A K3 sigma model with Z_2^8:M_{20} symmetry

The K3 sigma model based on the Z_2-orbifold of the D_4-torus theory is studied. It is shown that it has an equivalent description in terms of twelve free Majorana fermions, or as a rational conformal field theory based on the affine algebra su(2)^6. By combining these different viewpoints we show that the N=(4,4) preserving symmetries of this theory are described by the discrete symmetry group Z_2^8:M_{20}. This model therefore accounts for one of the largest maximal symmetry groups of K3 sigma models. The symmetry group involves also generators that, from the orbifold point of view, map untwisted and twisted sector states into one another.

3 The K3 sigma model as an su(2) 6 1 RCFT 9 3.1 Representations of the su(2) 6 L,1 ⊕ su(2) 6 R,1 current algebra 9 3.2 Spectrum of the K3 model in terms of representations of su(2) 6 1 ⊕ su (2)  Recently, Eguchi, Ooguri and Tachikawa made the intriguing observation [1] that the expansion coefficients of the elliptic genus of K3 can be naturally interpreted in terms of representations of the finite sporadic Mathieu group M 24 . This conjecture has now been established. In particular, the twining genera, i.e. the elliptic genera with the insertion of a group element g ∈ M 24 , have been determined combining different viewpoints [2][3][4][5].
The knowledge of all twining genera fixes the decomposition of every expansion coefficient in terms of M 24 representations, and it was shown in [6] (see also [7]) that the resulting multiplicities are indeed (non-negative) integers. Thus a consistent decomposition of all expansion coefficients in terms of M 24 representations is possible. More recently, evidence was also obtained that the same is true for the twisted twining genera [8,9], i.e. the analogues of Norton's generalised moonshine functions [10]. The ideas underlying Mathieu moonshine have also now been extended in other directions, see in particular [11][12][13][14][15][16][17][18].
Since the elliptic genus counts the net contribution of BPS states of string theory on K3, these results suggest that M 24 has a natural action on the BPS spectrum of these sigma models. Obviously, the simplest way to realise such an action would be if it came from a genuine symmetry of the full sigma model. The 'geometrical' symmetries of K3 surfaces were classified some time ago by Mukai [19] (with additional insights by Kondo [20]), who established that the holomorphic symplectic automorphism groups of K3 surfaces are all proper subgroups of the sporadic group M 23 (which stabilises one element in the standard representation of M 24 as permutation group on 24 symbols). Since the elliptic genus is constant along each connected component of the moduli space of N = (4,4) superconformal field theories at central charge (c, c) = (6, 6), one may then hope that the symmetries at different points in moduli space may be put together. This led two of us to suggest that an 'overarching symmetry group' based on the classical geometric symmetries of K3 non-linear sigma models could be defined in this manner [21]; indeed already under restriction to symmetries of Kummer K3s one obtains the group Z 4 2 : A 8 , which is a maximal subgroup of M 24 not contained in M 23 [22,23].
In order to enhance the symmetry group to M 24 , stringy symmetries of K3 sigma models may need to be included. Mukai's Theorem was generalised to the case of supersymmetry-preserving automorphism groups of non-linear sigma models on K3 [24]. Somewhat surprisingly it was found that the possible supersymmetry-preserving automorphism groups of K3 sigma models are not all subgroups of M 24 , but instead form groups that fit inside the Conway group Co 1 , which contains M 24 . In fact, the result of [24] gave a rather concrete description of the possible symmetry groups of all K3 sigma models. For example, it predicted the existence of a K3 sigma model with symmetry group G = 5 1+2 : Z 4 that was subsequently identified with a certain asymmetric Z 5 -orbifold of a torus theory that realises a K3 theory [25]. It also predicted the existence of a K3 sigma model with symmetry group Z 8 2 : M 20 , one of the largest maximal symmetry groups of K3 sigma models. It is the purpose of the present article to identify the microscopic realisation of this sigma model. As it turns out the relevant K3 sigma model can be described as the usual Z 2 -orbifold of a torus theory at the special D 4 -point, such that the bosonic theory before orbifolding has an so(8) 1 current symmetry, both for left-and right-movers. From this geometric viewpoint, the model is a nonlinear sigma model on the so-called tetrahedral Kummer K3 studied in detail in [21].
We should stress that the presence of the Z 8 2 : M 20 symmetry group is not entirely obvious in this description. In fact, Z 8 2 : M 20 contains A 5 , the alternating group of five symbols, and it is a priori not clear how this 5-fold permutation symmetry should arise from the Z 2 -orbifold of the D 4 -torus theory. The key idea behind our paper is that the bosonic theory has, after orbifolding, the chiral symmetry so(4) 1 ⊕ so(4) 1 ∼ = su(2) ⊕ 4 1 .
(1.1) Furthermore, the four free fermions of the supersymmetric torus theory give rise to the chiral symmetry that survives the orbifold projection. Taken together, the Z 2 -orbifold of the D 4 -torus theory therefore has the chiral symmetry [26] so(4) 1 ⊕ so(4) 1 ⊕ so(4) 1 ∼ = su(2) ⊕ 6 1 . (1.3) One of these su(2) 1 algebras can be identified with the R-symmetry of the N = 4 superconformal algebra that must remain invariant under the supersymmetry-preserving automorphisms. However, the other five factors may be permuted, and this is the origin of the A 5 symmetry of our torus orbifold. 1 Remarkably, in the description of the model as a torus orbifold, some generators of this A 5 symmetry mix states in the twisted and untwisted sectors. Our analysis also shows that the Z 2 -orbifold of the D 4 -torus theory can be alternatively realised as an asymmetric Z 4 -orbifold of the same D 4 -torus.
The paper is organised as follows. In Section 2 we discuss the Z 2 -orbifold of the D 4torus theory in detail, and explain how it may also be described in terms of twelve free Majorana fermions, both for the left-and the right-movers. In Section 3 we provide yet another description of the same theory, now as a rational conformal field theory (RCFT) based on the current algebra su(2) ⊕ 6 1 . This approach exhibits the aforementioned permutation symmetry most clearly; however, the structure of the N = (4, 4) supercharges is 1 Only A 5 rather than S 5 emerges since the states in the ( 1 4 , 1 2 ; 1 4 , 1 2 ) multiplet of the left-and rightmoving N = 4 algebras also have to be preserved, which requires the permutations to be even. rather involved from that viewpoint. We therefore construct the supersymmetry-preserving symmetries of our orbifold model using the orbifold description, and then translate them into the su(2) ⊕ 6 1 language, using the free fermion description as an intermediate step; this is done in Section 4. Actually, as is explained in Section 4.3, there are at least fifteen different ways in which one may write our K3 sigma model as a Z 2 -orbifold of a toroidal model -obviously, all these descriptions are equivalent, but differ in their distribution of states into the twisted and untwisted sector, respectively. Of these fifteen descriptions, five have the same expression for the four supercharges, and hence their supersymmetrypreserving symmetries can be directly combined. In Section 5 we analyse the structure of the resulting group, and demonstrate that it contains at least Z 8 2 : M 20 ; together with the results of [24] this then shows that the supersymmetry-preserving symmetry group of this model must be precisely equal to Z 8 2 : M 20 . Section 6 explains why our Z 2 -orbifold model has another realisation as an asymmetric Z 4 -orbifold of the same D 4 -torus theory. In fact, the Z 4 action can be described in terms of two consecutive Z 2 actions, namely the Z 2 -orbifold by T-duality (which is a consistent symmetry and leads again to the D 4 -torus theory), followed by the usual Z 2 inversion action. Finally, we close with some conclusions in Section 7. There are six appendices, where some of the more technical material has been collected.

K3 sigma model as Z -orbifold of the D 4 -torus model
In this section we describe the K3 sigma model based on the Z 2 -orbifold of the D 4 -torus model. In particular, we recall that it possesses an su(2) 6 L,1 ⊕ su(2) 6 R,1 (2.1) affine symmetry algebra 2 [26, proof of Thm. 3.7]. We also explain how it may be described in terms of twelve left-and twelve right-moving Majorana fermions.

