Non-geometric Calabi-Yau Backgrounds and K3 automorphisms

We consider compactifications of type IIA superstring theory on mirror-folds obtained as K3 fibrations over two-tori with non-geometric monodromies involving mirror symmetries. At special points in the moduli space these are asymmetric Gepner models. The compactifications are constructed from non-geometric automorphisms that arise from the diagonal action of an automorphism of the K3 surface and of an automorphism of the mirror surface. We identify the corresponding gaugings of N=4 supergravity in four dimensions, and show that the minima of the potential describe the same four-dimensional low-energy physics as the worldsheet formulation in terms of asymmetric Gepner models. In this way, we obtain a class of Minkowski vacua of type II string theory which preserve N=2 supersymmetry. The massless sector consists of N=2 supergravity coupled to 3 vector multiplets, giving the STU model. In some cases there are additional massless hypermultiplets.


Introduction
Geometric compactifications constitute only a subset of string backgrounds and have interesting generalisations to non-geometric backgrounds. Examples arise from spaces with local fibrations that have transition functions that include stringy duality symmetries. Spaces with torus fibrations and T-duality or U-duality transition functions are T-folds or U-folds [1], while those with Calabi-Yau fibrations and mirror symmetry transition functions are mirror-folds [1]. Such non-geometric spaces often have fewer moduli than their geometric counterparts, and the non-geometry typically breaks some of the symmetries, including supersymmetries, and provide an interesting tool for probing quantum geometry. Solvable worldsheet conformal field theories (CFT's) such as asymmetric orbifolds can arise at special points in the moduli space of a non-geometric background [2], allowing a complete analysis and important checks on general arguments.
Our focus here will be on mirror-folds of the type IIA superstring constructed from K3 bundles over T 2 with transition functions involving the mirror involution of the K3 surface. Previously Kawai et Sugawara have considered in [3] K3 mirror-folds with monodromies that, at least when the fiber is compact, break all supersymmetry; in the present work, we consider in contrast monodromies that preserve 8 supersymmetries, i.e. which preserve a quarter of the 32 supersymmetries of the type IIA string, or a half of the 16 supersymmetries of type IIA compactified on K3. As a result, we find interesting mirror-folds which give D = 4, N = 2 Minkowski vacua of type IIA superstring theory. As we shall see, particular examples give precisely the STU model of [4] at low energies.
Our constructions can be viewed as particular cases of reductions with a duality twist [2]. In such a construction, a theory in D dimensions with discrete duality symmetry G(Z) (e.g. T-duality or U-duality) is compactified on a d-torus with a G(Z) monodromy around each of the d circles, giving a string-theory generalization of Scherk-Schwarz reduction [5]. In many cases, the theory in D dimensions has an action of the continuous group G which is a symmetry of the low energy physics, but which is broken to a discrete subgroup in the full string theory. For a field φ transforming under G as φ → gφ the ansatz is of the form φ(x µ , y i ) = g(y)φ(x) (1.1) where y i , i = 1, . . . , d, are coordinates on T d and x µ , µ = 0, . . . , D − d − 1, are the remaining coordinates. With periodicities y i ∼ y i + 2πR i , the mondromies g(y i ) −1 g(y i + 2πR i ) must be in G(Z) for each i.
Of particular interest are the special cases in which there are points in the moduli space in D dimensions that happen to be fixed under the action of the monodromies. For example, consider a theory where the moduli space in D dimensions contains the moduli space for a 2-torus, i.e. SL(2, R)/U(1), identified under the action of the discrete group SL(2, Z). There are special points in the moduli space which are invariant under finite subgroups of SL(2, Z) isomorphic to Z r for r = 2, 3, 4, 6. If one considers reductions on a circle where the monodromy is in one of these Z r subgroups, then at any point in the moduli space invariant under the action of the monodromy, the reduction with a duality twist can be viewed as a Z r orbifold by a Z r twist together with a shift around the circle by 2πR/r [2]. From the effective field theory point of view, the moduli give scalar fields in D − 1 dimensions and the reduction gives a potential for these fields. At each fixed point in moduli space (i.e. at each point that is preserved by the monodromy) the potential has a minimum at which it vanishes, giving a Minkowski compactification [2]. In these cases, at the special points in moduli space there is both a stringy orbifold construction and a supergravity construction, giving complementary pictures of the same reduction. This extends to a reduction on T d to D − d dimensions with a duality twist on each of the d circles. The question of finding reductions giving Minkowski space in D − d dimensions becomes related to that of finding fixed points in moduli space preserved by a subgroup of G(Z).
Our starting point will be the theory in six dimensions obtained from compactifying IIA string theory on a K3 surface. The moduli space of metrics and B-fields on K3 gives the moduli space of K3 CFT's and is given by [6,7] O (4,20) O(4) × O (20) , is the perturbative duality group of twodimensional conformal field theories with K3 target spaces, which includes mirror symmetries. We then reduce to four dimensions on T 2 with an O(Γ 4,20 ) twist around each circle. Generically, such reductions break all supersymmetry; we will focus here on a class of monodromies admitting N = 2 vacua in four dimensions. From the string theory point of view, we will find them by considering points in the moduli space of sigma-models with K3 target spaces that are preserved by finite subgroups of O(Γ 4,20 ), and taking orbifolds by such automorphisms combined with shifts on the circles. The particular automorphisms of K3 CFTs that we will use are inspired by a worldsheet construction of asymmetric Gepner models presented in [8,9], following earlier works [10,11] (see [12,13] for later generalizations). They preserve only space-time supercharges from the worldsheet left-movers, and the asymmetry means that they are non-geometric in general.
From the supergravity point of view, the conventional reduction on T 2 without twists gives N = 4 supergravity coupled to twenty-two N = 4 abelian vector multiplets. The twisted reduction gives a gauged version of this supergravity with a non-abelian gauge group and a scalar potential. Vacua arise from minima of the potential, and of particular interest are theories with non-negative potentials and minima that give zero vacuum energy and a Minkowski vacuum. Our construction gives string theory compactifications whose supergravity limits are of this type, and moreover preserve N = 2 supersymmetry.
The relation between the asymmetric Gepner models that underlie these compactifications and gauged supergravities was suggested by one of the authors in [8] and explored by Blumenhagen et al. in [12] for a related but distinct class of models. In that work they considered quotients of Calabi-Yau compactifications by non-geometric automorphisms, rather than the freely-acting quotients involving torus shifts giving rise to fibrations over tori that we consider here. In the freely-acting quotient, the scale of (spontaneous) space-time supersymmetry breaking can be made arbitrarily small rather than being tied to the string scale as it would be for quotients with fixed points. This means that for our constructions, the gauged supergravity approach gives a good description of the low-energy physics. We will identify the gauging directly from geometric considerations (rather than from identifying the massless spectra of the four-dimensional theories) and analyse the worldsheet constructions of K3 mirror-folds with N = 2 supersymmetry from an algebraic geometry viewpoint. This approach will provide a powerful mathematical framework -that applies to Calabi-Yau three-folds as well -and give an explicit construction of the low-energy four-dimensional gauged supergravity.
We consider algebraic K3 surfaces defined as (the minimal resolution of) the zero-loci of quasi-homogeneous polynomials in weighted projective spaces P w 1 ,...,wn . These surfaces are characterized in particular by their Picard number ρ, the rank of their Picard lattice S (X) = H 2 (X, Z) ∩ H (1,1) (X) where 1 ρ 20. For the algebraic K3 surfaces of fixed ρ, the moduli space of CFTs factorizes into identified under a discrete subgroup, as we will review in section 2. The first factor is interpreted as the complex structure moduli space of the algebraic surface and the second as the complex Kähler moduli space. The definition of mirror symmetry for K3 surfaces is more subtle than for Calabi-Yau three-folds, as K3 is a hyperkähler manifold. For algebraic K3 surfaces, the notion of lattice-polarized mirror symmetry (LP-mirror symmetry) was introduced around 40 years ago by Pinkham [14] and independently by Dolgachev and Nikulin [15][16][17]; see the article by Dolgachev [18] for an introduction. In the special case considered by Aspinwall and Morrison [7], this amounts to an O(Γ 4,20 ) transformation that maps a K3 surface of Picard number ρ to one with Picard number 20 − ρ, interchanging the two factors in eq. (1.2). Another notion of mirror symmetry, perhaps more familiar to physicists, is the Berglund-Hübsch construction [19] (BH mirror symmetry), generalizing the Greene-Plesser construction of mirror Gepner models [20] to generic Landau-Ginzburg models with non-degenerate invertible polynomials. These two constructions (BH and LP mirror symmetry) overlap but do not always agree.
A key ingredient to reconcile these two approaches to K3 mirror symmetry, which will also play a central role in the present study of non-geometric automorphisms, is to consider non-symplectic automorphisms [21], which are automorphisms of the surface acting on the holomorphic two-form ω as σ ⋆ p : ω → ζ p ω, where e.g. ζ p = exp 2iπ/p. As we will review in section 2, at least when p is a prime number, the lattice-polarized mirror symmetry w.r.t. the invariant sublattice associated with the action of σ p , coincides with the Berglund-Hübsch mirror symmetry [22,23]. An important corollary that we will obtain is that the automorphism σ p of the surface on the one-hand and the corresponding automorphism σ T p of the (Berglund-Hübsch) mirror surface on the other hand act on sub-lattices of Γ 4,20 that are orthogonal to each other, denoted respectively T (σ p ) and T (σ T p ). It is expected that a similar statement is true for automorphisms of non prime order, and we expect the physics to work out in a similar fashion for such cases as well.
The non-geometric automorphisms of K3 CFTs that we study in this work correspond each to an O(Γ 4,20 ) transformation induced by a block-diagonal isometry in O(T (σ p ) ⊕ T (σ p )), the first block giving the isometry associated with the action of the automorphism σ p on the K3 surface, and the second block giving the isometry associated with the action of the automorphism σ T p on the mirror K3 surface. These isometries of the lattice Γ 4,20 can be thought of as mirrored automorphisms of K3 CFTs of order p. They can be decomposed as follows: where µ denotes the Berglund-Hübsch/lattice mirror involution. Here σ p is an order p large diffeomorphism of K3, µ maps the K3 to its mirror, σ T p is an order p large diffeomorphism of the mirror K3, and µ −1 maps the mirror K3 back to the original one.
Taking the quotient by two such automorphisms combined with shifts on the two one-cycles of a two-torus, at special points in the K3 moduli space fixed under the two automorphisms, gives the asymmetric Gepner models of [8]. We extend this construction to all points in moduli space using a reduction with duality twists. It is in general a difficult problem to find O(Γ 4,20 ) transformations that have fixed points in the K3 moduli space and so can lead to Minkowski vacua. We will show in section 3 that the fixed points of the monodromies that we consider correspond indeed precisely to the Gepner model construction of [8] on the worldsheet (which are Landau-Ginzburg points in the moduli space of K3 CFTs with enhanced discrete symmetry) and lead to four-dimensional theories with N = 2 Minkowski vacua.
The type IIA string theory compactified on K3 is non-perturbatively dual to the heterotic string compactified on T 4 [24]. The duality symmetry group O(Γ 4,20 ) acts on the heterotic side through isometries of the Narain lattice, containing T-duality transformations as well as diffeomorphisms and shifts of the B-field, and is often referred to as the heterotic T-duality group. The reduction from 6-dimensions on T 2 with O(Γ 4,20 ) monodromies round each circle provides twisted reductions of precisely the type introduced and studied in [2]. These reductions can be regarded in general as T-fold reductions of the heterotic string, with transition functions involving the T-duality group O(Γ 4,20 ) [1]. At the special points of the moduli space that are fixed-points of the twists, the construction reduces to a reduction of asymmetric orbifold type, with a quotient by elements of the T-duality group O(Γ 4,20 ) combined with shifts on T 2 [2]. Then the type IIA K3 mirrorfolds are dual to heterotic Tfolds, and at special points in the moduli space these become type IIA Gepner-type models and heterotic models of asymmetric orbifold type. Finding automorphisms with interesting fixed points is in general a difficult problem; the novelty here is that algebraic geometry leads us to a very interesting class of automorphisms that, when used in either the type IIA or the heterotic description, gives a rich class of models with N = 2 supersymmetry.
This work is organized as follows. The first half of the paper, up to subsection 3.2, is more algebraic-geometry oriented while the rest of the article deals with the physical aspects. In detail, section 2 provides the necessary mathematical background about K3 surfaces, mirror symmetry and K3 automorphisms and section 3 presents the mirrored automorphisms of mirror pairs of K3 surfaces and their relation with asymmetric Gepner model constructions. In section 4, we will consider the Scherk-Schwarz compactification of the 6-dimensional supergravity corresponding to type IIA compactified on K3 to obtain a four-dimensional gauged N = 4 supergravity. We show that for suitable choices of the twists, two gravitini become massive and two remain massless, giving vacua preserving N = 2 supersymmetry. In the final section, we will translate the O(Γ 4,20 ) monodromies defined in section 3 into the gauged supergravity framework, and obtain the moduli space of the low-energy theory.

