Orbifolds from $\boldsymbol{\mathrm{Sp}(4,\mathbb Z)}$ and their modular symmetries

The incorporation of Wilson lines leads to an extension of the modular symmetries of string compactification beyond $\mathrm{SL}(2,\mathbb Z)$. In the simplest case with one Wilson line $Z$, K\"ahler modulus $T$ and complex structure modulus $U$, we are led to the Siegel modular group $\mathrm{Sp}(4,\mathbb Z)$. It includes $\mathrm{SL}(2,\mathbb Z)_T\times\mathrm{SL}(2,\mathbb Z)_U$ as well as $\mathbb Z_2$ mirror symmetry, which interchanges $T$ and $U$. Possible applications to flavor physics of the Standard Model require the study of orbifolds of $\mathrm{Sp}(4,\mathbb Z)$ to obtain chiral fermions. We identify the 13 possible orbifolds and determine their modular flavor symmetries as subgroups of $\mathrm{Sp}(4,\mathbb Z)$. Some cases correspond to symmetric orbifolds that extend previously discussed cases of $\mathrm{SL}(2,\mathbb Z)$. Others are based on asymmetric orbifold twists (including mirror symmetry) that do no longer allow for a simple intuitive geometrical interpretation and require further study. Sometimes they can be mapped back to symmetric orbifolds with quantized Wilson lines. The symmetries of $\mathrm{Sp}(4,\mathbb Z)$ reveal exciting new aspects of modular symmetries with promising applications to flavor model building.


Introduction
Modular symmetries appear frequently in string theory. They might have applications in particle physics as discrete non-Abelian flavor symmetries. In the simplest case, these discrete modular symmetries descend from the modular group SL(2, Z) of a two-dimensional torus, on which two extra spatial dimensions have been compactified. The implementation within string theory requires some aspects of model building towards the SU(3) × SU(2) × U(1) Standard Model of particle physics. One of the key aspects is the desired presence of chiral fermions. This requires a twist of the torus. Then, chiral matter fields can be realized in the twisted sectors of the orbifold, located at the "fixed points" of the orbifold twist. Simplest examples correspond to the Z K orbifolds T 2 /Z K (K = 2, 3, 4, 6), where a full analysis has been performed recently [1][2][3][4]. They would correspond to six-dimensional string compactifications with an elliptic fibration.
In fact, in string theory a two-torus with background B-field is described by two moduli, the Kähler modulus T and the complex structure modulus U with modular symmetries SL(2, Z) T × SL(2, Z) U . This is accompanied by mirror symmetry which interchanges T and U . In the Z K orbifolds with (K > 2), the complex structure modulus is frozen to allow for the orbifold twist and in these cases we have one unconstrained modulus T . This is, however, not the case for the Z 2 orbifold, where we remain with two unconstrained moduli and manifest mirror symmetry.
In general, string theories have a much richer moduli structure as they require the compactification of six spatial dimensions. But even if we concentrate on a two-dimensional subsector, we have additional moduli in the form of gauge background fields (Wilson lines). The moduli structure of the simplest example of such a system is given by the Siegel modular group Sp(4, Z) with moduli T , U and one additional Wilson line modulus Z. This can be made manifest in the Narain lattice formulation [5]. In addition, this work is motivated by the recent bottom-up consideration of Sp(4, Z) flavor symmetries, see refs. [6,7]. Sp(4, Z) contains as subgroups SL(2, Z) T × SL(2, Z) U as well as mirror symmetry. From the string theory perspective, an application of Sp(4, Z) as modular flavor symmetry would again require some orbifolding to obtain chiral fermions. The first step in this direction is a classification of orbifolds from Sp(4, Z), which is the main purpose of the present paper. This is a generalization of the Z K orbifolds mentioned earlier. To perform the classification of Sp(4, Z) orbifolds, we realize that each inequivalent fixed point in the string moduli space (T, U, Z) that is left invariant by a subgroup of Sp(4, Z) corresponds to an inequivalent orbifold (or, in general, to a set of orbifolds). Hence, the fixed points of Sp(4, Z) correspond to string orbifolds. Then, chiral fermions could appear at the fixed points of these Sp(4, Z) orbifold actions on the extra-dimensional space. 1 A classification of the fixed points of Sp(4, Z) has been given by Gottschling [8][9][10] long ago. There are altogether 13 different cases: two with complex dimension 2, five with complex dimension 1 and six of dimension 0. The next step is the construction of those orbifolds that stabilize these fixed loci in moduli space. Our results are summarized in table 1. We identify the conventional (geometrical) twists on the moduli and, as a new mechanism, twists via mirror symmetry. As a result of this, asymmetric orbifolds appear frequently (although some of them are dual to symmetric orbifolds with specifically transformed moduli). A direct intuitive geometrical interpretation is often not available as the presence of Wilson lines and asymmetric twists introduce some "non-geometrical" aspects.
If we set the Wilson lines to zero, we obtain the T 2 /Z K examples discussed earlier: Section 3.1.1 in table 1 represents the symmetric Z 2 orbifold, section 3.2.1 the Z 4 orbifold, and section 3.2.2 the symmetric Z 3 orbifold (embedded in the Z 6 case). The orbifold of section 3.1.2 corresponds to an asymmetric Z 2 orbifold. In this case, however, we can define a duality transformation that maps it to a symmetric Z 2 orbifold with a quantized Wilson line (discussed in The paper is organized as follows. In section 2, we introduce the modular transformations of Sp(4, Z) on the string moduli (see eq. (8)) within the framework of Narain orbifold compactifications. Section 3 discusses case by case the stabilization of moduli by Sp(4, Z) orbifolds that lead to the results summarized in table 1. In addition, we explicitly construct for each orbifold the unbroken modular group G modular as a subgroup of Sp(4, Z) plus a CP-like transformation.
While these are important steps towards applications to the flavor problem of the Standard Model of particle physics, there are still many open questions. These will be mentioned in section 4, devoted to conclusions and outlook.

