T-duality orbifolds of heterotic Narain compactifications

To obtain a unified framework for symmetric and asymmetric heterotic orbifold constructions we provide a systematic study of Narain compactifications orbifolded by finite order T-duality subgroups. We review the generalized vielbein that parametrizes the Narain moduli space (i.e. the metric, the B-field and the Wilson lines) and introduce a convenient basis of generators of the heterotic T-duality group. Using this we generalize the space group description of orbifolds to Narain orbifolds. This yields a unified, crystallographic description of the orbifold twists, shifts as well as Narain moduli. In particular, we derive a character formula that counts the number of unfixed Narain moduli after orbifolding. Moreover, we develop new machinery that may ultimately open up the possibility for a full classification of Narain orbifolds. This is done by generalizing the geometrical concepts of Q-, Z- and affine classes from the theory of crystallography to the Narain case. Finally, we give a variety of examples illustrating various aspects of Narain orbifolds, including novel T-folds.


Introduction and conclusions
Since the early days of superstring theory, the heterotic string [1][2][3] has served as a promising candidate theory for a unified quantum description of particle physics as well as gravity, see e.g. [4] for a textbook introduction to string phenomenology. One of the main obstacles lies in the fact that the heterotic string is conventionally defined in a ten-dimensional space-time. Hence, six spatial dimensions have to be compactified in order to make contact to the observable four-dimensional world.
One possibility is to compactify on a six-dimensional (symmetric) toroidal orbifold [5,6] which is the quotient of a six-torus T 6 by some of its discrete isometries, see [7] for a full classification with N ≥ 1 supersymmetry in four dimensions. For example, one can use an Abelian rotational symmetry Z K and define the orbifold geometrically as the quotient space T 6 /Z K . Especially, in the presence of discrete Wilson lines [8] orbifold compactifications have been used to construct (minimal) supersymmetric extensions of the Standard Model (MSSM) from the heterotic string [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] 1 . These constructions can be considered to be promising directions to connect string theory to particle physics: Beside reproducing MSSM-like models, they offer an appealing geometrical interpretation, in which many properties of the elementary particles depend on their localization in extra dimensions [14,[33][34][35]. Unfortunately, these constructions generically leave a number of moduli, like the compactification radius R, unfixed.
A possibility to stabilize moduli is to generalize the construction of symmetric orbifolds to asymmetric ones: In this case one quotients the compactification space not only geometrically, but also by a genuine stringy symmetry [36]. The most famous example of such a symmetry of string theory is Tduality: In its simplest form, T -duality is a Z 2 transformation that identifies a string compactification on a circle with small radius R with another compactification on a circle with large radius 1/R. This is a full quantum duality on the string worldsheet as this can be described as field redefinitions in a path integral approach [37][38][39]. Now, in order to be able to perform the quotient by this T -duality transformation the radius R can no longer be a free parameter, but it has to be fixed at the so-called self-dual value R = 1 (in string units). This promotes the T -duality transformation R → 1/R to a symmetry of the theory. On the left-and right-moving coordinate fields X l and X r this T -duality transformation is realized by X l → +X l and X r → −X r . Hence, in general, such T -dualities act differently on the left-and right-moving degrees of freedom of the string and the resulting quotient spaces are often called asymmetric orbifolds [40]. Asymmetric orbifolds provide specific examples of non-geometric string backgrounds [41][42][43] or so-called T -folds [44,45]. More recently double field theory [46][47][48] was introduced as an attempt to obtain a setting with doubled geometry to describe such T -folds using geometrical tools inspired by a string field-theoretical description of the left-and right-moving string coordinates. Hence, asymmetric string constructions are of increasing interest in the connection to non-geometric flux backgrounds [49,50]. Various aspects of asymmetric orbifolds have been studied in the past [51][52][53][54][55][56][57][58][59] and with recent renewed interest [60,61] and in particular also in the context on non-supersymmetric constructions [62][63][64].

Main results
In this work we develop a generalized space group description of Narain orbifolds and utilize this formalism throughout this work to study various aspects of symmetric and asymmetric orbifolds in a unified fashion. To define the generalized space group, we first perform a concise investigation of the heterotic T -duality group: We decompose its generators into geometrical and non-geometric ones and use them to parametrize the maximal compact subgroup of the T -duality group. This is important, as the maximal compact subgroup contains the finite subgroups that can be used to build (a-)symmetric orbifolds. Hence, the generalized space group provides a unified framework to study symmetric and asymmetric orbifolds in a systematic manner.
We apply our understanding of the T -duality group to derive conditions for the stabilization of Narain moduli by orbifolding. This leads us to a closed character formula to count the number of unstabilized Narain moduli. In particular, this formula shows that all Narain moduli are fixed, if the left-and right-moving twists do not have any irreducible representations of the point group in common. We use our findings on moduli stabilization to formulate sufficient conditions for a Narain orbifold to exist crystallographically by reducing this question to the question whether certain Riccati equations admit solutions. Hence, using our generalized space group description one can check that a Narain orbifold exists at least crystallographically and one can identify the associated Narain torus that is compatible with the orbifold action.
Moreover, in this paper we lay the foundation for a classification of Narain orbifolds. Even though asymmetric orbifolds have been studied essentially since the birth of superstring theory, they have been analyzed so far essentially on a case-by-case basis. Based on our definition of the generalized space group we identify equivalence relations for Narain orbifolds. These equivalences extend the notations of Q-, Zand affine-equivalences from theory of crystallography to the Narain case leading to the notions of Narain Q-, Zand Poincaré-classes. This can be seen as a first step towards a classification of symmetric as well as asymmetric Narain orbifolds, which includes -besides the information on the six-dimensional compactification space -also the anti-symmetric Kalb-Ramond B-field, the (discrete) Wilson lines and the orbifold shift-vectors in a unified fashion.
Finally, we construct a non-trivial set of (two-dimensional and more general) Narain orbifolds by specifying their generalized space groups. We use these examples to illustrate many aspects of our study, like the stabilization of Narain moduli and the equivalence classes for Narain orbifolds.

Outlook
In this work we investigated necessary conditions for a Narain orbifold to exist. However, we ignored possible extra conditions coming from modular invariance, as they have been studied in the past, see e.g. [53]. However, it would be advantageous to check for full modular invariance on the level of the generalized space group and, ultimately, to incorporate modular invariance in the definition of generalized space groups such that generalized space groups yield modular invariant Narain orbifolds by construction.
Moreover, we can imagine various applications of our work: The space group formulation of Narain orbifolds allows for a systematic construction of large sets of examples in various dimensions and in both, the (D, D) case as well as the heterotic (D, D + 16) case. In addition, using our definitions of Narain Q-, Zand Poincaré classes one can unambiguously decide whether two Narain orbifold models are physically identical or not. This might proof to be very useful for systematic investigations and classifications for various reasons: First of all, in the traditional approach two (symmetric) orbifold models are often said to be equivalent if their massless matter spectra agree. However, this is neither necessary nor sufficient: For example, two different string constructions might possess identical massless spectra but different couplings, or the massless spectrum of a given toroidal orbifold compactification can be enhanced at specific points in its moduli space. Precisely here the Narain Poincaré classes would come to the rescue and decide for (in-)equivalence. However, our new definition of equivalence might be computationally very intensive and, hence, further studies might be necessary in order to apply it practically for large computer scans.
Second, having an unambiguous criterion for two Narain orbifolds to be inequivalent, our work can be used to classify Narain orbifolds, both symmetric and asymmetric ones. Such a classification would automatically include the orbifold twists and shifts as well as the background fields, i.e. the torus metric, the B-field and (discrete) Wilson lines.
Finally, one can use our definitions of Narain Q-, Zand Poincaré classes to decide whether a Narain orbifold is genuine asymmetric or only seemingly. Hence, our approach might be also very helpful in the study of non-geometrical backgrounds for string theory in general, since it has been proven to be quite difficult to obtain concrete, yet true, examples of such backgrounds.