Geometric description of the D 4 -torus model and its Z 2 -orbifold
A generic d = 4 bosonic torus model contains an affine u(1) 4 algebra of left-moving currents j k (z) = i∂φ k (z), k = 1, . . . , d = 4 and its right-moving counterpart (see Appendix A for our conventions and notations). At particular points in the moduli space (i.e. at special values for the metric and the B-field), the u(1) 4 current algebra is enhanced to a non-abelian affine algebra of rank 4. In the case where the lattice L underlying the torus T = R 4 /L is the D 4 -lattice L D 4 ⊂ R 4 , the B-field can be chosen such that the extended bosonic current algebra is so(8) 1 . To see this, consider the following basis of L D 4 , 2) 2 We use the notation su(2) n 1 := su(2) ⊕n 1 throughout. The indices L/R stand for left-and right-moving, respectively. as well as the B-field and set, for every pair of indices j, k ∈ {1, . . . , 4} with j = k, where e 1 , . . . , e 4 is the canonical orthonormal basis of R 4 . Then, the model contains states with momentum-winding (m ±j,±k , l ±j,±k ) ∈ L * D 4 ⊕ L D 4 , which have left and right u(1) 4 L charges (see Appendix A) Thus the momentum-winding fields V (Q ±j,±k ;0) (z) define twenty-four (1, 0)-fields which, together with j 1 (z), . . . , j 4 (z), form the standard so(8) 1 -current algebra. The right-moving so(8) 1 -current algebra is analogously obtained using the twenty-four (0, 1)-fields The full spectrum of the bosonic D 4 -torus model can be decomposed into representations of the left-and right-moving so(8) 1 algebras as where H L,0 is the left-moving so(8) 1 vacuum representation, while H L,v , H L,s and H L,c are the vector and the two spinor representations, respectively. The H R, * denote the corresponding right-moving representations. The vector representation H L,v ⊗ H R,v is generated by OPEs of the holomorphic and antiholomorphic currents with the windingmomentum fields while the spinor representations are generated by In particular, 4 k=1 Q k and 4 k=1 Q k are even for H L,s ⊗ H R,s and odd for H L,c ⊗ H R,c . Thus, the lattice of u(1) 4 L ⊕ u(1) 4 R charges (see (A.4)) of the D 4 -torus model equals The supersymmetric torus model is obtained by adjoining d = 4 free Majorana fermions ψ k (z), k = 1, . . . , 4, related to the U(1)-currents j k (z) by world-sheet supersymmetry, and their right-moving counterparts ψ k (z), k = 1, . . . , 4. These holomorphic fermions give rise to the affine symmetry .
Altogether, the affine symmetry (both for left-and right-movers) of the supersymmetric D 4 -torus model is In order to obtain a K3 sigma model we now consider a Z 2 -orbifold of this torus model. The group Z 2 acts in the usual manner on the bosonic degrees of freedom, i.e. it maps j k (z) → −j k (z),  k (z) → − k (z), k = 1, . . . , 4, and V λ → V −λ for all λ ∈ Γ 4,4 . In order for this to be a well-defined symmetry we need to choose our operators c λ of equation On the fermions, the group Z 2 acts as ψ k → −ψ k and likewise for the right-movers, ψ k → −ψ k , k = 1, . . . , 4. In particular, the orbifold leaves the su(2) L,1 ⊕ su(2) R,1 algebra in (2.11) invariant, since it is generated by the bilinear fermion combinations (2.12) -(2.15).

(2.19)
Since the central charge of the (left) current algebra (2.19) equals c = 6, the full K3 sigma model must be a rational theory with respect to this symmetry algebra. Hence we will be able to describe the whole theory quite succinctly in terms of an su(2) 6 L,1 ⊕ su(2) 6 R,1 RCFT; this will be further developed in Section 3. In order to understand the equivalence between that description and the D 4 -torus orbifold better, it is convenient to reformulate the torus orbifold in terms of free fermions.

Free fermion description of the D 4 -torus and its orbifold
The bosonic D 4 -torus model may be described in terms of eight free left-and right-moving Majorana fermions ψ j (z) and ψ j (z), j = 5, . . . 12, all of whose boundary conditions are coupled. In Appendix B, we state the correspondence in detail in terms of the four leftand four right-moving Dirac fermions which satisfy the OPEs By our choice of fermionisation (B.8), we identify the holomorphic U(1)-currents of the bosonic D 4 -torus model as All other generating fields of the theory as determined in Subsection 2.1 are expressed in terms of the Dirac fermions x k (z), x * k (z), x k (z), x * k (z), k = 1, . . . , 4, and the 'meromorphic building blocks' ξ ± k (z) = : exp ± i 2 φ k (z) : that are defined in Appendix B.
Next we observe that the Z 2 -action on the D 4 -torus model is induced by the transformation that leaves ψ 5 (z), . . . , ψ 8 (z) invariant, while mapping ψ k (z) → −ψ k (z) where k ∈ {9, . . . , 12}. In other words, we have x k (z) ↔ x * k (z), and analogously for the rightmoving fermions. Using the notations (2.5), the untwisted sector of the Z 2 -orbifold is hence generated by the Z 2 -invariant (1, 0)-fields with C-basis δ k e k : where ε j , δ k ∈ {±}. To describe the twisted sector, recall that the Z 2 -orbifolding of our eight Majorana fermions with coupled boundary conditions decouples effectively the boundary conditions of the fermions ψ 5 (z), . . . , ψ 8 (z) from those of ψ 9 (z), . . . , ψ 12 (z). For where the η ± k fields are introduced in Appendix B. Then the twisted ground states of our Z 2 -orbifold are described by those Ξ ± ε 1 ,...,ε 4 (z) for which an even number of the ε k are equal to +1.
Note that our expressions and normalisations of the U(1)-currents J 3,k (z) and j k (z), respectively, are different (compare (2.12) and (2.17) to (2.22) and (A.1)). In what follows, we shall use both choices of fermionisation conventions, since both of them are sometimes convenient. We will carefully distinguish the two choices in terms of our notations, not just for the U(1)-currents J 3,k (z) and j k (z), but also for the relevant Dirac fermions, which are denoted by χ k (z) or x k (z), respectively. This free fermion description is also convenient in order to determine the partition function of the theory and -by means of the elliptic genus -to confirm that it is a K3 model, see Appendix C. In fact, by the results of [27], the usual Z 2 -orbifold of every supersymmetric d = 4-dimensional torus model has the elliptic genus of K3 and thus is indeed a K3 model.

The N = (4, 4) supercurrents
The K3 sigma model possesses an N = (4, 4) superconformal symmetry on the world-sheet; the relevant supercharges can be most conveniently defined for the underlying supersym-metric D 4 -torus model. With notations as above, we define the complex fields Using the definition of the Dirac fermions χ k (z), k = 1, 2, see eq. (2.16), the holomorphic N = 4 supercurrents are then given by Indeed, it is straightforward to check that these fields satisfy the OPEs exhibiting ψ k (z) as superpartner of j k (z), k = 1, . . . , 4. Moreover, (2.28) -(2.31) obey the standard OPEs for the N = 4 supercurrents where T is the stress-energy tensor and J a,1 with (a = 3, +, −) are the su(2) 1 currents of (2.12) -(2.13). Therefore, the zero modes of these currents generate the SU(2) R-symmetry group of the N = 4 algebra. The free fermion description of our model of Subsection 2.2 allows us to express the supercurrents (2.28) -(2.31) in terms of the Dirac fermions χ k (z), χ * k (z), k = 1, 2, and the Majorana fermions ψ 5 (z), . . . , ψ 12 (z) by means of (2.22). However, for later use it is more convenient to introduce Dirac fermions in a completely symmetric way, as opposed to the construction in Subsection 2.2. Indeed, we extend (2.16) to the definitions instead of using the fields x k and x * k , k = 1, . . . , 4, of (2.20). Then one checks as well as 3 The K3 sigma model as an su(2) 6 1 RCFT The K3 model which we described in Section 2 can be obtained as a Z 2 × Z 2 -orbifold of the Gepner model (2) 4 [26, Thm. 3.7]. As such it is a rational CFT. In this section, we give a description of it as an su(2) 6 1 RCFT which turns out to be useful in order to determine the full symmetry group of this model.