Narain torus compactification and string moduli
To construct an orbifold in the Narain formulation of the heterotic string, we first discuss a general D-dimensional torus compactification with B-field and Wilson line backgrounds [11,12]. To do so, we have to impose torus boundary conditions on the D right-and D + 16 left-moving (bosonic) string modes (y R , y L ), respectively, The (2D + 16)-dimensional vector of integersN contains the winding numbers n ∈ Z D , the Kaluza-Klein numbers m ∈ Z D , and the gauge quantum numbers p ∈ Z 16 corresponding to the E 8 × E 8 (or SO(32)) gauge group of the supersymmetric heterotic string. Furthermore, E denotes the Narain vielbein. For the worldsheet one-loop string vacuum amplitude to be modular invariant, E has to satisfy Here, η is the Narain metric of signature (D, D + 16), g := α T g α g and the columns of the (16 × 16)-dimensional matrix α g contain the simple roots of E 8 ×E 8 (or Spin(32)/Z 2 ). Consequently, E spans a so-called Narain lattice: an even, integer, self-dual lattice of signature (D, D + 16). The "rotational" symmetries of the Narain lattice give rise to the so-called modular group of the Narain lattice, Oη(D, D + 16, Z) := Σ Σ ∈ GL(2D + 16, Z) withΣ TηΣ =η .
In order to understand the action of Oη(D, D + 16, Z), it is convenient to define the generalized metric of the Narain lattice, 2 and is the metric of the D-dimensional torus T D defined by the torus basis vectors contained as the columns of the D × D vielbein matrix e. In addition, B is the anti-symmetric background B-field, while the 16 × D matrix A gives rise to the Wilson lines along the D torus directions.
In the following, we will mainly set D = 2 and choose the two column vectors A i of the Wilson line matrix A as Then, we define the (dimensionless) string moduli as [15] T : The Kähler modulus T determines the B-field background, the overall size of the extradimensional two-torus, and is altered by the Wilson line parameters a i . The complex structure modulus U parameterizes the shape of the two-torus T 2 , while the Wilson line modulus Z depends on the parameters a i . The transformation of these moduli under a modular transfor-mationΣ ∈ Oη(2, 2 + 16, Z) can be computed by considering the generalized metric see for example ref. [16]. For the transformationsΣ ∈ {K S ,K T ,Ĉ S ,Ĉ T ,M ,Ŵ ( m ) ,Σ * } given in ref. [5] this yields

Narain orbifold compactification and string moduli
We can extend the Narain torus boundary conditions (1) by an orbifold action [13,14,[17][18][19][20][21] andN ∈ Z 2D+16 . The rotation matrix Θ denotes the so-called Narain twist. The set of all Narain twists generates the so-called Narain point group P Narain . If there exists a basis such that θ L = θ R ⊕ 1 16 for all twists, the orbifold is called symmetric, and otherwise it is called asymmetric. Moreover, the orbifold action (9) suggests to define transformations (Θ, EN ), which generate the so-called Narain space group S Narain . Then, an orbifold compactification (of worldsheet bosons) is fully specified by the choice of S Narain . Due to its right-left structure θ R and θ L in eq. (9), a Narain twist Θ has to satisfy the conditions Furthermore, the Narain twist has to map the Narain lattice to itself. Hence, it is convenient to define the Narain twist in the Narain lattice basiŝ such thatΘ is an integer matrix from GL(2D + 16, Z). In the Narain lattice basis, we denote the Narain point group byP Narain and the Narain space group byŜ Narain , where its elements are of the form (Θ,N ). Using the Narain lattice basis, the conditions (10) read Consequently,Θ ∈ Oη(D, D + 16, Z) has to be an element of the modular group of the Narain lattice that leaves the generalized metric H invariant, see eq. (7). In general, condition 12 fixes some of the moduli. In other words, the Narain twistΘ is a symmetry of the Narain lattice only for some special values of the moduli. In this case, we say that some moduli are stabilized geometrically by the orbifold action. We denote the modular group after orbifolding by G modular . It is given by those elements (Σ,T ) ∈Ŝ Narain withΣ ∈ Oη(D, D + 16, Z) and T ∈ Q 2D+16 that are outer automorphisms of the Narain space groupŜ Narain , cf. refs. [16,22].
In the case where the Narain space group is generated only by elements of the form (Θ, 0) and (1 2D+16 ,N ) withN ∈ Z 2D+16 , the modular group after orbifolding is given by 3 Then, one can compute the transformation of the moduli after orbifolding using the generalized metric eq. (7).
In addition to the modular group G modular (which is in general an infinite, discrete group), there exists the closely related finite modular group G fmg , which can play an important role in flavor physics [23][24][25]. This group appears in string theory as follows: On orbifolds, there are so-called twisted strings that are localized in extra dimensions at the fixed points of the orbifold action. They transform in general under a modular transformationΣ ∈ G modular nontrivially with a unitary matrix representation ρ r (Σ) of a finite modular group G fmg , for example G fmg ∼ = T for the T 2 /Z 3 orbifold [16,22,[26][27][28] and G fmg ∼ = (S T 3 × S U 3 ) ZM 4 for the T 2 /Z 2 orbifold (without CP) [3,4]. In addition, couplings Y in the superpotential become modular forms of the moduli. Hence, couplings Y also transform under a modular transformationΣ ∈ G modular in a unitary matrix representation ρ Y (Σ) of the finite modular group G fmg . However, in some cases ρ Y is not a faithful representation of G fmg . Then, one can compute the finite modular group G fmg of Y that is generated by the matrix representations ρ Y (Σ) of the modular forms such that [3,4].