Mathematical background
The moduli space of two-dimensional conformal field theories defined by quantizing non-linear sigma models on K3 surfaces is given by [6,7]: where O(Γ 4,20 ) is the isometry group of the even and unimodular (i.e. self-dual) lattice Γ 4,20 of signature (4,20): and Γ 3,19 is the K3 lattice which is isometric to the second cohomology group H 2 (X, Z) of a K3 surface X endowed with its cup product. Here U is the even unimodular lattice of signature (1,1)  An important sublattice of the K3 lattice Γ 3,19 of a K3 surface X is the Picard lattice 1 which is defined to be: The rank of this lattice (i.e. the rank of the corresponding Abelian group) ρ(X), or Picard number, is at least one for any algebraic K3 surface, and its signature is (1, ρ − 1). The transcendental lattice T (X) of an algebraic K3 surface X is defined to be the sub-lattice of H 2 (X, Z) ∼ = Γ 3,19 orthogonal to the Picard lattice: This lattice has signature (2, 20 − ρ). For sigma-model CFTs on algebraic K3 surfaces, the Picard lattice can be enlarged to the 'quantum Picard lattice': (2.7) One may think of the first factor as corresponding to the complex structure moduli space and the second one to the complex Kähler moduli space.
Mirror symmetry of Calabi-Yau three-folds exchanges their Hodge numbers h 2,1 and h 1,1 , and exchanges the complex structure and complex Kähler moduli spaces. For K3 surfaces the situation is different since (i) all K3 surfaces are diffeomorphic to each other and so have the same Hodge numbers and topology, and (ii) as these manifolds are hyperkähler, the complex structure and complex Kähler moduli are not unambiguously defined. For algebraic K3 surfaces there are two different notions of mirror symmetry that we will review in turn below, and these will both play a role in the construction of the non-geometric automorphisms we use in this paper.

Lattice-polarized mirror symmetry
A first notion of mirror symmetry of algebraic K3 surfaces is the lattice-polarized (LP) mirror symmetry discussed by Dolgachev in [18] and introduced by Pinkham [14], Nikulin and Dolgachev [15][16][17]. This construction uses the embedding of a given lattice M as a sub-lattice of the Picard lattice, which should be primitive. A primitive embedding of a lattice M into a lattice N, ι : M ֒→ N, is such that, viewing N as an Abelian group and ι(M) as a subgroup, the quotient N/ι(M) is a torsion-free Abelian group. Example 1. The Aspinwall-Morrison construction of mirror symmetry [7] is a particular instance of LP mirror symmetry. The authors considered an algebraic K3 surface X polarized by the whole Picard lattice, i.e. with M embedded primitively as ι(M) = S(X). One has then and M ∨ is the Picard lattice of the mirror surface. In other words, mirror symmetry is defined as the map where which exchanges the quantum Picard lattice and the transcendental lattice. Hence both factors in (2.7) are exchanged under this involution, which can be viewed as exchanging the complex structure and the complex Kähler moduli spaces of sigmamodel CFTs on the surface.

