Siegel modular flavor group and CP from string theory

We derive the potential modular symmetries of heterotic string theory. For a toroidal compactification with Wilson line modulus, we obtain the Siegel modular group $\mathrm{Sp}(4,\mathbb{Z})$ that includes the modular symmetries $\mathrm{SL}(2,\mathbb{Z})_T$ and $\mathrm{SL}(2,\mathbb{Z})_U$ (of the"geometric"moduli $T$ and $U$) as well as mirror symmetry. In addition, string theory provides a candidate for a CP-like symmetry that enhances the Siegel modular group to $\mathrm{GSp}(4,\mathbb{Z})$.


Introduction
Modular symmetries might play an important role for a description of the flavor structure in particle physics [1]. In string theory, modular transformations appear as the exchange of winding and momentum (Kaluza-Klein) modes in compactified extra dimensions, combined with a nontrivial transformation of the moduli. In the application to flavor symmetries, these moduli play the role of flavon fields that are responsible for the spontaneous breakdown of flavor and CP symmetries. While string theory requires six compact space dimensions with many moduli, the explicit discussion in flavor physics has, up to now, mainly concentrated on two compact extra dimensions and few geometric moduli (see e.g. ref. [2]). In the top-down discussion, this included i) the T 2 /Z 3 orbifold with Kähler modulus T (and frozen complex structure modulus U ) [3][4][5] subject to the modular group SL(2, Z) T and ii) the T 2 /Z 2 orbifold with T and U moduli with a corresponding modular group SL(2, Z) T × SL(2, Z) U combined with a mirror symmetry that interchanges T and U [6].
The present paper performs a next step towards a more exhaustive discussion of the "manymoduli-case". Our results are based on the observation that string theory includes more moduli beyond the (geometric) T -and U -moduli in form of Wilson lines connected to gauge symmetries in extra dimensions. Modular transformations act nontrivially on these Wilson lines and require a modified geometric interpretation. In the present paper, we illustrate this situation for compactifications on two-tori and the corresponding transformation of the Narain lattice in heterotic string theory. Our main results are: • Wilson line moduli lead to an enhancement of modular flavor symmetries, • for the case of two compactified dimensions, this leads to the appearance of the Siegel modular group Sp(4, Z), which includes SL(2, Z) T ×SL(2, Z) U as well as mirror symmetry, • a generalized geometric interpretation of the origin of these symmetries is given through an auxiliary Riemann surface of genus 2 (see figure 1) that combines the metric and gauge moduli in a common setting 1 , and • a candidate CP-like symmetry naturally appears in string models with two compact dimensions; interestingly, this symmetry also arises in a bottom-up discussion as an outer automorphism of the Siegel modular group, extending it to GSp(4, Z).  [8]. Their "third" modulus can thus be realized as a Wilson line in heterotic string theory. In addition, string constructions admit a CP-like symmetry, which appears at the same footing as all discrete (traditional and modular) symmetries. In section 4, we show that this CP-like symmetry also appears naturally from a bottom-up perspective: It corresponds to an outer automorphism of the Siegel modular group extending it to the general symplectic group GSp(4, Z). Conclusions and outlook are given in section 5. Finally, some technical details are discussed in two appendices.
2 The Siegel modular group Sp(2g, Z) The symplectic group over the integers Sp(2g, Z) (also called the Siegel modular group of genus g) is the group of linear transformations M which preserve a skew-symmetric bilinear form J, i.e.
Here, J is given as and 1 g is the g-dimensional identity matrix.
As reviewed in appendix A, there exists a natural action of Sp(2g, Z) on a symmetric g × g-dimensional matrix called Ω ∈ H g , where the Siegel upper half plane H g is defined as Hence, Ω contains g × (g + 1)/2 complex numbers that are called moduli. In more detail, one splits M ∈ Sp(2g, Z) into g × g-dimensional blocks A, B, C, and D as follows Then, M acts on Ω as Ω Note that ±M ∈ Sp(2g, Z) yield the same transformation eq. (5) of Ω.
In the following, we focus on g = 2. In this case, the moduli are encoded in a symmetric 2 × 2 matrix Ω whose components are denoted as As we will see explicitly in the following, T and U are two moduli associated with the modular group SL(2, Z) T × SL(2, Z) U , while Z is a new modulus that interrelates the two SL(2, Z) factors.