Paper outline
In Section 2 we recall the basics of the Narain description of heterotic torus compactifications with continuous Wilson lines A, the anti-symmetric Kalb-Ramond B-field and the metric G. In this section we exploit the fact that the moduli space of Narain compactifications is concisely described as the coset of the continuous T -duality group over its maximal compact subgroup and the discrete T -duality group O η (D, D + 16; Z).
Given this prominent roles of continuous and discrete T -duality groups, we reserve Section 3 to study their properties. In particular, we list a complete set of generators of O η (D, D + 16; R), which are chosen such that they parametrize the discrete T -duality group if their parameters are restricted to specific, quantized values. In addition, we give the non-linear transformations of the moduli G, B, A under arbitrary T -duality group elements.
After these preparations, Section 4 sets up a generalized space group description of Narain orbifolds involving combined shift-and twist-elements. In this section various properties of Narain orbifolds are uncovered. In particular, we show that the shifts of the generalized space group are quantized in the directions in which the twists act trivially. Moreover, we emphasize that the amount of preserved target-space supersymmetry is solely decided by the twists θ α r that acts on the right-moving sector.
Section 5 investigates two related questions: i) under what conditions does a Narain orbifold exist and ii) how many Narain moduli, G, B, A, are fixed. To facilitate this discussion the lattice basis is introduced in which the twists are represented by integral matrices ρ α ∈ O η (D, D + 16; Z). Some properties of these twists in the lattice basis can concisely be characterized using the generalized metric H and the associated Z 2 -grading Z. By exploiting the coset structure of the Narain moduli space, we show that a Narain orbifold exists provided that certain Ricatti equations, i.e. coupled matrix equations, have a solution. Deformations of such a solution correspond to the unconstrained moduli of a Narain orbifold. Using some results collected in Appendix A we derive a character formula to count their number.
All these results are used in Section 6 to lay the foundations for a classification of Narain orbifolds.
Given that the concepts of Q-, Zand affine-classes proved to be very useful for the classification of symmetric orbifolds, we extend these concepts to Narain orbifolds.
To illustrate the power of the generalized space group description of Narain orbifolds we study symmetric orbifolds in Section 7 in this language. Even though the main interest of Narain orbifolds lies in the construction of asymmetric orbifolds (or T -folds), we show in this section that the language of Narain orbifolds gives a convenient, unified description of the geometry and the (discrete) Wilson lines.
Finally, in Section 8 we employ the Narain Qand Z-classes to study two-dimensional Abelian Z K Narain orbifolds. We provide a large table with many examples of previously unknown twodimensional Narain orbifolds. By an explicit construction we show that it is possible to have a Z 12 two-dimensional Narain orbifold, while it is well-known that the largest order of Euclidean Z K twists is K = 6 in two dimensions. Moreover, Qand Z-classes are particularly useful to distinguish seemingly asymmetric from truly asymmetric orbifolds as we illustrate by various examples.

Heterotic Narain torus compactifications
This section reviews the Narain formulation of heterotic torus compactifications [76] and sets the notation used throughout this work. The moduli space can be described using the generalized vielbein E, which is parametrized by continuous Wilson lines A, the anti-symmetric Kalb-Ramond B-field and the metric G. This vielbein characterizes coordinate field boundary conditions as well as the momenta that appear in the representation of the Narain torus partition function as a lattice sum.

Worldsheet field content of the heterotic string
We parametrize the two-dimensional string worldsheet by (real) coordinates σ andσ, defined by where σ 0 and σ 1 denote the worldsheet time and space coordinate, respectively. Worldsheet fields that solely depend on σ orσ are called left-moving or right-moving fields, respectively. They are correspondingly labelled by a subscript l or r (or in capital letters L/R). The heterotic string is closed because of the identification (σ 0 , σ 1 ) ∼ (σ 0 , σ 1 + 1). Hence, (σ 0 , σ 1 ) are coordinates on a worldsheet cylinder for the freely propagating string. The heterotic string [1][2][3] is described by a conformal field theory on the worldsheet with 26 left-moving real bosonic fields and ten right-moving real bosonic and fermionic fields.
The easiest approach to connect this theory to particle physics in d dimensions (for example d = 4) is to perform a stepwise compactification: In the first step one compactifies the 16 surplus left-moving bosonic fields on a 16-dimensional torus in order to match the number of left-and right-moving bosonic fields to ten. The resulting theory corresponds to a ten-dimensional theory with a gauge group dictated by modular invariance of the string partition function. For example, in the case of ten-dimensional N = 1 supersymmetry the gauge group is fixed to either E 8 × E 8 or SO (32). Then, in a second step one compactifies on a D-dimensional space, for example on a Calabi-Yau or an orbifold. As a result one obtains a d-dimensional theory, where d + D = 10, e.g. 4 + 6 = 10. An alternative approach, which we use in this paper, is the so-called Narain construction, where the two-step compactification described above is performed in a single step compactification of the heterotic string directly to d dimensions, see Section 2.2.
In light-cone gauge two left-and right-moving uncompactified dimensions are gauge-fixed and, hence, eliminated. Thus, the heterotic string in light-cone gauge can be described by the following worldsheet fields: • As left-moving fields, there are 8+16=24 real bosonic fields. They are denoted by x µ l (σ) with µ = 2, . . . , d − 1 (µ = 0, 1 are chosen to be fixed in light-cone gauge) for the uncompactified and Y L (σ) for the compactified dimensions, respectively. Furthermore, we set where y l (σ) = y i l (σ) for i = 1, . . . , D live on the D-dimensional compactification space. In addition, y L (σ) = y I L (σ) for I = 1, . . . , 16 are often referred to as the gauge degrees of freedom.
Left-and right-moving bosonic fields can be combined to coordinate fields x µ (σ,σ) and X i (σ,σ) which parametrize the d uncompactified and D compactified dimensions, respectively, i.e.
Their classical equations of motion read which is solved by the general ansatz (2.3). Hence, collectively, we have 2D+16 compactified bosonic worldsheet fields Y nested in the following fashions: We define the following dimensions: D r = D l = D and D L = D l + 16 = D + 16. We will use the same notation as in eqn. (2.5) for other types of vectors. The separation (2.3) of the coordinate fields X i (σ,σ) into left-and right-moving coordinates y i l (σ) and y i r (σ) is unique up to a constant shift of the zero modes ξ i , i.e.
with ξ ∈ R D . This has important consequences for the number of worldsheet degrees of freedom: If one counts left-and right-movers y(σ,σ) ∈ R 2D independently there seems to be a doubling of degrees of freedom on the worldsheet compared to the coordinate fields X(σ,σ) ∈ R D , see eqn. (2.3). However, due to eqn. (2.6) there are only D independent zero-modes of y(σ,σ) that specify the position of the string and the numbers of worldsheet degrees of freedom are equal for X(σ,σ) and y(σ,σ).

Torus partition functions as Narain lattice sums
We consider torus compactifications T 2D+16 IΓ is a so-called 2D + 16-dimensional Narain lattice, which we will analyze in this section in detail. This will be of use when we discuss the more general case of Narain orbifolds later in Section 4.
In the case of a Narain torus, the closed string boundary conditions of the worldsheet fields are given by where s = 0, 1 parametrizes the different spin structures of the right-moving fermions ψ R , i.e. s = 0 yields the so-called Ramond sector and s = 1 the Neveu-Schwarz sector. Furthermore, L ∈ IΓ denotes a lattice vector of IΓ.
At one-loop the partition function Z full (τ,τ ) is given by the string vacuum-to-vacuum amplitude which corresponds to a worldsheet torus. This torus is defined by two periodicities of worldsheet fields: (σ 0 , σ 1 ) ∼ (σ 0 , σ 1 + 1) and (σ 0 , σ 1 ) ∼ (σ 0 + τ 2 , σ 1 + τ 1 ) for the string to close in the worldsheet-spatial and worldsheet-time directions, respectively. Here, τ = τ 1 + i τ 2 is the so-called modular parameter of the torus. Then, the full partition function Z full (τ,τ ) of the one-loop worldsheet torus can be factorized as follows (2.8) The individual partition functions are given by where q = e 2πi τ ,q = e −2πiτ and e d = (1, . . . , 1) denotes the d-dimensional vector with all entries equal to one. Here and in the following we often omit the dependencies on τ andτ for notational ease. In addition, η(τ ) denotes the Dedekind function and θ the theta-function. The vectors P are from the dual lattice IΓ * which is defined as P ∈ IΓ * if for any L ∈ IΓ. Here, we have introduced the Lorentzian inner product of lattice vectors as The metric η should not be confused with the Dedekind function η(τ ) that appears in partition functions; we assume that the reader understands from the context which is meant. The partition function Z ψ for the right-moving fermions can also be presented as a lattice sum, i.e. from (2.9b) we get where the lattice Γ ψ = Γ vec ⊕ Γ spin consists of the vectorial and spinorial weight lattices, given by R e 4 = odd} and Γ spin = {p R + 1 2 e 4 | p R ∈ Z 4 and p T R e 4 = even}. Furthermore, F is the target-space fermion number, i.e. F = 0 for p R ∈ Γ vec and F = 1 for p R ∈ Γ spin .
Eqn. (2.12) can also be obtained as follows: the eight real worldsheet fermions ψ R = (ψ µ R , ψ i R ) can be grouped in four complex fermions ψ R = (ψ m R , ψ a R ), where m = 1, . . . , d/2 − 1 and a = 1, . . . , D/2 correspond to the uncompactified and compactified dimensions, respectively. Then, one can bosonize the complex fermions. Consequently, the bosonized fermions carry momentum p R = (p m R , p a R ) and the associated partition function coincides with eqn. (2.12). The momentum p m R has an important target-space interpretation: A string state with p m R being integer or half-integer signals a target-space boson or fermion in d dimensions, respectively.