Representations of the su(2) 6
L,1 ⊕ su(2) 6 R,1 current algebra Let us begin by reviewing the representation theory of su(2) 1 . This algebra possesses only two irreducible highest weight representations, namely the vacuum representation [0], whose ground state has conformal weight 0 and is a singlet under the group SU(2) generated by the zero modes of the algebra, and the representation [1], whose ground states have conformal weight 1 4 and form an SU(2)-doublet. The fusion rules have the structure of a cyclic group of order 2, Therefore, the representation content of a model with su(2) 6 L,1 ⊕ su(2) 6 R,1 (left-and rightmoving) affine algebra 4 can be encoded in a subgroup 2) whose elements we denote by Occasionally, it will be useful to consider su(2) 6 1 as a direct sum of three so(4) 1 algebras. The algebra so(4) 1 ∼ = su(2) 2 1 has four irreducible highest weight representations that we denote by a pair [ab], a, b ∈ {0, 1}, of su(2) 1 labels. Equivalently, four Majorana fermions with coupled spin structures yield an su(2) 2 1 ∼ = so(4) 1 current algebra as in (2.12) -(2.15).
3.2 Spectrum of the K3 model in terms of representations of su(2) 6 1 ⊕ su(2) 6 1 In order to obtain the K3 sigma model we now have to perform an orbifold of this bosonic D 6 -torus model. In particular, we need to decouple the spin structures of the internal and external fermions (in order to describe the supersymmetric D 4 -torus model), and we have to perform the usual Z 2 -orbifold to obtain from the latter a K3 theory. In terms of the D 4 -torus model, the relevant orbifold group is therefore Z 2 × Z 2 with generators where F L,k , F R,k are the fermion number operators corresponding to ψ k , ψ k , k = 1, . . . , 12, respectively. Here g is the symmetry whose orbifold decouples the spin structures, while h induces the standard Z 2 -orbifold on the D 4 -torus model.
To implement the orbifold procedure, we now introduce the g-, h-and gh-twisted sectors and then project onto the invariant states in the untwisted and in the three twisted sectors. We focus on the NS-NS sector of our model first. In the g-twisted sector, the fields ψ k , ψ k , k = 5, . . . , 12 have integer modes. Thus, the g-twisted ground states form a representation of the Clifford algebra of the zero modes of these fields, i.e. they transform in spinor representations of the corresponding so(4) 1 algebras. Analogous considerations hold for the h-and gh-twisted sectors. Therefore, the tensorial properties of the various sectors with respect to the left and right-moving so(4) 3 L,1 ⊕ so(4) where T and S denote a tensor or a spinor representation of so(4) 1 as in (3.4), respectively. Finally, one has to project onto the representations that are invariant under both g and h. The invariant states have the same fermion numbers with respect to the three sets of fermions, which allows us to identify the corresponding parity operators with the total fermionic parity (−1) F L +F R , (3.12) In particular, the space of (g, h)-invariant fields contains a bosonic subspace (i.e. with positive total fermion parity) generated by the su(2) 6 L,1 ⊕ su(2) The representation content of this bosonic subspace corresponds therefore to the subgroup A bos of Z 6 2 × Z 6 2 , 14) The entire NS-NS spectrum of the orbifold theory is generated by the fusion of these representations with one fermionic representation (negative total fermion parity), for example the representation [11 11 11; 00 00 00] (3.15) which contains the holomorphic fields of weight 3 2 . The resulting subgroup of Z 6 2 × Z 6 2 describing the entire NS-NS spectrum of the theory is A bos ∪ A ferm , where accounts for the states of negative total fermion parity.
The R-R sector of our model is obtained by inverting the tensorial properties of the so(4) 3 L,1 ⊕ so(4) 3 R,1 representations with respect to the NS-NS sector, i.e. they are given by exchanging S ↔ T in (3.8) -(3.11). Since we still need to obey (3.12), the R-R spectrum thus consists of the states together with all representations that can be obtained by fusion with the NS-NS representations (3.13) and (3.15). With this description of the entire R-R spectrum of our orbifold, it is then straightforward to calculate the elliptic genus in terms of su(2) 6 1 characters. This is done in Appendix D.3, where we show that the elliptic genus reproduces indeed that of K3.
The structure of (3.14) and (3.16) reveals that the spectrum of the orbifold theory is invariant under simultaneous permutations of the six holomorphic and six anti-holomorphic su(2) 1 algebras. This is actually also a symmetry of the OPE, as is shown in Appendix D.2. Hence the group of symmetries of the model is (at least) (SU(2) 6 L × SU(2) 6 R ) : S 6 . Notice that neither the free fermion theory on the bosonic D 6 -torus, nor its orbifold by g, corresponding to the supersymmetric sigma model on the D 4 -torus, contain such an S 6 symmetry. Therefore, one cannot generate the whole group of symmetries of the K3 sigma model just by considering the transformations induced by the symmetries of the parent theories.
Our model contains 64 holomorphic fields of weight 3 2 , that generate several copies of the N = 4 superconformal algebra and, in particular, the four supercurrents (2.28) -(2.31). The corresponding su(2) L,1 algebra (whose zero modes generate the SU(2) R-symmetry group) is identified with the first factor of the full su(2) 6 L,1 affine algebra. Therefore, the group of symmetries preserving the four left and four right-moving supercurrents must be a subgroup of the stabiliser (SU(2) 5 × SU(2) 5 ) : S 5 of the first su(2) L,1 ⊕ su(2) R,1 factor.

Finite symmetries of the K3 sigma model
In [24], it was argued that the group G of symmetries of a non-linear sigma model on K3, preserving the N = (4, 4) superconformal algebra and the four R-R ground states that are charged under the R-symmetry, is always a subgroup of the Conway group Co 0 . Generically, G is a subgroup of Z 12 2 : M 24 ⊂ Co 0 , and the only exceptions are given by orbifolds of torus models by cyclic groups of order 3 or 5 [25].
In this paper we are interested in determining the group G of symmetries of the specific K3 sigma model described so far in three different ways: as the Z 2 -orbifold of the D 4 -torus model in Subsection 2.1; as a free fermion model in Subsection 2.2; and as an RCFT based on the vertex operator construction of the su(2) 6 L,1 ⊕ su(2) 6 R,1 affine algebra in Section 3. Each of these three descriptions exhibits some of the relevant finite symmetries in a natural way. Our first aim in this section is to represent all of these symmetries as elements of a subgroup of (SU(2) 6 L × SU(2) 6 R ) : S 6 , see the discussion at the end of the previous section. We first consider the discrete 'geometric' symmetries of the Z 2 -orbifold of the D 4 -torus model as a guide, and we use the free fermion description to express these symmetries (and new ones discovered in the process) in terms of a subgroup of (SU(2) 6 L × SU(2) 6 R ) : S 6 . We then turn to the su(2) 6 L,1 ⊕ su(2) 6 R,1 RCFT description of the K3 model to express generators of our symmetry group in a form that paves the way to the identification of a less obvious A 5 ⊂ S 6 symmetry group. This group is a factor of the group G of symmetries we are seeking. The full structure of G will then be studied in Section 5.

Symmetries from the Z 2 -orbifold of the D 4 -torus model
By construction, the Z 2 -orbifold of the bosonic D 4 -torus model has a geometric interpretation on the tetrahedral Kummer surface that is obtained by minimally resolving all the The symmetry group of that Kummer surface was studied in detail in [21]; in particular, the group of holomorphic symplectic automorphisms is the group T 192 ∼ = (Z 2 ) 4 : A 4 of order 192. The subgroup of type A 4 of T 192 is induced by those symmetries of the underlying torus that have a fixed point. The remaining symmetries in T 192 are generated by including the translational subgroup (Z 2 ) 4 (half-period shifts) which acts as a permutation group on the twisted ground states and leaves the untwisted sector invariant.
In this subsection we identify this group of symmetries, as well as some additional non-geometric generators, with a subgroup of (SU(2) 6 L × SU(2) 6 R ) : S 6 . In fact, we focus on the action of this group on the holomorphic and antiholomorphic currents generating the su(2) 6 L,1 ⊕ su(2) 6 R,1 algebra that survives the Z 2 -orbifold projection. Actually, we will only identify the group modulo the subgroup Z 6 2 × Z 6 2 of elements of (SU(2) 6 L × SU(2) 6 R ) : S 6 that act trivially on the currents.

Rotations
The group A 4 of rotations may be generated by the three following symmetries 5 which act on R 4 and induce symmetries of the D 4 -torus T D 4 that descend to the Kummer surface. Since the D 4 -torus model has a non-trivial B-field B given in (2.3), we need to ensure that these rotations induce symmetries on the conformal field theory. This is the case if and only if γ T k Bγ k = B for k ∈ {1, 2, 3}, and one immediately confirms that this latter condition is indeed obeyed.
We now give a description of the action of γ 1 and γ 2 in terms of the holomorphic fields in the free fermion model; this is sufficient in order to specify the symmetries in the form that is needed in Section 5. With the notations of Section 2 and Appendix B, see in particular (2.4), we observe that the lattice vectors l 1,2 , l 1,−2 , l 3,4 , l 3,−4 form a basis of R 4 . The symmetries γ 1 and γ 2 permute the eight lattice vectors l ±1,±2 , l ±3,±4 , inducing the following maps under the identification (B.9) These maps are induced by from which one obtains the actions on all (1, 0)-fields in the orbifold. Equivalently, for ψ 5 , . . . , ψ 12 we have The action on the superpartners of the four bosonic U(1) currents j k , k = 1, . . . , 4, is Note that in [21], a different choice of coordinates was used on the underlying torus. Also note that the minimal number of generators for A 4 is two. Indeed, Finally, in terms of the Dirac fermions (2.16) and (2.34), the γ 1 -and γ 2 -actions read, The transformation induced on the su(2) 6 1 currents by (2.12) -(2.15) and the analogous formulas for k = 3, . . . , 6 are The transformations γ 1 , γ 2 form a subgroup Z 2 2 of SU(2) 6 L × SU(2) 6 R . The symmetry γ 3 has a non-trivial image in S 6 and will not be needed to generate the group of discrete symmetries we are seeking.