In detail, SL(2, Z) U is contained in Sp(4, Z) because due to the defining condition M T (1). Then, we use eq. (5) and find that the moduli transform as Note that for Z = 0 we see that T and Z are invariant under SL(2, Z) U modular transformations, while U transforms as expected from SL(2, Z) U . Similarly, we can embed SL(2, Z) T into Sp(4, Z) via such that the moduli transform as using eq. (5). Let us remark that the modular S 2 transformations from SL(2, Z) T and SL(2, Z) U are related in Sp(4, Z), i.e. M (S 2 ,1 2 ) = −M (1 2 ,S 2 ) and the moduli transform as In addition, Sp(4, Z) contains a Z 2 mirror transformation As the name suggests, a mirror transformation interchanges T and U , i.e. using eq. (5) one can verify easily that Finally, Sp(4, Z) contains elements M (∆) with ∆ ∈ Z 2 . These elements are intrinsically tied to the modulus Z. They can be defined as Then, eq. (5) yields 3 The origin of the Sp(4, Z) Siegel modular group from strings It is well-known that compactifications of heterotic string theory on tori (and toroidal orbifolds) are naturally described in the Narain formulation [9][10][11]. There, one considers D right-and D + 16 left-moving (bosonic) string modes (y R , y L ) to be compactified as i.e. on an auxiliary torus of dimension 2D + 16. The 16 extra left-moving degrees of freedom give rise to an E 8 × E 8 (or SO(32)) gauge symmetry of the heterotic string. In more detail, the auxiliary torus corresponding to the identification (19) can be defined by the so-called Narain lattice that is spanned by the Narain vielbein E, a matrix of dimension (2D + 16) × (2D + 16). Here, n ∈ Z D gives the winding numbers, m ∈ Z D the Kaluza-Klein numbers and p ∈ Z 16 the gauge quantum numbers. As the one-loop partition function of the string worldsheet has to be modular invariant, the Narain lattice Γ has to be an even, integer and self-dual lattice with a metric η of signature (D, D + 16). This condition on Γ holds if the Narain vielbein E satisfieŝ Here, g := α T g α g is the Cartan matrix of the E 8 × E 8 gauge symmetry and α g denotes a matrix whose columns are the simple roots of E 8 × E 8 (or in the case of an SO(32) gauge symmetry, α g is a basis of the Spin(32)/Z 2 weight lattice).
It is convenient to define the so-called generalized metric of the Narain lattice in terms of the metric G := e T e (of the D-dimensional torus spanned by the geometrical vielbein e), the anti-symmetric B-field B and the Wilson lines A, where C := B + α 2 A T A and we use conventions similar to those of refs. [12,13], but replacing C by C T for later convenience. Note that due to eq. (21), the generalized metric H satisfies the condition The outer automorphisms of the Narain lattice are given by "rotational" transformations This is the general modular group of a toroidal compactification of the heterotic string. Ele-mentsΣ of Oη(D, D + 16, Z) act on the Narain vielbein E as [4] such that the Narain scalar product λ T 1 ηλ 2 is invariant for λ i ∈ Γ, i ∈ {1, 2}. In the following we take D = 2. Moreover, the (continuous) Wilson lines are chosen as and Wilson lines background fields A as cf. ref. [7]. Moreover, e 1 and e 2 are the two columns of the geometrical vielbein e, and φ denotes the angle enclosed by them. Note that the continuous Wilson lines a 1 and a 2 not only yield a new "Wilson line modulus" called Z but they also alter the definition of the Kähler modulus T . In contrast, the complex structure modulus U remains unchanged in the presence of Wilson lines.
In what follows, it will be important to for a general modular transformationΣ ∈ Oη(2, 3, Z).
where we have to change the conventions compared to refs. [3,4] due to the presence of Wilson lines and the resulting changes in the generalized metric eq. (22). Using eq. (28) we obtain as expected for a mirror transformation, see eq. (16) for the corresponding case in Sp(4, Z).
Modular group of the complex structure modulus. The general modular group Oη(2, 2 + 16, Z) contains a modular group SL(2, Z) U associated with the complex structure modulus U . It can be generated bŷ Then, we use eq. (28) in order to verify the Sp(4, Z) transformations of the moduli (T, U, Z) given in eq. (11).