Berglund-Hübsch mirror symmetry
The second notion of mirror symmetry, Berglund-Hübsch (BH) mirror symmetry, is not specific to K3 surfaces. It follows from the Greene-Plesser construction [20] of mirror Gepner models [25] discovered in physics, exchanging the vector and axial R-currents of the worldsheet (2, 2) superconformal field theories. It was generalized by Berglund and Hübsch [19] and Krawitz [26]; later Chiodo and Ruan proved in [27] that it coincides with cohomological mirror symmetry. Let us consider a K3 surface realized as the minimal resolution of a hypersurface in a weighted projective space P [w 1 w 2 w 3 w 4 ] with gcd (w 1 , . . . , (2.12) It is non-degenerate if the origin x 1 = · · · = x 4 = 0 is the only critical point and if the fractional weights w 1 /d, . . . , w 4 /d are uniquely determined by W . If furthermore the number of monomials equals the number of variables the polynomial is said to be invertible. By rescaling one can then put an invertible polynomial W in the form where the square matrix A W := (a i j ) is invertible. If 4 ℓ=1 w ℓ = d, the hypersurface {W = 0} in P [w 1 w 2 w 3 w 4 ] admits a minimal resolution X W which is a smooth K3 surface.
We denote by G W the Abelian group of all diagonal scaling transformations preserving the polynomial W : and SL W its subgroup containing elements of the form (µ 1 , . . . , µ 4 ) = e 2iπg 1 , . . . , e 2iπg 4 , which corresponds in physics to the group of supersymmetry-preserving symmetries. The group G W always contains the element j W with (g 1 = w 1 /d, . . . , g 4 = w 4 /d) generating a cyclic group J W of order d.
Let us consider a subgroup G ⊂ G W such that J W ⊆ G ⊆ SL W , and the quotient groupG = G/J W . The minimal resolution of the orbifold X W /G, denoted X W,G , is also a K3 surface, that we associate with the pair (W, G). In physics, an orbifold of a K3 sigma-model (or Landau-Ginzburg model) by a discrete group G satisfying the condition J W ⊆ G ⊆ SL W preserves all space-time supersymmetry.
We now introduce the Berglund-Hübsch mirror symmetry, which follows in the physical context from the isomorphism between the superconformal field theories associated with a pair of Landau-Ginzburg orbifolds, generalizing the original Greene-Plesser construction of mirror Gepner model orbifolds.
Definition 2. Let (W, G) be associated with the minimal resolution of X W /G, a smooth K3 surface. The pair (W T , G T ) is obtained as follows: The Berglund-Hübsch mirror surface of X W,G is then given by X W T ,G T , the minimal resolution of the orbifold X W T / G T .
Here X W T is the surface W T (x 1 , . . .x 4 ) = 0 and g = (g 1 , . . . , g 4 ) is an automorphism of X W T acting on the coordinatesx ℓ asx ℓ → exp (2πi g ℓ )x ℓ for ℓ = 1, 2, 3, 4 while h = (h 1 , . . . , h 4 ) specifies an automorphism of X W under which the coordinates scale as x ℓ → exp (2πi h ℓ ) x ℓ . It is straightforward to check that, if the pair (W, G) is associated with a smooth K3 surface obtained as the minimal resolution of X W /G, then the mirror pair (W T , G T ) is associated with a smooth K3 surface obtained as the minimal resolution of where d = lcm (w 1 , . . . , w 4 ). This polynomial is preserved by the symmetries under which any of the coordinates scales as x ℓ → exp (2πi w ℓ /d)x ℓ and the other coordinates are invariant. They generate the group of diagonal symmetries Consider the case in which we choose the group G to be J W = (w 1 /d, . . . , w 4 /d) . ThenG is the trivial group and X W,G = X W . The mirror surface is characterized by the same polynomial as W T = W , and the dual group G T is given by the elements As g ℓ = n ℓ w ℓ d for some integer n ℓ , we get the condition This means that G T = SL W in this case.

Non-symplectic automorphisms of prime order
The two notions of mirror symmetry of algebraic K3 surfaces do not necessarily agree. In particular, as was shown in [28], the LP mirror of a surface polarized by its whole Picard lattice is not always identical to the BH mirror of the same surface. There exists nevertheless a class of lattice-polarized mirror symmetries, in which the surface is polarized by a sub-lattice of the Picard lattice, that gives the same results as the Berglund-Hübsch construction, and that will be instrumental in our construction of non-geometric automorphisms. Let us first define a non-symplectic automorphism of order p of a K3 surface X as a diffeomorphism σ p : X → X of the surface acting on the holomorphic two-form where ζ p is a primitive p-th root of unity, i.e. such that ζ p generates a cyclic group isomorphic to Z/pZ, e.g. ζ p = exp(2iπ/p). 3 If p is a prime number then it is straightforward to see that 2 ≤ p ≤ 19 (see [21, Theorem 0.1]). The automorphism σ p : X → X acts on 2-forms through σ ⋆ p . Let S(σ p ) be the sub-lattice of H 2 (X, Z) invariant under the action of the isometry σ ⋆ p and T (σ p ) its orthogonal complement. The rank of the lattice S(σ p ) will be denoted by ρ p . As was shown by Nikulin [21], the invariant sublattice S(σ p ) is a subset of the Picard lattice, and both S(σ p ) and T (σ p ) are primitive sub-lattices of the K3 lattice. The following lemma was also proved in [21]: Lemma 1. Let σ p be an order p non-symplectic automorphism of a K3 surface X.
There exists a positive integer q such that the action σ ⋆ p on the vector space T (σ p ) ⊗C can be diagonalized as  where I q is the identity matrix in q dimensions and all integers n ∈ {1, p − 1} with gcd (n, p) = 1 appear once, i.e. all primitive p-roots of unity are eigenvalues and the corresponding eigenspaces are all of dimension q.
Remark 1. From [29, Proposition 9.3] and [21] the action of the non-symplectic automorphism on the K3 lattice is unique up to conjugation with isometries.
We will consider a particular type of hypersurface in weighted projective space admitting a non-symplectic automorphism of prime order p, whose non-degenerate invertible polynomial is of the form We will call such surfaces p-cyclic, following [23]. They admit the obvious order p non-symplectic automorphism σ p : x 1 → ζ p x 1 . By construction, the BH mirror of a p-cyclic surface has its defining polynomial of the form W T =x p 1 +f (x 2 ,x 3 ,x 4 ), therefore it also admits an order p automorphism σ T p :x → ζ px . The following theorem was proved for p = 2 by Artebani et al. [22] and for p ∈ {3, 5, 7, 13} by Comparin et al. [23]. 4 is the sub-lattice of the Picard lattice invariant under the action of the nonsymplectic automorphism σ p of prime order, with p ∈ {2, 3, 5, 7, 13}. Let X W T ,G T be its Berglund-Hübsch mirror, polarized by the invariant sublattice S(σ T p ) associated with the non-symplectic automorphism σ T p . Then X W,G and X W T ,G T belong to mirror families of K3 surfaces in the sense of lattice-polarized mirror symmetry.
We obtain from this theorem a simple corollary which will play an important role in the construction of non-geometric compactifications. We first define the quantum invariant sublattice as the orthogonal complement of T (σ p ) in the Γ 4,20 lattice, namely which has signature (2, ρ p ) with ρ p ρ(X). For a lattice L we denote by L R the real vector space L ⊗ R generated by its basis vectors.
is the invariant sublattice under the σ p action, with p ∈ {2, 3, 5, 7, 13}. and X W T ,G T its LP mirror, regarded as an S(σ T p )-polarized K3 surface, which is also its Berglund-Hübsch mirror following Theorem 1.
From the theorem we have We obtain then the orthogonal decomposition over R: Example 3. In order to illustrate this general construction, we consider first the self-mirror K3 surface X given by the hypersurface in the weighted projective space P [21,14,6,1] . It inherits the orbifold singularities of the ambient space, and these should be minimally resolved in order to obtain a smooth K3 surface. This K3 surface admits a non-symplectic automorphism of order 42, acting on z by z → e 2iπ/42 z and leaving the other coordinates invariant. Then this implies, by [21, Theorem 0.1], that the rank of the transcendental lattice is a multiple of the Euler function of 42, which is 12. Since the rank is necessarily less than 22, which is the rank of the K3 lattice, the rank is necessarily equal to 12 and the rank of the Picard lattice is then 10. Interestingly, by [28] the generic K3 surface in the weighted projective space P [21,14,6,1] has Picard lattice of rank 10, which is the same as the rank of the Picard lattice of the surface (2.26) of Fermat type. The Picard lattice S(X) is isometric to a self dual lattice of signature (1,9), which is Thus this surface is its own mirror in the sense of Aspinwall-Morrison, as we have The moduli space of complex structures associated with this surface corresponds to the set of space-like two-planes in R 3,19 orthogonal to the basis vectors of S(X), quotiented by O(T (X)), the group of isometries of the transcendental lattice: of real dimension 20. The hypersurface (2.26) admits several non-symplectic automorphisms of prime order. The hypersurface is p-cyclic for p = 2, 3, 7 and the corresponding automorphisms σ 2 , σ 3 , σ 7 are of order 2, 3 and 7. Their action is In all cases, the invariant sublattice S(σ p ) ⊂ Γ 3,19 is identified with the Picard lattice S(X) (see [23,29]), and S Q (σ p ) ∼ = S Q (X). Its orthogonal complement in Γ 4,20 is naturally the transcendental lattice of the surface, hence T (σ p ) ∼ = T (X). The action of σ 3 on the transcendental lattice (2.28) of the surface (2.26) is given, in the appropriate basis over C, by six copies of the companion matrix of the cyclotomic polynomial Φ 3 = 2 n=1 (z − e 2iπn/3 ), using Lemma 1. 6 Recall that by Remark 1 the action can be given in a unique way on the transcendental lattice. It splits into an action onto the U ⊕ U lattice and onto the E 8 lattice. For U ⊕ U we choose a lattice basis in which the lattice metric (or Gram The action of the isometry σ 3 on U ⊕ U is then given by the following matrix in O(U ⊕ U): For the E 8 lattice, choosing a lattice basis in which the lattice metric is The companion matrix of a polynomial of the form P (x) = a 0 + a 1 x + · · · + a n−1 , whose characteristic polynomial is P . the action of σ 3 on E 8 is given by the following matrix in O(E 8 ): Likewise, the action of σ 7 is given in the appropriate basis over C by two copies of the companion matrix of the cyclotomic polynomial Φ 7 = 6 n=1 (z − e 2iπn/7 ), and finally the action of σ 2 on T (X) is simply given by minus the identity matrix in twelve dimensions.
For this surface |SL W /J W | = 1 hence J W T = J W . Therefore, using either of the non-symplectic automorphisms of prime order, one finds that the surface X W,J W is its own Berglund-Hübsch mirror and, polarized by S(σ p ) ∼ = S(X), is also its own mirror in the sense of LP mirror symmetry.