Half-period shifts and more
The spectrum of the torus model can be naturally decomposed into eigenstates of the zero modes of the currents j k (z) = i∂φ k (z),  k (z) = i∂ φ k (z), k = 1, . . . , 4. For the D 4 -torus model, the possible eigenvalues are given by the charge lattice Γ = Γ 4,4 of (2.10).
In order to construct operators that commute with the orbifold action j k → −j k we consider an element (a; a) ∈  In particular, a shift by half a period 1 2 l, with l ∈ L D 4 , corresponds to a symmetry s (a;a) with (a; a) = 1 2 (Q(0, l); Q(0, l)) ∈ ( 1 2 Γ 4,4 )/Γ 4,4 , where we have used (A.3). The halfperiod shifts form a subgroup Z 4 2 of ( 1 2 Γ 4,4 )/Γ 4,4 and act by multiplication by (−1) m·l on all states of momentum m ∈ L * D 4 , independently of their winding numbers. The entire group Z 8 2 of symmetries (4.18) is generated by the half-period shifts of the D 4 -torus model together with the half-period shifts in the T-dual torus model. In the following, we will refer to all these symmetries generically as half-period shifts. For concreteness we choose a set of generators for ( 1 2 Γ 4,4 )/Γ 4,4 to be of the form Let us now describe the action of the elements s (a;a) on the su(2) 6 L,1 ⊕ su(2) 6 R,1 currents, using the free fermion description of the model. In this description, the zero mode of the U(1) current j k (z) = −i : ψ k+4 ψ k+8 :(z), k = 1, . . . , 4, (see (2.22)) is the generator of rotations in the plane spanned by ψ k+4 and ψ k+8 .
• The generators s v k , k = 1, . . . , 4 act by 19) while all the other fermions ψ l , ψ l with l ∈ {k + 4, k + 8} are fixed by s v k . For instance, using (2.34), one sees that s v 4 acts on the holomorphic fields by Thus, the induced action on the su(2) 6 L,1 holomorphic currents is 6 Therefore, s v 4 corresponds to a (34)(56) permutation acting simultaneously on the six left and the six right SU(2) factors in SU(2) 6 L × SU(2) 6 R . • The symmetry s u acts by a simultaneous 90-degree rotation in all planes (i+ 4, i+ 8), i = 1, . . . , 4, that is for all i = 1, . . . , 4. The induced action on the holomorphic currents is Therefore, s u corresponds to a (35)(46) permutation acting simultaneously on the six left and the six right SU(2) factors in SU(2) 6 L × SU(2) 6 R .
• The elements s v i +v j , 1 ≤ i < j ≤ 4, correspond to elements in SU(2) 6 L × SU(2) 6 R . In particular, s v 1 +v 2 acts on the holomorphic fields by (4.24) Therefore, the induced transformation on the holomorphic currents is (4.25) Furthermore, s v 2 +v 4 acts on the holomorphic fields by so that the induced action on the holomorphic currents is • The symmetries s w ij act on the left-moving currents in the same way as s v i +v j , 1 ≤ i < j ≤ 4, while they leave the right-moving currents fixed. In particular, on the leftmoving currents, s w 12 acts as s v 1 +v 2 as given in (4.25), while s w 24 acts as s v 2 +v 4 according to (4.27). Furthermore, s w 23 +w 14 and s w 12 +w 24 +w 14 act trivially on all currents, so they must correspond to elements in the center of SU (2)

Quantum symmetry
Apart from the geometric symmetries, our K3 model has a quantum symmetry Q of order 2 that acts by −1 on the twisted sector and fixes the untwisted sector. In the free fermion description, Q acts by −1 on the h-and gh-twisted sectors and trivially on the g-twisted and untwisted sectors. By  4.2 Symmetries in the su(2) 6 L,1 ⊕ su(2) 6 R,1 RCFT As is manifest from the description of the model as an su(2) 6 L,1 ⊕ su(2) 6 R,1 RCFT in Section 3, the group of symmetries of our K3 model is (SU(2) 6 L × SU(2) 6 R ) : S 6 , see Appendix D.2 for a detailed proof. We are ultimately interested in identifying the finite subgroup of (SU(2) 6 L × SU(2) 6 R ) : S 6 that preserves the N = (4, 4) superconformal algebra (2.28) -(2.31), and that fixes the four R-R ground states which transform as doublets under the left-and right-moving SU(2) R-symmetries. In this section, we describe a number of symmetries in terms of the su(2) 6 L,1 ⊕ su(2) 6 R,1 RCFT. In Section 5 we then prove that these symmetries generate the symmetry group G = Z 8 2 : M 20 . Some elements in the group (SU(2) 6 L × SU(2) 6 R ) : S 6 of symmetries of the su(2) 6 L,1 ⊕ su(2) 6 R,1 RCFT obviously leave the four supercharges invariant. In particular, this is true for the central elements t i , t i , i = 1, . . . , 6, of SU(2) 6 L × SU(2) 6 R which fix the su(2) 6 L,1 ⊕ su(2) 6 R,1 currents and act on their representations by Since (2.35) -(2.38) implies that the holomorphic supercurrents of the N = (4, 4) superconformal algebra transform in the representation [11 11 11; 00 00 00], it follows that the subgroup which fixes the N = (4, 4) superconformal algebra is generated by where we recall from (3.14) and (3.16) that the spectrum of the K3 model contains only representations with a i = b i or with a i = b i +1, so that t i t j and t i t j corresponds to the same symmetry of the K3 model. Therefore, we obtain a group Z 5 2 of symmetries preserving the N = (4, 4) algebra.
In the R-R sector, the elements t 1 t i , 1 < i ≤ 6, act by multiplication with (−1) on the four charged R-R ground states in the representation [10 00 00; 10 00 00] in (3.17), corresponding to the N = (4, 4) supermultiplet ( 1 4 , 1 2 ; 1 4 , 1 2 ). In order to preserve these states, we should compose t 1 t i with the symmetry (−1) R that acts by multiplication with (−1) on the R-R sector and trivially on the NS-NS sector. Note, however, that the symmetry acts trivially on all the states in the spectrum. Therefore, the subgroup of Z 6 2 ×Z 6 2 preserving the N = (4, 4) superconformal algebra and the 'charged' R-R ground states is Z 4 2 , and it is generated by t 2 t j , j = 3, 4, 5, 6 . This group contains the quantum symmetry Q of the Z 2 -orbifold of the D 4 -torus model, which by (4.28) is given by Let us next consider the symmetries that are induced from the D 4 -torus model in the affine algebra description. We focus our attention on those symmetries which in Section 5 are shown to generate the symmetry group G = Z 8 2 : M 20 . As we have seen above, the half-period shifts s v 4 in (4.21) and s u in (4.23) correspond to the permutations of the currents and of the corresponding representations.
The half-period shifts s v 2 +v 4 and s v 1 +v 2 and the rotations γ 1 , γ 2 act as left-right symmetric SU(2) 6 L × SU(2) 6 R transformations. They are determined by their action on the currents (see eqs. (4.25), (4.27), (4.12) -(4.14), (4.15) -(4.17)), up to elements in the centre Z 6 2 × Z 6 2 of SU(2) 6 L × SU(2) 6 R . It is convenient to define the SU(2) matrices which act by conjugation on su(2) 1,L and su(2) 1,R . The unusual notations are motivated by the particular representations of these matrices which we discover in Section 5. Here we only note that with these four matrices, is realised as the group of unit quaternions. We also observe a natural symmetry of order 3 on SU (2), induced by the cyclic permutation of (1, ω, ω). It is the inner automorphism µ(ω) of the quaternion algebra which is given by conjugation with 1 We denote by ρ L,R : SU(2) 6 → SU (2) We observe that the commutator of any two such elements is in the centre Z 6 2 × Z 6 2 of SU(2) 6 L × SU(2) 6 R . More precisely, it is always a product of an even number of symmetries t i t i , which, by the discussion above, acts trivially on all the states in the theory. Thus, the four symmetries in (4.39) effectively generate an abelian subgroup Z 4 2 of the group of symmetries preserving the N = (4, 4) superconformal algebra.
The geometric symmetries above are left-right symmetric elements of SU(2) 6 L ×SU(2) 6 R . However, it is clear that the action of the purely left-moving or purely right-moving SU(2) 6 will also preserve the N = (4, 4) superconformal algebra and the four R-R ground states that are charged under the R-symmetry. Two examples of such purely left-moving transformations are s w 12 and s w 24 . Thus, if we define the embedding into the left SU(2) 6 L factor, we obtain four additional symmetries s w 24 = ρ L (00 ωω ωω) , that form a subgroup of the left-moving SU(2) 6 L , and that have no geometric interpretation. The commutators of these elements are again in the centre Z 6 2 of the left SU(2) 6 L , but, in general, they act non-trivially on the states of the theory. Therefore, the resulting group is non-abelian.

