, chirality corresponds to the parity of D-brane dimensionalities and the lattices Γ 4,4 even and Γ 4,4 odd are identified with the even and odd integral cohomology groups H even (T 4 , Z) and H odd (T 4 , Z), and the bilinear form is the cup product.
This 'democracy' among the three lattices is characteristic of four-dimensional torus models and is related to the triality automorphism of Spin (4,4), which permutes the vector and the two irreducible spinor representations.

Groups and models
In this section, we classify all possible groups of symmetries G preserving the N = (4, 4) superconformal algebra and describe the corresponding torus models. The main idea is to focus on the action of G on the lattice Γ 4,4 even of even dimensional D-brane charges. By the triality described in section 3.2, this analysis is sufficient to reconstruct the action of G on the lattice Γ 4,4 w-m of winding-momenta and on all the fields of the theory.

Symmetries of torus models
Let G be the group of symmetries of a torus model that preserve the small N = (4, 4) superconformal algebra. A discussed in section 3.1, the representation of G over the space Π even of R-R ground states with positive chirality is given by a group G 1 ∼ = G 0 /(−1) of orthogonal transformations of Π even that fix the subspace Π + even with (−1) F R +1 = +1. Furthermore, G 1 must act by automorphisms on the lattice Γ 4,4 even of even D-brane charges. Conversely, let G 1 ⊂ SO + (Γ 4,4 even ) ⊂ SO + (4, 4) be a group of automorphisms of the lattice of even D-brane charges that fix the subspace Π + even ⊂ Π even . Thus, G 1 acts faithfully by SO(4) transformations on the space Π − even of R-R ground fields with (−1) F R +1 = −1. The spin cover Spin(4) of this SO(4) group is the SU(2) A ×SU(2)Ã group generated by the zero modes of the currents A 3 0 , A ± 0 ,Ã 3 0 ,Ã ± 0 . Therefore, the group G 1 is the representation on the R-R sector of some group G 0 ⊂ SU(2) L × SU(2) R , acting on the NS-NS-fields as in (2.30)-(2.33). The group G 0 is a isomorphic to a double (spin) cover of G 1 , i.e. to the preimage of G 1 ⊂ SO(Γ 4,4 even ) ⊂ SO(4, 4) under the spin covering Spin(4, 4) → SO(4, 4), and acts on the space Π w-m of winding-momenta in one of the irreducible spin representations. By the triality construction described in the previous section, since G 1 acts by automorphisms on the lattice Γ 4,4 even , then G 0 must act by automorphisms on the lattice Γ 4,4 w-m of winding momenta. Therefore, (U(1) 4 L × U(1) 4 R ).G 0 is a group of symmetries of the torus model that preserves the N = (4, 4) algebra.
Thus, we have found an alternative description of the groups of symmetries G considered in section 2. In fact, as will be discussed below, there are two more equivalent characterizations of these groups: Theorem 2 Let G 1 be a subgroup of SO(4, R) and let Z 2 .G 1 ⊆ Spin(4) denote its preimage under the spin covering homomorphism Spin(4) → SO(4). The following properties are equivalent: , is the group of symmetries of a supersymmetric non-linear sigma model on T 4 that preserves a small N = (4, 4) superconformal algebra.
3. G 1 = SO 0 (Λ), where Λ is an even positive-definite lattice of rank at most four and SO 0 (Λ) ⊆ SO(Λ) is the group of automorphisms of Λ that act trivially on the discriminant group Λ * /Λ (Λ * denotes the dual of Λ, see appendix B). 4. G 1 is the subgroup of W + (E 8 ), the group of even Weyl transformations of the E 8 lattice, fixing a sublattice of rank at least four.
The equivalence of (1) and (2) has been discussed above. Let us first show that (2) ⇔ (3). Let G 1 be a subgroup of SO + (Γ 4,4 even ) that leaves the subspace Π + even pointwise fixed. Let us denote by Γ G 1 ⊆ Γ 4,4 even the sublattice of vectors fixed by G 1 and by Γ G 1 its orthogonal complement in Γ 4,4 Since Π + even ⊆ Γ G 1 ⊗R, it follows that Γ G 1 is an even negative-definite lattice of rank at most 4. Notice that Γ G 1 and Γ G 1 are primitive mutually orthogonal sub-lattices of Γ 4,4 even and the direct sum Γ G 1 ⊕Γ G 1 has maximal rank 8. Then, by the standard 'gluing' construction (see appendix B for details), there is an isomorphism (Γ G 1 ) * /Γ G 1 ∼ = → (Γ G 1 ) * /Γ G 1 of discriminant groups that reverses the induced discriminant form and such that denotes the image of x in the corresponding discriminant group. Since G 1 acts trivially on the discriminant group (Γ G 1 ) * /Γ G 1 , then it must act trivially also on (Γ G 1 ) * /Γ G 1 [31]. We conclude that G 1 is a group of automorphisms of the positive-definite even lattice Λ ∼ = Γ G 1 (−1) of rank (at most) four that fixes the discriminant group Λ * /Λ. Vice-versa, given any even positive-definite lattice Λ of rank at most four, there is a primitive embedding of Λ(−1) in Γ 4,4 even and one can always find a positive definite four dimensional subspace Π + even ⊂ Γ 4,4 even ⊗ R such that Λ(−1) = Γ 4,4 even ∩ (Π + even ) ⊥ . As explained in appendix B, the automorphisms of Λ that fix the discriminant group Λ * /Λ extend to automorphisms of Γ 4,4 even that fix pointwise the orthogonal space Π + even . This proves the equivalence of (2) and (3).