Modular invariance
The full partition function is required to be modular invariant: At one-loop the worldsheet has the topology of a torus with modular parameter τ . Not all τ ∈ C with Im(τ ) > 0 parametrize inequivalent worldsheet tori. Because of conformal symmetry tori related by the modular transformations give the same physics. T and S generate the modular group PSL(2, Z). Invariance of the partition function (2.8) under T and S transformations requires that ∀ P ∈ IΓ : 1 2 P T η P ≡ 0 and IΓ * = IΓ , (2.14) where a ≡ b means that a and b are equal up to some integer. These conditions tell us that IΓ is an even self-dual lattice with signature (D, D + 16); the so-called Narain lattice. Note that vectors P ∈ IΓ can be redefined as for U ∈ O(D; R) × O(D + 16; R) without changing the partition function (2.8).

Narain lattices
We analyse the conditions (2.14) in more detail. To do so, we may parametrize a general lattice vector P ∈ IΓ as 16) in terms of an invertible matrix E. This matrix E is called the generalized vielbein of the Narain lattice IΓ as its columns correspond to 2D + 16 basis vectors of the lattice IΓ. The components of the vector N can be interpreted as winding numbers m ∈ Z D , Kaluza-Klein numbers n ∈ Z D and gauge lattice numbers q ∈ Z 16 . From the vielbein E we can define the Narain metric η as Then, the scalar product of two vectors P i = E N i ∈ IΓ for i = 1, 2 is given by Hence, the lattice IΓ is even if Note that an even lattice is automatically integral, i.e. P T η P = N T ηN ∈ Z. Therefore, the Narain metric η is a symmetric, integer matrix with even entries on the diagonal and signature (D, D + 16). The dual lattice IΓ * is spanned by the dual vielbein E * which is defined as so that for a given P = E * N ∈ IΓ * we have P T ηP ≡ 0 for all P = E N ∈ IΓ. By comparing this equation with (2.17) one infers that the dual basis is given by  Consequently, det η = ±1 and we see from eqn. (2.17) that the volume of the unit cell spanned by the vielbein E is given by vol(IΓ) = ± det E = 1. It is often convenient to choose a special representation of the Narain metric. If not stated otherwise we will use where g is the metric of an even, self-dual 16-dimensional lattice. (Throughout this paper we use a hatted notation to refer to objects that are naturally defined in the lattice basis.) We choose it to be the Cartan matrix of E 8 × E 8 and write g = α T g α g where the columns of α g are the 16 simple root vectors of E 8 × E 8 . The explicit expression for α g is given by The columns of α(E 8 ) represent the eight simple roots α I (E 8 ), I = 1, . . . , 8, of the exceptional Lie algebra E 8 . They can be chosen as follows (2.25)

The Narain moduli space
Given the choice of a Narain metric η in eqn. (2.23) it is natural to look for a corresponding generalized vielbein E, which yields this Narain metric E T η E = η. We see that a particular solution R to equation (2.17) is given by The general solution to (2.17) can be written in terms of this particular solution as so that consequently, In the following we want to identify which transformations U and E in eqn. (2.27) map between physically inequivalent theories and which do not. Therefore, we will identify the moduli space of heterotic Narain constructions. To do so, we define 2 Then, one can absorb U into a redefinition of E by defining E as Therefore, E contains the parameters (i.e. the moduli) that continuously deform the Narain lattice with vielbein R to Narain lattices with vielbeins R E, which are in general physically inequivalent but share the same Narain metric η. However, not all vielbeins E are physically inequivalent: Consider two vielbeins E, E for two Narain lattices IΓ, IΓ satisfying (2.17). Under what condition(s) do these backgrounds describe the same Narain lattice IΓ = IΓ? This happens when for each point P ∈ IΓ there is a unique point P ∈ IΓ which is identical to it: In the parametrization (2.16) this amounts to In the remainder of this paper we will use this conjugation with R to switch between Oη and O η group elements.
which is unphysical as discussed above. Hence, the Narain lattices IΓ and IΓ are the same if there exists a rotation matrix U such that M ∈ GL(2D + 16; Z). Moreover, we assumed that both E and E give the same Narain metric η, see (2.17). This implies that the matrix M is actually an element of the so-called T -duality group O η (D, D + 16; Z), i.e.
is given by provided that the constraints u T r u r = u T l u l + u T Ll u Ll = 1 D , u T lL u lL + u T L u L = 1 16 and u T l u lL + u T Ll u L = 0 are fulfilled. As we have already seen above, often the closely related matrix Hence, E = E(e, B, A) is parametrized by the Narain moduli e, B and A, where e is the D-dimensional vielbein of the D-torus with metric G = e T e. A is a 16 × D matrix, whose i-th column contains the Wilson line which is associated to the i-th basis vector in e and, finally, B denotes the anti-symmetric Kalb-Ramond B-field. In summary, we can specify the most general form of the generalized vielbein E with Narain metric η = E T η E as given in eqn. (2.23). It reads

Equivalent Narain metrics
One may encounter different Narain metrics, say η and η from GL(2D + 16; Z), such that In this case one cannot immediately compare the moduli in E and E , because their hatted versions E and E lie in two different moduli spaces. Since we are talking about two representations of the same Narain lattice we have Hence, we assume in the following that the Narain metric η is given by eqn. (2.23).

Coordinate fields and momenta
Consider the generalized vielbein in its most general form, i.e. E = U R E M , and choose U = 1 and M = 1, see eqn. (2.40). Then, a Narain lattice vector P is represented as It can be thought of to describe both: On the one hand, L ∈ IΓ defines the periodicity for the compactification on a Narain lattice, see eqn. (2.7). On the other hand, P ∈ IΓ gives the conjugate momentum, see eqn. (2.10).
The matrix R induces the change of right-and left-moving coordinate fields, y r , y l and y L , to D mixed fields X,X and the remaining 16 left-moving gauge coordinates X g . This relation thus defines which combination of right-and left-moving degrees of freedom are interpreted as the physical coordinates X and which as the dual coordinatesX. The torus periodicities, read in terms of the coordinates X, their dualsX and gauge coordinates X g On-shell the right-and left-moving coordinate fields, y r , y l , have anti-holomorphic and holomorphic mode expansions for a string with boundary condition (2.46) given by respectively. Using the change of coordinate field basis (2.45), we see that the conventional coordinate field X and its dual X have the expansions The term linear in the worldsheet space variable σ 1 of X gives the D-dimensional winding modes, i.e.
The term linear in the worldsheet time variable σ 0 of X corresponds to the D-dimensional momentum which is given by As expected, for the dual coordinate X the roles of momentum and winding are interchanged.

The T -duality group
This section is devoted to exhibit a number of properties of the T -duality group. In particular, we develop a convenient basis for this group and parametrize its maximal compact subgroup. In addition, we show that the non-linear transformations of the Narain moduli is a consequence of the coset structure in which the generalized vielbein E lives.  Table 1. These matrices are chosen such that if we restrict the parameters to be from Z rather than R, these matrices have only integral entries.

Decomposition of the generalized vielbein
As a first application of the matrices of Table 1, we decompose the generalized vielbein (2.39) as a product Table 1. Here, the index i = e, B, A labels the matrix M i and each matrix M i depends on the corresponding kind of Narain moduli e, B and A. This parametrization will turn out to be very useful throughout this paper.

Coset decomposition of the T -duality group
In Section 2.4 we recalled that the moduli space of Narain compactifications can be described geometrically as a coset space (2.35). This already shows the central role that the coset space plays in our discussion and therefore we expand on this property in some detail here. The generalized vielbein E is an element of the coset O(D; R)×O(D + 16; R)\O η (D, D + 16; R) .

(3.2)
This means that any element H ∈ O η (D, D + 16; R) can be decomposed as where the specific standard form (2.39) of the generalized vielbein E lies inside the coset (3.2) and ). This equation (3.4) will be used frequently throughout this paper, for example, when we discuss T -duality transformations of Narain moduli in Section 3.3 and when we analyze the stabilization of Narain moduli in generalized orbifolds in Section 5. (with the additional requirement that 1 2 ∆A T ∆A and 1 2 ∆α T ∆α are integer matrices). The elements listed in the first two rows generate the geometric subgroup G geom of the duality group. The elements on the third row correspond to true T -duality elements that invert one or all radii. Note the difference between α g and ∆α: α g contains the simple roots of E 8 ×E 8 and is used in the definitions of M W (∆W ), M A (∆A) and M α (∆α), while ∆α is the parameter of M α (∆α).

Compact subgroup in the coset decomposition
In what follows, we consider eqn.
which will recur frequently throughout the rest of this work. Next, we compute the products of matrices contained in eqn. where each matrix is given in its 3 × 3-block structure, e.g. U M is given in eqn. (2.38). The result is set equal which yields 3 × 3 = 9 equations from eqn. (3.4). By doing so, we can solve for the blocks of U M = R U M R −1 as defined in eqn. (2.38) and obtain for arbitrary u r ∈ O(D; R). We have checked explicitly that these equations give a matrix U such that the conditions (2.36) are satisfied. Let us remark one observations from eqn. (3.10a): M 12 = 0 is a necessary condition for u r = u l . In other words, if M 12 = 0 then u r = u l . In addition, let us stress that these equations (3.10) will become very important later in the context of Narain orbifolds where U becomes the orbifold twist Θ, for example in Section 5.2. Furthermore, we identify the following three expressions from eqn. (3.4), which we use in the following discussion.

