Modular flavor symmetries of three-generation modes on magnetized toroidal orbifolds
aa r X i v : . [ h e p - t h ] J a n EPHOU-21-001
Modular flavor symmetries of three-generation modes onmagnetized toroidal orbifolds
Shota Kikuchi, Tatsuo Kobayashi, and Hikaru Uchida
Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
Abstract
We study the modular symmetry on magnetized toroidal orbifolds with Scherk-Schwarzphases. In particular, we investigate finite modular flavor groups for three-generationmodes on magnetized orbifolds. The three-generation modes can be the three-dimensionalirreducible representations of covering groups and central extended groups of Γ N for N = 3 , , , , ,
16, that is, covering groups of ∆(6( N/ ) for N = even and central exten-sions of P SL (2 , Z N ) for N =odd. We also study anomaly behaviors. Introduction
The origin of the flavor structure such as quark and lepton masses and their mixing angles is oneof the most significant mysteries in particle physics. Non-Abelian discrete flavor symmetries [1–6] such as S N , A N , ∆(3 N ), and ∆(6 M ) for the three generations of quarks and leptons areattractive candidates to realize the flavor structure. However, in order to obtain the realisticmasses and mixing angles of the quarks and leptons, the complicated vacuum alignment ofgauge singlet scalars, the so-called flavons, is required.The geometries of compact spaces predicted in higher dimensional theories such as super-string theory can be candidates of the origin of the flavor structure. (See Refs. [7, 8].) Forexample, a torus and its orbifold have the complex structure modulus τ , which decides theshape of the torus and the orbifold. There is the modular symmetry Γ ≡ SL (2 , Z ) as wellas ¯Γ ≡ SL (2 , Z ) / Z as the geometrical symmetry on a torus and some of orbifolds. Underthe modular transformation, chiral zero-modes on the torus and the orbifolds, correspondingto the flavors of quarks and leptons, are transformed. That is, the modular symmetry canbe regarded as the flavor symmetry. In addition, Yukawa coupligns as well as higher ordercouplings can be functions of the modulus τ and then they also transform under the modulartransformation since they can be obtained by overlap integrals of the zero-mode profiles on thetorus and the orbifolds. Instead of flavons, a vacuum expectation value of the modulus τ breaksthe flavor symmetry, and characterizes the flavor structure. These features are different fromones in the conventional flavor models. The modular transformation behavior of zero-modeswas investigated in magnetized D-brane models [9–15] and heterotic orbifold models [16–20].(See also Refs. [21–23].) In particlular, on magnetized T with the magnetic flux M , there are M -number of chiral zero-modes [24] and in recent work [13], it was shown that the zero-modeswith M = even and vanishing Scherk-Schwarz (SS) phases behave as modular forms of weight1 / M -dimensional representations of the finite modular subgroup e Γ M , which is the quadruple covering group of Γ M . There also exists the modular symmetryon the magnetized T / Z (t)2 twisted orbifold. The number of zero-modes on the magnetized T / Z (t)2 twisted orbifold was investigated in Refs. [25–28]. Similarly, in Ref. [14], it was shownthat zero-modes on the magnetized T × T with the magnetic fluxes M ( i ) ( i = 1 ,
2) on T i andits orbifolds behave as modular forms of weight 1 and they transform under the finite modularsubgroup Γ ′ M (1) ,M (2) ) , which is the double covering group of Γ M (1) ,M (2) ) . The number ofzero-modes was investigated in Ref. [15]. The modular transformation for Yukawa couplingshas also studied in Ref. [15]. Thus, it is important to study the modular flavor symmetries,particularly in magnetized orbifold models.Furthermore, the finite modular subgroups Γ N for N = 2 , , , S , A , S , A , respectively [29]. Similarly, Γ ′ N for N = 3 , , T ′ , S ′ , A ′ ,respectively [30]. These results are well motivated for realistic model buildings. In particular, Both of modulus on T i , τ i , are identified each other, i.e. τ = τ ≡ τ . Such moduli identification can berealized by certain three-form fluxes [23] or Z (p)2 permutation.
1n Ref. [29], three-dimensional irreducible representations are studied in the finite modularsubgroups and it was shown that three-dimensional irreducible representations appear onlyin the finite modular subgroups: Γ ≃ P SL (2 , Z ) ≃ A , Γ ≃ S , Γ ≃ P SL (2 , Z ) ≃ A ,Γ ≃ P SL (2 , Z ), Γ ⊃ ∆(96), and Γ ⊃ ∆(384). Recently, the bottom-up approach of modelbuilding with the modular flavor symmetries was studied extensively for Γ N [31] and for itscovering groups [30, 32].In this paper, we study modular flavor groups of the three-generation modes on magnetizedorbifolds. We find that the three-generation modes are the three-dimensional representationsof corresponding covering groups and central extended groups of the above finite modularsubgroups provided in Ref. [29].This paper is organized as follows. In section 2, we review the modular symmetry onmagnetized T and T / Z (t)2 twisted orbifold without the SS phases. In section 3, we studythe modular symmetry on magnetized T and T / Z (t)2 twisted orbifold with the SS phases.We can consider the modular symmetry of not only wavefunctions with the magnetic flux M =even and the vanishing SS phases but also ones with the magnetic flux M =odd and thecertain SS phases. In section 4, we show the specific modular flavor groups for three-generationmodes on magnetized T / Z (t)2 twisted orbifold with the SS phases. We can find the three-generation modes are the three-dimensional representations of the quadruple covering groupsand Z central extended groups of the corresponding modular flavor groups provided in Ref. [29].We also extend the analyses to the modular symmetry on magnetized T / Z (t )2 × T / Z (t )2 orbifold and the Z (p)2 permutation orbifold, i.e. ( T × T ) / ( Z (t)2 × Z (p)2 ) orbifold in sections 5 and6. We can obtain three-dimensional representations of all the double covering groups of Γ N for N = 4 , ,