The proof that (3) ⇔ (4) is completely analogous, with the lattice Γ 4,4 replaced by E 8 . Let Λ be an even positive definite lattice of rank at most 4 and let SO 0 (Λ) denotes the group of (positive determinant) automorphisms of Λ that act trivially on the discriminant group Λ * /Λ. By a theorem by Nikulin (Theorem 1.12.4 of [31]), any even positive definite lattice with rank at most four can be primitively embedded in E 8 , the unique 8-dimensional positive-definite even unimodular lattice. Furthermore, every automorphism g ∈ SO 0 (Λ) can be extended to an automorphism of E 8 that fixes the orthogonal complement Λ ⊥ of Λ in E 8 . Therefore, SO 0 (Λ) is a subgroup of the group G 1 ⊂ W + (E 8 ) that fixes the sublattice Λ ⊥ ⊂ E 8 .
Conversely, suppose that G 1 is the subgroup of W + (E 8 ) that stabilises pointwise a sublattice Λ ⊥ ≡ (E 8 ) G 1 of rank at least 4. Then, G 1 acts faithfully as a group of automorphisms of the orthogonal complement Λ of Λ ⊥ . Furthermore, since G 1 fixes (Λ ⊥ ) * /Λ ⊥ , by the gluing construction it must also fix the discriminant group Λ * /Λ of its orthogonal complement. We conclude that G 1 ∼ = SO 0 (Λ).

Characterization of the groups of symmetries
In this section, we classify all possible groups G 1 satisfying the equivalent properties (3) and (4) of theorem 2.
Let Λ be an even positive definite lattice of rank r ≤ 4 and let SO 0 (Λ) the group of positive determinant automorphisms that act trivially on the discriminant group Λ * /Λ. By theorem 2, there is a torus model such that G 1 ∼ = SO 0 (Λ). We recall that Λ can always be primitively embedded in the E 8 lattice and that every g ∈ SO 0 (Λ) extends to an automorphismĝ ∈ W + (E 8 ) that fixes the orthogonal complement of Λ [31]. Thus, Λ can always be regarded as a primitive sublattice of E 8 and we can label each g ∈ SO 0 (Λ) by the W + (E 8 ) conjugacy class ofĝ. The group W + (E 8 ) contains 11 conjugacy classes whose elements fix a sublattice of rank at least 4 [32], see table 1.  Table 1: Conjugacy classes of elementsĝ ∈ W + (E 8 ) that fix a sublattice of rank at least 4. For each class, we list the the eigenvalues η 1 , η 2 , η −1 1 , η −1 2 of the corresponding g ∈ G 1 on the space Π − even , the rank of (Γ 4,4 even ) g and the determinant of (1 − g) on (Γ 4,4 even ) g . The corresponding symmetry ±g 0 ∈ G 0 of the torus model is determined up to a sign. We list the eigenvalues ±ζ L , ±ζ R , ±ζ −1 L , ±ζ −1 R (each with multiplicity 2) for the action of ±g 0 over the fermions ψ a ,ψ a and the orders o(±g 0 ) of g 0 and −g 0 .
Let λ 1 , . . . , λ r be a basis of generators for Λ and Q ij := λ i · λ j be the corresponding Gram matrix. Then g ∈ SO 0 (Λ) acts by where g ij is an integral matrix satisfying Let u 1 , . . . , u r be the dual basis of generators of Λ * , given by so that u i · λ j = δ ij . The action of g extends to Λ ⊗ Q by linearity, so that . . , r , and the condition that g acts trivially on Λ * /Λ is equivalent to In particular, which puts a non-trivial constraint on the possible Gram matrices Q, provided that rk(1 − g) = r. The determinant det(1 − g) and the rank rk(1 − g) for each possible conjugacy class in W + (E 8 ) are given in table 1. For each W + (E 8 )-conjugacy class of g, there are only few isomorphism classes of lattices Λ of rank r = rk(1 − g) and satisfying (4.3) (see [33]). In the following sections, we consider a case by case analysis of these lattices. First of all, in section 4.3 we consider the case where G 1 has no symmetry g with rk(1 − g) = 4. In this case, we argue that the whole group G 0 is induced by geometric automorphisms of the target space in a suitable geometric interpretation of the CFT as a non-linear sigma model on T 4 .