Further symmetries from other fermionisation choices
Apart form the quantum symmetries t 2 t j , j = 3, 4, 5, 6, all symmetries which we have constructed so far have some geometric origin, for example by restricting a geometrically induced symmetry to its left-moving part as in (4.41). In this subsection we obtain additional symmetries by making different fermionisation choices, thus giving rise to other symmetries. More precisely, we regroup the summands of our su(2) 6 L,1 ⊕ su(2) 6 R,1 current algebra into so(4) 1 pieces in a different fashion. To do so, we remark that taking the orbifold of our K3 model by the order two quantum symmetry Q = Q 3456 of (4.33) recovers the original D 4 -torus model. On the other hand, by conjugating Q by elements in (SU(2) 6 L × SU(2) 6 R ) : S 6 , we obtain 15 different symmetries The orbifold by any of these symmetries is a D 4 -torus model equivalent to the one we considered previously. Furthermore, the five symmetries Q ijkl with 1 < i < j < k < l preserve the four supercharges as well as the four R-R ground states in the ( 1 4 , 1 2 ; 1 4 , 1 2 ) supermultiplet. Thus, we have five different descriptions of our K3 model (considered as an N = (4, 4) superconformal model) as a Z 2 -orbifold of a D 4 -torus model. Notice that all these torus models are actually equivalent to one another, in the sense that we can identify the fields of any two of them so that both the OPEs and the N = (4, 4) superconformal algebra are preserved. Nevertheless, these descriptions are different in the sense that the fields of the torus orbifold are associated to different fields of the K3 model -in particular, we have a different splitting between the twisted and untwisted sector of the model. As a consequence, the symmetries of the K3 model, induced by the geometric symmetries of one underlying D 4torus description, might correspond to symmetries mixing twisted and untwisted states in a different description. In this subsection, we show that this is indeed the case.
Analogously to (2.34), we arrange the twelve Majorana fermions into six Dirac fermions χ j as (4.45) Formally, this description is completely analogous to that of the original D 4 -torus model. However, the six Dirac fermions transform in different representations of the su(2) 6 1 current algebra. Correspondingly, the expression of the su(2) 6 L,1 -currents in terms of the Dirac fermions is different, namely (for the Cartan torus) The four supercharges G ± , G ′ ± are invariant under Q ′ = Q 2345 , so they are preserved by the orbifold projection and form an N = (4, 4) superconformal algebra in the torus model. The explicit expression of the supercharge G + in the free fermion description of the new torus model is found to be (see Appendix E.2 for the detailed calculation) so that In particular,  1 (z) generates the rotations in the plane spanned by ψ 9 and 1 2 (− ψ 5 (z) + ψ 6 (z) + ψ 7 (z) + ψ 8 (z)). The half-shift symmetry α, corresponding to a 180-degree rotation in this plane, acts on the Dirac fermions by and  In terms of currents, this corresponds to α : (4.61) In terms of (SU(2) 6 L × SU(2) 6 R ) : S 6 , the half-period shift α that we just found corresponds therefore to the permutation α p,T = (25)(34) (4.62) of the currents and their representations, followed by a left-right symmetric SU(2) 6 L × SU(2) 6 R transformation  In [24] the N = (4, 4) preserving symmetries of K3 sigma models (fixing the states in the ( 1 4 , 1 2 ; 1 4 , 1 2 ) supermultiplet) were classified. It follows from the proof of the main theorem in [24] (see case 1 in Appendix B.3 of [24]) that one of the maximal groups of symmetries is Z 8 2 : M 20 , i.e. there is no bigger symmetry group that contains Z 8 2 : M 20 . In the following we show that the group of N = (4, 4) preserving symmetries of our orbifold model contains Z 8 2 : M 20 ; together with the above result, this establishes that the group of these symmetries is precisely equal to Z 8 2 : M 20 .
Recall that the group Z 12 2 : M 24 is a maximal subgroup of the Conway group Co 0 , which is the group of automorphisms of the Leech lattice, the unique 24-dimensional even selfdual lattice with no vectors of squared length 2 [29, Chapter 10]. Furthermore, Z 12 2 : M 24 has a standard 24-dimensional real representation, where Z 12 2 acts by certain changes of signs of the basis vectors x 1 , . . . , x 24 , and M 24 ⊂ S 24 acts by permutations of these vectors. More precisely, consider Z 24 ⊂ R 24 with x 1 , . . . , x 24 ∈ Z 24 the standard basis of R 24 . Let G 24 ⊂ F 24 2 = (Z/2Z) 24 denote the extended binary Golay code 7 G 24 , a 12-dimensional subspace of F 24 2 whose elements have 0, 8, 12, 16 or 24 non-zero coordinate entries. Then g ∈ G 24 ∼ = Z 12 2 acts by flipping the signs of those x k for which g k = 0. The Mathieu group M 24 , by definition, is the subgroup of S 24 that preserves G 24 ⊂ F 24 2 . In this section, we show that the group of symmetries of the K3 model described in Sections 2 and 3 is the subgroup of Z 12 2 : M 24 that fixes four basis vectors (say, x 1 , x 2 , x 3 , x 4 ) in the standard representation of that group. The choice of four arbitrary distinct vectors (a tetrad ) determines a decomposition of the basis into the disjoint union of six tetrads where T 1 = {x 1 , x 2 , x 3 , x 4 }, such that the union of any two distinct tetrads T i ⊔ T j , 1 ≤ i < j ≤ 6, corresponds to an element of length 8 (octad ) in the Golay code. The subgroup of Z 12 2 : M 24 that preserves the tetrad T 1 := {x 1 , x 2 , x 3 , x 4 } pointwise is Z 8 2 : M 20 . Here Z 8 2 is the subgroup of Z 12 2 ∼ = G 24 whose elements have empty intersection with T 1 , and M 20 ∼ = Z 4 2 : A 5 is the semidirect product of a group Z 4 2 ⊂ M 24 that fixes T 1 pointwise and all six tetrads setwise, and the group A 5 of even permutations of the tetrads T 2 , . . . , T 6 .
Let G be the group of symmetries of our K3 model which is generated by (i) the rotations γ 1 , γ 2 of Subsection 4.1.1; (ii) the half-period shifts s v 1 +v 2 , s v 2 +v 4 , s v 4 and s u of Subsection 4.1.2; (iii) the central symmetries t i t j , 1 < i < j ≤ 6, and the asymmetric symmetries s w 24 , s w 12 , γ L 1 and γ L 2 of Subsection 4.2; and finally (iv) the new symmetry α of Subsection 4.3. We will show that the representation of G on the 24-dimensional space of R-R ground states is exactly the standard representation of Z 8 2 : M 20 . In particular, this establishes that Z 8 2 : M 20 ⊆ G, and hence by the argument above, that G is actually the full N = (4, 4) preserving symmetry group of our orbifold model.
The space of R-R ground states is naturally split into six four-dimensional subspaces, where the i-th subspace transforms as a (2, 2)-representation under the i-th left-right 7 Hereafter, we use the less precise term 'Golay code' to designate G 24 as there is no ambiguity. SU(2) L × SU(2) R factor, and trivially under the other factors of SU(2) 6 L × SU(2) 6 R . In each subspace we choose a basis of states as It is useful to arrange this basis of ground states into an array of six columns and four rows, where each column represents a tetrad and the states in each tetrad are ordered as |1 , |2 , |3 , |4 from top downwards: This array corresponds to the Miracle Octad Generator (MOG) arrangement of the Golay code (see [29,Chapter 11]).
Our proof that G ∼ = Z 8 2 : M 20 consists of three steps. First we show, using the information gathered in Subsection 4.2, that the group G ′ generated by the symmetries t i t j in the centre of SU(2) 6 L × SU(2) 6 R , by the half-period shifts s v 1 +v 2 , s v 2 +v 4 , and by the rotations γ 1 and γ 2 , is isomorphic to the subgroup Z 8 2 of the Golay code acting by sign changes in the standard representation of Z 8 2 : M 20 . Then we adjoin to G ′ the purely left-moving symmetries s w 12 , s w 24 , γ L 1 and γ L 2 ; we obtain a group G ′′ that is identified with the subgroup Z 8 2 : Z 4 2 of Z 8 2 : M 20 that fixes each tetrad setwise. Finally, we show that, by adjoining to G ′′ the half-period shifts s v 4 , s u and the symmetry α, we obtain the group G ∼ = Z 8 2 : M 20 .

5.1
The subgroup G ′ ∼ = Z 8 2 of Z 8 2 : M 20 Let us consider the subgroup Z 4 2 of the centre Z 6 2 × Z 6 2 of SU(2) 6 L × SU(2) 6 R generated by the ten elements where t i acts as in (4.29). These symmetries change the sign of all the states in the tetrads T i and T j , while leaving the other states fixed. We represent the elements of this group pictorially as Next we consider the group Z 4 2 of symmetries that is generated by the geometric symmetries s v 2 +v 4 , s v 1 +v 2 , γ 1 and γ 2 as given in (4.39); this is a subgroup of left-right symmetric elements of SU(2) 6 L × SU(2) 6 R . Note that the matrices 0, 1, ω,ω that were introduced in (4.35) act on each tetrad by 8 ρ L,R (0), ρ L,R (1), ρ L,R (ω), ρ L,R (ω) ∈ SU(2) L × SU(2) R . We write F 4 := {0, 1, ω,ω}, and for x, y ∈ F 4 we define x + y ∈ F 4 by One checks that the resulting rules of addition 1+ω = ω+1 =ω, 1+ω =ω+1 = ω, ω+ω =ω+ω = 1, 0+x = x+0 = x, x+x = 0 (5.5) agree with those of the finite field F 4 with four elements. 9 Observe that the inner automorphism µ(ω) in (4.37) of the underlying quaternion algebra corresponds to multiplication by ω on F 4 , and thus it equips F 4 with the multiplication law of the field with four elements. Then the elements of the group Z 4 2 that is generated by s v 2 +v 4 , s v 1 +v 2 , γ 1 and γ 2 are given in terms of vectors in F 6 4 , where the group law for the abelian group is given by component-wise sum of the six digits. For example, s v 2 +v 4 • γ 1 = ρ L,R (00 ωω ωω + 0ωω0 ω1) = ρ L,R (0ω 1ω 0ω) .