Example 4. Let us consider the hypersurface
in the weighted projective space P [12,8,3,1] . In this case |SL W /J W | = 2 and there are two choices of G with J W ⊆ G ⊆ SL W , either G = J W or G = SL W . Berglund-Hübsch mirror symmetry provides then the mirror pair (W, J W ) and (W, SL W ). The surface defined by eq. (2.34) admits a non-symplectic automorphism of order 3 acting as σ 3 : x → e 2iπ/3 x, while keeping the other variables fixed. As was shown in [23], the surface X W,J W is polarized by the invariant lattice S(σ 3 ) = E 6 ⊕ U and the transcendental lattice is contained in the lattice T (σ 3 ) = E 8 ⊕ A 2 ⊕ U ⊕ U. Then rank S(X) ≥ 8. On the other hand, the surface also admits a nonsymplectic automorphism of order 24 acting by z → e 2iπ/24 z that gives (see Lemma 1) rank T (X) = 8 or 16. The second case contradicts the previous inequality so that ρ X = rank S(X) = 14.
It was shown in [23] that the S(σ 3 )-polarized X W,J W and the S(σ T 3 )-polarized surface X W,SL W form a LP mirror pair. Indeed we have: First, for the surface (W, J W ), the action of σ 3 on T (σ 3 ), is given as follows (by Remark 1 the action is unique up to conjugation by isometries). On the A 2 lattice, by taking the lattice metric (Gram matrix) in a lattice basis in which the lattice metric is The surface (2.34) admits also a non-symplectic automorphism of order 2, acting as σ 2 : w → −w. Following [22], the invariant lattice of σ 2 is of rank six and isometric to S( The previous theorem indicates that the S(σ 2 )-polarized surface X W,J W and the S(σ T 2 )-polarized surface X W,SL W form a LP mirror pair. One can check that (W, SL W ) admits an order two 3 Non-geometric automorphisms of K3 sigma-models In this section we define mirrored automorphisms of sigma-model CFTs with K3 target spaces, combining the action of non-symplectic automorphisms of a surface and of its mirror, and study the corresponding isometries of the lattice Γ 4,20 . This construction is inspired by the non-geometric string theory compactifications that were obtained in [8] as asymmetric orbifolds of Gepner models.

Mirrored automorphisms and isometries of the Γ 4,20 lattice
We consider a p-cyclic K3-surface X, associated with a given non-symplectic automorphism σ p of prime order. The BH mirror of this surface admits also a nonsymplectic automorphism σ T p of the same order; by the theorem 1, these two surfaces, polarized by the invariant sublattices with respect to σ p and σ T p respectively, are also LP mirrors. The automorphism σ p has an action on the vector space T (σ p ) R while the automorphism σ T p acts on the vector space T (σ T p ) R . By the corollary 1 these vector spaces are orthogonal to each other in Γ R 4,20 . Inspired by physical considerations that will be illustrated in the next subsection, we will consider a mirrored automorphism that combines σ p and σ T p into a nongeometrical automorphismσ p of the CFT defined by quantizing the non-linear sigma model with a p-cyclic K3 surface target space. To prove that its action on the lattice Γ 4,20 is well-defined we will need the following proposition: The equality of the determinants follows from the fact that the lattice Γ 4,20 is unimodular (see [30,Corollary 2.6]). By properties of K3 surfaces (see [21,Theorem 0.1] and [30,Lemma 2.5]) the automorphisms σ p and σ T p act trivially on the discriminant groups of T (σ T p ) and of T (σ p ) so that we can extend the diagonal action by σ p and σ T p to the whole lattice Γ 4,20 .
Corollary 1 and Proposition 1 allow us to define 'mirrored automorphisms' of CFTs with p-cyclic K3 surfaces target spaces in the following way: Definition 3. Let (X W,G , ) be a p-cyclic K3 surface polarized by the invariant sublattice S(σ p ) and (X W T ,G T ,  T ), polarized by S(σ T p ), its LP mirror, with p ∈ {2, 3, 5, 7, 13}.
By Corollary 1 and Proposition 1 the diagonal action by (σ p , σ T p ) on the lattice T (σ T p ) ⊕ T (σ p ) can be extended to an isometry of the lattice Γ 4,20 , that we associate with the action of a CFT automorphism denotedσ p , that we name mirrored automorphism.
The action of the mirrored automorphismσ p on the vector space Γ R 4,20 For other values of p, including the non-prime cases, the physical construction suggests that similar non-geometric automorphisms acting as in eq. (3.1) can be defined. However the mathematical classification of these automorphisms is not yet complete (see [31]).
The isometry of the lattice Γ 4,20 induced byσ p is not in the geometric group O(Γ 3,19 ) ⋉ Z 3,19 and so is non-geometrical. The automorphismσ p acts aŝ where µ denotes the BH/LP mirror involution which maps the K3 to its mirror; µ −1 maps the mirror K3 back to the original one, σ p is a diffeomporphism of the original K3 and σ T p is a diffeomorphism of its mirror. Due to Proposition 1 the lattice isometry induced byσ p generates the order p isometry subgroup Being of finite order, it is conjugate to a subgroup of the maximal compact subgroup Explicitly, the action of the non-geometric automorphismσ p on Γ 4,20 , hence on the CFT with a K3 target space, is obtained by considering the geometrical action of σ p on T (σ p ) and, for the BH mirror surface, the action of σ T p on T (σ T p ). The lattice Γ 4,20 is an over-lattice of index p k := | det T (σ p )|, k a non-negative integer, of the sum T (σ p ) ⊕ T (σ T p ) (recall that T (σ p ) and T (σ T p ) are p-elementary lattices i.e. the discriminant groups are sums (Z/pZ) ⊕k , k as before). Then to construct Γ 4,20 one should add to the generators of the lattice T (σ p )⊕T (σ T p ) exactly k classes of the form (a + b)/p with a/p in the discriminant group of T (σ p ) and b/p in the discriminant group T (σ T p ), and we ask also that ((a + b)/k) 2 ∈ Z. The action of the isometry (σ p , σ T p ) on these classes is then obtained by linearity (over the rationals). We get in this way a set of generators of Γ 4,20 on which we have an isometryσ p of order p which is induced by the isometry (σ p , σ T p ) on T (σ p ) ⊕ T (σ T p ), i.e. the restriction of σ p to that lattice is equal to (σ p , σ T p ) 8 .
Given that, for any given p-cyclic K3 surface, all the relevant sublattices T (σ p ) and T (σ T p ) have been tabulated in [22,23], explicit forms of the Γ 4,20 isometries can be determined from lattice theory for any given example, see e.g. [32]. The corresponding matrices can be diagonalized on C according to lemma 1 and are characterized respectively by a set of rank(T (σ p )) angles and a set of rank(T (σ T p )) angles; these angles will be discussed further in the following sections.
Example 5. We consider the self-mirror surface (2.26) already discussed in example 3. The action of the geometrical automorphism σ 3 on the K3 lattice Γ 3,19 has been described there, see eqs. (2.31,2.33). The lattice Γ 4,20 admits an orthogonal decomposition into the invariant quantum lattices S Q (σ p ) and S Q (σ T p ) -or equivalently into T (σ T p ) and T (σ p ) -corresponding respectively to the quantum Picard lattice S Q (X) and the transcendental lattice T (X) of this surface.
As the surface and its mirror are isomorphic to each other, it is straightforward to define the action of the mirrored CFT automorphismσ 3 . It has a diagonal action on duplicating the action of σ 3 on the transcendental lattice that was studied in subsection 2.3. The action ofσ 3 on the sigma-model CFT associated with the surface (2.26) is therefore given by the block-diagonal 24 × 24 integer matrix: the two generators to a := e + f and b := e − f , in this way the lattice 2 ⊕ −2 has index two in U .
One takes now the isometric involution on 2 ⊕ −2 which acts as ι := (id, − id). The discriminant group of 2 , resp. −2 , is generated by a/2, resp. b/2. One considers now the class w := (a + b)/2, which has square w 2 = 0 ∈ Z; the lattice generated by a and (a + b)/2 has determinant 1 and it is in fact isometric to U . To see this one takes the generators w and v := (a − (a + b))/2 = (a − b)/2 and the induced involutionι acts exchanging v and w; in particular it cannot be put in a diagonal form over Z.
The action ofσ 3 is then induced from the block-diagonal 24 × 24 integer matrix  where the various matrices M U ⊕U This matrix is an isometry of the lattice The latter is a sublattice of index 3 of Γ 4,20 (see Proposition 1) and more precisely the lattice E 6 ⊕ A 2 is a sublattice of index 3 of E 8 . Now the isometries of order three σ ⋆ 3 and (σ T 3 ) ⋆ have no fixed vectors on A 2 , respectively E 6 and act trivially on the discriminant groups. They can be then combined (see Definition 3) to give an isometry of order three on E 8 without fixed vectors, but up to isometry there is only one such isometry on E 8 which is given by equation (2.33). So the action ofσ 3 on the sigma-model CFT associated with this surface is given by the matrixM 3 in eq. (3.5), as for the previous example.