The lattices Λ admitting symmetries g ∈ SO 0 (Λ) with rk(1 − g) = 4 and the W + (E 8 ) conjugacy classes of the associated liftsĝ are given in table 2. The corresponding groups G 1 = SO 0 (Λ) and their double covering G 0 cannot be described as automorphisms of the target space in any geometric interpretation of the model. A detailed description of the entries of this table is provided in section 4.4. Table 2 is derived as follows. When 1 − g has rank rk(1 − g) = 4 = r, then the lattice Λ is uniquely determined, up to isomorphism, by the W + (E 8 ) conjugacy class ofĝ. Indeed, in this case, Λ is the orthogonal complement in E 8 of the sublattice fixed byĝ and for any two elementsĝ,ĝ ′ ∈ W + (E 8 ) in the same conjugacy class, the corresponding lattices Λ and Λ ′ are related by a W + (E 8 ) transformation and are therefore isomorphic.
For each g ∈ SO 0 (Λ) with rk(1 − g) = 4, the conjugacy class ofĝ in W + (E 8 ) can be determined, in most cases, by the eigenvalues of g on Λ. The only exceptions are the classes 2A and 2E, that have the same eigenvalues. To distinguish these cases, it is sufficient to Lattice Λ det Q W + (E 8 ) classes  Finally, we argue that each of the six lattices in table 2 corresponds to a unique point in the moduli space M T 4 . In other words, there is a unique torus model (up to dualities) realizing each of the symmetry groups of table 2. In fact, the group G 1 of a torus model is isomorphic to SO 0 (Λ), with Λ one of the entries of table 2, if and only if the lattice S := Γ 4,4 even ∩ Π − even is isomorphic to Λ(−1). Vice versa, each sublattice S ⊂ Γ 4,4 even isomorphic to Λ(−1) is associated with a unique torus model with symmetry group G 1 ∼ = SO 0 (Λ), defined by setting Π − even = S ⊗ R. Thus, we have to show that any two sublattices S, S ′ ⊂ Γ 4,4 even with S ∼ = Λ(−1) ∼ = S ′ are related by a O(Γ 4,4 even ) transformation, so that the associated torus models are related by some T-duality. Let K and K ′ be the orthogonal complements in Γ 4,4 even of S and S ′ , respectively. Then, by the standard gluing construction, there are isomorphisms γ : S * /S → K * /K and γ ′ : (S ′ ) * /S ′ → (K ′ ) * /K ′ between the discriminant groups such that the induced discriminant forms are inverted (see appendix B) and such that the lattice Γ 4,4 even is given by where [v] and [w] are the images of v and w in the corresponding discriminant groups. The existence of the isomorphisms γ and γ ′ implies, in particular, that K and K ′ must be positive definite lattices of rank 4 with the same determinant as Λ. In all six cases of table 2, one finds that γ, γ ′ exist if and only if K ∼ = Λ ∼ = K ′ [33]. Furthermore, the isomorphisms γ and γ ′ satisfying (4.4) are unique up to automorphisms of S, S ′ , K, K ′ . It follows that we can always find isomorphisms f S : S * → (S ′ ) * and f K : K * → (K ′ ) * such that the induced isomorphisms of discriminant groups satisfy By (4.5), this implies that f S ⊕ f K : S * ⊕ K * → (S ′ ) * ⊕ (K ′ ) * restricts to an automorphism of Γ 4,4 even and, by construction, this automorphism maps S to S ′ .

Models with purely geometric symmetry group
In this section, we will show that G 1 contains no elements with rk(1 − g) = 4 if and only if the whole group G 0 = Z 2 .G 1 is induced by target space automorphisms in a suitable geometric interpretation of the torus model. We will also classify such purely geometric groups; analogous results were obtained in [27]. Suppose that G 1 ⊂ SO + (Γ 4,4 even ) contains only elements g in class 1A, 2B or 3A of W + (E 8 ), so that rk(1 − g) < 4 (see table 1). Let Γ G 1 ⊂ Γ 4,4 even be the sublattice fixed by G 1 and Γ G 1 its orthogonal complement. The lattice Λ := Γ G 1 (−1) is an even positive definite lattice of rank at most 4 and such that SO 0 (Λ) ∼ = G 1 ; furthermore, there is no vector of Λ fixed by all elements of G 1 . We will show that under the above condition for G 1 , Λ has rank at most three.