16, i.e. covering groups of ∆(6 N ′ ) with N ′ = N/
2, and Z central extended groupsof Γ N for N = 3 , ,
7, i.e. Z extensions of P SL (2 , Z N ). In section 7, we conclude this study. InAppendix A, we review that the SS phases can be replaced by the Wilson lines (WLs) throughgauge transformation and we show that the modular transformation for them are consistent.In Appendix B, we also show that the Z N SS phases are related to the Z N shift modes. InAppendix C, we prove that e ∆(6 M ), which is the quadruple covering group of ∆(6 M ), canbe obtained from e Γ M . In Appendix D, we express three-dimensional modular forms obtainedfrom the wavefunctions on magnetized orbifolds. T and T / Z (t)2 twistedorbifold without the Scherk-Schwarz phases In this section, we review the modular symmetry on magnetized T and T / Z (t)2 twisted orbifoldwithout the Scherk-Schwarz (SS) phases.First, we review the moular symmetry of T [33–36]. A two-dimensional torus T can beconstructed as T ≃ C / Λ, where Λ is a two-dimensional lattice spanned by lattice vectors e k ( k = 1 , τ ≡ e /e (Im τ > C as u and one of T as z ≡ u/e , so that z + 1 and z + τ are identified with z . The metric on T is given by ds = 2 h µν dz µ d ¯ z ν , h = | e | (cid:18) (cid:19) , (1)and then the area of T is A = | e | Im τ .Here, we can consider the same lattice spanned by the following lattice vectors transformedby SL (2 , Z ) ≡ Γ, (cid:18) e ′ e ′ (cid:19) = (cid:18) a bc d (cid:19) (cid:18) e e (cid:19) , γ = (cid:18) a bc d (cid:19) ∈ SL (2 , Z ) ≡ Γ . (2)The SL (2 , Z ) is generated by S = (cid:18) − (cid:19) , T = (cid:18) (cid:19) , (3)and they satisfy the following algebraic relations, Z ≡ S = − I , Z = S = ( ST ) = I . (4)Under the SL (2 , Z ) transformation, the complex coordinate of the torus z and the complexstructure modulus τ are transformed as γ : ( z, τ ) → ( γ ( z, τ )) = (cid:18) zcτ + d , aτ + bcτ + d (cid:19) . (5)The above transformation for the modulus τ is called the (imhomogeneous) modular transfor-mation and also ¯Γ ≡ Γ / {± I } is called the (imhomogeneous) modular group since τ is invariantunder Z = − I .We define the principal congruence subgroup, Γ( N ), of level N byΓ( N ) ≡ (cid:26) h = (cid:18) a ′ b ′ c ′ d ′ (cid:19) ∈ Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) a ′ b ′ c ′ d ′ (cid:19) ≡ (cid:18) (cid:19) (mod N ) (cid:27) . (6)Then, the modular forms, f ( τ ), of the (integral) weight k for Γ( N ) is the holomorphic functionsof τ which transform under the modular transformation in Eq. (5) as f ( γ ( τ )) = J k ( γ, τ ) ρ ( γ ) f ( τ ) , J k ( γ, τ ) = ( cτ + d ) k , γ ( τ ) = aτ + bcτ + d , γ = (cid:18) a bc d (cid:19) ∈ Γ . (7)Here, ρ ( γ ) denotes the unitary representation of the quotient group Γ ′ N ≡ Γ / Γ( N ) satisfyingthe following algebraic relations, ρ ( Z ) = ρ ( S ) = ( − k I , ρ ( Z ) = ρ ( S ) = [ ρ ( S ) ρ ( T )] = I , ρ ( Z ) ρ ( T ) = ρ ( T ) ρ ( Z ) , (8) ρ ( T ) N = I . (9)3or even weight k , in particular, ρ ( γ ) becomes the unitary representation of the quotient groupΓ N ≡ ¯Γ / ¯Γ( N ), where ¯Γ( N ) ≡ Γ( N ) / {± I } for N = 1 , and ¯Γ( N ) ≡ Γ( N ) for N >
2. Notethat Γ N for N = 2, 3, 4, and 5 are isomorphic to S , A , S , and A , respectively [29], and alsoΓ ′ N for N = 3, 4, and 5 are isomorphic to the corresponding double covering groups: T ′ , S ′ , and A ′ , respectively [30]. In what follows, we review the wavefunctions of ( z, τ ) on a magnetizedtorus and then review their behavior as modular forms under the modular transformation inEq. (5).First, let us review the wavefunctions, particularly the zero-mode wavefunctions of the two-dimensional spinor, on the torus with U (1) magnetic flux [24]. Here, we do not consider theWLs or the SS phases. In the next section, we will study the case with the non-vanishing SSphases . The U (1) magnetic flux is given by F = πiM Im τ dz ∧ d ¯ z, (10)which satisfies the quantization condition, (2 π ) − R T F = M ∈ Z . This flux is induced by thevector potential, A ( z ) = πM Im τ Im (¯ zdz ) . (11)This vector potential transforms under lattice translations as A ( z + 1) = A ( z ) + d (cid:18) πM Im τ Im z (cid:19) = A ( z ) + dχ ( z ) , (12) A ( z + τ ) = A ( z ) + d (cid:18) πM Im τ Im¯ τ z (cid:19) = A ( z ) + dχ ( z ) , (13)which correspond to U (1) gauge transformation. Thereby, the two-dimensional spinor with U (1) unite charge q = 1, ψ ( z, τ ) = (cid:18) ψ + ( z, τ ) ψ − ( z, τ ) (cid:19) , (14)should satisfy the following boundary conditions, ψ ( z + 1 , τ ) = e iχ ( z ) ψ ( z, τ ) = e πiM Im z Im τ ψ ( z, τ ) , (15) ψ ( z + τ, τ ) = e iχ ( z ) ψ ( z, τ ) = e πiM Im¯ τz Im τ ψ ( z, τ ) . (16)Under these boundary conditions, we can solve the zero-mode Dirac equation, i Dψ ( z, τ ) = 0 , (17) Since Z = − I ∈ Γ( N ) for N = 1 , ρ ( Z ) = I should be satisfied and then the modular weight k should beeven. The WLs can be replaced by the SS phases [26]. We review it and also show the consistency in terms ofthe modular symmetry in Appendix A ψ + ( z, τ ) ( ψ − ( z, τ )) has | M | -number of degenerate zero-modes when M is positive(negative). In what follows, we consider the positive flux M . The j -th zero-mode wavefunctionon the torus with the flux M is expressed as ψ j,MT ( z, τ ) = (cid:18) M A (cid:19) / e πiMz Im z Im τ ϑ (cid:20) jM (cid:21) ( M z, M τ ) , ∀ j ∈ Z M = { , , , ..., M − } , (18)where ϑ denotes the Jacobi theta function defined as ϑ (cid:20) ab (cid:21) ( ν, τ ) = X l ∈ Z e πi ( a + l ) τ e πi ( a + l )( ν + b ) . (19)We