-25 -
The basis (5.2) consists of simultaneous eigenvectors for the (left-right symmetric) matrices ρ L,R (0), ρ L,R (1), ρ L,R (ω), ρ L,R (ω) ∈ SU(2) L × SU(2) R , with the following eigenvalues Therefore, each element in the group Z 4 2 acts by sign flips, and it is easy to construct the precise eigenvalues using the rules (5.9). For example, by (4.39) (5.10) We are now ready to make the connection with the Golay code. A standard construction of the Golay code makes use of the hexacode, which is a particular 3-dimensional subspace of F 6 4 given by all words ab cd ef ∈ F 6 4 which obey (5.8) [29,Chapter 11]. Hence the 15 elements of F 6 4 of the form (5.7), together with 00 00 00, are exactly all elements (words) in the hexacode having 0 as first digit. From each word in the hexacode, one can construct various elements of the Golay code, first by using the replacement rules (5.9), and then flipping the signs of any even number of columns. 10 We conclude that the group G ′ ∼ = Z 8 2 generated by t i t j , 1 < i < j ≤ 6, together with s v 2 +v 4 , s v 1 +v 2 , γ 1 and γ 2 , is exactly the subgroup of the Golay code with empty intersection with the first tetrad. In other words, G ′ can be identified with the normal subgroup Z 8 2 in Z 8 2 : M 20 , G ′ ∼ = Z 8 2 := t i t j , 1 < i < j ≤ 6, s v 2 +v 4 , s v 1 +v 2 , γ 1 , γ 2 ⊂ Z 8 2 : M 20 . (5.11)

5.2
The group Z 8 2 : Z 4 2 fixing the tetrads setwise In this subsection, we enlarge the group G ′ ∼ = Z 8 2 described in the previous subsection by adjoining the symmetries s w 24 , s w 12 , γ L 1 and γ L 2 , defined in (4.41). We will show that the resulting group G ′′ can be identified with the subgroup Z 10 To be precise, this way one obtains only half of the Golay code, namely those words of 'even parity' in MOG terminology, but this half contains all the elements with empty intersection with the first tetrad. Notice that ρ s L is different from ρ L,R defined in (5.9): the two substitutions are related by a cyclic permutation of the symbols (1, ω,ω). According to (4.41) and these rules, the generators s w 24 , s w 12 , γ L 1 and γ L 2 can be represented as Notice that by our analysis of Subsection 5.1 the sign flips ρ s L (00 ωω ωω) = ρ L,R (00ωωωω) , ρ s L (00ωωωω) = ρ L,R (00 11 11) , ρ s L (0ωω0 ω1) = ρ L,R (01 10ωω) , ρ s L (0ωω1 0ω) = ρ L,R (0ω 1ω 0ω) (5.14) are elements of G ′ ∼ = Z 8 2 . Therefore, the group generated by G ′ and by the symmetries s w 24 , s w 12 , γ L 1 and γ L 2 can be equivalently obtained by adjoining to G ′ the pure permutations These symmetries have order 2 and commute with each other, so that they form an abelian group Z 4 2 of permutations of the 24 R-R ground states that preserve each tetrad setwise. Each non-trivial element of this group is associated with codewords from the hexacode of the form (5.7) through the rules (5.12). By the results of Subsection 5.1, eq. (5.7) lists all non-zero codewords of the hexacode whose first entry is zero. Hence by [29,Ch. 11,Sect. 9], the group generated by s p w 24 , s p w 12 , γ L,p 1 and γ L,p 2 is exactly the subgroup of M 24 that preserves the tetrads setwise and fixes the first tetrad pointwise. Therefore, the group generated by G ′ together with s w 24 , s w 12 , γ L 1 and γ L 2 can be identified with the group Z while α acts by a permutation followed by a sign flip, By composing α with the element ρ L,R (01ωω 10) ∈ G ′ ∼ = Z 8 2 , one obtains a pure permutation α p = .
Using the explicit description of the involutions of M 24 that fix the first tetrad T 1 (see [29, Ch. 11, Sect. 9]), it is clear that the permutations s u , s v 4 and α p are elements of M 20 ⊂ M 24 . 16) and every word in the generators s u , s v 4 and α p which does not permute the six factors of SU (2) is an element of G ′ . One also verifies that for every g ∈ G ′′ ∼ = Z 8 2 : Z 4 2 , the conjugates s v 4 gs −1 v 4 , s u gs −1 u and α p g(α p ) −1 belong to G ′′ . Therefore, G ′′ is a normal subgroup of G and 6 A special symmetry g of order four in G = Z 8 2 : M 20 In [25] it was observed that for those K3 sigma models that are abelian torus orbifolds, the corresponding quantum symmetry (whose orbifold leads back to the torus model) is never an element of M 24 . This result was obtained by studying the 42 Co 0 conjugacy classes that define possible symmetries of K3 sigma models. Of these 42 conjugacy classes, 31 certainly have a trivial multiplier as the trace over the 24-dimensional representation is non-zero; then it is consistent to orbifold by the cyclic group that is generated by the relevant symmetry, and one can analyse (by calculating the elliptic genus) whether the resulting orbifold is a K3 or a toroidal sigma model. It was found that the symmetries which lead to a toroidal model do not have a representative in M 24 , see section 4 of [25]. For the remaining 11 Co 0 conjugacy classes it was on the other hand not obvious whether the corresponding symmetry obeys the level-matching condition, i.e. whether it is consistent to orbifold by it. (The level-matching condition is equivalent to the statement that a certain multiplier phase of the twining genus is trivial.) In all but one case, the elliptic genus of the putative orbifold did not make sense (i.e. did not agree with either that of K3 or the four-torus), thus suggesting that the orbifold was in fact inconsistent. However, there was one class, the 4D conjugacy class of Co 0 , for which the putative orbifold gave rise to the elliptic genus of the four-torus, thus indicating that the orbifold may in fact be consistent. As we shall see below, this suspicion is indeed correct, as the 4D generator of Co 0 can be identified with an order 4 symmetry of our K3 sigma model.
In addition, this orbifold turns out to induce an equivalence between different descriptions of the D 4 -torus model underlying our K3 sigma model, which could be relevant in our quest for a field theoretic explanation of Mathieu moonshine. Indeed, we have already remarked in Section 4.3 that there are at least fifteen different ways in which one may write our model as a Z 2 -orbifold of the D 4 -torus model. As will be substantiated further down, it turns out that orbifolding by the symmetry of order four that we identify with the generator of the 4D conjugacy class of Co 0 yields a 'new' D new 4 -torus model. The latter is equivalent to the original D 4 -torus model as an N = (4, 4) superconformal field theory.

The g -orbifold of the K3 model
Let us consider the symmetry g of our K3 sigma model defined by This symmetry has order 4, its trace over the 24-dimensional representation of R-R ground states is 0, and its square is the quantum symmetry Q = t 3 t 4 t 5 t 6 , whose trace over the 24-dimensional representation is −8. These properties identify g as an element in the conjugacy class 4D in the Conway group Co 0 , as discussed in Section 4 of [25]. In the following we want to show that the orbifold by this group element is indeed consistent and leads to a toroidal superconformal field theory.
As is explained in Appendix D.3, the elliptic genus of our model can be written in terms of su(2) 1 characters as in (D.31). Thus we can calculate the twining genus, i.e. the elliptic genus with the insertion of the group element (6.1), by inserting the various operators into the su(2) 1 traces (D.9) and (D.10). With the help of the identities Tr [1] ( we obtain the twining genera φ e,g (τ, z) = φ e,g 3 (τ, z)

By the Riemann bilinear identities
this can be rewritten as is the standard weak Jacobi form of weight −2 and index 1. Using the modular properties of the theta functions, it is easy to see that φ e,g is a Jacobi form for Γ 0 (4) with trivial multiplier. Therefore, the orbifold of the K3 model by g is expected to be consistent, since the level matching condition for the twisted sector is satisfied. In fact, φ e,g = φ e,g 3 equals the M 24 -twining genus φ 2B of class 2B, so that φ g,e = φ g 3 ,e = −φ g,g 2 = −φ g 3 ,g 2 , (6.7) φ e,g = φ e,g 3 = −φ g 2 ,g = −φ g 2 ,g 3 , (6.8) Furthermore, since g 2 is the quantum symmetry Q and the orbifold by Q is a torus model, one has φ e,e + φ e,g 2 + φ g 2 ,e + φ g 2 ,g 2 = 0 . (6.10) It follows that the orbifold of the K3 model by g has vanishing elliptic genus so that it defines a torus model, as predicted in [25].