Let Λ (2) denote the sublattice of Λ generated by all its elements of squared length 2 (roots). Suppose first, by absurd, that Λ (2) has rank four. All root lattices of rank 4 are listed in Table 2, so that Λ (2) must be one of them. Now, Λ has the same rank as Λ (2) and its vectors have integral scalar product with all elements of Λ (2) so that Λ (2) ⊆ Λ ⊆ (Λ (2) ) * . Furthermore, all the vectors of Λ of squared norm two are contained in Λ (2) . For all root lattices Λ (2) in table 2, the only sublattices of (Λ (2) ) * satisfying these properties are the root lattices themselves, so that necessarily Λ = Λ (2) . But this contradicts the assumption that G 1 has only elements in the classes 1A, 2B and 3A. It follows that Λ (2) has rank at most three and we denote byΛ its orthogonal complement in Λ.
For each g ∈ SO 0 (Λ), g = 1, let Λ g ⊂ Λ be the sublattice fixed by g and Λ g its orthogonal complement in Λ. By eq.(4.3) and table 1, Λ g has rank rk(1 − g) = 2 and determinant 3 (for g in class 3A) or 4 (for g in class 2B), so that the only possibilities for the Gram matrix of Λ g , up to isomorphism, are In both cases, the lattice Λ g is generated by its elements of squared length 2, so that Λ g ⊆ Λ (2) for all g ∈ G 1 . It follows that, for all g ∈ G 1 ,Λ is orthogonal to Λ g , so that it must be fixed by g. However, by construction, there is no sublattice fixed by all g ∈ G 1 , so thatΛ = 0 and Λ has the same rank as Λ (2) and thus at most three. Since Γ G 1 = Λ(−1) is a primitive sublattice of Γ 4,4 even with rank r ≤ 3, then by Corollary 1.13.4 of [31] the orthogonal complement Γ G 1 is isomorphic to the orthogonal sum Γ s,s ⊕ Λ, where Γ s,s is the even unimodular lattice of signature (s, s), with s := 4 − r ≥ 1. Therefore, if G 1 contains no elements g with rk(1 − g) = 4, then there exists a pair of primitive null vectors λ 1 , λ 2 ∈ Γ 4,4 even with λ 1 • λ 2 = 1 that are fixed by G 1 . As described in section 3.2, two such elements of Γ 4,4 even correspond to two maximal isotropic sublattices Λ 1 , Λ 2 ⊂ Γ 4,4 w-m , with Λ 1 ⊕ Λ 2 ∼ = Γ 4,4 w-m , that are fixed (setwise) by G 0 = Z 2 .G 1 . In turn, Λ 1 , Λ 2 define a geometric interpretation of the CFT as a non-linear sigma model on some torus T 4 = R 4 /L (see section 2), such that the lattice L and the B-field are preserved by G 0 . With respect to this geometric interpretation, G 0 acts by SU(2) transformations (2.31),(2.33) with g L = g R on the complex coordinates Z (1) , Z (2) on T 4 defined by (2.11).
The groups of automorphisms of complex tori of complex dimension 2 have been studied by Fujiki [28]. Upon a suitable choice of the B-field, each such automorphism induces a symmetry of the corresponding non-linear sigma model, preserving a N = (4, 4) superconformal algebra [27]. In particular, G 0 corresponds to the group of fixed-point automorphisms of the torus. In [28], the action of these geometric automorphisms on the integral second cohomology group H 2 (T 4 , Z) is considered; this is the exact geometric analogue of the action of G 1 = G 0 /(−1) on even D-brane charges. More precisely, for a given geometric interpretation of the CFT, the lattice Γ 4,4 even can be identified with the even integral cohomology group H even (T 4 , Z). In the interpretation where G 1 is purely geometric, the components of H even (T 4 , Z) of degrees zero and four are fixed, so that (Γ 4,4 even ) G 1 can be identified with H 2 (T 4 , Z) G 1 . Our analysis, therefore, naturally leads to the same result as in Lemma 3.3 and Theorem 6.4 of [28]. The list of possible lattices Λ = (Γ 4,4 even ) G 1 (−1), their rank, the group G 1 ∼ = SO 0 (Λ) and its Z 2 -central extension G 0 are given in table 3. Notice that, since Λ has rank at most three, G 1 is actually a subgroup of SO(3) and its even ) G 1 with rk Λ < 4. For each such lattice, we report the rank, the group G 1 = SO 0 (Λ) of automorphisms fixing the discriminant group Λ * /Λ, the double covering G 0 = Z 2 .G 1 and the order and generators of G 0 ⊂ SU (2) × SU (2). We use the notation e r := exp(2πir), r ∈ Q, and ω = 1 2 (−1 + i + j + k).