take the following normalization condition, Z T dzd ¯ z (cid:16) ψ j,MT ( z, τ ) (cid:17) ∗ ψ k,MT ( z, τ ) = (2Im τ ) − / δ j,k . (20)Now, we can see the wavefunctions for ∀ j in Eq. (18) behave as modular forms of weight1 / under the modular transformation in Eq. (5) [13] as follows. In order to see that, we firstintroduce the double covering group of Γ, e Γ ≡ { [ γ, ǫ ] | γ ∈ Γ , ǫ ∈ {± }} . (21)The generators are given by e S ≡ [ S, , e T ≡ [ T, , (22)and they satisfy the following algebraic relations, e Z ≡ e S , e Z = e S = ( e S e T ) = [ I , − , e Z = e S = ( e S e T ) = [ I , ≡ I , e Z e T = e T e Z. (23)Note that the modular transformation in Eq. (5) does not change under replacing γ ∈ Γ with e γ ≡ [ γ, ǫ ] ∈ e Γ. We also introduce the congruence subgroup, e Γ( N ) ≡ { [ h, ǫ ] ∈ e Γ | h ∈ Γ( N ) , ǫ = 1 } . (24)Then, the modular forms, f ( τ ), of the (half integral) weight k/ e Γ( N ) transform under themodular transformation as f ( e γ ( τ )) = e J k/ ( e γ, τ ) e ρ ( e γ ) f ( τ ) , e γ ∈ e Γ , (25) e J k/ ( e γ, τ ) = ǫ k J k/ ( γ, τ ) = ǫ k ( cτ + d ) k/ , k ∈ Z , (26) See in detail e.g. [34, 37, 38]. e ρ ( e γ ) is the unitary representation of the quotient group e Γ N ≡ e Γ / e Γ( N ), which is thedouble covering group of Γ ′ N , satisfying the following algebraic relations, e ρ ( e Z ) = e ρ ( e S ) = e πik/ I , (27) e ρ ( e Z ) = e ρ ( e S ) = [ e ρ ( e S ) e ρ ( e T )] = e πik I , (28) e ρ ( e Z ) = e ρ ( e S ) = [ e ρ ( e S ) e ρ ( e T )] = I , (29) e ρ ( e Z ) e ρ ( e T ) = e ρ ( e T ) e ρ ( e Z ) (30) e ρ ( e T ) N = I . (31)Here, we take ( − k/ = e − πik/ . On the other hand, the wavefunctions for ∀ j in Eq. (18)transform under the modular transformation as ψ j,MT ( e γ ( z, τ )) = e J / ( e γ, τ ) M − X k =0 e ρ T ( e γ ) jk ψ k,MT ( z, τ ) , e γ ∈ e Γ , (32) e ρ T ( e S ) jk = e πi/ √ M e πi jkM , e ρ T ( e T ) jk = e πi j M δ j,k , (33)where e ρ T ( e γ ) satisfies Eqs. (27)-(31) with k/ / N = 2 M although I jk = δ j,k inEq. (27) is modified into δ M − j,k , derived from ψ j,MT ( e Z ( z, τ )) = ψ j,MT ( − z, τ ) = ψ M − j,MT ( z, τ ) . (34)Note that the above modular transformation for the wavefunctions without the SS phasescan be valid only if the magnetic flux M is even because of the consistency of the boundaryconditions in Eqs. (15) and (16) under the T transformation. That is, the wavefunctions afterthe T transformation satisfy ψ ( z + τ + 1 , τ + 1) = e πiM Im(¯ τ +1) z Im τ ψ ( z, τ + 1) , (35)while the wavefunctions before the T transformation satisfy ψ ( z + τ + 1 , τ ) = e − πiM e πiM Im(¯ τ +1) z Im τ ψ ( z, τ ) . (36)In the next section, however, we will show that when we take the SS phases into account, wecan also consider the modular transformation for wavefunctions with the flux M =odd. Thus,the wavefunctions on T with the magnetic flux M ∈ Z and vanishing the SS phases behaveas the modular forms of weight 1 / e Γ(2 M ) and then they transform as M -dimensionalrepresentations of e Γ M .Finally, we also review the zero-mode wavefunctions on the magnetized T / Z (t)2 twistedorbifold without the SS phases [25] and the modular transformation for them [13]. (See alsoRef. [9, 10].) T / Z (t)2 twisted orbifold can be obtained by further identifying Z (t)2 twisted point − z with z . Note that the modulus τ is not restricted by Z (t)2 twist orbifolding, which means6e can also consider the modular transformation on the T / Z (t)2 twisted orbifold. Then, thewavefunctions on the magnetized T / Z (t)2 twisted orbifold should also satisfy the followingboundary condition, ψ j,MT / Z (t) m ( − z, τ ) = ( − m ψ j,MT / Z (t) m ( z, τ ) , m ∈ Z (t)2 , (37)in addition to the boundary conditions on the magnetized T in Eqs. (15) and (16). Actually,their boundary conditions are satisfied by the following linear combination of the wavefunctionson the magnetized T as ψ j,MT / Z (t) m ( z, τ ) = N j (t) (cid:16) ψ j,MT ( z, τ ) + ( − m ψ j,MT ( − z, τ ) (cid:17) , (38)where N j (t) denotes the normalization factor determined by the normalization condition inEq. (20). Since the wavefunctions on the T without the SS phases satisfy Eq. (34), ones onthe T / Z (t)2 twisted orbifold without the SS phases can be expanded by ψ j,MT / Z (t) m ( z, τ ) = N j (t) M − X k =0 ( δ j,k + ( − m δ M − j,k ) ψ k,MT ( z, τ ) , N j (t) = (cid:26) / j = 0 , M/ / √ . (39)In this case without the SS phases, there are M/ Z (t)2 -even ( m = 0) modes and M/ − Z (t)2 -odd ( m = 1) modes for M ∈ Z . Furthermore, under the modulartransformation, these transform similarly as Eq. (32) replacing Eq. (33) with e ρ T / Z (t)02 ( e S ) jk = N j (t) N k (t) e πi/ √ M cos (cid:18) πjkM (cid:19) , e ρ T / Z (t)02 ( e T ) jk = e πi j M δ j,k , (40) e ρ T / Z (t)12 ( e S ) jk = N j (t) N k (t) ie πi/ √ M sin (cid:18) πjkM (cid:19) , e ρ T / Z (t)12 ( e T ) jk = e πi j M δ j,k , (41)where e ρ T / Z (t) m ( e γ ) for each m ∈ Z satisfies Eqs. (27)-(31) with k/ / N = 2 M although I jk = δ j,k in Eq. (27) is modified into ( − m δ j,k , derived from Eq. (37). Thus, boththe Z (t)2 -even and odd mode wavefunctions on the T / Z (t)2 twisted orbifold with the magneticflux M ∈ Z and vanishing the SS phases behave as the modular forms of weight 1 / e Γ(2 M ). Then, they transform as ( M/ M/ − e Γ M for Z (t)2 -even and odd modes, respectively. It means the representations on the magnetized T can be decomposed into smaller representations on the magnetized T / Z (t)2 twisted orbifold. T and T / Z (t)2 twistedorbifold with the Scherk-Schwarz phases In this section, we review the wavefunctions on magnetized T and T / Z (t)2 twisted orbifoldwith the SS phases [26] and then we study the modular symmetry for them.7he wavefunctions on T with the flux M and the SS phases ( α , α ) (0 ≤ α , α < satisfy the following boundary