Stabilizing moduli by Sp(4, Z) orbifolds
In the following, we consider all inequivalent fixed points of Sp(4, Z) in the Siegel upper halfplane H 2 , as listed in table 2 of ref. [6]. For each fixed point τ f ∈ H 2 , we explicitly construct an orbifold compactification in the Narain formulation by specifying a Narain point groupP Narain . Then, the string moduli are stabilized geometrically by the orbifold action in agreement with the fixed point τ f . To do so, we focus on the moduli (T, U, Z) of a D = 2 subsector of a full six-dimensional string compactification. In more detail, for each fixed point τ f , we consider the stabilizer groupH := H/{±1 4 } from appendix D of ref. [6], where We take the generators ofH and write them in terms of Sp(4, Z) basis elements using the notation of ref. [5]. Then, we apply the dictionary eq. (14) between Sp(4, Z) and Oη(2, 2+16, Z) in order to translate the stabilizer groupH into a subgroupP Narain of Oη(2, 2 + 16, Z). By construction,P Narain maps the Narain lattice in D = 2 to itself. Consequently, we can utilizê P Narain as a Narain point group to define a Narain space groupŜ Narain without roto-translations. It turns out that the moduli (T, U, Z) of the resulting (a)symmetric orbifold are fixed by the orbifold action due to condition (12). Hence, for each fixed point τ f of Sp(4, Z) listed in ref. [6], we verify that the string moduli (T, U, Z) are fixed accordingly, i.e.
using an appropriate orbifold compactification. In addition, we use the dictionary between Sp(4, Z) and Oη(2, 2 + 16, Z) to translate both, the normalizer N (H) from ref. [6] and the CP transformation from ref. [7] into Oη(2, 2 + 16, Z). We check explicitly that the resulting transformations are outer automorphisms of the corresponding Narain space group. Hence, for each orbifold we identify the modular group G modular , including a CP-like transformation, and compute the transformations of the unfixed moduli with respect to the modular generatorŝ Σ ∈ G modular .
Finally, let us remark that the Narain point groupsP Narain that we construct for each inequivalent fixed point of Sp(4, Z) are the "maximal" point groups that one can use for the given fixed point. In other words, one can also consider a subgroup ofP Narain as the point group of an orbifold. For example, we will encounter a Z 6 Narain point group for a specific fixed point of Sp(4, Z). In this case, also the Z 3 subgroup of Z 6 can serve as a point group that will geometrically stabilize the moduli in the same way as the Z 6 orbifold.

Orbifolds with moduli spaces of dimension 2
According to ref. [6], there are two inequivalent subspaces of complex dimension 2 that are left invariant by subgroups of Sp(4, Z). As we show in the following, they can be implemented in string theory by the compactification on Z 2 orbifolds.

Symmetric Z 2 orbifold
Let us consider the point in the Siegel upper half-plane H 2 , i.e. a complex two-dimensional subspace of H 2 given by τ 3 = 0. In the following, we will explicitly construct a string compactification such that the string moduli are fixed accordingly. To do so, we first have to consider the stabilizerH of τ f .
For the point eq. (18) the stabilizer isH ∼ = Z 2 , generated by see ref.
[6, table 2 and appendix D]. Using the notation of ref. [5], this element h is given by the square of a modular S transformation of the modulus τ 2 , From the dictionary eq. (14) we know that this Sp(4, Z) element corresponds to the Oη(2, 2 + 16, Z) elementΘ As a remark, in Oη(2, 2 + 16, Z) we have (K S ) 2 = (Ĉ S ) 2 . So, we could equally writeΘ = (Ĉ S ) 2 . This Narain twistΘ defines a Z 2 Narain point groupP Narain of a symmetric Z 2 orbifold. The moduli (T, U, Z) in the generalized metric H are constrained by the invariance condition (12) underΘ. As a result, we find a 1 = a 2 = 0. Hence, the Wilson line modulus Z has to vanish, Z = 0. On the other hand, the moduli are unconstrained, as it has been known from ref. [3,4], for example. In other words, the Narain lattice is mapped to itself under the Narain twistΘ eq. (21) only if the Wilson line modulus is trivial Z = 0, while the Kähler modulus T and the complex structure modulus U can vary freely. One says that Z has been stabilized geometrically by the orbifold action. This is in agreement with τ f under the identification τ 1 = U , τ 2 = T and τ 3 = Z = 0. In other words, the complex two-dimensional subspace with τ 3 = 0 described in ref. [6] can be constructed in string theory by the compactification on a symmetric Z 2 orbifold with vanishing Wilson line.
Next, we translate the normalizer from ref. [6] into Oη(2, 2 + 16, Z) and check that each generator is an outer automorphism of the Narain space group, see eq. (13). In this way, we identify the modular group (without CP) of the symmetric Z 2 orbifold, where the Z 2 quotient ensures the relation ( see section 3 of ref. [16]. Then, we use the generalized metric (7) in order to identify the transformation of the moduli (T, U ) and we obtain the transformations (8a)-(8e) with Z = 0, while Z = 0 is invariant under all of these transformations. As a remark, for the symmetric Z 2 orbifold, the finite modular group G fmg is known explicitly from the transformation matrices ρ r (Σ) of twisted strings with respect to modular transformationsΣ. It is given by [3,4] where the ZM 2 mirror symmetry of the moduli acts as a ZM 4 symmetry on the twisted strings of the Z 2 orbifold. Moreover, the four twisted strings localized at the four orbifold fixed points transform as r = 4 1 of the finite modular group [144, 115].
Next, we consider table 1 of ref. [7] and translate their CP transformation into Oη(2, 2 + 16, Z). We obtainĈ One can verify easily thatĈP is an outer automorphism of the Narain space group, i.e.
Furthermore, we use the generalized metric (7) and confirm the transformation of the string moduli (T, U ), while Z = 0 is invariant underĈP, see also ref. [34].