spin cover G 0 is a subgroup of SU(2). This SU (2) is diagonally embedded in the group SU(2) L × SU(2) R acting as in (2.30)-(2.33), that is G 0 ⊂ SU(2) L × SU(2) R is generated by elements of the form (g L , g R ) = (g, g) for some g ∈ SU(2). The torus models with a given symmetry group G 1 , associated with a lattice Λ = (Γ 4,4 even ) G 1 (−1) of rank r, form a sublocus of dimension 4(4 − r) in the moduli space. Indeed, this is the dimension of the Grassmannian parametrizing the choice of a four dimensional positive-definite space Π + even within the space (Γ 4,4 even ) G 1 ⊗ R of signature (4, 4 − r).

Models with non-geometric symmetries
As discussed in section 4.2, there are six torus models that admit symmetries preserving the N = (4, 4) algebra and with no geometric interpretation. They are characterized by the condition that Γ 4,4 even ∩ Π − even ∼ = Λ(−1), where Λ is one of the six lattices in table 2. In this section, we provide some more details about the lattices Λ, the groups G 1 = SO 0 (Λ) and the corresponding torus models. For each of these six cases, we first describe the lattice Λ: we provide the matrix Q ij := λ i · λ j for a standard choice of generators λ 1 , . . . , λ 4 of Λ, the determinant det Q, the structure of the discriminant group Λ * /Λ and the values of the induced discriminant form q : Λ * /Λ → Q/2Z on its generators. We also provide a definition of Λ ⊂ R 4 as a lattice in the space of quaternions H ∼ = R 4 . Next, we describe the group G 1 ∼ = SO 0 (Λ) of automorphisms of Λ with positive determinant and that act trivially on the discriminant group. In all the cases we are interested in, Λ is the root lattice of some Lie algebra and SO 0 (Λ) is the group W + (Λ) of even Weyl transformations. Since G 1 is a finite subgroup of SO(4) ∼ = (SU(2) × SU(2))/(−1, −1), it can be described as , depending on whether −1 ∈ G 1 or not (see appendix A). We classify the elements of G 1 depending on their order and conjugacy class in W + (E 8 ) and for each such class we indicate the number of elements in G 1 .
The group G 0 is the preimage of G 1 under the covering map Spin(4) → SO(4) and it can be easily determined as described in appendix A. We provide a set of generators (g L , g R ) of G 0 ⊂ SU(2) × SU(2) in the form of a pair of quaternions. The corresponding on the lattice Λ = (Γ 4,4 even ) G 1 (−1). Next, we provide a description of the winding-momentum lattice Γ 4,4 w-m as a lattice in R 4,4 ∼ = Π L ⊕ Π R ∼ = H ⊕ H. In other words, we denote the elements of Γ 4,4 w-m as pairs of quaternions (q L ; q R ) ∈ Γ 4,4 w-m , where q L ∈ Π L ∼ = H, q R ∈ Π R ∼ = H, so that the action of (g L , g R ) ∈ G 0 ⊂ SU(2) × SU(2) is simply given by left multiplication (g L , g R ) : (q L ; q R ) → (g L q L ; g R q R ) (q L , q R ) ∈ Γ 4,4 w-m , (g L , g R ) ∈ G 0 , as in eq.(2.39). The action of (g L , g R ) on the fields of the theory is given by (2.30)-(2.33). We also describe the sublattices Γ 4,4 w-m ∩Π L and Γ 4,4 w-m ∩Π R of purely holomorphic and purely anti-holomorphic winding-momenta. If these sublattices have positive rank, then the chiral algebra of the model is extended with respect to the one of a generic torus model. Finally, we provide a possible geometric interpretation of the CFT as a non-linear sigma model with target space R 4 /L and B-field B. The information about L and B is given in the form of a matrix whose columns l 1 , . . . , l 4 form a set generators for L ⊂ R 4 , the quadratic from Q ij = l i · l j and the B-field B in the form of a real antisymmetric matrix, such that Here l 1 * , . . . , l 4 * is the basis of L * dual to l 1 , . . . , l 4 , given by l i * = (Q −1 ) ij l j .
If the CFT has a group of symmetries G preserving the N = (4, 4) superconformal algebra, then for each g ∈ G one can define the twining genus which depends only on the conjugacy class of g in G. For a non-linear sigma model on T 4 or K3, φ g is a weak Jacobi form of weight 0 and index 1 under a suitable congruence subgroup of SL(2, Z), which depends on g.