The D new 4 -torus model and the interpretation of the orbifold action
The preceding arguments show that the g -orbifold of our K3 model, denoted K3/ g , is a torus model. Since we have the full K3 model under control, we can work out not just the elliptic genus of the orbifold by g , but the full partition function. This then allows us to determine the actual torus model. As we shall see it is again a D 4 -torus model, which we call the D new 4 -torus model, although it is equivalent to the original one as an N = (4, 4) superconformal field theory.
The calculation of the partition function is somewhat technical, and we only sketch some of the relevant steps in Appendix F.2. The final answer, eq. (F.14), is however rather simple, and it agrees precisely with the partition function of the D 4 -torus model. Since this is the only torus model with this partition function, as one confirms by observing that the underlying bosonic torus model is the only torus model at central charge (c, c) = (4, 4) which possesses a current algebra of dimension 28, it follows that the model agrees with the D 4 -torus model.
We can therefore write schematically 12) and note that, since g 2 = Q, it is possible to construct the g -orbifold of the K3 model in two steps. The first one yields K3/ g 2 = D 4 , (6.13) while the second step involves taking the orbifold of the D 4 -torus model by the order two symmetryḡ induced by g on that model; thus we have (6.14) These steps can be clearly identified at the level of the partition function, as shown in Appendix F.2.
Since the group g is abelian, we can reverse the orbifold K3/ g = D new 4 , and hence write our K3 sigma model as a Z 4 -orbifold of the D new 4 -torus model. Denoting byg the generator of that orbifold, we thus have It is natural to seek an interpretation ofg, i.e. of the generator of the 'quantum symmetry' associated to g as a symmetry of the D new 4 -torus model. As before, we may perform also the g -orbifold in two steps. The g 2 -orbifold of the D new 4 -torus model yields our original D 4 -torus model, i.e.
andg induces on it the usual Z 2 -orbifold action, i.e. the one described in Section 2. In fact,g 2 turns out to agree with the T-duality generator of the D new 4 -torus model, i.e. with the operator that inverts the signs of all left-moving oscillators, with a trivial action on the right-movers. This is known to be a symmetry of a D 4 -torus model with extended so(8) 1 -symmetry, as follows from the analysis of [30].
Thus we have shown that our K3 sigma model can also be written as an asymmetric Z 4 torus orbifold, and that the corresponding quantum symmetry is the one associated to the 4D conjugacy class of Co 0 .

Conclusions
In this paper we have considered a superconformal field theory that describes a K3 sigma model with one of the largest maximal symmetry groups, namely Z 8 2 : M 20 . In particular, we have found different descriptions for this model: as a Z 2 -orbifold of the D 4 -torus model, as a theory of 12 left-and right-moving Majorana fermions, and as a rational conformal field theory based on the chiral algebra su(2) ⊕6 1 . By combining these different viewpoints various properties of this model have become manifest. This may prove useful in order to understand the origin of the M 24 symmetry in the elliptic genus of K3.
A result of our work is a very explicit description of all symmetries of the sigma model on the tetrahedral Kummer surface. In [23] two of us have highlighted a 45-dimensional vector space of states V CFT 45 that are generic to all standard Z 2 -orbifold CFTs on K3 and which govern the massive leading order of the elliptic genus of K3. On V CFT 45 , the combined geometric symmetries of all such theories generate an action of the maximal subgroup Z 4 2 : A 8 of M 24 , as is shown in [23]. Using the description of symmetries for the special model studied in the present paper, it will be possible to investigate the action on V CFT 45 for symmetries that are 'non-geometric' from the viewpoint of the tetrahedral Kummer surface.
Recall from [26] that our K3 sigma model also possesses a Gepner-type description as Z 2 × Z 2 -orbifold of the well-known model (2) 4 , and that this implies invariance under Greene-Plesser mirror symmetry of our model [31]. This symmetry is not contained in the group Z 8 2 : M 20 investigated in the present paper, as mirror symmetry is an automorphism of the N = (4, 4) superconformal algebra, but it does not preserve it pointwise. It might be interesting to determine the action of mirror symmetry in this model.
There are also other special points in the moduli space of K3 sigma models that would be interesting to construct. For example, there should be a K3 sigma model with M 21 symmetry group, and it would be very interesting to find an explicit description for it. However, it is clear that it cannot be a standard torus orbifold.
The Z 8 2 : M 20 theory we have considered should possess an interesting exactly marginal deformation that breaks the symmetry group to M 20 . One should expect that, at least generically, this deformation will break the large chiral symmetry of our K3 sigma model to the N = (4, 4) superconformal algebra. Thus the resulting deformed models should possess an M 20 symmetry while at the same time exhibiting only the 'minimal' number of BPS states. 11 These deformed models may therefore play an important role in understanding the algebraic reasons underlying Mathieu Moonshine. 11 Note that the appearance of fermionic BPS states -that contribute with the 'wrong' sign to the elliptic genus -is always associated to an extension of the chiral algebra, because under spectral flow any 'fermionic' BPS state transforming in the (h = 1 4 , l = 1 2 ) representation of the right-movers is mapped to the right-moving NS vacuum, see [32,33] for details.

A Conventions and notations for torus models
In this appendix we fix our conventions and notations for supersymmetric torus models. A real d-dimensional torus may be described as T = R d /L, where L ⊂ R d is a lattice of maximal rank. We denote by L * ⊂ R d its dual lattice, using the standard Euclidean metric to define inner products and to identify R d with (R d ) * . We shall usually use standard Cartesian coordinates, with e 1 , . . . , e d ∈ R d the standard basis of R d . In order to describe a torus theory one must also fix a Kalb-Ramond B-field in terms of a skew-symmetric d × d-matrix B with real entries. The field content of the bosonic torus model is then generated by • d real left-moving U(1)-currents j k (z) = i∂φ k (z), k = 1, . . . , d, which obey the OPEs (A.1) The notation for the right-moving currents is analogous, with 12  k (z) = i∂ φ k (z). The mode expansions of the left-moving currents are • Winding-momentum fields associated with vectors (m, l) ∈ L * ⊕L. In order to define them, we set We also introduce operators c λ for each λ ∈ Γ which obey with a suitable 2-cocycle ǫ(λ, µ) ∈ {±1}. In other words, 13 ∀ λ, µ, ν ∈ Γ : ǫ(λ, µ) = (−1) λ 2 µ 2 (−1) λ·µ ǫ(µ, λ) , ǫ(λ, µ)ǫ(λ + µ, ν) = ǫ(λ, µ + ν)ǫ(µ, ν) .
Then for (Q; Q) ∈ Γ, the fields obey the OPEs where ∼ only indicates the most singular terms, and V (Q;Q) (z, z) has conformal di- z) is the vacuum field. The toroidal model is uniquely determined by its charge lattice Γ ⊂ R d,d by means of (A.1), (A.4) -(A.8), independently of its geometric interpretation on the torus T = R d /L with B-field B. Different choices for the cocycle ǫ satisfying (A.7) are related by a redefinition of the fields V λ , see, for example, [34] for details.
In the corresponding supersymmetric torus model, in addition to the fields listed above, we adjoin • d 'external' free Majorana fermions ψ k (z), k = 1, . . . , d, which are related to the U(1)-currents by world-sheet supersymmetry and which obey the OPEs . (A.10) The right-moving Majorana fermions are denoted by ψ k (z). The mode expansions of the left-moving fermions are Then the partition function of the supersymmetric torus model is Here, the trace is taken over the full Hilbert space of the theory, F L +F R is the total fermion number operator accounting for the external fermions, J 0 , J 0 denote the zero modes of the U(1)-currents of the left-and the right-moving N = 2 superconformal algebras as in (A.12), and the central charges are c = c = 3d 2 . Moreover, η(τ ) is the Dedekind eta function and ϑ k (τ, z), k = 1, . . . , 4, are the Jacobi theta functions described in Appendix F. The first factor in the partition function (A.13) accounts for the external fermions, while the second factor accounts for the contributions from the bosonic torus model. If Γ = Γ d,d as in (2.10), its partition function can be written in terms of theta functions as thus suggesting a free fermionic description of this bosonic torus model.