Asymmetric Z 2 orbifold
The second fixed point in the list of ref. [6] is given by In this case, the stabilizerH ∼ = Z 2 is generated by the mirror element in the notation of ref. [5]. SinceM ∈ Oη(2, 2+16, Z) denotes the corresponding mirror element in the string constructing, we choose a Narain twist see eq. (157) in the appendix. Consequently, we construct an asymmetric Z 2 orbifold with mirror symmetryM as Narain twist. Then, eq. (12) yields In this case, one can check that the expectations for the corresponding fixed point τ f are fulfilled, and Z = −a 2 + U a 1 . Hence, the two-dimensional subspace with τ 1 = τ 2 and an unconstrained τ 3 , as described in the bottom-up construction of ref. [6], can be obtained in string theory from the compactification on an asymmetric Z 2 orbifold using mirror symmetry as orbifold twist.
Note that a physical torus must satisfy G 11 > 0. Thus, eq. (32) constrains the Wilson line to a 2 1 < 1. Let us remark that the Narain twistΘ given by the mirror transformationM also acts on the 16 gauge degrees of freedom of the heterotic string. This will induce some gauge symmetry breaking, cf. refs. [35,36].
Next, we consider the normalizer given in ref. [6]. We use our dictionary eq. (14) and obtain the group We verify that this group gives rise to the rotational outer automorphisms of the Narain space group. Hence, G modular is the modular group of the asymmetric Z 2 orbifold. Using eq. (7) we find that the two independent moduli (T, Z) transform as under the generators of the modular symmetry G modular of this asymmetric orbifold.
Finally, we consider the CP transformation for the fixed point τ f with τ 1 = τ 2 in table 1 of ref. [7]. Using our dictionary to Oη(2, 2 + 16, Z), this transformation corresponds tô Hence,ĈP is an outer automorphism ofŜ Narain . From the generalized metric (7) we compute the transformation of the string moduli (T, Z), resulting in for this asymmetric Z 2 orbifold.
SinceΘ andΘ are related in eq. (43) by conjugation withB ∈ Oη(2, 2+16, Z), the two Narain point groups belong to the same Narain Z-class [13]: they describe the same physics but in different duality frames. Now, we useΘ :=M and simplify the Narain twistΘ from eq. (43), in agreement with eq. (42). As a result, we have shown that the asymmetric Z 2 orbifold with a Narain twistΘ given by mirror symmetry is equivalent to a symmetric Z 2 orbifold with Narain twistΘ . Then, the new Narain twistΘ constrains the generalized metric eq. (12) such that the Wilson lines have to be fixed, Consequently, the moduli (T, U, Z) of the symmetric Z 2 orbifold read as expected for τ f . The modular group of this orbifold can be obtained from eq. (35) using the basis changeB, where we have defined Note that these transformations satisfy the relations Then, we use eq. (7) to compute the modular transformations of the moduli (T, U ), while Z = 1 /2 is invariant under all of these transformations. Using the relationŝ we observe that the transformations follow from eqs. (51). Thus, there exists an alternative set of generators of G modular given bŷ Finally, we analyze CP for the fixed point τ f with τ 3 = 1 /2. Using the basis changeB and CP =Σ * from eq. (37), we obtain Thus,ĈP is an outer automorphism of the Narain space group of the symmetric Z 2 orbifold with discrete Wilson line. The generalized metric (7) transforms underĈP such that we find while Z = 1 /2 is invariant under the transformation withĈP .

Orbifolds with moduli spaces of dimension 1
Following ref. [6], there are five inequivalent subspaces of complex dimension 1 that are left invariant by subgroups of Sp(4, Z). In the following, we implement them explicitly in string theory by the compactification on various symmetric and asymmetric orbifolds.

Symmetric Z 4 orbifold
Consider the fixed point In this case, the stabilizerH ∼ = Z 4 is generated by using the notation of ref. [5]. This Sp(4, Z) element corresponds in string theory to a modular S transformation of the complex structure modulus. Thus, we choose a Narain twist Θ :=Ĉ S ∈ Oη(2, 2 + 16, Z) .
As a result, we construct a symmetric Z 4 orbifold with Narain twistΘ =Ĉ S . In order to satisfy eq. (12) we have to set Hence, we have to fix the Wilson lines to zero Z = 0, the complex structure modulus to U = i, while the Kähler modulus T remains unconstrained, as expected for the values of τ f in this case.
The normalizer is given by see ref. [6]. Compared to ref. [6], we use M (S,1 2 ) and M (1 2 ,S) as generators of N (H) instead of . The corresponding group in Oη(2, 2 + 16, Z) reads which is the modular group of the symmetric Z 4 orbifold. Note that Z R 4 allows for a geometrical interpretation as a π /2 sublattice rotation, assuming that this D = 2 orbifold is a subsector of a full six-dimensional string compactification, see for example section 3 of ref. [2].
Consequently, Z R 4 is an R-symmetry, as our notation explicitly indicates. Note that the order of this geometrical Z 4 sublattice rotation will generically be larger than four due to fractional R-charges of twisted strings, cf. refs. [1,2,[38][39][40][41][42]. Using the generalized metric (7), we confirm that the Kähler modulus T transforms under the generators of the modular group G modular as expected, i.e.
Finally, we consider the CP transformation for the fixed point τ f with τ 1 = i and τ 3 = 0 given in table 1 of ref. [7]. Using our dictionary eq. (14) from GSp(4, Z) to Oη(2, 2 + 16, Z), the corresponding CP transformation of this symmetric Z 4 orbifold is given bŷ Thus, it is an outer automorphism ofŜ Narain . We applyĈP to the generalized metric (7) and find that U = i and Z = 0 are invariant, while the Kähler modulus transforms as