Symmetries of Landau-Ginzburg mirror pairs
The Gepner models arise at special points in the moduli space (2.1) of sigma-model CFTs on K3 surfaces. These Gepner points play a special role in the present context as some of them are fixed under the action of the mirrored automorphisms defined in the previous subsection.
A Gepner model for a K3 surface [25] is a (4, 4) superconformal field theory obtained as the infrared fixed point of a (2,2) Landau-Ginzburg orbifold [11,33] with Fermat type superpotential quotiented by the order K = gcd(k 1 , . . . , k 3 ) diagonal Z K symmetry j W , acting on the chiral superfields as and realizing the projection onto integral R-charges. We will refer to this orbifold as the diagonal Z K orbifold. The twisted sectors of this orbifold are labelled by γ ∈ {0, . . . , K − 1} and will be referred to as γ-twisted sectors. The Landau-Ginzburg orbifold/Gepner model with superpotential (3.8) has a quantum Abelian symmetry [34] which is not present in the large-volume limit of the sigma-model. In the diagonal Z K orbifold theory, the quantum symmetry acts on a field in the γ-twisted sector of the model as: (3.10) At the infrared fixed point, the superconformal field theory obtained from this model is an orbifold of a product of N = 2 minimal model CFTs, as every single-field Landau-Ginzburg model with superpotential W = X k ℓ flows to a super-coset CFT SU(2) k ℓ /U(1) k ℓ [35]. These Gepner models lead to IIA superstring theory compactifications in six dimensions with N = (1, 1) supersymmetry, or, compactifying further on a two-torus, to N = 4 supersymmetry in four-dimensions.

Asymmetric orbifolds
We will now explain the relation between the mirrored automorphisms introduced in subsection 3.1 and the non-geometric orbifolds of Gepner models presented in [8,9], following earlier works [10,11].
We consider the Gepner model corresponding to the Landau-Ginzburg orbifold of superpotential (3.8) and assume that p := k 1 is a prime number. The theory admits the order p symmetry σ p : Z 1 → e 2iπ/p Z 1 .
(3.11) Quotienting the Gepner model by this automorphism alone would break all spacetime supersymmetry. Indeed one can see that in the corresponding orbifold theory (see [8] for details): • all the worldsheet operators corresponding to space-time supercharges are charged under the Z p symmetry hence are projected out of the spectrum, • the b-twisted sectors of the Z p orbifold by the symmetry (3.11) contain states with non-integer left and right U(1) R -charges whenever b = 0.
One observes that one can define a subgroup of the quantum symmetry group of the model (3.8), isomorphic to Z p , generated by: One can then modify the orbifold of the LG orbifold/Gepner model (3.8) by the symmetry (3.11) that we described above by adding a specific discrete torsion keeping the space-time supercharges coming from the left-moving sector in the spectrum.
Starting from the Gepner model, one defines the Z p orbifold projection by assigning to every state in the theory a chargê where Q p is the charge of the given state under the action of σ p and Q q p is the charge under the quantum symmetry σ q 3 , and by projecting onto states withQ p ∈ Z. This discrete torsion has also an effect in the twisted sectors b = 0 of the new Z p orbifold.
In those sectors the diagonal Z K orbifold projection is modified, as one projects onto states withQ K ∈ Z, whereQ (3.14) The charge assignments (3.13) and (3.14) are related to each other by modular invariance. One can check, by inspecting the one-loop partition function, that the Z p orbifold projection w.r.t. the chargeQ p keeps all space-time supercharges from the leftmovers, while none of the space-time supercharges from the right-movers is invariant. Furthermore the diagonal Z K projection w.r.t. the chargeQ K keeps only states with integer left R-charge. Hence space-time supersymmetry from the left-movers on the worldsheet is preserved by this orbifold with discrete torsion. Notice that one could have used the charge Q p − Q q p instead, in which case the invariant space-time supercharges come from the right-movers.
Under mirror symmetry, the right R-charges in every N = 2 minimal model are mapped to minus themselves. As a consequence, mirror symmetry exchanges the geometrical automorphism σ p with its quantum counterpart σ q p . In view of the discussion in section 2, a generator σ q p of an order p subgroup of the quantum symmetry of a Landau-Ginzburg orbifold superconformal field theory is identified with a non-symplectic automorphism σ T p of the corresponding BH mirror K3 surface. Hence, the mirrored automorphisms introduced in subsection 3.1 correspond precisely, at the Gepner points in the moduli space, to the orbifolds with discrete torsion described here.
The two-fold choice of discrete torsion in the definition of the Landau-Ginzburg model symmetriesQ p := Q p ± Q q p that we have noticed above corresponds, in the language of subsection 3.1, to the possibility of pairing the action of σ p either with the action of σ T p or of its inverse.

Fractional mirror symmetry
We have described in the previous subsection orbifolds of Gepner models with discrete torsion, that preserve all space-time supersymmetry from the left-movers, and none from the right-movers at first sight. As discussed in [9], they belong to a more general family of quotients of Gepner models by non-symplectic automorphisms of the corresponding K3 surfaces with discrete torsion. This construction leads generically to non-geometric compactifications, i.e. that do not belong to the moduli space of compactifications on smooth manifolds, preserving N = 2 supersymmetry in four dimensions (after further compactification on T 2 ). Similar constructions exist for Calabi-Yau three-folds, leading to N = 1 in four dimensions. However, in the specific case of an orbifold of a p-cyclic K3 surface (or more generically of a p-cyclic CY manifold) by a non-symplectic automorphism of order p as considered in the present work, the twisted sectors b = 0 contain right-moving operators that, despite having non-integral right R-charge, have the properties of generators of space-time supersymmetry. Interestingly, the worldsheet model for the non-geometric compactification is actually isomorphic as a (2, 2) superconformal field theory to the original Calabi-Yau model in this case.
This isomorphism implies the existence of quantum symmetries, called fractional mirror symmetries in [9], between geometric and non-geometric compactifications in the Landau-Ginzburg regime. Outside of the Gepner point such symmetry extends to a map between a (2, 2) non-linear sigma-model on a Calabi-Yau manifold and a non-geometric worldsheet model. A linear-sigma model description [36] was proposed in [9], generalizing the ideas of [37].
In the following, we will focus on freely-acting orbifolds combining this type of K3 non-geometrical orbifold with a shift along a circle; in this case the accidental isomorphism and corresponding restoration of N = 4 space-time supersymmetry do not play a role, as the b-twisted sectors with b = 0 will only contain massive states from the space-time point of view.