conditions, ψ α ,α ( z + 1 , τ ) = e πiα e iχ ( z ) ψ α ,α ( z, τ ) = e πiα e πiM Im z Im τ ψ α ,α ( z, τ ) , (42) ψ α ,α ( z + τ, τ ) = e πiα e iχ ( z ) ψ α ,α ( z, τ ) = e πiα e πiM Im¯ τz Im τ ψ α ,α ( z, τ ) , (43)instead of Eqs. (15) and(16). Then, the j -th zero-mode wavefunction is expressed as ψ ( j + α ,α ) ,MT ( z, τ ) = (cid:18) M A (cid:19) / e πiMz Im z Im τ ϑ (cid:20) j + α M − α (cid:21) ( M z, M τ ) , ∀ j ∈ Z M . (44)Note that Eq. (18) corresponds to Eq. (44) with ( α , α ) = (0 , T transformation satisfy ψ α ′ ,α ′ ( z + τ + 1 , τ + 1) = e πiα ′ e πiM Im(¯ τ +1) z Im τ ψ α ,α ( z, τ + 1) , (45)while the wavefunctions before the T transformation satisfy ψ α ,α ( z + τ + 1 , τ ) = e πi ( α + α − M/ e πiM Im(¯ τ +1) z Im τ ψ α ,α ( z, τ ) . (46)Thus, in order to see the modular symmetry, particularly the T symmetry, of the wavefunctions, α ′ ≡ α + α − M/ α ′ ≡ α (mod 1) is required under the T transformation. Under the S transformation, similarly, α ′ ≡ α (mod 1), α ′ ≡ − α (mod 1)are required. Then, the modular transformation in Eqs. (32) and (33) are deformed as ψ ( j + α ′ ,α ′ ) ,MT ( e γ ( z, τ )) = e J / ( e γ, τ ) M − X k =0 e ρ T ( e γ ) jk ψ ( k + α ,α ) ,MT ( z, τ ) , e γ ∈ e Γ , (47) e ρ T ( e S ) jk = e πi/ √ M e πi (( j +1) k +(1 − α ′ ) α ) /M δ α ′ ,α δ − α ′ ,α , (48) e ρ T ( e T ) jk = e πi ( j + α ′ )( j − α ′ + x ) /M δ j,k δ α ,α ′ δ α ′ − α ′ + x/ ,α , (49)where x ≡ M (mod 2) and e ρ T ( e γ ) satisfies Eqs. (27)-(30) with k/ / I jk inEq. (27) is modified into e − πi ( j + α ′ ) /M δ M − j − ,k δ − α ′ ,α δ − α ′ ,α , derived from ψ ( j + α ,α ) ,MT ( e Z ( z, τ )) = ψ ( j + α ,α ) ,MT ( − z, τ ) = e − πi ( j + α ) /M ψ ( M − ( j + α ) , − α ) ,MT ( z, τ ) . (50) The wavefunction on the magnetized T ≃ C / Λ with the Z N SS phases is related to the Z N -eigenmodewavefunction on the magnetized Z N full shifted orbifold of e T ≃ C / e Λ ( e Λ = N Λ) withouth the SS phases [13,39],as shown in Appendix B. The analyses for the wavefunctions on the magnetized T with the ( Z N ) SS phasesare consistent with ones for the wavefunctions on the magnetized e T / Z N full shifted orbifold without the SSphases in Ref. [13]. α , α ) transform into oneswith the different SS phases ( α ′ , α ′ ). Conversely, when M is even, only the wavefunctions with( α , α ) = (0 ,
0) are closed under the modular transformation. This case is reviewed in previoussection. Similarly, when M is odd, only the wavefunctions with ( α , α ) = (1 / , /
2) are closedunder the modular transformation. In this case, e ρ T ( e T ) satisfies e ρ T ( e T ) M = e πi/ I , e ρ T ( e T ) M . (51)Thus, the wavefunctions on T with the magnetic flux M ∈ Z + 1 and the SS phases ( α , α ) =(1 / , /
2) behave as the modular forms of weight 1 / e Γ(8 M ) and then they transform as M -dimensional representations of e Γ M .Furthermore, we consider the magnetized T / Z (t)2 twisted orbifold with the SS phases . Inthis case, we can only consider the Z SS phases, ( α , α ) = ( ℓ / , ℓ / ℓ , ℓ ∈ Z ), whichare derived from 1 − α ≡ α (mod 1) , − α ≡ α (mod 1) . (52)The wavefunctions on the magnetized T / Z (t)2 twisted orbifold with the Z SS phases can beexpanded by ones on the magnetized T in Eq. (44) as ψ ( j + ℓ , ℓ ) ,MT / Z (t) m ( z, τ ) = N ( j + ℓ , ℓ ) (t) M − X k =0 (cid:16) δ j,k + ( − m e − πi ( j + ℓ ) ℓ /m δ M − j − ℓ ,k (cid:17) ψ ( k + ℓ , ℓ ) ,MT ( z, τ ) , (53)where we use Eq. (50) instead of Eq. (34). Then, the modular transformation for the wave-functions in Eq. (53) is similarly obtained by replacing Eqs. (48) and (49) with e ρ T / Z (t)02 ( e S ) jk = N ( j + ℓ , ℓ ) (t) N ( k + ℓ , ℓ ) (t) e πi/ √ M e πi ( kℓ ′ − jℓ ) cos (cid:18) π (cid:18) j + ℓ ′ (cid:19) (cid:18) k + ℓ (cid:19) /M (cid:19) δ ℓ ′ ,ℓ δ ℓ ′ ,ℓ , (54) e ρ T / Z (t)02 ( e T ) jk = e πi (cid:18) j + ℓ ′ (cid:19)(cid:18) j − ℓ ′ + x (cid:19) /M δ j,k δ ℓ ′ ,ℓ δ ℓ ′ − ℓ ′ + x,ℓ , (55) e ρ T / Z (t)12 ( e S ) jk = N ( j + ℓ , ℓ ) (t) N ( k + ℓ , ℓ ) (t) ie πi/ √ M e πi ( kℓ ′ − jℓ ) sin (cid:18) π (cid:18) j + ℓ ′ (cid:19) (cid:18) k + ℓ (cid:19) /M (cid:19) δ ℓ ′ ,ℓ δ ℓ ′ ,ℓ , (56) e ρ T / Z (t)12 ( e T ) jk = e πi (cid:18) j + ℓ ′ (cid:19)(cid:18) j − ℓ ′ + x (cid:19) /M δ j,k δ ℓ ′ ,ℓ δ ℓ ′ − ℓ ′ + x,ℓ . (57) Similarly, the wavefunctions on the magnetized T / Z (t)2 twisted orbifold with the SS phases are related toones on the magnetized e T / Z twisted and full shifted orbifold without the SS phases in Ref. [13].
9n particular, when M =even and ( α , α ) = (0 , M =odd and ( α , α ) = (1 / , / e ρ T / Z (t)02 ( e S ) jk = N ( j + , ) (t) N ( k + , ) (t) e πi/ √ M e πi ( k − j ) cos (cid:18) π (cid:18) j + 12 (cid:19) (cid:18) k + 12 (cid:19) /M (cid:19) , (58) e ρ T / Z (t)02 ( e T ) jk = e πi ( j + ) /M δ j,k , (59) e ρ T / Z (t)12 ( e S ) jk = N ( j + , ) (t) N ( k + , ) (t) ie πi/ √ M e πi ( k − j ) sin (cid:18) π (cid:18) j + 12 (cid:19) (cid:18) k + 12 (cid:19) /M (cid:19) , (60) e ρ T / Z (t)12 ( e T ) jk = e πi ( j + ) /M δ j,k , (61)where Eqs. (58)-(61) for each m ∈ Z satisfy Eqs. (27)-(30), and (51) with k/ / I jk = δ j,k in Eq. (27) is modified into ( − m δ j,k , derived from Eq. (37). Note that there are( M − / Z (t)2 -even ( m = 0) modes and ( M +1) / Z (t)2 -odd ( m = 1) modeswhen M =odd and ( α , α ) = (1 / , / Z (t)2 -even and odd mode wavefunctions onthe T / Z (t)2 twisted orbifold with the magnetic flux M ∈ Z + 1 and the SS phases ( α , α ) =(1 / , /