One can also construct the orbifold of the conformal field theory by the cyclic group g ∼ = Z N , by introducing the g i -twisted sectors, i = 1, . . . , N − 1, and including in the spectrum of the theory only the g-invariant subspace of all twisted and untwisted sectors. The orbifold is a consistent CFT, provided that the level-matching condition is satisfied, i.e. that the spin h−h of the g i -twisted fields takes values in i N Z, for all i ∈ Z. By construction, the orbifold is a N = (4, 4) superconformal algebra at central charge c =c = 6, so that, if consistent, it must be a non-linear sigma model on T 4 or K3. These two cases can be distinguished by the elliptic genus of the orbifold, which is given by the formula where φ g i ,g j is defined as the g j -twining trace over the g i -twisted sector In fact, in order to distinguish between a torus and a K3 model, it is sufficient to consider the value of the elliptic genus at z = 0 (the Witten index), which is a constant equal to the Euler number of the target space φ(τ, z = 0) = 0 for a torus model, 24 for a K3 model.
In the rest of this section, we will compute the twining genera and determine the kind (either torus or K3 model) of the orbifold theory for all possible symmetries g ∈ G of a non-linear sigma model on T 4 . First of all, if g acts trivially on the right-moving fermionsψ a , a = 1, . . . , 4, then the twining genus φ g vanishes for the same reasons as for the elliptic genus. Furthermore, since the fermions are included in the spectrum of the orbifold theory, also the orbifold elliptic genus φ orb vanishes. This implies that if g is a finite subgroup of U(1) 4 × U(1) 4 , then the orbifold by this group, if consistent, gives again a torus model. Therefore, it is sufficient to focus on the elements g ∈ G with non-trivial image g 0 ∈ G 0 modulo U(1) 4 × U(1) 4 . It is convenient to label each g 0 ∈ G 0 by the W + (E 8 ) class of the corresponding g ′ ∈ G 1 ∼ = G 0 /Z 2 and by the sign of the eigenvalues of the lift g 0 when acting on the fermions ψ a ,ψ a , a = 1, . . . , 4. For example, +3A denotes an element g 0 which is the lift to G 0 of some g ′ ∈ G 1 in the W + (E 8 ) class 3A, with eigenvalues +ζ L , +ζ −1 L , +ζ R , +ζ −1 R as listed in table 1. For the classes acting asymmetrically between left-and right-movers, we use a prime, as in ±4A ′ , to denote the lift g 0 where the left-moving eigenvalues ±ζ L , ±ζ −1 L and the right-moving ones ±ζ R , ±ζ −1 R are exchanged with respect to the ones listed in table 1.
For a given g ∈ G, whose corresponding image g 0 ∈ G 0 has eigenvalues ζ L , ζ −1 L , ζ R , ζ −1 R , the twining genus is given by where the contribution from the 16 R-R ground states is the contribution from the left-moving fermionic and bosonic oscillators is , and the contribution from winding-momentum is Here, Λ L := Γ 4,4 w-m ∩ Π L and v ≡ ( v L , v R ) ∈ (Γ 4,4 w-m ⊗ R)/Γ 4,4 w-m determines an element of U(1) 4 L × U(1) 4 R as in eq.(2.23). The choice of v amounts to choosing a lift g ∈ G of g 0 ∈ G 0 . In the derivation of eq.(5.1), we also used the fact that ξ g can be chosen so that ξ g (λ) = 1 for all λ fixed by g 0 . Notice that φ w−m g (τ ) is the only factor of φ g which potentially depends on the choice of v. When the lattice Λ L contains no non-zero vectors fixed by g 0 , we obtain φ w−m g (τ ) = 1 and the dependence on v cancels. The twining genus φ g can be written in terms of the eta function and theta functions as  in G 0 , the dimension of the untwisted R-R ground sector H U , the dimensions of the g-invariant twisted sectors H g k , k = 1, . . . , N − 1, N = o(g), the total dimension of the space H of R-R ground states in the orbifold (which is always 24 for a K3 model), the trace of the quantum symmetry Q g over H and the Co 0 class of Q g , as reported in [26].
where r L ∈ Q/Z is such that ζ L = e 2πir L and From eqs.(5.2) and (5.3), we obtain In particular, when ζ L = 1 or ζ R = 1, the Witten index φ g (τ, 0) vanishes. Using where N is the order of g, we can easily compute the Witten index of the orbifold theory.