Modular group of the Kähler modulus. In addition to SL(2, Z) U , due to mirror symmetry eq. (29) there exists a modular group SL(2, Z) T associated with the Kähler modulus T .
It can be defined byK where ∆A is a 16 × 2-dimensional matrix with integer entries. Since g is the Cartan matrix of an even lattice (of E 8 × E 8 or Spin(32)/Z 2 ), the 2 × 2 matrix 1 2 ∆A T g ∆A is integer. We focus on shifts ∆A in the directions of a 1 and a 2 . Hence, we definê for , m ∈ Z. By doing so, we will focus in what follows on a subgroup Oη(2, 3, Z) of Oη(2, 2 + 16, Z). Then, using the transformation (28) of the generalized metric, we obtain Applying eq. (28) toΣ * gives rise to a CP-like transformation of the moduli. This string result on CP can also be understood from a bottom-up perspective as we will see in section 4.
As a remark, there exist further Oη(2, 2 + 16, Z) transformations not present in Sp(4, Z): One can perform Weyl reflections in the 16-dimensional lattice of E 8 × E 8 (or Spin(32)/Z 2 ), see for exampleM W (∆W ) in ref. [12].  in the Narain formulation of the heterotic string. In the last column we list the transformation of the moduli, computed in two ways: i) using eq. (5) for Sp(4, Z), and independently ii) using eq. (28) for Oη(2, 3, Z). The CP-like transformation M * will be defined in section 4.

CP as an outer automorphism of Sp(4, Z)
We have seen in the previous section that a CP-like transformation appears naturally in (toroidal) string compactifications. As we shall see in this section in a bottom-up discussion, this transformation does not belong to Sp(4, Z) but corresponds to an outer automorphism of Sp(4, Z) that, once included, enhances Sp(4, Z) to the general symplectic group GSp(4, Z).
We define a transformation where M * is given by In order to see the physical meaning of M * , we apply eq. (38) to various elements of Sp(4, Z):  [19,20]. Indeed, as explained in appendix A, the action of GSp(4, Z) on Ω can be defined in analogy to the action of GL(2, Z) on one modulus, cf. ref. [3,21,22] and ref. [23].
Explicitly, for an element of the general symplectic group we find the transformation rules which confirms our expectation for a CP-like transformation.

Conclusions and Outlook
The potential (traditional and modular) flavor symmetries of string theory compactifications are determined through the outer automorphisms of the Narain lattice. For the heterotic string, the modular symmetries are a subgroup of Oη(D, D + 16, Z), where D is the dimension of the relevant compact space, i.e. D ≤ 6. As a starting point, we have concentrated in this paper on a D = 2 sublattice of compact six-dimensional space. Apart from the Kähler and complex structure moduli T and U , we include a Wilson line modulus Z and arrive at the modular symmetry group Oη(2, 3, Z). We show that this group is closely related to the Siegel modular group Sp(4, Z), which has been studied intensively in the mathematical literature. The (complex) three-dimensional moduli space of Sp(4, Z) can be visualized through an auxiliary Riemann surface of genus 2 (see figure 1). Our top-down construction allows for a physical interpretation of the recent bottom-up discussion of ref. [8]: Their "third" modulus τ 3 (apart from τ 1 = U and τ 2 = T ) can be understood as a Wilson line modulus Z of compactified (heterotic) string theory.
Beyond these results, an important open task is to make contact with realistic models of "flavor" including chiral fermions. With this purpose, the T 2 toroidal compactification can be altered by a Z K orbifolding, i.e. T 2 /Z K . This orbifolding results in the appearance of finite Siegel modular groups, where chiral matter transforms in representations thereof. In addition, a Z K orbifolding can break the Siegel modular group Sp(4, Z): The geometrical Z K rotation that acts on the two-torus has to be embedded into the 16 degrees of freedom of the gauge symmetry due to worldsheet modular invariance of the string partition function. It is known that a shift embedding yields discrete Wilson lines [24], such that the Wilson line modulus Z is frozen at some discrete value. In this case, the Siegel modular group Sp(4, Z) is broken. For a Z K orbifold with K = 2 the unbroken subgroup from Sp(4, Z) is the modular group SL(2, Z) T of the Kähler modulus T , while for K = 2 one finds SL(2, Z) T × SL(2, Z) U combined with a mirror symmetry that interchanges T and U , see ref. [6]. On the other hand, a rotational embedding into the 16 gauge degrees of freedom gives rise to continuous Wilson lines [25,26], where the Wilson line modulus Z remains as a free modulus. Hence, one expects that a two-dimensional Z 2 orbifold with rotational embedding yields the full Sp(4, Z) Siegel modular group.