Worldsheet construction of non-geometric backgrounds: summary
The mirrored automorphisms described in subsections 3.1 for the geometry and 3.2 for the field theory are the building blocks of non-geometric compactifications of type IIA superstring theory, whose vacua correspond to the non-geometric freely-acting orbifolds of [8] that we will now summarize briefly.
The starting point is a Gepner model for a K3 surface as described in subsection 3.2. Consider the tensor product of this superconformal field theory with the free c = 3 superconformal theory with a two-torus target space of coordinates Y 1 , Y 2 . We consider a freely-acting supersymmetry-breaking Z k 1 × Z k 2 orbifold of this K3 × T 2 superconformal model generated by As it is, this orbifold breaks all space-time supersymmetry for the reasons given in the previous subsection. We add to each of these two freely-acting orbifold actions a discrete torsion of the same type as in eqs. (3.13,3.14) above. As described there the discrete torsion is such that all these models have integral left-moving R-charges but non-integral rightmoving ones, hence a type IIA superstring theory built upon one of these models will have a four-dimensional Minkowski vacuum with N = 2 space-time supersymmetry, all space-time supercharges being obtained from the left-moving Ramond ground states.
The two gravitini obtained from the right-moving Ramond sector are indeed massive in the Z k 1 × Z k 2 orbifold theory. Because of the freely-acting nature of this orbifold, no massless states could possibly arise from the corresponding twisted sectors. If one chooses an orthogonal two-torus of radii R 1 and R 2 , the masses squared of the two massive gravitini of broken N = 4 supersymmetry are [8]: (3.16) The massless spectra of all these models were computed in [9]. 9 It was found there that the massless states are identified with a subset of the chiral rings of the K3 SCFT, containing states built out of the identity operator in the SU(2) k 1 /U(1) k 1 and SU(2) k 2 /U(1) k 2 minimal models; we will consider the corresponding moduli spaces in subsection 5.2 from another perspective.
In about half of the possible constructions, this subset is empty and hence all the moduli of the original K3 SCFT have become massive; the only remaining massless moduli are the T and U moduli of the two-torus and the axio-dilaton modulus S, that are part of space-time vector multiplets. It gives the N = 2 four-dimensional ST U supergravity model at low energies (compared to the inverse size of the torus). In the remaining constructions, some of the K3 moduli survive and appear in the low energy theory in massless hypermultiplets. We now turn to the second part of this article, where we analyse these constructions from the low-energy four-dimensional viewpoint.

N = gauged supergravity from duality twists
In this section we study the supergravity dimensional reduction that corresponds to the stringy construction considered in the previous sections. We have considered type IIA superstring theory compactified on K3 × T 2 identified under certain automorphisms. This requires being at a point in the K3 moduli space which is a fixed point under the automorphisms. (These fixed points were found in the last section from Landau-Ginzburg orbifolds.) This construction is extended to general points in moduli space by a compactification with duality twists [2]. We will here discuss the supergravity limit of this, which is a dimensional reduction of Scherk-Schwarz type [5]. We consider then type IIA superstring theory compactified on K3 to 6 dimensions and then further compactified on T 2 with duality twists with non-geometric monodromy. In the supergravity limit, compactifying IIA supergravity on K3 gives N = (1, 1) supergravity in six dimensions coupled to 20 vector multiplets, and this has a duality symmetry O(4, 20) × R. Then further compactifying on T 2 with an O(4, 20) monodromy round each circle gives a Scherk-Schwarz reduction of the supergravity, resulting in a gauged N = 4 supergravity in four dimensions. This construction has been discussed extensively in the supergravity literature; see e.g. [5,[38][39][40] and references therein. For our string theory constructions, the monodromies are required to be in the duality group O (Γ 4,20 ), i.e. the isometry group of the lattice of total cohomology of the K3 surface as was discussed in section 2; in the physics literature it is often refered to as O(4, 20; Z).
We will focus here on the case with monodromies that are in the O(4) × O(20) subgroup of O (4,20) as it is for these compact monodromies that fixed points in the moduli space corresponding to Minkowski minima of the D = 4 supergravity scalar potential are possible. As we shall see, some interesting features arise for these special cases, and will give some vacua that break the N = 4 supersymmetry in four dimensions to N = 2. We will summarize the supergravity results here, and give more details elsewhere.
In this section we will consider the supergravity reduction with monodromies in the continuous group O (4,20) and in the following section we will consider the consistent type IIA superstring theory compactifications that arise from the discrete monodromies constructed in section 3.

Twisted reduction on T 2
The starting point in six dimensions is N = (1, 1) supergravity coupled to 20 vector multiplets, and this has a rigid duality symmetry G = O(4, 20) (and a further rigid symmetry consisting of constant shifts of the dilaton). There is also a local symmetry which in the bosonic sector is H = O(4)×O (20). In extending to the fermionic sector, the local symmetry is actually a double cover of this, H s = P in (4)  We now turn to dimensional reduction on T 2 with twists in G, giving rise to a gauged supergravity in four dimensions. Consider first the untwisted case. Simple dimensional reduction (with no twists) on T 2 gives rise to N = 4 supergravity coupled to 22 vector multiplets in four dimensions. The massless Abelian theory has G = SL(2) × O (6,22) global symmetry, and a local symmetry P in(6) × O (22), acting on the bosonic sector through O(6)×O (22). The N = 4 supergravity multiplet contains the vielbein, four gravitini ψ i µ , six graviphotons A m µ , four spin-half fermions χ i and a complex scalar τ , which takes value in SL(2)/SO (2). The SL(2) acts as usual through fractional linear transformations τ → (aτ + b)/(cτ + d).  O(6, 22) preserves a metric η M N of signature (6,22) and for the supergravity theory we can choose a basis in which this is the diagonal metric (I 6 , −I 22 ). However, in the next section when we apply the supergravity analysis to string theory, we will take η M N to be a lattice metric on Γ 4,20 ⊕ U ⊕ U.