In particular, the orbifold is a torus model if g 0 is in one of the classes while the orbifold is a K3 model when g 0 is in any of the other classes. In the latter case, the level-matching condition is always satisfied, while for the classes (5.4) it puts non-trivial constraints on the possible lifts g ∈ G of g 0 ∈ G 0 . In [26], a similar analysis was performed for non-linear sigma models on K3: all possible symmetries of such models were classified according to whether the corresponding orbifold is a torus or a K3 model. As shown in [23], each symmetry of K3 models can be labeled by a conjugacy class in the Conway group Co 0 , which is determined by its action on the 24-dimensional space of R-R ground states. For each symmetry, the 'nature' of the corresponding orbifold theory only depends on its Co 0 class [26].
We can now compare the analysis of [26] with the results of the present paper. More precisely, suppose that g ∈ G is a symmetry of a torus model C T 4 , such that the corresponding orbifold theory is a K3 model C ′ K3 . This K3 model always contains a 'quantum symmetry' Q g that acts by e 2πik N on the g k -twisted sector. The original CFT C T 4 can be reobtained by taking the orbifold of C ′ K3 by Q g . Since we are able to compute the dimension dim H g k = 1 N N i=1 φ g k ,g i (τ, 0) of the space of g-invariant g k -twisted R-R ground states, we can compare the eigenvalues of the quantum symmetries Q g with the eigenvalues expected for a Conway class in the 24-dimensional representation. The results, summarized in table 4, match perfectly with the expectations from [26]. 4 Our argument implies that there are K3 models that are orbifolds of non-linear sigma models on T 4 by symmetries of order 2, 3, 4, 5, 6, 8, 10, 12. The torus model with a symmetry of order 5, together with its orbifold K3 model, was studied in [26]; it turns out that this is the model of section 4.4.2 with symmetry group G 0 = I 120 × I 120Ī 120 . Furthermore, the torus model based on the lattice Λ D 4 of section 4.4.1 was considered in [34], where it was stressed that a symmetry in the class −4A exists. Its orbifold is the K3 model with the 'largest symmetry group' Z 8 2 : M 20 [34].

Conclusions
In this paper we consider supersymmetric non-linear sigma models with target space a four dimensional torus T 4 and classify all possible group of symmetries of such models that preserve the N = (4, 4) superconformal algebra. This results extends the analysis of [23] on non-linear sigma models on K3 to include all known conformal field theories with N = (4, 4) superconformal symmetry at central charge 6.
One of the immediate by-products of our investigation is the description of new interesting torus models with a large amount of discrete symmetries. The new results concern mainly the models admitting left-right asymmetric symmetries with no geometric interpretation. It turns out that there are only few isolated points in the moduli space where such symmetries are realized.
Some of the corresponding torus models are the usual suspects: the simplest example is the product of four orthogonal circles at the self-dual radius, with vanishing B-field (section 4.4.6). Another well-known example is the model with target space R 4 /D 4 , where D 4 is the root lattice of the so(8) algebra (section 4.4.1); the discrete symmetries and some of the orbifolds of the latter model were recently investigated in [20,34]. The chiral algebra in these two theories is much larger than the one at a generic point in the moduli space.
On the contrary, some of the new models considered in section 4 have the smallest possible chiral algebra for a non-linear sigma model on T 4 . These cases include, in particular, the model with a symmetry of order 5, considered in [26]: we discover that its group of symmetries G 0 is actually isomorphic to the binary icosahedral group I 120 , and that the lattice of winding-momenta is most easily described in terms of the icosian ring of unit quaternions (section 4.4.2). Another interesting model has a symmetry group G 0 isomorphic to the binary dihedral group D 24 (6) of order 24 and its lattice of winding-momenta is based on the Eisenstein integers (section 4.4.5).
Our results can find some useful applications to the study of toroidal and K3 superstring compactifications. In particular, it would be interesting to understand how this investigation is related to the groups of symmetries of heterotic strings compactified on T 4 , since the supersymmetric side of these models is analogous to the chiral half of the type II models we consider.
In the context of the Mathieu moonshine phenomenon, the analysis of the present paper can lead to a generalization of the results of [35]. In this work, some Siegel modular forms were constructed as the multiplicative lifts of the twisted-twining genera of generalized Mathieu moonshine [11]. Many of these modular forms admit an interpretation as partition functions for 1/4 BPS states in four dimensional CHL models with 16 space-time supersymmetries. In [35], it was observed that there exists a particular modular transformation, induced by electric-magnetic duality in the four dimensional CHL model, that exchanges the multiplicative lifts of twining genera in two distinct K3 models, related by an orbifold. An obvious generalization of the Siegel modular forms of [35] can be obtained by taking the multiplicative lifts of twining genera in torus models, such as the ones considered in section 5 of the present paper. In this case, one expects the 'electric-magnetic duality' to relate these forms to the multiplicative lifts of the twining genera in exceptional K3 models, i.e. the K3 models admitting a group of symmetries not in M 24 [26]. It would be very interesting to investigate in detail this instance of electric-magnetic duality and to understand its physical meaning.