Symmetric Z 6 orbifold
The next fixed point in the list of ref. [6] is given by where ω := exp( 2πi /3). In this case, the stabilizerH ∼ = Z 6 is generated by one element that we can write in the notation of ref. [5] as Hence, we choose a Narain twist This Narain twist defines a symmetric Z 6 orbifold. In order to satisfy eq. (12), we have to fix the string geometry as follows This results in U = ω and Z = 0 , while the Kähler modulus T is unconstrained, as expected by comparing to the value of τ f in the corresponding bottom-up construction of ref. [6].
In this case, the normalizer reads see ref. [6]. Compared to ref. [6], we use M (S,1 2 ) as generator of N (H) instead of M (S 3 ,1 2 ) . The corresponding group in Oη(2, 2 + 16, Z) gives the modular group of the symmetric Z 6 orbifold, where (Ĉ S ) 3Ĉ T generates a geometrical Z 6 rotation (and (Ĉ S ) 3Ĉ T =Θ −1 ). As a sublattice rotation of a six-dimensional orbifold compactification this gives rise to an R-symmetry, whose order will generically be larger than six due to fractional R-charges of twisted strings [1,2,[38][39][40][41][42]. Using the generalized metric (7), we compute the transformations of the Kähler modulus T under the generators of the modular group G modular and obtain the expected results, i.e.
Finally, we consider the CP transformation from table 1 of ref. [7] that leaves the fixed point τ f with τ 1 = ω and τ 3 = 0 invariant. Using our dictionary eq. (14) from GSp(4, Z) to Oη(2, 2 + 16, Z), the corresponding CP transformation is given bŷ Hence,ĈP is an outer automorphism of the Z 6 Narain space group. Using eq. (7) for the CP-like transformationĈP, we find that U = ω and Z = 0 are invariant, while the Kähler modulus T transforms as TĈ Let us briefly remark that one can define a symmetric Z 3 orbifold by taking just the Z 3 subgroup of the stabilizerH ∼ = Z 6 as Narain point group, i.e. consider a Z 3 Narain point group P Narain generated by the Narain twistΘ 2 , whereΘ is given in eq. (68). This also fixes U = ω and Z = 0, yields the same modular group G modular and the same CP transformation.
Hence, the moduli (T, U, Z) of this asymmetric Z 2 × Z 2 orbifold read so that only G 12 and G 22 are free, in agreement with τ 1 = τ 2 and τ 3 = 0.
The normalizer N (H) is generated by four elements given by Here, compared to ref. [6], we use M (S,S) as generator of N (H) instead of M (S 3 ,S 3 ) . Translated to Oη(2, 2 + 16, Z) this yields the modular group of the asymmetric Z 2 × Z 2 orbifold. Note that we find PSL(2, Z) instead of SL(2, Z) be-causeK SĈS is of order 2 (instead of 4). Then, the modulus T transforms nontrivially under PSL(2, Z), while it is invariant under (K S ) 2 andM .
According to table 1 of ref. [7] the fixed point τ f with τ 1 = τ 2 and τ 3 = 0 allows for a CP transformation given bŷ Hence,ĈP is an outer automorphism of this asymmetric Z 2 × Z 2 Narain space group. Using eq. (7) for the CP-like transformationĈP, we find that Z = 0 is invariant, while the modulus T = U transforms as TĈ
This is similar to eq. (79) but with a non-trivial Wilson line a 2 = 0. Then, the moduli read so that only G 12 and G 22 are free. This confirms the expectation for τ f in this situation.
In order to identify the modular group G modular of this orbifold, we consider the normalizer see ref. [6]. We use the dictionary eq. (14) to translate this into Oη(2, 2 + 16, Z). We find where we have defined Since these transformations are outer automorphisms of the Narain space group, they give rise to the modular group of this asymmetric Z 2 × Z 2 orbifold with discrete Wilson line. Using the generalized metric eq. (7), we identify their action on the modulus T and obtain Finally, for the fixed point τ f with τ 1 = τ 2 and τ 3 = 1 /2 we identify CP from table 1 of ref. [7] asĈ Hence,ĈP is an outer automorphism of this asymmetric Z 2 × Z 2 Narain space group. Using eq. (7), it leaves Z = 1 /2 invariant, while the modulus T = U transforms as
In this case, the moduli read as expected for the given values of τ f .
According to ref. [6] and using the Sp(4, Z) generators defined in ref. [5], the normalizer N (H) can be generated by four elements Using the dictionary eq. (14), the corresponding group in Oη(2, 2 + 16, Z) reads where we defined As the group G modular consists of outer automorphisms of the asymmetric S 3 Narain space group, this yields the modular group of this orbifold. Then, using eq. (7) the Kähler modulus T transforms under the generators of G modular as while T is invariant underM andŴ ( 0 1 ) (K S ) 2 . Finally, for the fixed point τ f with τ 1 = τ 2 and τ 3 = τ 1/2 we translate CP from table 1 of ref. [7] into Oη(2, 2 + 16, Z), yieldinĝ Consequently,ĈP is an outer automorphism of this asymmetric S 3 Narain space group. We use eq. (7) to show thatĈP acts as