2) behave as the modular forms of weight 1 / e Γ(8 M ). Then, they transform as( M − / M + 1) / e Γ M for Z (t)2 -even and odd modes,respectively. We show the number of the Z (t)2 -eigenmodes, N m ( M ), which have the modularsymmetry and the finite modular subgroups in Tables 1 and 2. M Z (t)2 -even: N ( M ) M + 1 2 3 4 5 Z (t)2 -odd: N ( M ) M − e Γ M e Γ e Γ e Γ e Γ Table 1: The number of the Z (t)2 -even ( m = 0) modes, N ( M ), and the Z (t)2 -odd ( m = 1)modes, N ( M ), on the T / Z (t)2 twisted orbifold with M =even and ( α , α ) = (0 , M Z (t)2 -even: N ( M ) M − Z (t)2 -odd: N ( M ) M +12 e Γ M e Γ e Γ e Γ e Γ Table 2: The number of the Z (t)2 -even ( m = 0) modes, N ( M ), and the Z (t)2 -odd ( m = 1) modes, N ( M ), on the T / Z (t)2 twisted orbifold with M =odd and ( α , α ) = (1 / , / Modular flavor groups of three-generation modes onmagnetized T / Z (t)2 twisted orbifold As mentioned in introduction, in Ref. [29], three-dimensional representations can be obtainedfrom the specific finite modular subgroups: Γ ≃ A , Γ ≃ S , Γ ≃ A , Γ ≃ P SL (2 , Z ),Γ ⊃ ∆(96), and Γ ⊃ ∆(384). In this section, we show that the three-generation modes onthe magnetized T / Z (t)2 twisted orbifold shown in Tables 1 and 2 are the representations of thecorresponding covering or central extended groups of the modular flavor groups. T / Z (t)2 twisted orbifold with magnetic flux M = even and vanish-ing Scherk-Schwarz phases In this subsection, we show the modular flavor groups of the three-generation modes on the T / Z (t)2 twisted orbifold with M =even and ( α , α ) = (0 , Z (t)2 -even modes with M = 4 and the Z (t)2 -odd modeswith M = 8. They are the representations of e Γ and e Γ , respectively. In the following, weshow they are the representations of the subgroups, e ∆(96) and e ∆(384), which are the quadruplecovering groups of ∆(96) and ∆(384), respectively.First, Γ N satisfy S = ( ST ) = T N = . (62)On the other hand, ∆(96) ≃ ( Z × Z ′ ) ⋊ Z ⋊ Z ≃ ∆(48) ⋊ Z and ∆(384) ≃ ( Z × Z ′ ) ⋊ Z ⋊ Z ≃ ∆(192) ⋊ Z satisfy a M = a ′ M = b = c = , ( M = 4 , , (63) aa ′ = a ′ a, cbc − = b − , bab − = a − a ′− , ba ′ b − = a, cac − = a ′− , ca ′ c − = a − , where a ( ′ ) , b , c denote the generators of Z ( ′ ) M ( M = 4 , Z , Z , respectively [2, 3, 40]. In orderto obtain ∆(96) and ∆(384) from the above algebra (62) for N = 8 and 16, respectively, thefollowing relation, ( S − T − ST ) = (64)should be also satisfied. Actually, we can show that if S and T satisfy Eq. (64) in addition toEq. (62) for N = 2 M, M ∈ Z , the following generators , a = ST ST , a ′ = ST S − T − , b = T M +3 ST M , c = ST M − ST M − , (65) See Refs. [2,3,40] to see the algebraic relations for the generators of each non-Abelian discrete flavor group. For N = 2 M , M = 2(2 s − s ∈ Z , similarly, the generators, a = ST ST , a ′ = ST S − T − , b = T M ST M ,and c = ST M ST M , satisfy Eq. (63). M )from Γ M by satisfying the additional relation in Eq. (64). Similarly, e Γ M satisfy Eqs. (27)-(31)with k/ / N = 2 M . If Eq. (64) is also satisfied, especially for M ∈ Z , the followinggenerators, a = ST S T , a ′ = ST S − T − , b = T M +3 S M − T M , c = ST M − ST M − , (66)satisfy a M = a ′ M = b = c = , (67) aa ′ = a ′ a, cbc − = b − , bab − = a − a ′− , ba ′ b − = a, cac − = a ′− , ca ′ c = a − , which means the generators in Eq. (66) are ones of e ∆(6 M ) ≃ ( Z M × Z M ) ⋊ Z ⋊ Z ≃ ∆(3 M ) ⋊ Z , where a ( ′ ) , b , c denote ones of Z ( ′ ) M , Z , Z , respectively. (We give the proof inAppendix C.) In other words, we can obtain e ∆(6 M ), especially for M ∈ Z , from e Γ M bysatisfying the additional relation in Eq. (64).Let us study the case of the three-generation modes on the T / Z (t)2 twisted orbifold with M = 4 , α , α ) = (0 , S and T transformation matrices for the Z (t)2 -even modeswith M = 4 are given by S = e πi/ √ √ −√ −√ , T = e πi/ − , (68)and ones for the Z (t)2 -odd modes with M = 8 are given by S = e πi/ √ √ −√ −√ , T = e πi/ e πi/ − . (69)Note that here and hereafter (as well as in section 6), we omit ρ . Both of the above S and T matrices are the same forms as S = e iθ √ √ −√ −√ , T = e iθ e iθ − , ∀ θ , , ∈ R , (70)and we can check that Eq. (70) satisfies Eq. (64) in general. Thus, the three-generation Z (t)2 -even modes with M = 4 and Z (t)2 -odd modes with M = 8 are transformed under the modulartransformation as the three-dimensional representations of e ∆(96) and e ∆(384), respectively .We also comment on the modular flavor anomaly. As discussed in Ref. [22,42], the transfor-mation g can be anomalous if det( g ) = 1. Then, let us see the anomaly of the modular flavor See also Ref. [11]. e ∆(6 M ). From Eqs. (27)-(31) with k/ / N = 2 M , (66), and (67), we canobtain det( a ) = det( a ′ ) = det( b ) = 1 , det( c ) = det( T ) M +3 , det( c ) = 1 . (71)Actually, both of Eqs. (68) and (69) satisfy Eq. (71) and det( c ) = e πi/ . Thus, only Z sym-metry, generated by c , can be anomalous and then ∆(48) and ∆(192) remain anomaly-free,respectively. T / Z (t)2 twisted orbifold with magnetic flux M = odd and theScherk-Schwarz phases ( α , α ) = (1 / , / In this subsection, we show the modular flavor groups of the three-generation modes on the T / Z (t)2 twisted orbifold with M =odd and ( α , α ) = (1 / , / Z (t)2 -odd modes with M = 5 and the Z (t)2 -evenmodes with M = 7.First, the S and T transformation matrices for the Z (t)2 -odd modes with M = 5 are givenby S = ie πi/ √ (cid:0) π (cid:1) e πi/ sin (cid:0) π (cid:1) √ e πi/ e − πi/ sin (cid:0) π (cid:1) (cid:0) π (cid:1) −√ e πi/ √ e − πi/ −√ e − πi/ , T = e πi/ e πi/ e πi/ , (72)which satisfy Eqs. (27)-(30) and (51) with k/ / I in Eq. (27) with ( − m =1 I = − I . When we define the following generators, a = ST , b = ST , c = T , (73)from the above S and T in Eq. (72), they satisfy a = b = ( ab ) = c = , ac = ca, bc = cb, (74)which mean they are the generators of A × Z . Thus, the three-generational Z (t)2 -odd modeswith M = 5 are transformed under the modular transformation as the three-dimensional rep-resentations of A × Z .Next, the S and T transformation matrices for the Z (t)2 -even modes with M = 7 are givenby S = 2 e πi/ √ cos (cid:0) π (cid:1) e πi/ cos (cid:0) π (cid:1) e πi/ cos (cid:0) π (cid:1) e − πi/ cos (cid:0) π (cid:1) cos (cid:0) π (cid:1) − e πi/ cos (cid:0) π (cid:1) e − πi/ cos (cid:0) π (cid:1) − e − πi/ cos (cid:0) π (cid:1) cos (cid:0) π (cid:1) , T = e πi/ e πi/ e πi/ (75) The anomalous symmetry which is the discrete subsymmetry of U (1) can be cancelled by the Green-Schwarzmechanism. k/ /