B Fermionisation of the bosonic D 4 -torus model
In this appendix we provide some details on the fermionisation procedure needed in Subsection 2.
where ζ ∈ C is a primitive eighth root of unity with ζ 2 = i, (see e.g. [28, (12.67), (12.68)]). In addition, In (2.20), we introduce the four Dirac fermions where according to (2.21), obey (A.1), such that j k (z) = i∂φ k (z) (and analogously on the right hand side) allows us to identify, up to appropriate cocycle factors, Here, we have formally introduced the 'meromorphic factors' ξ ± 1 (z), . . . , ξ ± 4 (z) with ξ ± k (z) := : exp ± i 2 φ k (z) :. By democratically distributing phases between the holomorphic and antiholomorphic part, such that for the c = c = 1 theory of the free Dirac fermion x k (z), the two R-R-ground states are created by ξ ± k (z)ξ ± k (z). In Subsection 2.1 we describe the bosonic d = 4-dimensional D 4 -torus model with charge lattice (2.10). Its left-moving so(8) 1 current algebra is generated by the U(1)currents j 1 (z), . . . , j 4 (z) together with the twenty-four (1, 0)-fields V (Q ±j,±k ;0) (z) specified by (2.5). All winding-momentum fields can be generated from the ( 1 2 , 1 2 )-fields V (Q;Q) (z, z) listed in (2.9) by taking OPEs with holomorphic currents. Using (B.6) we can thus give a complete list of generating fields in terms of the free fermion data. One checks that indeed the following identifications are compatible with the respective OPEs: • four (1, 0)-fields generating the Cartan subalgebra of so(8) 1 : • twenty-four (1, 0)-fields corresponding to the roots of D 4 for 1 ≤ j < k ≤ 4, (B.9) • for the ( 1 2 , 1 2 )-fields V (Q;Q) (z, z), with d = 4, as one confirms by means of the product formulas for the Jacobi theta functions given in (F.2). The superpartners ψ k (z), k = 1, . . . , 4, of the four left-moving bosonic currents j k (z) (together with their analogs in the right-moving sector) are uncorrelated with H D 4 −torus , i.e. they contribute a tensor factor H ferm to the space of states H ferm ⊗ H D 4 −torus of our supersymmetric torus theory. As is explained in Appendix A, the U(1)-currents (A.12) of the left-and the right-moving N = 2 superconformal algebras in this model are obtained from H ferm , such that the decomposition of H ferm ⊗ H D 4 −torus into NS-NS and R-R sectors in the usual sense is governed by H ferm . So, for example, the R-R ground states of the full D 4 -torus model come from the sector where the ψ k , k = 1, . . . , 4, (and their rightmoving counterparts) have Ramond boundary conditions, while ψ k , k = 5, . . . , 12, (and the corresponding right-movers) have Neveu-Schwarz boundary conditions. Since there are four fermionic zero modes ψ k,0 , k = 1, . . . , 4, there are four left-and four right-moving ground states in this sector, which account for the sixteen R-R ground states of the supersymmetric torus theory. These states have conformal weight h = h = 1 4 . The Z 2 -orbifolding described in Subsection 2.1 acts as ψ k → −ψ k for k = 1, . . . , 4, while its action on the fermions ψ k , k = 5, . . . , 12, as established in Subsection 2.2, leaves ψ k , k = 5, . . . , 8, invariant and flips the sign of ψ k , k = 9, . . . , 12. The action on the rightmoving fermions is identical. Thus in the orbifold model, the twelve free (left-moving) 14  The sixteen twisted R-R ground states appear in the sectors (NS, NS, R) and (NS, R, NS). The full partition function therefore equals The R-R-sector with the inclusion of the total fermion number operator that only acts on the fermions from H ferm contributes to this by By definition, the elliptic genus of any N = (2, 2) SCFT is It can thus be obtained from Z R (τ, z) by inserting y = 1 and leaving y untouched. Hence for our Z 2 -orbifold model, which agrees with the elliptic genus of K3, thus confirming that our orbifold model is indeed a K3 theory.

D
Properties of su(2) n L,1 ⊕ su(2) n R,1 RCFTs In this appendix, we collect some properties of the special toroidal models at central charge c = c = n with n ∈ N which enjoy an extended su(2) n L,1 ⊕ su(2) n R,1 symmetry.
D.1 Vertex operator construction for su(2) n L,1 ⊕ su(2) n R,1 For n ∈ N, the su(2) n L,1 ⊕ su(2) n R,1 affine algebra can be realised as a model of n free bosons Y k (z,z) := Y k (z) + Y k (z), k = 1, . . . , n, compactified on an n-dimensional real torus. The Cartan generators are chosen as in accord with (2.17). Now consider any CFT with central charges c = c = n that possesses an su(2) n L,1 ⊕ su(2) n R,1 current algebra. As explained in Appendix A, the remaining field content of the model is generated by winding-momentum fields V (Q;Q) (z, z) as in (A.8) with charge vectors (Q; Q) in the charge lattice Γ ⊂ R n,n with quadratic form (A.5), where n = d. The simple structure of the representations of su(2) 1 as discussed in Subsection 3.1 allows us to elucidate further the form of the charge lattice Γ for the models with su(2) n L,1 ⊕ su(2) n R,1 current algebra. For the vacuum representation [0 · · · 0; 0 · · · 0] of su(2) n L,1 ⊕ su(2) n R,1 , the charge vectors take values in the lattice More generally, the primary field for a representation [a 1 · · · a n ; b 1 · · · b n ] of su(2) n L,1 ⊕ su(2) n R,1 corresponds to V λ (z,z) with λ = 1 √ 2 (a 1 , . . . , a n ; b 1 , .
and all the states in this representation have momenta in the translated lattice λ + Γ 0 . Consistency of our model requires that the charge lattice Γ is an even integral selfdual lattice with In order to include fermionic states in this description, one drops the condition that the lattice Γ is even and includes vectors λ with odd λ 2 . In this case, the Z 2 -graded locality condition for the vertex operators is satisfied by requiring (A.7), where the factor (−1) λ 2 µ 2 is trivial if Γ is even, i.e. if the model is purely bosonic.
Summarising, the spectrum of the theory is completely determined by specifying the charge lattice Γ or, equivalently, the abelian group which, in the notation of Subsection 3.1, simply yields the subgroup A ⊂ Z n 2 ×Z n 2 describing the representation content of the model with respect to the su(2) n L,1 ⊕ su(2) n R,1 algebra.
D.2 S 6 as a symmetry group of the su(2) 6 L,1 ⊕ su(2) 6 R,1 RCFT In Subsection 3.2, we consider the spectrum of our K3 model in terms of representations of the su(2) 6 L,1 ⊕ su(2) 6 R,1 affine algebra. According to (3.13) and (3.15), the spectrum is symmetric under a group S 6 that permutes simultaneously the various su(2) 1 factors in the left and right sectors, and this symmetry preserves the fusion rules. In this appendix, we prove that these transformations are compatible with the OPEs of the primary fields of the current algebra, and therefore define genuine symmetries of the CFT.
D.3 Elliptic genus for the su(2) 6 L,1 ⊕ su(2) 6 R,1 RCFT In this appendix, we derive the elliptic genus of our model, which we have already determined in (C.8), but this time using the su(2) 1 description.
To do so, we need to determine those R-R states that are BPS with respect to the right N = 4 superconformal algebra. These states are all contained in the representations (3.17) - (3.19) This formula exactly reproduces the elliptic genus of K3, as expected. Here, the sign (−1) F L +F R is positive for the representations (3.17) - (3.19), and negative for (D.28) -(D.30). The overall factor 2 takes into account the fact that in the spectrum of the model, each of these representations of su(2) 6 L,1 is tensored with two distinct right-moving states of conformal weight 1 4 that form a doublet under the diagonal right-moving SU(2)-symmetry.
Notice that only the twisted sectors of our Z 2 × Z 2 -orbifold contribute to the elliptic genus, since the untwisted R-R sector contains no BPS states (the right conformal weight of these states is at least 3 4 ). This is not in contradiction with the description of the theory in terms of a non-linear sigma model on the resolution of T D 4 /Z 2 , because the untwisted sector in the Z 2 -orbifold of the D 4 -torus model corresponds to the untwisted sector together with the g-twisted sector of the Z 2 × Z 2 -orbifold of the free fermion theory describing the bosonic D 6 -torus model. In Section 3 we construct our K3 model as a Z 2 × Z 2 -orbifold of a free fermion model with so(12) L,1 ⊕ so(12) R,1 symmetry, i.e. as a Z 2 × Z 2 -orbifold of a bosonic D 6 -torus model. This model can be described in terms of the vertex operator construction for the su(2) 6 L,1 ⊕ su(2) 6 R,1 RCFTs in Appendix D.1 by means of a charge lattice Γ ⊂ Γ * 0 , see (D.4). A basis for Γ is given by and similarly for the right-moving γ 1 , . . . , γ 6 . The associated vertex operators correspond to the holomorphic free fermions for i = 1, . . . , 6, and analogously the right-moving fermions correspond to V ±γ 1 , . . . , V ±γ 6 . The lattice Γ is the orthogonal sum Γ L ⊕ Γ R of a purely 'left-moving' (that is, Q = 0) and a purely 'right-moving' (Q = 0) lattice. From now on, we will focus on the left-moving component Γ L .

(E.6)
This definition amounts to a choice of phases whose compatibility with our previous choices is ensured below in (E.8) -(E.19) by implementing appropriate phase factors c(λ). The cocycle ǫ determining the OPE of these fields is defined by ǫ(γ i , γ j ) := +1 for i ≤ j , −1 for i > j , (E.7) together with linearity conditions analogous to (D.23).