Orbifolds with moduli spaces of dimension 0
Finally, there are six inequivalent subspaces of complex dimension 0 that are left invariant by subgroups of Sp(4, Z) [6]. As before, for each fixed point the corresponding orbifold is constructed by embedding the stabilizerH from Sp(4, Z) into Oη(2, 2 + 16, Z). This yields a Narain point groupP Narain in D = 2. In all cases discussed here, it turns out thatP Narain gives rise to an asymmetric orbifold, whose moduli are stabilized geometrically, as expected.
Note that the normalizers are given in terms of the stabilizers, N (H) = H, as there are no free moduli. Hence, embedding N (H) into Oη(2, 2 + 16, Z) just gives G modular ∼ =P Narain . If the D = 2 orbifold constructed here is a subsector of a full D = 6 orbifold, G modular yields a traditional flavor symmetry, as a transformation is called modular only if some modulus transforms nontrivially. In addition, we consider CP from ref. [7] and confirm that these transformations are also unbroken in the corresponding string constructions. As the string moduli are stabilized geometrically, one cannot move in moduli space away from the CPconserving point. Hence, CP cannot be broken spontaneously by the moduli in these cases.

Asymmetric Z 5 orbifold
The next fixed point in the Siegel upper half-plane is given by where ζ := exp ( 2πi /5). In this case, the stabilizerH ∼ = Z 5 is generated by using the Sp(4, Z) generators of ref. [5]. Then, we define the corresponding Narain twist Θ ∈ Oη(2, 2 + 16, Z)Θ (111) Note that h 5 = −1 4 , which is identified with +1 4 inH. Also the Narain twistΘ is of order 5 such thatΘ defines a Z 5 Narain point group of an asymmetric Z 5 orbifold. This orbifold action is compatible with the generalized metric eq. (12) if Consequently, all moduli are fixed by the orbifold action to the values in agreement with the fixed point τ f ∈ H 2 of Sp(4, Z).
Finally, we translate the CP generator of this case from table 1 of ref. [7] into Oη(2, 2+16, Z) and obtain Furthermore, the geometrically stabilized moduli (T, U, Z) are invariant under this CP transformation. Its a trivial fact that CP cannot be broken spontaneously by the moduli of this Z 5 orbifold sector since all moduli are completely fixed according to eq. (113).

Asymmetric S 4 orbifold
Taking the fixed point whereη := 1 3 (1 + 2 √ 2 i), the stabilizerH ∼ = S 4 is generated by two elements We write these Sp(4, Z) elements in terms of the generators of ref. [5] Then, we define the corresponding Narain twistŝ Note that (h 2 ) 4 = −1 4 is of order 8 in H (and of order 4 inH). The corresponding Narain twistΘ 2 is also of order 4.Θ 1 andΘ 2 obey the relations Hence,Θ 1 andΘ 2 generate a S 4 Narain point group of an asymmetric S 4 orbifold. This orbifold action constrains the generalized metric eq. (12) such that Consequently, all moduli of this orbifold are stabilized geometrically, Hence, the subspace τ 1 = τ 2 =η and τ 3 = 1 2 (η − 1) can be implemented in string theory using an asymmetric S 4 orbifold.
We obtain a CP transformation for this asymmetric S 4 orbifold by translating the corresponding case from table 1 of ref. [7] into Oη(2, 2 + 16, Z). This yieldŝ for i ∈ {1, 2}. Then, we confirm that the geometrically stabilized string moduli (T, U, Z) given in eq. (121) are invariant under this CP transformation.

Asymmetric (Z
The fixed point is stabilized byH ∼ = (Z 4 × Z 2 ) Z 2 , generated by three elements We write these Sp(4, Z) elements in terms of the generators of ref. [5] Hence, they define a (Z 4 × Z 2 ) Z 2 Narain point group of an asymmetric (Z 4 × Z 2 ) Z 2 orbifold. This orbifold action fixes all components of the generalized metric eq. (12) as follows Consequently, all moduli are fixed in agreement with τ 1 = τ 2 = i and τ 3 = 0.
Finally, we consider the fixed point τ f with τ 1 = τ 2 = i and τ 3 = 0 in table 1 of ref. [7] and translate the corresponding CP transformation into Oη(2, 2 + 16, Z). Hence, we confirm explicitly that the outer automorphism of the Narain space group for i ∈ {1, 2, 3}, leaves the geometrically stabilized moduli (T, U, Z) given in eq. (129) invariant. Hence, we have identified a CP-like transformation of this orbifold theory.

Asymmetric S 3 × Z 6 orbifold
Next, we consider the fixed point Its stabilizerH ∼ = S 3 × Z 6 is generated by three elements h 1 and h 2 generate S 3 , while h 3 is the generator of Z 6 . These Sp(4, Z) elements can be written in terms of the generators of ref. [5] and we obtain Using the dictionary eq. (14), we can define the corresponding Narain twistŝ Hence, they generate the permutation group S 3 . Furthermore, the order 6 Narain twistΘ 3 commutes with both,Θ 1 andΘ 2 . Consequently, the three Narain twistsΘ 1 ,Θ 2 andΘ 3 generate an S 3 × Z 6 Narain point groupP Narain , which we use to define an asymmetric S 3 × Z 6 orbifold. Next, we construct the Narain lattice of this orbifold by demanding invariance of the generalized metric eq. (12) under the three Narain twists. This yields So, we see that all moduli (T, U, Z) are stabilized and their values read Thus, the invariant subspace τ 1 = τ 2 = ω and τ 3 = 0 discussed in the bottom-up construction of ref. [6] can be constructed explicitly in string theory using an asymmetric orbifold compactification of D = 2 dimensions with Narain point groupP Narain ∼ = S 3 × Z 6 .