Transformation of Narain moduli
Using the coset decomposition discussed above, we can derive the transformation properties of the Hence, we are able to identify the transformation properties of e, G + C T and A under general T -duality transformations from eqn. (3.2). We find where u T r u r = 1 D . These transformations can be expanded out (by taking the anti-symmetric part of G + C T to solve for B ) and we obtain the transformations of the moduli G, B, A, i.e.

Specific elements of the T -duality group
Next, we discuss various elements and subgroups of the group O η (D, D + 16; R) in detail and analyze their actions on the Narain moduli G, B, A. The parametrizations of these subgroups can be found in Table 1 and their most important products are given in Table 2.

The geometric subgroup
The elements M e , M W , M A and M B as listed in Table 1 generate a subgroup of O η (D, D+16; R) which we denote by G geom (R). This is the largest T -duality subgroup, that still admits a standard geometrical interpretation, hence the name: geometric subgroup. In more detail, all elements M geom ∈ G geom (R) can be cast to the form Then, using the results of Section 2.4 we see that the generalized vielbein (3.1) transforms under M geom as is given in the standard form (3.1). Furthermore, in eqn. (3.16) we have used various group multiplication properties as given in Table 2 to compute the product E(e, B, A) M geom (analogously, one could have used the general transformations (3.13) and (3.14) for M = M geom to derive eqn. (3.16)).
Notice that under a M W (∆W )-transformation the form of the generalized vielbein is not strictly preserved. Nevertheless, it is of the correct form such that it can be absorbed by the choice of In the following, we give details for various elements of the T -duality group. We start with the four generators M e , M W , M A and M B of the geometric subgroup G geom (R) and use eqns.  of the Wilson lines, while G and B remain invariant.
In the case of the discrete T -duality group we define Hence, ρ W is an automorphism of the E 8 × E 8 root lattice spanned by α g .  Wilson line shifts are generated by M A (∆A) with α −1 g ∆A ∈ M 16×D (R). Indeed, we obtain Hence, transformations of the Wilson lines A are accompanied by a B-field transformation, while the metric G is kept invariant. Furthermore, we find where we remark that Wilson line shifts and B-field shifts commute, see Table 2.

Non-geometric elements
In the following, we give details for non-geometric elements of the T -duality group. We use eqns. (3.14) in order to compute the transformation of moduli.

T -duality inversions
We can define Z 2 involutions where i denotes the standard basis vector in the i-th torus direction. The element I (±i) can be written as conjugation of a reflection in the i-th left-or right-moving direction as I (±i) = R −1 I (±i) R using Therefore, all the elements I (±i) can be obtained from I (±1) by conjugation with an appropriate change of basis element M e (∆K). The element I (−i) induces a T -duality inversion along the i-th torus direction. We can preform the T -duality inversion in all torus directions simultaneously by as given in Table 1. Using the general results (3.14) we find for this element Inverted B-field shifts M β (∆β) Even though the following two elements M β (∆β) and M α (∆α) can be obtained by combining the Band A-shifts with the inversion element I, we list them explicitly as they are important in the context of non-geometry. Inverted B-field shifts, often referred to as β-transformations, are generated by with ∆β T = −∆β ∈ M D×D (R). The β-transformations of the metric, B-field and gauge backgrounds take the form .

Inverted Wilson line shifts M α (∆α)
Finally, by inverting the Wilson line shifts M A we obtain . The inversion of changes of bases, i.e. I M e (∆K) I and I M W (∆W ) I, just become changes of bases again. Hence, they do not give us novel transformations. For completeness we nevertheless list them in Table 2 Table 1 and the matrix R defined in eqn. (2.26). Then, we obtain By comparing this with eqn. (2.37) one can read off the expressions for the submatrices and u r = ∆θ. One can verify that these expressions satisfy the constraints (2.36).
In addition, for a given element U ∈ O(D; R) × O(D + 16; R) one can use eqn. (3.36) to decompose where we assumed that u + is invertible.

Generalized space groups of Narain orbifolds
In this section we introduce the generalized space group for heterotic Narain orbifolds and discuss some of its properties. In particular, we define orbifold projections to characterize quantization conditions of the generalized shift vectors and state the conditions to preserve N = 1 supersymmetry.

Heterotic Narain orbifolds
Next, we consider orbifolds of the heterotic Narain lattice construction denoted by Here, the 2D + 16-dimensional torus T 2D+16 IΓ is specified by a Narain lattice IΓ. In addition, the Narain point group P is defined as a (sub-)group of the rotational symmetries of IΓ, as we will see later in eqn. (4.13). Hence, the Narain point group P is finite. The generators of P are (2D + 16) × (2D + 16) matrices and they are denoted by Θ α , for α = 1, . . . , N P . K α is the order of Θ α . In more detail, for each generator Θ α , the order K α is the smallest non-negative integer such that Θ Kα α = 1. Elements of P are often called twists. In the following, a generic twist will be denoted by Θ and K gives its order.
To define the compactification of the heterotic string on a Narain orbifold [40,53], the main idea is to generalize the boundary conditions (2.7) of the 2D + 16-dimensional right-and left-moving coordinate-vector Y to for all elements Θ ∈ P and L ∈ IΓ. In general, for each twist Θ there is a so-called generalized shift V Θ associated to it, which will be discussed in detail later. Importantly, the twists Θ are not allowed to mix right-and left-moving fields in eqn. (4.2). Hence, for all Θ ∈ P we demand Consequently, we find the conditions for all generators Θ α of the Narain point group.
Furthermore, we call a Narain orbifold symmetric [5,6], if there is a basis such that all generators Θ α ∈ P are simultaneously of the form If such a basis does not exist, then the corresponding Narain orbifold is asymmetric. Even though this definition of symmetric orbifolds involves a choice of basis, this property is in fact basis independent. Nevertheless, in a given basis it might be difficult to see whether a Narain orbifold is symmetric or asymmetric: One can bring a symmetric twist Θ sym into a seemingly asymmetric twist Θ asym = However, the conjugation with U can neither change the orders of θ r and Θ L , nor the two finite groups which are generated by either θ α r or Θ α L .

Generalized space group
It has been proven to be very convenient to employ a space group formulation of the heterotic string on symmetric orbifolds, especially in the context of classifications [7]. This language can be extended to Narain orbifolds naturally. The generalized space group S associated to a Narain orbifold is defined as being generated by the elements 1, L and Θ α , V α for all L ∈ IΓ and Θ α ∈ P , where V α , a vector with 2D + 16 components, is the so-called generalized shift which is associated to the twist Θ α . Conversely, we demand that for all space group elements of the form (1, L ) ∈ S it follows that L ∈ IΓ. So, the Narain lattice IΓ is the subgroup of S that contains all pure translations of S. Note that a generator (Θ α , V α ) is a generalized roto-translation if V α = 0, see [7]. These generators build the so-called Narain orbifold group O, which is defined modulo lattice translations. Hence, just as P, the Narain orbifold group O is a finite group.
A general space group element g = (Θ, λ) ∈ S is defined to act on Y as Consequently, the unit element of S is given by (4.8) The inverse element g −1 of g = (Θ, λ) ∈ S reads Furthermore, two elements g = (Θ, λ) and g = (Θ , λ ) are multiplied as Hence, the generalized space group S is in general non-Abelian even if the Narain point group P is Abelian.
For orbifolds, each sector of string states is characterized by a boundary condition (4.2) and, thus, by the so-called constructing element g = (Θ, λ) ∈ S, where λ = V Θ + L and L ∈ IΓ. Only those elements g ∈ S that commute with the constructing element g yield projections and, hence, give rise to non-vanishing contributions to the twisted sector partition function. This only happens when ΘΘ = Θ Θ and (1 − Θ)λ = (1 − Θ )λ . (4.11)

Conditions on the twists Θ α
Furthermore, we choose L ∈ IΓ and consider Thus, the lattice IΓ is a normal subgroup of S and the Narain point group P has to consist of automorphisms of the Narain lattice, i.e.
In addition, we have to impose eqn. (4.4) on the twist generators Θ α . It is is interesting to pause here and reflect on the possible orders of twists for a given number of dimensions D Γ for general orbifolds associated to a lattice Γ. As is well-known [87], if the order K satisfies then there exists at least one lattice Γ with rotational symmetry of order K. Here, φ(K) is the Euler φ-function and this bound does not take into account that one can build point groups as direct sums of lower dimensional cases. However, in the current paper we are not working with a general lattice Γ in D Γ dimensions, but with Narain lattices Γ = IΓ with D IΓ = 2D + 16. Hence, contrary to the Euclidean case, it is not guaranteed that there exists a Narain lattice for each order K satisfying the bound (4.14).