Discrete modular flavor symmetries correspond to the finite Siegel modular groups Γ g,n (here, of genus g = 2). For the Z 2 orbifold we have n = 2 and Γ 2,2 is isomorphic to S 6 , the permutation group of six elements [8]. This includes the finite modular group S 3 × S 3 as well as mirror symmetry obtained in ref. [6], where only the moduli T and U associated with SL(2, Z) T × SL(2, Z) U had been considered and the Wilson line modulus was set to Z = 0. Finally, we have shown in a general study that, in addition to these modular flavor symmetries, there is a natural appearance of a CP-like transformation predicted in string theory. As discussed in sec. 4, from a bottom-up perspective, it can be understood as an outer automorphism of the Siegel modular group extending it to GSp(4, Z). A full discussion of the symmetries of the Z 2 orbifold, including CP, will be subject of a future publication [27]. A Symplectic groups Sp(2g, Z) and modular transformations Let us review some aspects of the symplectic group Sp(2g, Z) and its relation to modular transformations (see e.g. [28,29] for further details). The symplectic group Sp(2g, Z) can be defined by considering an auxiliary genus-g Riemann surface T g and its symmetries as follows: The genus-g surface has 2g nontrivial 1-cycles denoted by (β i , α j ) for i, j ∈ {1, . . . , g}, see figure 1 for the cases g = 1 and g = 2. These cycles form the canonical basis of the homology group H 1 (T g , Z) ∼ = Z 2g . The holomorphic 1-forms ω i build the dual cohomology basis, which one can choose to satisfy α j ω i = δ ij and β j ω i = β i ω j . In these terms, the skew-symmetric form J in eq. (1) is interpreted as the intersection numbers ( β α ) ∩ ( β α ) of the 2gdimensional vectors of 1-cycles ( β α ) = (β 1 , . . . , β g , α 1 , . . . , α g ) T , such that α i ∩ α j = β i ∩ β j = 0 and −(α i ∩ β j ) = β i ∩ α j = δ ij . Now, one transforms the 1-cycles ( β α ) The A consequence of the Torelli theorem for Riemann surfaces is that the genus-g surface T g is determined by the complex g-dimensional torus, which can be defined as the quotient of C g divided by a complex lattice. This lattice is given by the columns of the g × 2g period matrix of T g , By choosing a basis in which α j ω i = δ ij , one can always rewrite Π g , such that where we have defined the g × g complex modulus matrix Ω, such that Ω T = Ω. Clearly, the transformations ( β α ) → M ( β α ) induce transformations on the modulus matrix Ω in eq. (47). Restricting further to Im Ω > 0, we arrive at the modular space of the genus-g compact surface T g , H g = Ω ∈ C g×g | Ω T = Ω, Im Ω > 0 .  Z). b) A compact Riemann surface of genus 2 T 2 and its four basis 1-cycles (β 1 , β 2 , α 1 , α 2 ) T . The Siegel modular group Sp(4, Z) is the modular symmetry group of T 2 . As discussed in ref. [7], setting the Wilson line modulus Z defined in eq. (27c) to Z = 0 splits the genus 2 surface into two separated two-tori. Note that these auxiliary surfaces must not be mistaken as compactification spaces.
Consider the g = 1 case. We observe that M are 2 × 2 integer matrices with unit determinant, i.e. they describe the modular group Sp(2, Z) ∼ = SL(2, Z) of a T 2 torus. Given the holomorphic 1-form ω = dz and the nontrivial 1-cycles α and β, shown in figure 1a), the period matrix of T 2 is given by The last equation arises from the choice α ω = 1 and the definition of the modulus τ := β ω.
We now let the SL(2, Z) element M = a b c d act on the 1-cycle vector (β, α) T . This implies that the period matrix transforms as By demanding that the holomorphic 1-form transforms under M as ω = ω(cτ + d) −1 , we normalize the transformed period matrix, which then becomes This allows us to identify the standard modular transformation τ → (aτ + b)(cτ + d) −1 .