Asymmetric S 3 × Z 2 orbifold
The next fixed point in the Siegel upper half-plane from the list of ref. [6] is given by In this case, the stabilizerH ∼ = S 3 × Z 2 is generated by three elements Next, we map these Sp(4, Z) elements into the Narain construction using ref. [5] and define the following Narain twistŝ The Narain twistsΘ 1 andΘ 2 satisfy the relations Hence, they generate the permutation group S 3 . Furthermore, the order 2 Narain twistΘ 3 commutes with both,Θ 1 andΘ 2 . Thus, the Narain twistsΘ 1 ,Θ 2 andΘ 3 generate an S 3 × Z 2 Narain point group. The resulting asymmetric S 3 × Z 2 orbifold restricts the generalized metric eq. (12) to a unique form, given by This orbifold is similar to the S 3 orbifold with moduli given in eq. (101), but with the additional constraints G 22 = α and B 12 = 0. As a consequence, the moduli (T, U, Z) have to take the values as expected from the bottom-up discussion in ref. [6].
Also in this case, we can use table 1 of ref. [7] to identify a CP-like transformation of the asymmetric S 3 × Z 2 orbifold, for all generators i ∈ {1, 2, 3} of this S 3 × Z 2 Narain point group.

Asymmetric Z 12 orbifold
Finally, we consider the fixed point This point is stabilized byH ∼ = Z 12 , which is generated by an element h ∈ Sp(4, Z) that we can write in terms of the generators defined in ref. [5] as follows This Sp(4, Z) element can be mapped to a Narain twistΘ ∈ Oη(2, 2+16, Z) using the dictionary eq. (14), and we obtainΘ : which is of order 12. Hence,Θ generates a Z 12 Narain point group and, consequently, an asymmetric Z 12 orbifold in D = 2 dimensions, cf. ref. [44] and section 8 of ref. [13]. The generalized metric H needs to be invariant, see eq. (12), which fixes H to As a consequence, the moduli (T, U, Z) have to take the values Finally, we translate the CP transformation from table 1 of ref. [7] for the case τ 1 = U = ω, τ 2 = T = i and τ 3 = Z = 0 into Oη(2, 2 + 16, Z). We find Thus,ĈP is an outer automorphism of the Narain space group of this asymmetric Z 12 orbifold.