Orbifold projections of IΓ
In general, a twist Θ ∈ P of order K acts as the identity in some directions of Y while it acts as a Z K twist on others. To identify these directions, we define projection operators for each twist Θ ∈ P: The projection operators P Θ and P Θ ⊥ project a vector onto the directions in which Θ acts trivially and non-trivially, respectively. In detail, we define the projectors with the properties Then, any vector λ ∈ R 2D+16 can be decomposed into two vectors λ Θ and λ Θ ⊥ according to and Θ λ Θ = λ Θ . The final relation clarifies the use of the subscript : It defines the directions which are left invariant by Θ. Moreover, it is important to realize that the projected Narain lattice IΓ Θ = P Θ IΓ is in general not Narain. In detail, even if IΓ and Θ IΓ are Narain lattices, see eqn. (4.13), the normalisation 1/K in the projection operator P Θ in eqn. (4.15) can make IΓ Θ non-Narain. A Narain lattice is said to be factorized w.r.t. the orbifold twists when for all twists Θ ∈ P. In this case, obviously, all projected Narain lattices are themselves Narain again.

Quantization of the generalized shifts V α
For each Narain point group generator Θ α of order K α we consider the generator (Θ α , V α ) of the generalized space group S. Then, its K α -th power reads (where P α = P Θα ) without summation over α. Consequently, we have to demand the condition That is, the shift V α needs to be quantized in units of K α in the directions where Θ α acts trivially, i.e. V α is given by The same procedure can be applied to some arbitrary element Θ ∈ P of order K with associated element (Θ, V Θ ) ∈ S. This yields As a remark, for example in the case when Θ k α has a fixed torus for 0 < k < K α (i.e. when Θ k α has more invariant directions than Θ α ) eqn. (4.22) gives stronger quantization conditions on the shift V α than eqn. (4.21).
Various choices for V α correspond to the same Narain orbifold. Indeed, one can shift the origin, i.e.
and hence transform the generalized shifts

Preserving at least N = 1 target-space supersymmetry
To enable the discussion on target-space supersymmetry we first need to recall a few facts about supersymmetry on the worldsheet. By construction the heterotic string has (1, 0) worldsheet supersymmetry. Hence, we can identify the worldsheet supercurrent where ψ R = (ψ i R ) are the real worldsheet fermions of the D compactified dimensions and u Rr is a D × D matrix. For each twist Θ α , the space group action (4.7) is defined to be accompanied by a transformation of ψ R as where θ α R ∈ O(D; R). Since the first term ψ µ R∂ x µ in eqn. (4.24) is orbifold invariant the worldsheet supercurrent T F has to be orbifold invariant as well. Consequently, we need to require that the twists on the right-moving coordinates y r and on the right-moving fermions ψ R are identified: θ α R = u Rr θ α r u −1 Rr . Given that the properties of target-space fermions are determined by the right-moving momentum p R associated to these right-moving fermions, as given eqn. (2.12), the question of target-space supersymmetry is only affected by the transformations generated by θ α R in the right-moving sector. In particular, target-space supersymmetry is independent of the choice one makes for Θ α L . Only if one restricts oneself to symmetric orbifolds, for which θ α L = θ α l ⊕ 1 16 and θ α l = θ α r = θ α R with u Rr = 1 D , see eqn.
Here, we introduced the so-called twist vector φ α R = (φ m α R , φ a α R ) as the vector of phases corresponding to θ α R , such that θ α R acts as using the complex indices defined below eqn. (2.12). In fact, the last condition of eqn. (4.27) only needs to be imposed mod integers (i.e. ≡) and this specific choice fixes the unbroken supercharges for d = 4 and φ a α R = 0 to be represented as ±( 1 2 , 1 2 , 1 2 , 1 2 ).

Moduli stabilization in Narain orbifolds
As we have seen in the previous section, the space group description of Narain orbifolds is naturally formulated using the twist Θ and the generalized vielbein E. On the other hand, the question about moduli stabilization and classification, in particular, are more conveniently discussed in the so-called lattice basis in which the twist is encoded by an integral matrix ρ. Therefore, we begin this section with a discussion of Narain orbifolds in the lattice basis. Beside the integral twist matrices ρ, we introduce the generalized metric H and a closely related Z 2 -grading Z. After that we investigate under which conditions Narain orbifolds exist and derive restrictions on the Narain moduli that have to be imposed in order to be compatible with the orbifold action. In particular, we derive a character formula that counts the dimension of the orbifold Narain moduli space.

Narain orbifolds in the lattice basis
Twists and shifts in the lattice basis We have seen in eqn. (4.13) that each point group generator Θ α has to map a Narain vector E N to another Narain vector E N = Θ α E N , see eqn. (2.16). It follows that N = ρ α N , where we define ρ α as Here, we used E = U R E and we absorbed U in the definition of Θ The matrices ρ α represent the generating twists Θ α in the so-called lattice basis. They have to be invertible over the integers (i.e. ρ α ∈ GL(2D + 16; Z)) because each ρ α has to map an integer vector N one-to-one to another integer vector N . Furthermore, they inherit the following conditions while twists Θ α ∈ P are given in the so-called coordinate basis. The lattice basis will be of special importance for the classification of Narain orbifolds later in Section 6.1. Moreover, the space group generators (Θ α , V α ) and (1, L) ∈ S can be represented in the lattice basis as In other words, condition (5.4) states that the generators ρ α and the generalized metric H have to be compatible. The generalized metric is given explicitly by as follows from its definition (5.5) .

A Z 2 grading
The compatibility condition (5.4) of the orbifold twists in the lattice basis can also be represented as  The second expression in this equation is obtained using I = R −1 η R, as given in Table 1, and the relation R T R = η I. Explicitly, Z is given by The constraints (5.7), which the generalized metric satisfies, translate to the following conditions on Z: Z T η Z = η and Z 2 = 1 .

On the existence of Narain orbifolds for a given point group
Assume a given finite point group P ⊂ O η (D, D + 16; Z) with generators ρ α in the lattice basis. We want to understand these generators ρ α as the crucial ingredient in the definition of a Narain orbifold. Therefore, we have to address the following question: Under which condition does a corresponding Narain orbifold exist? In terms of the terminology introduced in Section 4 this can be phrased as follows: When does a Narain lattice exist, such that all generators Θ α of the corresponding group P in the coordinate basis satisfy (4.4) and are symmetries of this lattice (4.13)?
In the following, we will answer this question in the lattice basis. Then, the conditions on Θ α translate to conditions (5.2) and (5.4) on ρ α ∈ P. In fact, eqn. (5.2) is fulfilled by assumption (i.e P ⊂ O η (D, D + 16; Z) and finite). Thus, it remains to show that eqn. (5.4) is fulfilled, i.e. we have to find a generalized metric that is compatible with all generators ρ α . Consequently, a Narain orbifold with given point group P exists if one finds Narain moduli G, B and A that are invariant under ρ α ∈ P.
If such a generalized vielbein exists, then generically, not all the moduli of the Narain torus compactification are still free; some Narain moduli are stabilized. Thus, we can use our discussion on the transformation properties of Narain moduli under general T -duality transformations in Section 3.3 in order to derive conditions for moduli stabilization.
To address these questions, we study the existence of both a twist (where θ α r is the u r part of the matrix Θ α as defined in eqn. (2.38)) that determine the Narain moduli uniquely already. Expanding out eqn. (5.13), we obtain where ρ α r := e −1 θ α r e. (Note that there is a redundancy between e and θ α r , which reflects the fact that the vielbein e is not uniquely determined by the metric G.) It is sufficient to solve only these three matrix equations (5.14) in order to find a solution of all nine equations (5.12) because of the coset decomposition (3.4): Indeed, we can alternatively obtain the set of coupled equations (5.14) by comparing eqn.
for each generator ρ = ρ α of the point group P. These conditions can be thought of as algebraic Riccati equations (see e.g. [89]) which constrain some and sometimes even all the moduli G, B and A. Hence we have reduced the existence question of Narain orbifolds to the question whether these Riccati equations admit real solutions.

Mapping from the lattice basis to the coordinate basis
Assume we are given a finite point group P ⊂ O η (D, D + 16; Z) with generators ρ α in the lattice basis and we want to know a compatible Narain lattice as well as the twists Θ α in the coordinate basis.
To obtain this data we can perform the following steps: First, we find a solution to eqns. (5.17), i.e. find orbifold invariant moduli G, B and A. After that we make a choice for a geometrical vielbein e such that e T e = G. By doing so, we have obtained a generalized vielbein E = R E(e, B, A), which is compatible with P in the sense of eqn. (5.12). Finally, we compute the twists in the lattice basis: Using the geometrical vielbein e we can determine the right-moving twists θ α r = e ρ α r e −1 , where ρ α r is given by eqn. (5.14b). Consequently, we can compute the blocks of Θ α from eqn. (3.10), i.e.
where M α i for i = 1, 2, 3 are defined in eqn. (3.8) setting M = ρ α . This method we will be exemplified in Section 8 where we discuss a number of two-dimensional Narain orbifolds.
However, the converse is in general not true. In Section 8 we provide examples for both cases: In Section 8.2 we list several Narain orbifolds that are necessarily symmetric because ( ρ α ) 12 = 0 and in Section 8.4 we give one Narain orbifold that is symmetric even though ( ρ α ) 12 = 0.