The 22 vector multiplets in four dimensions each contain a vector
A gauged version of this supergravity (with electric gauge group) can be obtained by choosing a subgroup K of the rigid G = O(6, 22) symmetry (of dimension 28 at most) and promoting it to a local symmetry, using a minimal coupling to the 28 vector fields already in the theory. 10 For this to work, the vector representation of O(6, 22) must be the adjoint representation of K ⊂ O (6,22). The gauging of the four-dimensional supergravity is completely specified by the structure constants t M N P of the gauge group K (M, N = 1, . . . 28) which satisfy the Jacobi identity and the constraint that K preserves the O(6, 22) invariant metric η M N which is the condition that t M N P = η M Q t N P Q is completely antisymmetric. Supersymmetry then requires the addition of a scalar potential V , together with fermion mass terms given by and in terms of certain scalar-dependent tensors A ij 1 , A ij 2 , A j 2ai , A ij ab . Here i, j = 1, . . . 4 are SU(4) indices. Supersymmetry and gauge invariance put strong restrictions on the subgroups K that can be gauged, and fixes the form of the scalar potential and the tensors A ij 1 , A ij 2 , A j 2ai , A ij ab ; see [41][42][43] and references therein. In particular, the scalar potential is We now turn to the twisted reduction of the six-dimensional supergravity on T 2 to obtain a gauged N = 4 supergravity. We consider a rectangular torus for simplicity, with coordinates y i , i = 1, 2, with y 1 ∼ y 1 + 2πR 1 and y 2 ∼ y 2 + 2πR 2 . Then the complex Kähler modulus is T = iR 1 R 2 and complex structure modulus is U = iR 2 /R 1 .
We introduce twists around each of the two circles as described in the introduction. A field ψ(x µ , y i ) (where y i are coordinates on T 2 and x µ , µ = 0, . . . , 3, are the coordinates of the four-dimensional space-time) is taken to depend on y through G transformations. Specifically, suppose ψ(x µ , y i ) transforms in a representation of G, ψ → R[g]ψ under a rigid transformation h ∈ H R . Then the Scherk-Schwarz ansatz is giving the D = 6 field ψ(x µ , y i ) as the transformation of a D = 4 field ψ 0 (x) under a y-dependent G transformation g 1 (y 1 ) around the first cycle and a y-dependent G transformation g 2 (y 2 ) around the second cycle. The two G transformations are required to commute, g 1 (y 1 )g 2 (y 2 ) = g 2 (y 2 )g 1 (y 1 ) , and the y-dependence is taken to be exponential, so that Then the monodromies are (g 1 (0)) −1 g 1 (2πR 1 ) = e 2πR 1 N 1 , (g 2 (0)) −1 g 2 (2πR 2 ) = e 2πR 2 N 2 (4.6) for two commuting elements N 1 , N 2 of the Lie algebra of G, [N 1 , N 2 ] = 0. In the sixdimensional supergravity, the only fields transforming under G are the vector fields A and the scalar fields, represented by the vielbeinV. These then get non-trivial y dependence, while the fermions and graviton do not, as they are singlets under G. This picture depends on using the formalism in which the local H symmetry is not fixed. Choosing a physical gauge for the local H symmetry would mean that a G transformation must be accompanied by a compensating H transformation that act on the fermions through an H s transformation, so that in this gauge the fermions also get y dependence, and this requires the choice of a lift of the twist in H to one in the double cover H s . Full details of the reduction for the bosonic sector are given in [44], and here we will just quote the results needed, mostly following the notation of [44]. The O(4, 20) invariant metric is η IJ where I, J = 1, . . . 24; for the supergravity theory we can choose a basis in which this is the diagonal metric (I 4 , −I 20 ). However, in the next section when we apply the supergravity analysis to string theory, we will take η IJ to be the metric on the lattice Γ 4,20 . The generators of the gauge group K can be combined into a O (6,22) vector T M as (4.7) The Lie algebra of K is then [44] [T M , where the structure constants of the gauge group are and all other structure constants are zero. Then the only non-vanishing commutators are The monodromies in O(4) × O (20) are then e 2πR 1 N 1 = cos θ 1 sin θ 1 − sin θ 1 cos θ 1 ⊗ · · · ⊗ cos θ 12 sin θ 12 − sin θ 12 cos θ 12 , e 2πR 2 N 2 = cosθ 1 sinθ 1 − sinθ 1 cosθ 1 ⊗ · · · ⊗ cosθ 12 sinθ 12 − sinθ 12 cosθ 12 . (4.11) We choose a basis in which the angles θ 1 , θ 2 andθ 1 ,θ 2 specify monodromies in the O(4) factor and the remaining angles specify monodromies in O (20).
These twists will in general give masses to fields that are charged under U(1) 12 ⊂ O(4) × O (20). For a state with charges q i (i, j = 1, . . . 12) under U(1) 12 , the mass m will be given by (4.12) Using this formula, the masses of all fields can be found by finding the charges q i . The 28 vector fields are in the 28 of O (6,22) and this decomposes into (4, 1) + (1, 24) under O(2, 2) × O (4,20). A twist with all angles non-zero makes the vectors in the (1, 24) representation massive and leaves the (4, 1) vectors massless. The 24 vector fields in the (1, 24) representation can be written as 12 complex vector fields A i , where A i has charge q i = 1 and q j = 0 for j = i. Then A i has mass m given by (4.13) If the angles θ i ,θ i are both zero for some i, then the vector A i will remain massless.  The axion and dilaton in SL(2)/U(1) are also uncharged and remain massless, so the scalars in All other scalars become generically massive, with masses given by eq (4.12). This formula indicates also that, for a given set of charges {q i , i = 1, . . . , 12}, some values of the angles θ i ,θ i can lead to accidentally massless scalars.
It will be useful to decompose the indices i = 1, . . . 12 = (M, A) into indices M = 1, 2 labeling the Cartan subalgebra of O(4) and indices A = 3, . . . 12 labeling the Cartan subalgebra of O (20), so that the charges are q i = (q M , q A ). Then, for example, the 80 real scalars in the (1,1,4,20) representation take values in the coset O(4, 20)/O(4)×O (20) and can be written in terms of complex scalars φ M A , ρ M A where the scalar φ N B has charges q i = (q M , q A ) where q M = δ M N and q A = δ AB while ρ N B has charges q i = (q M , q A ) where q M = δ M N and q A = −δ AB . Then φ M A has mass squared and ρ M A has mass squared (4.16) Then for generic twists with all angles non-zero, the massless bosonic fields consist of the graviton, 4 vector fields in the 4 of O(2, 2) and 6 scalars in the coset space [SL(2)/U(1)] 3 ; this is precisely the bosonic sector of the STU model [4].
We now turn to the fermion mass terms (4.1), (4.2). At the origin, V = I 28 , the mass matrices of the model simplify considerably to give [45] A ij where G m are the 't Hooft matrices used to convert an SO(6) vector index to an antisymmetric pair of Spin(6) = SU(4) indices. The first matrix A 1 gives direct access to the fraction of supersymmetry preserved by the vacuum, as it provides the mass term for the gravitini given by where A ij 1 is a complex symmetric matrix. The mass matrix for ψ µi is where A 1ij = (A ij 1 ) * . This is a Hermitian matrix whose eigenvalues are (after a calculation using formulae from [45]) (m 1 ) 2 and (m 2 ) 2 , both with degeneracy two, where Similarly, the masses of the spin-1/2 fields can be found by calculating the tensors appearing in the mass formulae (4. The fermions in the 3 × (2, 1, 1) representation will all get mass m 1 given by (4.20) while those in the 3 × (1, 2, 1) representation will all get mass m 2 given by (4.21). The remaining fermions will all be massive for generic angles.
We see that something special happens if θ 1 = θ 2 andθ 1 =θ 2 so that m 2 is zero or θ 1 = −θ 2 andθ 1 = −θ 2 so that m 1 is zero. In either case, there are 2 massless gravitini and six massless spin-half fields. In this case the vacuum breaks the N = 4 local supersymmetry to N = 2 supersymmetry, and the massless fields fit into the N = 2 supergravity multiplet with three massless vector multiplets, which is just the spectrum of the STU model. Unlike most occurrences of the STU model in string theory, in the present case it does not occur as a truncation of a richer theory, but describes the whole low-energy sector of the theory.
For generic angles, both m 1 and m 2 are non-zero therefore all the fermions become massive and all supersymmetry is broken.  (20) and hence SU(2) 2 survives as an R-symmetry in the low-energy theory. The surviving SU(2) is the R-symmetry for the unbroken N = 2 supersymmetry.

Massive multiplets
In generic models, the remaining states now organize themselves in massive multiplets. The massive states that are singlets under SO (20) are:

Accidental massless multiplets
In certain models, a fraction of the BPS hypermultiplets are neutral under the monodromy and are therefore massless. From the mass formula (4.12), a supergravity field with charges q i (i, j = 1, . . . 12) under U(1) 12 will be massless if

Further accidental massless multiplets from KK modes
There can be further accidental massless multiplets from Kaluza-Klein modes [2]. For a trivial reduction without monodromy on the two circles with coordinates y 1 , y 2 , each field has a mode expansion of the form φ(x µ , y 1 , y 2 ) = n 1 ,n 2 e in 1 y 1 /R 1 +in 2 y 2 /R 2 φ n 1 ,n 2 (x) (4.29) with a sum over integers n 1 , n 2 . The mode φ n 1 ,n 2 (x) then has mass m with For a reduction with duality twists of the type discussed above, this formula is modified for fields that are charged under U(1) 12 ⊂ O(4) × O (20). For a mode φ n 1 ,n 2 (x) with charges q i (i, j = 1, . . . 12) under U(1) 12 , the mass m will be given by the following modification of (4.12): (4.31) In the truncated supergravity theory, the condition for massless states were (4.27) and (4.28). Now we see that the condition that the full Kaluza-Klein spectrum contains massless modes is that For the gravitini, there is a similar modification of the mass formulae. The gravitini KK modes will include massless spin-3/2 fields if θ 1 + θ 2 = 0 mod 4π andθ 1 +θ 2 = 0 mod 4π , or θ 1 − θ 2 = 0 mod 4π andθ 1 −θ 2 = 0 mod 4π . These are for gravitini modes that are periodic in y 1 , y 2 ; the conditions for antiperiodic ones would be slightly different.

Compactifications with non-geometric monodromies
We will now apply the supergravity framework developed in the last section to the non-geometric compactifications analysed in sections 2 and 3 from the algebraic geometry and string theory viewpoints. In string theory, the non-compact symmetry groups O (4,20) and O (6,22) are broken to the discrete subgroups preserving the charge lattice [2]. In particular, O(4, 20) is broken to the group O(Γ 4,20 ) preserving the lattice Γ 4,20 , and we choose the natural basis in which the metric η IJ is the metric on the lattice Γ 4,20 given in section 2. As a result, the mass parameters introduced in the twisted reduction now take discrete values. Our aim here is to find the nongeometric type IIA compactifications, consisting of K3 fibrations over two-tori with non-geometric twists, in the sense of [2], that at fixed points of the twist reproduce the orbifold constructions of [8] summarized in section 3.
We start with a (p 1 , p 2 )−cyclic K3 surface, i.e. a hypersurface in a weighted projective space defined by a polynomial of the form where p 1 and p 2 are prime numbers. As we have seen, such surface admits two non-symplectic automorphisms σ p 1 and σ p 2 generating an automorphism group isomorphic to Z p 1 × Z p 2 . Using Definition 3 one can associate to them non-geometric automorphismsσ p 1 ,σ p 2 generating a subgroup O(σ p 1 ) × O(σ p 2 ) of the duality group O(Γ 4,20 ), isomorphic to Z p 1 × Z p 2 ; see eqs. (3.1,3.3). In subsection 3.3 we defined orbifold compactifications consisting of identifying the IIA superstring theory on K3 × T 2 under the action ofσ p 1 combined with a shift of 2πR 1 /p 1 for the first one-cycle of the torus, andσ p 2 combined with a shift of 2πR 2 /p 2 for the second one-cycle of the torus. This is all defined at a particular point in the moduli space of CFTs on K3 that is a fixed point under these transformations, corresponding to the (p 1 , p 2 )-cyclic K3 surface at a Landau-Ginzburg point.
The reduction with a duality twist construction gives a way to extend this to all points in moduli space, and then the supergravity analysis of section 4 gives the resulting low energy effective field theory. The twisted reduction gives a fibration of K3 surfaces over T 2 with two non-geometric monodromies in O(Γ 4,20 ) associated with the one-cycles of the torus: • An order p 1 monodromy belonging to O(σ p 1 ), associated with the non-geometric automorphismσ p 1 , for the first one-cycle of the torus.
• An order p 2 monodromy belonging to O(σ p 2 ), associated with the non-geometric automorphismσ p 2 , for the second one-cycle of the torus.
Similarly, the order seven non-geometric automorphism would give a monodromy matrixM 7 which can be brought to the diagonal form (4.11) with the twelve angles θ i given by exp(2irπ/7), for r = 1, ..., 6, each with degeneracy 4.
To specify the reduction, we choose the monodromy e 2πN 2 for the other circle from another automorphism, e.g. that resulting from σ 3 or σ 7 , and this gives the structure constants t 2I J .
In full generality, using Lemma 1 in section 2, the GL(24; Z) matrices associated with the non-geometric automorphisms can be diagonalized over C, or equivalently can be written as elements From these angles, the full structure of the effective supergravity theory can be read off, as seen in section 4. The scalar potential of the supergravity admits minima at the fixed points of the automorphisms that reproduce the four-dimensional physics obtained from the asymmetric Gepner models considered in section 3.
The stringy compactifications discussed in this work have Minkowski vacua that preserve N = 2 supersymmetry as we will show below, and have three massless vector multiplets S,T and U associated respectively to the axion-dilaton and to the T 2 moduli. About half of the corresponding asymmetric Gepner models, for instance the self-mirror surface (2.26) withσ 3 andσ 7 monodromies, give just N = 2 STU supergravity at low energies, while in the other cases the low-energy theory contains some additional massless hypermultiplets, depending of the choice of K3 surfaces and of automorphisms; the associated moduli space will be discussed in subsection 5.2.