A somehow related open question concerns the existence of any kind of moonshine phenomenon for non-linear sigma models on T 4 . By analogy with the K3 case, one would expect the corresponding twining genera to reproduce some of the functions computed in section 5. 5 The symmetries considered in the present paper can be described in terms of topological defects preserving a small N = (4, 4) superconformal algebra. The orbifold procedure relating the corresponding torus models to K3 has also a natural interpretation in terms of defects [36]. At the moment, only the topological defects of d-dimensional torus models that preserve the u(1) d ⊕ u(1) d current algebra have been classified [37]. In the same spirit, it is natural to ask whether the classification of this paper includes all possible defects preserving the N = (4, 4) algebra or if more possibilities exist. In the latter case, one might obtain new explicit descriptions of K3 models arising from the associated generalized orbifolds of torus models [38][39][40].
A Discrete subgroups of Spin(4) and SO (4) For the classification of the discrete subgroups of Spin(4) ∼ = SU(2) × SU(2) and SO(4) ∼ = Spin(4)/(−1, −1) we follow closely the treatment of [41]. We recall that SU(2) can be described as the group of unit quaternions SU(2) = {a + bi + cj + dk ∈ H | a 2 + b 2 + c 2 + d 2 = 1} , and its discrete subgroups are G ⊂ SU (2) |G| Name Image in SO (3)  where e r = exp(2πir), Notice that C n ⊂ SO (3), with odd n, admits both a double and a single cover in SU (2) wherer denotes the complex conjugate of r. A generic discrete G ⊂ SU(2) × SU(2) is a subgroup of a direct product group G ⊆ L × R. Any such subgroup G can be defined in terms of two surjective homomorphisms α : L → F , β : R → F , from L and R onto the same abstract group F . More precisely, G is the group of pairs (l, r) ∈ L × R such that l and r have the same image in F G = {(l, r) ∈ L × R | α(l) = β(r)} , for some suitable α, β. In particular, L 0 := ker α and R 0 := ker β are (isomorphic to) the normal subgroups of G generated by elements of the form (l, 1) ∈ G and (1, r) ∈ G, respectively, so that L 0 × R 0 ⊆ G ⊆ L × R .
For example, when |F | = 1, we have L 0 = L, R 0 = R and G is simply the direct product L × R. At the opposite extreme, when L = F = R and α, β are the identity maps, then L 0 = 1 = R 0 and G is also isomorphic to F . The groups of purely geometric symmetries of torus models are of this kind. In most cases, giving the groups L, R and F is sufficient to determine the homomorphisms α and β uniquely (up to conjugation of L and R in SU(2) and up to automorphisms of F ). This justifies the notation L × F R for the corresponding group G. When there are inequivalent choices for α, β, we denote the 'most obvious' group by L × F R and the other ones by some decorations, such as L × FR . The precise structure of each group will be clear from its list of generators. For example, I 120 × I 120 I 120 denotes the group where both α, β are the identity map, while I 120 × I 120Ī 120 denotes the group where α is the identity map and β is a non-trivial outer automorphism of I 120 . The two groups are actually isomorphic as abstract groups (and isomorphic to I 120 ), but they are not conjugated within SU(2) × SU (2).

B Lattices
In this appendix, we review some general properties of even lattices and their automorphisms, in particular the 'gluing' construction (see [42] and [31] for more details). For any lattice Λ ⊂ R n , we denote by Λ * its dual and by Λ(−1) the lattice obtained by changing the sign of the quadratic form. We define the determinant of Λ to be the determinant | det Q| of the Gram matrix Q ij := λ i ·λ j , for some basis λ 1 , . . . , λ r of generators of Λ. The determinant is independent of the choice of the basis. If Λ is integral, and in particular if it is even, then Λ ⊂ Λ * and we can define the discriminant group Λ * /Λ. This is a finite abelian group of order | det Q|.
Any automorphism g ∈ O(Λ) induces an automorphism of the dual Λ * and of the discriminant group, preserving the discriminant form q Λ . We denote by O 0 (Λ) the subgroup of automorphisms acting trivially on the discriminant group and by SO(Λ) and SO 0 (Λ) the corresponding subgroups of orientation-preserving automorphisms. If Λ and Λ ⊥ are two primitive mutually orthogonal sublattices of the even unimodular lattice Γ, as above, then any element g ∈ O 0 (Λ) extends to an automorphism of Γ [31]. Indeed, g obviously extends to an automorphism of Λ * ⊕ ⊥ (Λ ⊥ ) * by (v, w) → (g(v), w) and since [v] = [g(v)] for all v ∈ Λ * this automorphism preserves the sublattice Γ.