Dimensionality of the Narain orbifold moduli space
Assuming that a Narain orbifold exists, i.e. assuming that we have found a generalized vielbein E 0 that satisfies eqn. (5.12), we want to determine the number of unconstrained Narain moduli. In other words, we want to count the number of moduli perturbations δH that can deform the associated generalized metric H 0 such that H 0 + δH remains invariant under the Narain orbifold action.
To address this question, we make use of the results from Appendix A and set H = P. Then, the tangent space to the orbifold-invariant moduli space is given by M P = δm P = P P δm , (5.20) where the projection operator P P is defined in eqn. (A.11). The moduli deformations δH, can be parametrized as follows where δB = δB + 1 2 δA T A 0 − 1 2 A T 0 δA, δG = δe T e 0 + e T 0 δe. According to eqn. (A.13) the dimension of the orbifold-invariant Narain moduli space, i.e. the number of moduli, is determined by where we have introduce the right-and left-characters The number of fixed moduli is given by D(D + 16) − dim(M P ). In particular, all Narain moduli are frozen if dim(M P ) = 0. In this case, the Narain orbifold moduli space M P is a point (or a set of disjoint points). This happens when the right-and left-characters (5.23) are orthogonal. In light of this, we can use the property that characters of irreducible representations form an orthonormal basis to analyze eqn. (5.22). In detail, for two (complex) irreducible representations µ and ν of the finite point group P we have This can be used to construct some situations with all moduli fixed, i.e. dim(M H ) = 0: • If the matrix representations of θ r and Θ L are both irreducible, they have to be different, since the former is D-dimensional while the latter is (D + 16)-dimensional, and hence, their characters are orthogonal.
• If the representations of θ r and Θ L are reducible, one can decompose them into irreducible ones as where the irreducible representations θ r µ and Θ L ν are in general complex. Hence, if and only if θ r and Θ L do not contain any irreducible representation in common, again the characters χ r and χ L are orthogonal. An particular example of this is obtained, when Θ L = 1 and θ r does not contain any trivial one-dimensional representations of P.

A T -fold constructed as an asymmetric Z 2 Narain orbifold
To illustrate the various results, we conclude this section by considering a simple but instructive construction of a T -fold: We define an asymmetric Z 2 Narain orbifold by choosing see Table 1. First, we identify a specific example of a compatible Narain lattice using the Z 2 grading Z.
Then, we will use the discussion from Section 5.2 to see that this is actually the most general solution.
Finally, we confirm this by counting the number of unstabilized Narain moduli using Section 5.4.
To find a compatible Narain lattice, we notice that Z = I is a valid Z 2 grading satisfying eqn. (5.8).
In fact, all Narain moduli are stabilized in this case as we are going to show next. We use eqn. (3.8) with M = I, which yields  The fact that all Narain moduli are stabilized in this example is also easy to understand using the number of unstabilized Narain moduli dim(M P ), see eqn.

Towards a classification of Narain orbifolds
In this section we would like to lay the foundations for a classification of inequivalent Narain orbifolds. In general, the key to a classification of any structure is to identify those transformations that relate (or even define) equivalent structures. These transformations can be used to define equivalence relations that consequently give rise to equivalence classes. For the classification of D-dimensional -geometrical -orbifolds the structure turns out to be the space group and the equivalence relations are based on the notions of Q-, Zand affine-classes [7]. In this section we show that extending these notions to generalized space groups is the key for a classification of Narain orbifolds.
In more detail, for the classification of Narain orbifolds we identify three main structures: (i) the integral Narain point group P of finite lattice automorphisms, (ii) an associated Narain lattice IΓ (given by a geometrical torus with metric G, a B-field and Wilson lines A) that is compatible with the point group and, finally, (iii) the full generalized space group S, which fully specifies a Narain orbifold as we have seen in Section 4. The main purposes of this section are to define equivalences for these three structures, namely Narain Q-, Zand Poincaré-equivalences, together with their associated equivalence-classes and to analyze their interpretations.

Narain Qand Z-classes
For the definition of Narain Qand Z-classes we need to describe the Narain point group in the lattice basis, where P ⊂ O η (D, D + 16; Z), see Section 5.1. Then, one only has to consider integral finite order elements ρ α ∈ P. Since Narain Qand Z-classes are analogously defined, we take the field F to be either Qand Z and begin with the definition of F-equivalence: Two  Note that if two point groups are from the same Z-class they are also from the same Q-class, because if M ∈ GL(2D + 16; Z) then M ∈ GL(2D + 16; Q). But the converse is not true, i.e. two point groups from the same Q-class can be in inequivalent Z-classes.

Interpretation of Narain Qand Z-classes
To prepare the interpretation of the Narain Qand Z-classes, let us assume that two Narain point groups P and P are from the same F-class, where the field F is either Q or Z. Then, there exists a matrix M ∈ GL(2D + 16; F) such that for each generator ρ α ∈ P there is a generator ρ α ∈ P with Now, consider a Narain lattice spanned by a generalized vielbein E, such that E is compatible with all generators ρ α and insert eqn. (6.3), i.e.
Consequently, we find Hence, we can interpret eqn. (6.5) as follows: If P is a symmetry of a Narain lattice with generalized vielbein E and Narain metric η then P is a symmetry of a Narain lattice with generalized vielbein E = E M and Narain metric η = M T η M . Furthermore, we notice that both point groups have the same geometrical action Θ α which corresponds to both ρ α and ρ α . In other words, the corresponding point groups P and P in the coordinate basis are identical (up to a trivial basis change) for point groups from the same F-class. Consequently, the question of symmetric or asymmetric orbifolds, the number of unbroken supersymmetries in d uncompactified dimensions and the number of invariant Narain moduli eqn. (5.22) are also equal. This is independent of the choice for the field F to be Q or Z.
Next, we have to distinguish between these two Narain classes: Let us first consider the case That is, even though we have seen in eqns. (6.4) and (6.5) that the Narain point groups P and P are identical in the same F-class, their generators Θ α and Θ α = U −1 B Θ α U B can look different, for example, one is symmetric and the other looks asymmetric. This is the case if one chooses the corresponding Narain lattices as different points, specified by (e, B, A) and (e , B , A ), in the same representation of the Narain moduli space, i.e. with the same U in eqn. (6.8). As an example for eqn. (6.8), we will discus two F-equivalent Z 3 point groups P (1) and P (2) in Section 8.4, where the point group P (1) is symmetric while P (2) looks asymmetric due to a non-trivial transformation U B .

Narain Poincaré-classes
As final type of equivalence transformations, we want to generalize affine transformations (F, λ) of Euclidean D-dimensional orbifolds (with linear mapping F ∈ GL(D; R) and translation λ ∈ R D ) to the Narain case. Importantly, the (2D + 16)-dimensional Narain lattice is equipped with a metric η with signature (D, D + 16), which has to be preserved by any transformation. Hence, it is essential for the Narain case to restrict affine transformations in 2D + 16 dimensions to Poincaré transformations (F, λ) of the Narain lattice, where F ∈ O η (D, D + 16; R) and λ ∈ R 2D+16 . Therefore, we need to introduce Poincaré-classes instead of affine classes in order to describe Narain orbifolds.
This might give the impression that Poincaré transformations of Narain orbifolds are more restrictive than affine transformations of ordinary Euclidean orbifolds. This is not the case since  Table 1 with ∆K ∈ GL(D; R). Consequently, Poincaré-classes generalize the notion of affine classes to Narain orbifolds.
In light of this, we define the following equivalence relation: Consider two Narain orbifolds, i.e. two space groups S (1) and S (2) with point groups in the same Z-class. Two such Narain space groups are defined to be equivalent if there exists a Poincaré transformation (F, λ) with F ∈ O η (D, D + 16; R) and λ ∈ R 2D+16 such that More explicitly, in terms of the generators (Θ (κ)α , V (κ)α ) and (1, L (κ) ) of the space groups S (κ) for κ = 1, 2 this reads

Interpretation of Narain Poincaré-classes
First of all, we show that two generalized space groups from the same affine class correspond to the same Narain orbifold but possibly at different points in the moduli space. To see this, let us denote the generalized vielbeins that specify the Narain lattices from the respective generalized space E(e (κ) , B (κ) , A (κ) ) is given in eqn. (3.1). Since L (κ) = E (κ) N (κ) are related by the transformation (6.10), a Poincaré transformation (F, λ) of the corresponding generalized vielbeins E (1) and E (2) is given by where we assume without loss of generality that we do not perform a discrete T -duality transformation (i.e. N (2) = N (1) ). This can be rewritten as where  (1) and A (2) = A (1) . In this case, also the left-and right-moving mass formulae of the heterotic string stay the same. So far we only gave an interpretation of the first equivalence relation in eqns. (6.10). The second relation tells us that the orbifold twists can take various guises by conjugation with F ∈ O η (D, D+16; R).
The third equivalence relation in eqns. (6.10) can be interpreted by resorting to the decomposition mentioned in Section 4.5.