Gravitini masses and supersymmetry
As we have seen, the isometry induced by the action of the non-geometrical automor-  2) factor acts as an order p rotation in the space-like two-plane in the vector space generated by T (σ p ) while the second O(2) factor acts as an order p rotation in the space-like two-plane in the vector space generated by T (σ T p ). 13 12 More explicitly, there exists a positive integer q such that the complex numbers {exp iθ i , i = 1, . . . , 12} are given by the primitive p roots of unity, q times each. 13 For instance, for the order three automorphism studied in example 3, each O(2) generator comes from the O(2, 2; Z) generator given by eq. (2.31) after O(2, 2; R) conjugation.
The parts of the monodromies in O(2) × O(2) ⊂ O(4) transformations are specified by the angles θ 1 , θ 2 andθ 1 ,θ 2 . These are then Here ε i = 0 if there is no discrete torsion. If there is discrete torsion, then ε i ∈ {−1, 1} corresponding to the two possible choices of discrete torsion for each cycle, as seen from the corresponding worldsheet description in subsection 3.2.
We then draw the following conclusions which are in accord with the Gepner model description of the vacua that was obtained in [8] (see in particular around eq. (4.3) in [8]): • For 'geometric' non-symplectic automorphisms of K3 surfaces, which have vanishing discrete torsion ε i = 0, all gravitini become massive and so all the spacetime supersymmetry is broken.
• The two non-geometric twists preserve the same N = 2 ⊂ N = 4 space-time supersymmetry if ε 1 = ε 2 = ±1. Then two of the gravitini remain massless, while the other two acquire an equal mass. In the worldsheet description, ε 1 = ε 2 means that the same choice of discrete torsion was made for both of the corresponding Gepner model freely acting orbifolds, so that they both preserve space-time supercharges from either the left-movers or the right-movers.
Then only the non-geometric twists with discrete torsion that pair the non-symplectic automorphisms with the corresponding automorphisms acting on the mirror K3 surfaces can be compatible with N = 2 vacua in four dimensions. To conclude, there is a perfect agreement between the gauged N = 4 supergravity and the worldsheet construction. Note finally that the mass scale of the spontaneous supersymmetry breaking N = 4 → N = 2 is set by the (inverse of the) volume of the two-torus [8] and can be taken to be much smaller than the string mass scale. Therefore it makes sense to analyse the model within the four-dimensional supergravity framework (while ten-dimensional supergravity would be inappropriate for these non-geometric constructions).
For p > 2, B p is of complex dimension rank(T (σ p ))/(p − 1) − 1 and is isomorphic to a complex ball. 15 For p = 2 one gets a Hermitian symmetric space of complex dimension rank(T (σ p )) − 2.
We now consider the non-geometric automorphismσ p constructed from σ p as defined in section 3. To understand its action on the CFT moduli space one has to look also at the mirror surface X W T ,G T which admits an action of the non-symplectic automorphism σ T p . In the same way as before, we define 16 B T p = {z ∈ P(T T ζp ) , (z, z) = 0, (z,z) > 0} , (5.12) and The moduli space of K3 surfaces with non-symplectic action by σ T p is then given by Γ T p \B T p . To summarize, by using the description of the period map for K3 surfaces, the K3 surface with non-symplectic automorphism σ p has period in Γ p \B p , and the mirror K3 surface has period in Γ T p \B T p . By using the definition ofσ p (see Definition 3), we expect that the hypermultiplet moduli space of CFTs invariant under the action of the non-geometric automorphismσ p is obtained by the direct product of these two spaces:M Interestingly, this dimension is the same for every automorphism σ p of a given prime order p, regardless of the rank of the corresponding invariant lattice S(σ p ). We have checked this result against some of the string theory spectra computed in [8] and found agreement. With two monodromy twists associated with the two one-cycles of the two-torus, one should consider the intersection of the corresponding moduli spaces, which is easier to do case by case. For instance, for the self-mirror K3 surface (2.26) twisted by the non-geometric monodromiesσ 3 andσ 7 , this intersection is just a point 17 and so there are no massless hypermultiplets in the low energy supergravity and we obtain just the N = 2 STU supergravity model (provided that ε 1 = ε 2 so that the two automorphisms preserve the same half of the supersymmetry).

Conclusion
In this work we have constructed a new class of N = 2 four-dimensional nongeometric compactifications of type IIA superstring theories, that consist of K3 fibrations over two-tori with non-geometric monodromies which lead in most cases to pure N = 2 STU supergravity with no hypermultiplets at low energies.
The monodromies correspond to non-geometric automorphisms that we have obtained by pairing a non-symplectic automorphism of a K3 surface with a nonsymplectic automorphism of the mirror surface. We have demonstrated that the action of such an automorphism can be lifted to an isometry of the lattice Γ 4,20 , i.e. an element of the duality group O(Γ 4,20 ) of CFTs on K3 surfaces, and hence leads to a well-defined string theory compactification. We have shown that the fixed loci of these automorphisms are given by asymmetric Gepner model orbifolds, considered recently in [8]. The new understanding of these non-geometric backgrounds in terms of mirrored automorphisms should apply to non-geometric automorphisms of Calabi-Yau three-folds as well (except naturally the lattice-related aspects).
We have analysed the models from the four-dimensional N = 4 gauged supergravity perspective valid at low energies. The matrices corresponding to the Γ 4,20 isometries that we have constructed provide directly the structure constants which parametrise the gauged supergravities obtained from twisted reductions of K3 × T 2 , and we have showed that the minima of the superpotential preserve N = 2 supersymmetry in four dimensions, as expected from the string theory constructions of such vacua. In some cases, hypermultiplets remain massless in the four dimensional theories; we have obtained, using results from the mathematical literature, the corresponding hypermultiplet moduli space, whose dimension agrees with the string theory predictions.
We plan to provide more details on the four-dimensional gauged supergravity construction in a companion paper. In particular we will analyse the scalar manifold of the low-energy theory in order to show explicitly that the hypermultiplet moduli space predictions from algebraic geometry are verified, and check that all the consistency conditions of gauged supergravity are met for these particular gaugings.
The duality between the type IIA string theory compactified on K3 and the heterotic string compactified on T 4 [24] gives a heterotic dual to our constructions consisting of a toroidal reduction of the heterotic string with monodromy twists, that gives an asymmetric orbifold construction at fixed points; such models were introduced in [2]. Particular examples of heterotic asymmetric orbifolds are given by CHL compactifications [47]; the latter correspond, on the type IIA side, to symplectic automorphisms of K3 surfaces. Here, algebraic geometry leads us to a particularly interesting class of constructions that correspond to non-symplectic K3 automorphisms on the type IIA side and preserve N = 2 supersymmetry. The corresponding heterotic orbifolds will be discussed further in a forthcoming publication.