Conclusion and Outlook
In the present work we have initiated the discussion of flavor symmetries of the Siegel modular group Sp(4, Z) from a top-down perspective. In string theory, Sp(4, Z) describes properties of the moduli T and U of a two-torus compactification as well as a Wilson line Z, as can be derived from the Narain lattice construction. This can be visualized through the moduli of a Riemann surface of genus 2, which in the case of a vanishing Wilson line splits in two separate tori describing the T and U moduli independently.
The road to understand the relevance of Sp(4, Z) for flavor symmetries of the Standard Model requires several steps. In a first step, we have to insist on the presence of chiral matter fields, which can be achieved by an orbifold twist. In the case of the previously discussed twotorus with vanishing Wilson lines, we had identified the possible orbifolds as those with twists Z K and K = 2, 3, 4, 6, with fixed points of the complex structure modulus U at the boundaries of the fundamental domain of SL(2, Z) U . The generalization to Sp(4, Z) then requires the classification of those orbifolds that lead to the fixed surfaces of Sp(4, Z) in the Siegel upper half plane. These include two surfaces of complex dimension 2, five of complex dimension 1 and six of dimension 0. We have identified the 13 corresponding orbifolds explicitly and summarize our results in tables 1 and 2. In contrast to the previously discussed cases, we often find asymmetric orbifolds, which appear, for example, once we mod out the mirror symmetry (which interchanges T and U ). For each orbifold, we obtain the unbroken modular group G modular including CP and the associated moduli transformations. With these results, we have completed the first step towards the understanding of the flavor structure of Sp(4, Z).
In a second step, we would then have to analyze the properties of these orbifolds in detail. The symmetric orbifolds can be understood easily as they have a simple geometric interpretation. They extend the previously discussed cases Z K with K = 2, 3, 4, 6. The construction in section 3.1.1 corresponds to the Z 2 orbifold with complex moduli T an U and vanishing Wilson line. The cases with K = 3, 4, 6 require a fixing of the complex structure modulus U , which is addressed in section 3.2.1 for Z 4 and section 3.2.2 for Z 3 and Z 6 . The case discussed in section 3.1.2 corresponds to an asymmetric orbifold with two complex moduli T = U and Wilson line Z. As a direct geometric interpretation is lost in this case, an understanding of its properties requires further investigations. To regain the standard geometric picture it can be mapped to a symmetric Z 2 orbifold with moduli T and U and a quantized Wilson line Z = 1 /2 as shown in section 3.1.3. This reinterpretation of an asymmetric orbifold as a symmetric orbifold with specifically transformed moduli is not always possible. Some of the other asymmetric orbifolds of table 1 cannot be mapped to symmetric orbifolds and need a more detailed analysis. Further studies should include the consideration of these asymmetric orbifolds towards the construction of models with the particle content of the Standard Model of particle physics.
In a third step, one would have to discuss for each orbifold the discrete flavor symmetries in the full eclectic picture of refs. [45,46]. This includes the traditional flavor group as well as the unbroken finite Siegel modular group that originates from the unbroken subgroup G modular of Sp(4, Z) as given in section 3. The finite Siegel modular groups are denoted by Γ 2,N = Sp(4, Z)/Γ 2 (N ), where Γ 2 (N ) is the principal congruence subgroup of Sp(4, Z) with genus 2 and level N . This generalizes the homogeneous finite modular groups Γ 1,N = Γ N of SL(2, Z), discussed for example in ref. [47]. For N = 2 we have Γ 2,2 = Sp(4, 2) ∼ = S 6 of order 720, while Γ 2,3 = Sp(4, 3) has already 51,840 elements [48]. These groups are huge, but they are usually not fully realized because of the orbifold twist that was introduced to get chiral matter from the torus. The task here would be to determine the unbroken finite Siegel modular groups for the 13 orbifold cases of table 1 given the modular transformations determined in section 3. This is beyond the scope of the present paper. Some clues can be found from the previously discussed cases with vanishing Wilson lines. The symmetric Z 2 orbifold from section 3.1.1 (with Z = 0) is known to have a finite modular group (S 3 × S 3 ) Z 4 (from Γ T 2 × Γ U 2 combined with mirror symmetry [3,4]) which nicely fits into Γ 2,2 = S 6 for level N = 2. It is not clear yet whether Γ 2,2 is also the relevant group for the asymmetric Z 2 orbifold discussed in section 3.1.2, although this seems plausible. On the other hand, in the case of the Z 3 orbifold with vanishing Wilson line, the finite modular group was found to be T = Γ 3 of level N = 3 [16,22,[26][27][28]. This leads to the conjecture that for the case described in section 3.2.2, the finite modular group would descend from the finite Siegel modular group Γ 2,3 , where we have confirmed that Γ 2,3 contains T . Thus, the result of this third step would be the determination of the finite Siegel modular flavor symmetry as well as the traditional flavor symmetry for each of the orbifolds given in table 1.
Once this has been achieved, the ultimate step to establish the full connection to bottomup constructions is to determine the representations of the matter fields with respect to the full eclectic flavor group G efg . Chiral fields tend to correspond to twisted fields located at the fixed points of the orbifold twist. In the Z 3 case, for example, we have three fixed points and matter fields transform as triplet representations of the traditional flavor symmetry ∆(54) ⊂ G efg and as a 1 ⊕ 2 of the finite modular group T ⊂ G efg . This shows, among others, that twisted fields need not correspond to irreducible representations of the finite modular group. So far, determining the representations of matter fields under the discrete flavor symmetries requires explicit computations of string vertex operators and their associated operator product expansions. This or the identification of a simpler method remains to be explored in detail for each of the orbifolds under consideration. This underlines that top-down model building with modular flavor symmetries has just begun to unfold its various possibilities. More work is needed in order to finally bridge the gap to bottom-up constructions.
The consideration of the finite Siegel modular flavor symmetry from a bottom-up perspective has been pioneered recently in ref. [6]. They considered the case with two unconstrained moduli: T = U and a Wilson line Z. Superficially, this would look like case 3.1.2 in our table 1, but this interpretation is not necessarily correct. By choosing the moduli at T = U by hand, ref. [6] imposes S 4 × Z 2 as finite Siegel modular subgroup of Γ 2,2 ∼ = S 6 . In addition, some matter fields are postulated to build triplet representations of S 4 . In our picture, the moduli can be stabilized at T = U by an asymmetric Z 2 orbifold discussed in section 3.1.2. In principle, this orbifold also determines the unbroken finite Siegel modular group and the corresponding representations of matter fields. Unfortunately, their determination is technically involved and the results are currently not available. Hence, a detailed correspondence between ref. [6] and the top-down approach has yet to be clarified. To obtain a better geometric interpretation, one could reformulate this case as a symmetric orbifold with a quantized Wilson line as shown in section 3.1.3. This might help to make contact to the discussion in the bottom-up approach of ref. [6]. We hope to report on the resolution of these questions in a future publication.
Finally, we stress that the results from our present endeavor may have interesting applications also in the study of other top-down scenarios. For example, in the context of magnetized toroidal compactifications [49][50][51][52][53][54][55] one typically derives the flavor properties of the models from the modular properties associated with the complex structure of a two-torus, disregarding the modular behavior of the Kähler and Wilson line moduli also present in the construction. It would be interesting to study how our considerations change the conclusions in these cases.

A Remark on mirror symmetry
Note that in Sp(4, Z) the mirror transformation M × can be expressed as Let us denote the original definition from ref. [5] of mirror symmetry byM . Then, we use the dictionary eq. (14) between Sp(4, Z) and Oη(2, 2 + 16, Z) and map the right-hand side of eq. (153) into the modular group Oη(2, 2 + 16, Z) of the string setup and definê Crucially, the new mirror transformationM differs from the original definition ofM , This is contrary to our expectation from eq. (153), as one would associate M × ∈ Sp(4, Z) witĥ M ∈ Oη(2, 2 + 16, Z) using the dictionary eq. (14) of ref. [5]. However, the generalized metric transforms identically underM andM , i.e. using e q. (7) defined in eq. (154) as the generator of mirror symmetry instead ofM that was defined in ref. [5]. Note that the new mirror transformationM also acts nontrivially on the 16 gauge degrees of freedom of the heterotic string as a Z 2 reflection.