Symmetric orbifolds as Narain orbifolds
The main objective of our study in this paper is to set up a framework to investigate asymmetric orbifolds. Nevertheless, it is very instructive to apply the Narain formalism also to symmetric orbifolds [5,6]: It provides us with a unified view on both, geometric moduli and Wilson lines [8].
Moreover, this case can be used to illustrate the power of the T -duality group approach in the investigation of moduli stabilization. For concreteness and simplicity, we only consider symmetric Z K orbifolds in this section. Extending the discussion is straightforward, yet beyond the scope of the present paper.

Symmetric Z K orbifolds
The Narain point group of a symmetric Z K orbifold is generated by a single twist Θ of order K and the associated generator of the generalized space group is given by (Θ, V ). For the orbifold to be symmetric, we choose the twist Θ to be of the form given in eqn. (4.5). Thus, we obtain for Θ k , k = 1, . . . , K, see Table 1 and using θ T θ = 1 D . Using the definition (5.1) of the integral matrix ρ we can subsequently obtain an expression for ρ k , which can be further evaluated with the help of the multiplication Table 2 for T -duality group elements. This yields where we definedθ = e −1 θ e (7.3a) Since ρ is an integral matrix,θ, ∆B k and ∆A k all have to be constant, i.e. moduli-independent, matrices. As a cross-check, let us confirm that for k = K we obtain ρ K = 1: Indeed, in this case we getθ K = 1 D , ∆A K = 0 and ∆B K = 0 and consequently, ρ K = M e (1 D ) = 1, as required. Furthermore, we find from eqn. (7.2) that ρ is an element of the discrete geometric subgroup G geom (Z) ⊂ Oη(D, D + 16; Z), see eqn. (3.15) with ∆W = 1 16 .
The twist Θ is in general accompanied by a shift V T = (V T r , V T l , V T L ), see eqn. (4.6). As we have seen in Section 4.5, the shift is quantized, i.e. KV Θ = E N V ∈ IΓ. It is instructive to analyze this in more detail for the case that θ rotates in all D compact dimensions. Then, the projection operator eqn. (4.15) reads and we obtain the condition This is solved by where Λ E 8 ×E 8 denotes the root lattice of E 8 × E 8 and we used eqn. (2.44). Hence, V L is the gauge shift vector of order K known to the symmetric orbifold literature, e.g. [13,52]. Furthermore, we can set V r = V l = 0 by shifting the origin using the transformation (4.23).

Moduli stabilization in symmetric Z K orbifolds
The fact that even for symmetric Z K orbifolds a certain number of moduli, G, B and A, become constrained, can be inferred in two ways: First of all, the conditions (7.3) can be obtained from eqns. (7.2), as shown above by using the fact that for symmetric orbifolds the twist ρ is an element of the geometric subgroup G geom (Z) ⊂ Oη(D, D + 16; Z). A second derivation of eqn. (7.3) follows from the general discussion in Section 5.2, which is valid for both, symmetric and asymmetric orbifolds: To see this, we use and in addition we have ρ r = e −1 θ r e =θ. Consequently, the Narain moduli are constrained by eqns. (5.16), which are equivalent to eqns. (7.3). Thus, we found two equivalent ways to derive the conditions (7.3) for Narain moduli stabilization in the case of symmetric orbifolds.
We start with fixing moduli in the metric G. From eqn. (7.3a) and Θ T Θ = 1 we obtain the which fixes some of the moduli, as is well-known. The general solution to eqn. (7.10) for a givenθ can be parametrized as where G 0 is some symmetric positive definite matrix, for example G 0 = 1 D . Now, it is easy to demonstrate that some metric moduli remain unconstrained for symmetric orbifolds: at least we can scale G 0 with an arbitrary positive factor, while eqn.(7.10) stays fulfilled. Next, we consider the Wilson lines. If θ rotates in all nθ n , (7.12) and the Wilson lines are uniquely determined from ∆A k in eqn. (7.3c), e.g. from k = 1 Consequently, the Wilson lines A are completely frozen as they have to be discrete, i.e. quantized in units of 1/K in the directions where θ acts non-trivially. As a further consequence of eqn. (7.3c) we see that two Wilson lines (i.e. two columns of A) have to be identical up to some trivial ∆A k if the corresponding columns in the geometrical vielbein e are mapped to each other byθ k . Finally, the B-field is constrained by the condition (7.3b) combined with eqn. (7.9). In analogy to eqn. (7.11) the general solution of this equation can written as where B 0 is an arbitrary anti-symmetric matrix (for example, B 0 = 0) and B P is a particular solution to eqn. (7.14). For example, in D = 2 the anti-symmetric 2 × 2 matrix B contains a single modulus. It is subject to eqn. (7.14), i.e. (7.16) where det(θ) = ±1. Thus, for det(θ) = 1 we obtain ∆B ! = 0 and the single B-field modulus in B is unconstrained and B P = 0. On the other hand, B is stabilized at B P = 1 2 ∆B if det(θ) = −1.

Number of moduli in symmetric Z K orbifolds
We can compute the number of (real) unstabilized moduli for symmetric Z K orbifolds for general K using the results of Section 5.4. To do so, we assume for simplicity D = 6 and K = 2. Furthermore, where δ a,b = 1 if a ≡ b and δ a,b = 0 otherwise. For example, for Z 3 we take φ 1 R = φ 2 R = 1 3 and obtain dim(M Z 3 ) = 6 + 2 × 0 + 4 × (1 + 0 + 1 + 1) = 18. As is well-known, these 18 (real) moduli correspond to 9 complex structure moduli, see e.g. [90].

Two-dimensional Abelian Narain orbifolds
In this section, we study examples of generalized space groups of Narain orbifolds with Abelian Narain point groups Z K in two dimensions. Many of them correspond to previously unknown two-dimensional Narain orbifolds. We collect them in a comprehensive table. Furthermore, to illustrate various aspects of the theory developed in previous sections, we describe some of these two-dimensional Z K Narain orbifolds in more detail. For example, by an explicit construction we show that it is possible to have Z 12 two-dimensional Narain orbifolds, while it is well-known that for Euclidean orbifolds in D = 2 the largest order of a twist is K = 6. Moreover, the Qand Z-classes are used to distinguish seemingly asymmetric from truly asymmetric orbifolds.

(D, D)-Narain orbifold formalism
To prepare the discussion of various illustrative examples of two-dimensional Narain orbifolds, we briefly restrict the Narain orbifold formalism to the case where η has signature (D, D): Then, in analogy to Section 5.2 we know that the Z K Narain orbifold exists.
If the matrix-block ρ 12 is zero the orbifold is symmetric (i.e. θ r = θ l ) and a necessary (but not sufficient) condition for the orbifold to be asymmetric is ρ 12 = 0, as can be seen from eqn. (8.6).

Qand Z-classes of two-dimensional Z K Narain orbifolds
Following the discussion of the last section we focus on two-dimensional Narain orbifolds with point groups P ⊂ O η (2, 2; Z), generated by a single twist ρ of order K.
To initiate this investigation, we give a brief discussion on the possible orders following Section 4.3: For Narain orbifolds with D = 2 we have to set D Γ = 2D = 4. Then, eqn. (4.14) yields the following list of possible orders K ∈ { 1, 2, 3, 4, 5, 6, 8, 10, 12 } . (8.9) In contrast, for two-dimensional symmetric orbifolds we have D Γ = D = 2 which yields only K ∈ {1, 2, 3, 4, 6}. Indeed, as we discuss in the following, we found examples for K = 12. They are genuine asymmetric because twists of order 12 are not possible for D Γ = 2. On the other hand, we did not find any examples for K = 5, 8 and 10 in the scan of two-dimensional Narain orbifold we performed for this paper.
In Table 3 we list a number of Abelian Z K Narain orbifolds of order K, which we constructed explicitly in our scan. For each Narain point group P ⊂ O η (2, 2; Z) this table displays the following data in the various columns: 1. column labels the inequivalent orbifolds and characterizes the orbifold as symmetric or asymmetric; 2. column gives a representation of the generating twist ρ of order K in the lattice basis; 3. column displays the corresponding right-twist θ r ; 4. column displays the corresponding left-twist θ l ; 5. column indicates the relation between these twists; 6. column gives a choice of the geometrical vielbein e; 7. column gives to resulting metric as G = e T e;

column gives the anti-symmetric B-field.
A couple of further comments about the conventions of this table are in order: Our labelling conventions for inequivalent Narain orbifolds are as follows. The inequivalent Q-classes of a given order K are enumerated by a Roman number R=I,II,. . . as Z K -R. Furthermore, when we give inequivalent Z-classes within a given Q-class, we enumerate them with n = 1, 2, 3 as Z K -R-n. In fact, only the Q-class Z 2 -II is subdivided into three inequivalent Z-classes. Furthermore, the given right-and left-twists depend on our choice for the geometrical vielbein e and on the Narain moduli G and B.
To describe all these two dimensional Narain orbifolds in detail would lead to a lengthy discussion. Therefore, we focus in the following subsections on a number of striking features of some of these orbifolds instead. Before, doing so we make a couple of observations: First of all, we see that the number of asymmetric orbifolds greatly outweighs the number of symmetric orbifolds. This might imply that there exist many more asymmetric Narain orbifolds than symmetric ones. Most of the asymmetric orbifolds constructed in the past have twists that are trivial for either the left-or the rightmoving sectors, like the Z 3 -II and Z 3 -III orbifolds. In our scan we also encountered such examples, but again it seems that the majority of asymmetric orbifolds are not of this type: Most of them have non-trivial left-and right-moving twists simultaneously. In fact, there are even cases where the orders of the left-and right-moving twists are co-prime: the Z 6 -IV and Z 6 -VII Narain orbifolds. Since their orders are coprime, all their characters are orthogonal. Using the results of Section 5.4 this immediately implies that all moduli are stabilized for these orbifolds.