2. They also satisfy( S − T − ST ) = . (76)When we define the following generators, a = ST , b = S T , c = T (77)from the above S and T in Eq. (75), they satisfy a = b = ( ab ) = ( a − b − ab ) = c = , ac = ca, bc = cb (78)which mean they are the generators of P SL (2 , Z ) × Z . Thus, the three-generational Z (t)2 -evenmodes with M = 7 are transformed under the modular transformation as the three-dimensionalrepresentations of P SL (2 , Z ) × Z .Similarly, we comment on the anomaly of those modular flavor groups. From Eqs. (27)-(30)with k/ /
2, (51), (73), and (74) as well as (77) and (78), we can obtaindet( a ) = det( b ) = 1 , det( c ) = det( e πi/ I ) , det( c ) = 1 . (79)Actually, Eq. (72) as well as Eq. (75) satisfy Eq. (79). Thus, in both cases, only Z symmetry,generated by c , can be anomalous and then A and P SL (2 , Z ) remain anomaly-free. T × T T × T , where both of the modulus on T i ( i = 1 , τ i , are identified each other, i.e. τ = τ ≡ τ .(See Ref. [14].) First, let us consider the modular transformation for the wavefunctions on the T / Z (t )2 × T / Z (t )2 with the magnetic flux M ( i ) =even and the SS phases ( α ( i )1 , α ( i )2 ) = (0 , M ( i ) =odd and the SS phases ( α ( i )1 , α ( i )2 ) = (1 / , /
2) on each T i / Z (t i )2 .The wavefunctions transform under the modular transformation asΨ j (1) j (2) ,M (1) M (2) (t ) m (t ) m ( γ ( z , z , τ ))= J ( γ, τ ) N m ( M (1) ) X k (1) =0 N m ( M (2) ) X k (2) =0 ρ (t ) m (t ) m ( γ ) ( j (1) j (2) )( k (1) k (2) ) Ψ k (1) k (2) ,M (1) M (2) (t ) m (t ) m ( z , z , τ ) (80) m ∈ Z (t )2 , m ∈ Z (t )2 , γ ∈ ΓΨ j (1) j (2) ,M (1) M (2) (t ) m (t ) m ( z , z , τ ) = ψ ( k (1) + α (1)1 ,α (1)2 ) ,M (1) T / Z (t1) m ( z , τ ) ψ ( k (2) + α (2)1 ,α (2)2 ) ,M (2) T / Z (t2) m ( z , τ ) (81) ρ (t ) m (t ) m ( S ) ( j (1) j (2) )( k (1) k (2) ) = e ρ T / Z (t1) m ( e S ) j (1) k (1) e ρ T / Z (t2) m ( e S ) j (2) k (2) , (82) ρ (t ) m (t ) m ( T ) ( j (1) j (2) )( k (1) k (2) ) = e ρ T / Z (t1) m ( e T ) j (1) k (1) e ρ T / Z (t2) m ( e T ) j (2) k (2) , (83)14here e ρ T / Z (t i ) mi ( e γ ) ( i = 1 ,
2) correspond to Eqs. (40)-(41) for M ( i ) =even and ( α ( i )1 , α ( i )2 ) = (0 , M ( i ) =odd and ( α ( i )1 , α ( i )2 ) = (1 / , / ρ (t ) m (t ) m ( γ ) satisfiesEq. (8) with k = 1, where ρ ( Z ) = − I is replaced by ρ (t ) m (t ) m ( Z ) = − ( − m + m I , and alsosatisfies ρ ( T ) M (1) ,M (2) ) = I , ( M (1) = 2 s (1) , M (2) = 2 s (2) ) , (84) ρ ( T ) M (1) ,M (2) ) = I , ( M (1) = 4 s (1) , M (2) = 2 s (2) − , (85) ρ ( T ) M (1) ,M (2) ) = − I , ρ ( T ) M (1) ,M (2) ) = I , ( M (1) = 2(2 s (1) − , M (2) = 2 s (2) −
1) (86) ρ ( T ) lcm( M (1) ,M (2) ) = e πi M (1)+ M (2)4gcd( M (1) ,M (2)) I , ( M (1) = 2 s (1) − , M (2) = 2 s (2) − ⇒ ρ ( T ) N = I , N = lcm( M (1) , M (2) ) ( M (1) + M (2) ∈ Z )2lcm( M (1) , M (2) ) ( M (1) + M (2) ∈ Z )4lcm( M (1) , M (2) ) ( M (1) + M (2) ∈ Z ) , (87)corresponding to Eq. (9), where s (1) , s (2) ∈ Z and we omit the Z (t)2 indices since the above rela-tions are independent of them. Thus, the wavefunctions on the magnetized T / Z (t )2 × T / Z (t )2 orbifold behave as the modular forms of weight 1 for Γ(4lcm( M (1) , M (2) )) (in general) and thenthey transform as N m ( M (1) ) N m ( M (2) )-dimensional representations of Γ ′ M (1) ,M (2) ) , where N m i ( M ( i ) ) ( i = 1 ,
2) denote the number of zero-mode wavefunctions on T i / Z (t i )2 .We can further consider the Z (p)2 permutation orbifold if M (1) = M (2) = M , α (1) i = α (2) i = α i ( i = 1 , m = m = m . The Z (p)2 permutation means the transformation of thecomplex coordinate of T / Z (t )2 × T / Z (t )2 : ( z , z ) → ( z , z ), and then the Z (p)2 permutationorbifold can be considered by identifying z and z . Hence, the wavefunctions on the Z (p)2 permutation orbifold of T / Z (t )2 × T / Z (t )2 , i.e. ( T × T ) / ( Z (t)2 × Z (p)2 ) orbifold, are expressedas Ψ j (1) j (2) ,M ( t ) m ( p ) n ( z , z , τ ) = N j (1) j (2) (t , p) (cid:16) Ψ j (1) j (2) ,MM (t) m (t) m ( z , z , τ ) + ( − n Ψ j (1) j (2) ,MM (t) m (t) m ( z , z , τ ) (cid:17) m ∈ Z (t)2 , n ∈ Z (p)2 , j (1) ≥ j (2) , N j (1) j (2) (t , p) = (cid:26) / j (1) = j (2) )1 / √ j (1) > j (2) ) , (88)and they satisfy the following boundary condition,Ψ j (1) j (2) ,M ( t ) m ( p ) n ( z , z , τ ) = ( − n Ψ j (1) j (2) ,M ( t ) m ( p ) n ( z , z , τ ) , (89)in addition to ones in Eqs. (15), (16), and (37). Thus, we can obtain N m ( M )( N m ( M ) + 1) / Z (p)2 -even ( n = 0) modes and N m ( M )( N m ( M ) − / Z (p)2 -odd ( n =1) modes. We show the number of ( Z (t)2 twist, Z (p)2 permutation)-eigenmodes, N ( m,n ) ( M ) = lcm( a, b ) denotes the least common multiple of a and b , and gcd( a, b ) denotes the greatest common divisorof a and b . When m + m = 1, Γ ′ N is replaced by Γ N even though the modular weight k = 1. Note that this situationdoes not appear on modular forms and actually the wavefunctions vanish at z = z = 0. m ( M )( N m ( M ) + ( − n ) /
2, which have the modular symmetry in Tables 3 and 4. Under themodular transformation, the wavefunctions in Eq. (88) similarly transform as Eq. (80) replacingEqs. (82) and (83) with ρ (t) m (p) n ( γ ) ( j (1) j (2) )( k (1) k (2) ) = 2 N j (1) j (2) (t , p) N k (1) k (2) (t , p) (cid:0) ρ (t) m ( γ ) ( j (1) j (2) )( k (1) k (2) ) + ( − n ρ (t) m ( S ) ( j (1) j (2) )( k (1) k (2) ) (cid:1) , (90)where it satisfies Eq. (8) with k = 1 and also satisfies ρ (t) m (p) n ( T ) M = I , ( M ∈ Z ) , (91) ρ (t) m (p) n ( T ) M = i I , ρ (t) m (p) n ( T ) M = − I , ρ (t) m (p) n ( T ) M = I , ( M ∈ Z + 1) , (92)corresponding to Eq. (9). Thus, the wavefunctions on the ( T × T ) / ( Z (t)2 × Z (p)2 ) orbifold withthe magnetic flux M ∈ Z and the SS phases ( α , α ) = (0 ,