Two equivalent asymmetric Z 12 Narain orbifolds
With our first two examples we want to illustrate that we are able to construct genuine asymmetric orbifolds using the formalism for Narain orbifolds exposed in this paper. Concretely, we define two Z 12 label twist ρ twist θr twist θ l relation vielbein e metric G B-field  For each inequivalent orbifold it gives important data that characterizes Narain orbifolds, like the twists in both, the lattice and the coordinate basis and the values of the (frozen) moduli.
Narain point groups P (1) and P (1) in D = 2, each being generated by an element ρ (1) , ρ (2) ∈ O η (2, 2; Z) of order 12. In each case, we determine the corresponding Narain lattice and the twist Θ which is given by its action on right-and left-movers, θ r and θ l , respectively. As there is no symmetric Z 12 orbifold in D = 2 (i.e. there is no two-dimensional lattice with rotational symmetry of order 12), these orbifolds must be genuine asymmetric 3 . Moreover, to emphasize that the use of Z-classes is extremely powerful to investigate whether two orbifolds are distinct, we show that these two Z 12 point groups are in fact equivalent by giving an explicit O η (D, D; Z) matrix that relates the two twists in the lattice basis.
The first asymmetric Z 12 orbifold example has a non-vanish B-field B (1) = 0: We choose and obtain from eqn. (8.5). Then we follow the procedure outlined in Section 5.3 to find that all Narain moduli are stabilized and take the form while the twist Θ (1) is given by and θ (1)l = θ 5 (1)r . (8.13) This precisely corresponds to the data given for the Z 12 -I orbifold in Table 3. 3 Such asymmetric Z12 orbifolds were studied in the past [51,91].
An equivalent description of this asymmetric Z 12 -I orbifold has no B-field at all (B (2) = 0). For this case we take (8.14) The stabilized Narain moduli are now given by with the twist Θ (2) is given by To show explicitly that these two Z 12 orbifolds are Z-equivalent (and consequently also Q-equivalent), we observe that we can relate the two Z 12 generators,  Here, we used that both generators ρ (1) and ρ (2) are defined with respect to the same Narain metric η. Hence, the corresponding Narain point groups P (1) and P (2) are identical up to the discrete Tduality transformation with M , i.e. these point groups lie in the same Z-class. In other words, we have described the same asymmetric Z 12 orbifold in two different duality frames, once with and once without B-field.

Exposing a seemingly asymmetric Z 3 Narain orbifold
It might happen that one uses a description, i.e. choice of duality frame, in which a given Narain orbifold appears to be asymmetric. Consider for example a two-dimensional Z 3 Narain orbifold defined by the twist in the lattice basis. We use the subscript (a) to refer to this seemingly asymmetric orbifold: It is not obviously a symmetric orbifold, as it does not meet the sufficient condition ( ρ) 12 = 0 for being a symmetric Narain orbifold formulated in Section 5.3. Since in this case, eqns. where parameters R (a) and w (a) are unconstrained. Furthermore, the twist Θ (a) is specified by and θ (a)l = θ 2 (a)r .
Since θ (a)r = θ (a)l , this seems to indicate that this an asymmetric Narain orbifold. However, it is equivalent to the symmetric orbifold Z 3 -I of Table 3: To see this, we describe this symmetric Z 3 -I orbifold (labelled with a subscript (s)) in some detail: The defining twist in the lattice basis is given by In this case, ρ (s)r acts cryptographically on e, i.e. the first column e 1 of e is mapped to the second column e 2 and e 2 is mapped to −e 1 − e 2 . Furthermore, the Narain moduli are given by where R (s) and b (s) are unconstrained. Thus, the vielbein e (s) spans the root lattice of SU(3) multiplied by an arbitrary radius R (s) . Furthermore, the twist Θ (s) is specified by Clearly, these two descriptions look very different: The parametrization of the moduli does not seem to be alike, since, for example, in case (a) the B-field is fixed while in case (s) it is a modulus. Moreover, the twist seems to be asymmetric for case (a) but symmetric for case (s). However, their Narain point groups P (s) and P (a) belong to the same Z-class (and consequently also to the same Q-class); they are equivalent up to a discrete T -duality transformation.
Explicitly, the discrete T -duality transformation that relates P (s) and P Note that det(u l ) = +1 but det(u r ) = −1. This corresponds to the matrix U B from eqn. (6.8) that maps the symmetric twist from point group P (s) to the seemingly asymmetric twist from point group P (a) . Let us close this subsection with the comment that for Narain orbifolds of order 3, we were able to distinguish between three Q-classes, where each Q-class contains only a single Z-class. In the nomenclature of Table 3 the two-dimensional Narain orbifold Z 3 -I is a symmetric orbifold, while the other two, Z 3 -II and Z 3 -II, are asymmetric. In fact, they are each others mirrors in the sense that their θ l and θ r are interchanged.

Symmetric Z 2 Narain orbifolds from inequivalent Z-classes
For the examples considered so far, we found that each Narain Q-class contains just a single Narain Z-class. This might convey the impression that the notion of Z-classes for Narain orbifolds is obsolete.
To emphasize that this is not the case, we consider two symmetric Z 2 Narain point groups in D = 2 next. Both correspond geometrically to the Möbius strip, where the B-field is either turned on or off.
We will show that even though these two Narain point groups belong to the same Narain Q-class, they live in different Narain Z-classes, hence they are physically inequivalent. Consider the symmetric Z 2 -II-2 Narain orbifold of Table 3  In this case, the Narain moduli are given by This orbifold geometrically corresponds to the Möbius strip, see Figure 1.
Another symmetric Z 2 orbifold has a non-vanishing B-field: For this Z 2 -II-3 Narain orbifold in Table 3  In this case, the Narain moduli are given by Note, that the metric G (2) is identical to G (1) from the case above; the only difference is that we now have a non-vanishing B-field. The conjugation of the generator ρ (1) with M B (∆B) in eqn. (8.36) tells us that these two Narain point groups belong to the same Q-class. However, it turns out that they are from different Zclasses: There is no M ∈ O η (D, D; Z) that can relate ρ (1) to ρ (2) . Since the transformation (8.36) is a conjugation with a discrete fractional B-field transformation, the Z-classes under investigation can be used to parametrize the inequivalent choices for the B-field for the given geometrical setting. As can be inferred from Table 3 we identified three inequivalent Z-classes for the Q-class Z 2 -II, where Z 2 -II-1 and Z 2 -II-2 both have vanishing B-field but are based on inequivalent lattices.

A Moduli deformations and the generalized metric
Choose a specific generalized metric H 0 . Next, consider the finite group of all discrete T -duality transformations that leaves this generalized metric invariant and choose a subgroup H thereof. Then, the general question, which we are addressing in this section, reads: What infinitesimal moduli deformations are allowed such that the deformed generalized metric stays invariant under all transformations from H? We will answer this question in three steps. First, we define the group H in Appendix A.1. Second, we parametrize all infinitesimal moduli deformations in Appendix A.2. Third, in Appendix A. 3 we restrict them to the ones which are compatible with the action of H. In addition, in Appendix A.4 we derive a closed expression which counts the number of moduli that are compatible with the action of H. We use the results form this appendix in Section 5.4, where we set H = P, i.e. equal to the point group in the lattice basis. By doing so, we identify the moduli in Narain orbifolds.
A.1 T -duality transformations that leave a generalized metric invariant Next, we consider the perturbations of the generalized metric δH = δE T E 0 + E T 0 δE to first order.
Using eqn. (A.4) one can see that the constraint (η −1 (H 0 + δH)) 2 = 1 from eqn. (5.7) is fulfilled. In fact, we may write δh = δe T + δe, where δe = 1 2 δh + 1 2 δu with Hence, the infinitesimal moduli are uniquely identified by δm, i.e. δm encodes the deformations of the metric δG, the B-field δB and the Wilson lines δA. This can be stated explicitly as follows. We can determine δe by using eqn. (A.6) with E 0 = R E and the expression forÊ given in eqn. (2.39). Thereby we directly confirm that δu D and δu D+16 are anti-symmetric and we derive that δm is given at linear order in the moduli perturbations δG, δB and δA as given in eqn. (5.21), using (e 0 + δe) −1 ≈ e −1 − e −1 0 δe e −1 0 . Here, we interpret δm as a vector with D(D + 16) components using the standard tensor product notation ⊗. To solve this condition we introduce the projection operator P H that projects the moduli perturbations on their H-invariant subspace, i.e.