0) behave as the modular forms ofweight 1 for Γ(2 M ) and then they transform as N ( m,n ) ( M )-dimensional representations of Γ ′ M ,as shown in Table 3. Similarly, the wavefunctions with the magnetic flux M ∈ Z + 1 and theSS phases ( α , α ) = (1 / , /
2) behave as the modular forms of weight 1 for Γ(4 M ) and thenthey transform as N ( m,n ) ( M )-dimensional representations of Γ ′ M , as shown in Table 4. M N (0 , ( M ) ( M + 2)( M + 4) / N (0 , ( M ) M ( M + 2) / N (1 , ( M ) M ( M − / N (1 , ( M ) ( M − M − / ′ M Γ ′ Γ ′ Γ ′ Γ ′ Table 3: The number of ( Z (t)2 twist, Z (p)2 permutation)-eigenmodes, N ( m,n ) ( M ), on the ( T × T ) / ( Z (t)2 × Z (p)2 ) orbifold with M =even and ( α , α ) = (0 , M N (0 , ( M ) ( M − M + 1) / N (0 , ( M ) ( M − M − / N (1 , ( M ) ( M + 1)( M + 3) / N (1 , ( M ) ( M + 1)( M − / ′ M Γ ′ Γ ′ Γ ′ Γ ′ Table 4: The number of ( Z (t)2 twist, Z (p)2 permutation)-eigenmodes, N ( m,n ) ( M ), on the ( T × T ) / ( Z (t)2 × Z (p)2 ) orbifold with M =odd and ( α , α ) = (1 / , / T × T .16 Modular flavor groups of three-generation modes onmagnetized orbifolds of T × T ( T × T ) / ( Z (t)2 × Z (p)2 ) orbifold Firstly, we consider the three-generation modes on the magnetized ( T × T ) / ( Z (t)2 × Z (p)2 )orbifold in Tables 3 and 4.As shown in Table 3, we can obtain four models with three-generation modes on the ( T × T ) / ( Z (t)2 × Z (p)2 ) orbifold with M =even and ( α , α ) = (0 , M, m, n ) = (2 , , , , , , , , ′ M and also can be repesentations of∆ ′ (6 M ), which are the double covering groups of ∆(6 M ), similar as shown in subsection 4.1.Namely, if Eq. (64) is also satisfied in addition to Eqs. (8) and (9) with k = 1 and N = 2 M ,the following generators, a = ST ST , a ′ = ST S − T − , b = T M +3 S M − T M , c = ST M − ST M − , ( M = 4 s ) (93) a = ST ST , a ′ = ST S − T − , b = T M S M T M , c = ST M ST M , ( M = 2(2 s − s ∈ Z , satisfy a M = a ′ M = b = c = , (95) aa ′ = a ′ a, cbc − = b − , bab − = a − a ′− , ba ′ b − = a, cac − = a ′− , ca ′ c − = a − , which means the generators in Eq. (93) are ones of ∆ ′ (6 M ) ≃ ( Z M × Z M ) ⋊ Z ⋊ Z ≃ ∆(3 M ) ⋊ Z , where a ( ′ ) , b , c denote ones of Z ( ′ ) M , Z , Z , respectively. Actually, all of thefollowing S and T transformation matrices for ( M, m, n ) = (2 , , , , , , , ,
1) satisfy Eq. (64) since they form as Eq. (70).The S and T transformation matrices for ( M, m, n ) = (2 , ,
0) are given by S = i √ √ −√ −√ , T = i − . (96)The S and T transformation matrices for ( M, m, n ) = (4 , ,
1) are given by S = − i √ √ −√ −√ , T = e πi/ e πi/ − . (97)The S and T transformation matrices for ( M, m, n ) = (6 , ,
0) are given by S = − i √ √ −√ −√ , T = e πi/ i − . (98) When M = 1 ,
2, Eq. (64) is automatically satisfied by considering Eq. (8). (See in detail Appendix C.) S and T transformation matrices for ( M, m, n ) = (6 , ,
0) are given by S = i √ √ −√ −√ , T = e πi/ e πi/ − . (99)Note that since the T matrix in Eq. (98) also satisfies T = e πi/ I , this can be the Z generator, d = T , which commutes with all the generators in Eq. (94) and also the generators a and a ′ in Eq. (94) satisfy a = a ′ = . Thus, the three-generation modes for ( M, m, n ) = (2 , , , , , , , ,
1) are transformed under the modular transformation as the three-dimensional representations of S ′ ≃ ∆ ′ (24), ∆ ′ (96), S ′ × Z , and ∆ ′ (384), respectively.We also comment on the anomaly of those modular flavor groups. From Eqs. (8) with k = 1,(9) with N = 2 M , (93), (94), and (95), similarly, we can obtaindet( a ) = det( a ′ ) = det( b ) = 1 , det( c ) = ( det( T ) M +3 ( M = 4 s )det( T ) M +6 ( M = 2(2 s − , det( c ) = 1 . (100)All of Eqs. (96)-(99) satisfy Eq. (100) and det( c ) = i . In Eq. (98), det( d ) = det( T ) = 1 is alsosatisfied. Thus, in all cases, only Z symmetry, generated by c , can be anomalous and then A ≃ ∆(12), ∆(48), A × Z , and ∆(192) remain anomaly-free.As shown in Table 4, we can obtain four models with three-generation modes on the ( T × T ) / ( Z (t)2 × Z (p)2 ) orbifold with M =odd and ( α , α ) = (1 / , / M, m, n ) = (3 , , , , , , , , S and T transformation matrices satisfyEqs. (8) and (92) with k = 1. First, from the following S and T transformation matrices for( M, m, n ) = (3 , , S = − i e πi/ e πi/ e − πi/ − e πi/ e − πi/ − e − πi/ , T = e πi/ e πi/ e πi/ , (101)we can obtain the following generators, a = ST , b = ST, c = T (102)satisfying a = b = ( ab ) = c = , ac = ca, bc = cb (103)which mean the generators in Eq. (102) are ones of A × Z . Thus, the three-generationmodes, ( M, m, n ) = (3 , , A × Z . 18econd, from the following S and T transformation matrices for ( M, m, n ) = (5 , , S = 4 i A √ e πi/ AB e πi/ B √ e − πi/ AB B − A −√ e πi/ ABe − πi/ B −√ e − πi/ AB A , T = e πi/ e πi/ e πi/ , (104) A = cos (cid:16) π (cid:17) , B = cos (cid:18) π (cid:19) , we can obtain the following generators, a = ST , b = ST, c = T (105)satisfying a = b = ( ab ) = c = , ac = ca, bc = cb (106)which mean the generators in Eq. (105) are ones of A × Z . Thus, the three-generationmodes, ( M, m, n ) = (5 , , A × Z .Third, similarly, from the following S and T transformation matrices for ( M, m, n ) =(5 , , S = − i A − B ) −√ e πi/ ( A + B ) −√ e πi/ ( A + B ) −√ e − πi/ ( A + B ) A − e πi/ ( B + 1) −√ e − πi/ ( A + B ) e − πi/ ( B + 1) A − ,A = sin (cid:16) π (cid:17) , B = sin (cid:18) π (cid:19) ,T = e πi/ e πi/ e πi/ (107)we can obtain the generators in Eq. (105) satisfying Eq. (106). Thus, the three-generationmodes, ( M, m, n ) = (5 , , A × Z .Fourth, from the following S and T transformation matrices for ( M, m, n ) = (7 , , S = 4 i AD − B − e πi ( A + BC ) − e πi ( AB + CD ) − e − πi ( A + BC ) AB − C e πi ( B + AC ) − e − πi ( AB + CD ) e − πi ( B + AC ) BD − A ,A = cos (cid:16) π (cid:17) , B = cos (cid:18) π (cid:19) , C = cos (cid:18) π (cid:19) , D = cos (cid:18) π (cid:19) ,T = e πi/ e πi/ e πi/ , (108)19hich also satisfy Eq. (76), we can obtain the following generators, a = ST , b = S T , c = T (109)satisfying a = b = ( ab ) = ( a − b − ab ) = c = , ac = ca, bc = cb (110)which mean the generators in Eq. (109) are ones of P SL (2 , Z ) × Z . Thus, the three-generationmodes, ( M, m, n ) = (7 , , P SL (2 , Z ) × Z .Finally, we also comment on the anomaly of those modular flavor groups. From Eqs. (101)-(109), and also Eqs. (8) and (92) with k = 1, we can obtaindet( a ) = det( b ) = 1 , det( c ) = det( i I ) = − i, det( c ) = 1 . (111)Thus, in all the above cases, only Z symmetry, generated by c , can be anomalous and then A , A and P SL (2 , Z ) remain anomaly-free.Therefore, on the magnetized ( T × T ) / ( Z (t)2 × Z (p)2 ) orbifold, we can obtain three-dimensionalrepresentations of all the double covering groups of Γ ≃ S , Γ ⊃ ∆(96), and Γ ⊃ ∆(384)for even magnetic fluxes and Z central extended groups of Γ ≃ P SL (2 , Z ) ≃ A , Γ ≃ P SL (2 , Z ) ≃ A , Γ ≃ P SL (2 , Z ) for odd magnetic fluxes. T / Z (t )2 × T / Z (t )2 orbifold Finally, we consider the three-generation modes on the magnetized T / Z (t )2 × T / Z (t )2 orbifold,where T / Z (t )2 and T / Z (t )2 are not identified. In order to obtain N m ( M (1) ) N m ( M (2) ) = 3on the magnetized T / Z (t )2 × T / Z (t )2 orbifold, we can only consider N m ( M (1) ) = 3 and N m ( M (2) ) = 1. Then, from Tables 1 and 2, we can consider twelve patterns, listed inTable 5. The corresponding finite modular subgroups which can be found by considering Z = − ( − m + m and Eqs. (84)-(87) are also listed in Table 5. The S and T transforma-tion matrices for the Z (t)2 -odd modes with M = 1 as well as the Z (t)2 -odd modes with M = 4are given by S = e πi/ , T = e πi/ , (112)and ones for the Z (t)2 -even modes with M = 3 are given by S = e πi/ , T = e πi/ , (113) There is an exception in Eq. (85); Eq. (85) for M (1) = 4 singlet mode, that is Z (t )2 -odd mode of M (1) = 4,corresponds to Eq. (87) with M (1) = 1. N m ( M (1) ) = 3 modes are expressed in section 4. Then, we can find the specificmodular flavor groups as shown in Table 5. We also show the anomaly-free groups of them inTable 5.( M (1) , m : M (2) , m ) finite modular subgroup modular flavor group anomaly-free group(4 , ,
1) Γ ∆(96) ∆(96)(4 , ,
1) Γ ∆(96) ∆(96)(4 , ,
0) Γ ′ ∆ ′ (96) × Z ∆(48) × Z (8 , ,
1) Γ ′ ∆ ′ (384) ∆(192)(8 , ,
1) Γ ′ ∆ ′ (384) ∆(192)(8 , ,
0) Γ ∆(384) × Z ∆(384) × Z (5 , ,
1) Γ ′ A × Z A (5 , ,
1) Γ ′ A × Z A (5 , ,
0) Γ A × Z A × Z (7 , ,
1) Γ P SL (2 , Z ) P SL (2 , Z )(7 , ,
1) Γ P SL (2 , Z ) P SL (2 , Z )(7 , ,
0) Γ ′ P SL (2 , Z ) × Z × Z P SL (2 , Z ) × Z Table 5: The flavor groups of the three-generation modes, ( M (1) , m : M (2) , m ), which satisfy N m ( M (1) ) = 3 and N m ( M (2) ) = 1, on the magnetized T / Z (t )2 × T / Z (t )2 orbifold. Theanomaly-free subgroups are also shown. We have studied the modular symmetry of wavefunctions on magnetized orbifolds: T / Z (t)2 twisted orbifold, T / Z (t )2 × T / Z (t )2 twisted orbifold, and the Z (p)2 permutation orbifold, i.e. ( T × T ) / ( Z (t)2 × Z (p)2 ) orbifold, with the Scherk-Schwarz phases. It has been found that we can con-sider the modular symmetry of not only wavefunctions with the magnetic flux M =even andthe vanishing SS phases ( α , α ) = (0 ,
0) but also ones with the magnetic flux M =odd and theSS phases ( α , α ) = (1 / , / T / Z (t)2 twistedorbifold with the magnetic flux M = 4 , e ∆(96), e ∆(384), which are quadruple covering groups of ∆(96), ∆(384), respectively. Among them,only Z symmetries can be anomalous and then ∆(48), ∆(192) are anomaly free, respectively.Note that since the anomalous Z symmetry is discrete subgroup of U (1), it can be canceled bythe Green-Schwarz mechanism. The three-generation modes on the magnetized T / Z (t)2 twistedorbifold with the magnetic flux M = 5 , A × Z , P SL (2 , Z ) × Z , respectively. Among them, only Z symmetries can be anomalous and then A and P SL (2 , Z ) are anomaly free, respectively. Similarly, the three-generation modes on the21agnetized ( T × T ) / ( Z (t)2 × Z (p)2 ) orbifold are the corresponding three-dimensional representa-tions of the double covering groups of Γ N for N = 4 , ,
16 and Z central extended groups of Γ N for N = 3 , ,
7, provided in Ref. [29]. Among them, only Z symmetries can be anomalous andthen ∆(3 M ) for N = 2 M = 4 , , A for N = 3, A for N = 5, P SL (2 , Z ) for N = 7 areanomaly free. We have also showed the specific modular flavor groups of the three-generationmodes on the other distinguishable magnetized T / Z (t )2 × T / Z (t )2 orbifolds in Table 5.Our results on flavor symmetries of three generations are useful to understand quarks andlepton masses and their mixing angles. Also, anomaly behaviors are useful. (See e.g. [43].) Wewould investigate the realistic model building considering the obtained modular flavor groupsin magnetized orbifold models elsewhere. Acknowledgement
T. K. was supported in part by MEXT KAKENHI Grant Number JP19H04605. H. U. wassupported by Grant-in-Aid for JSPS Research Fellows No. 20J20388.
A Scherk-Schwarz phases and Wilson lines
Here, we show that the Scherk-Schwarz (SS) phases can be converted into the Wilison lines(WLs) through gauge transformation [26] and also that the modular transformations for themare consistent each other.First, let us consider the following gauge transformation, e ψ α ,α ( z, τ ) = e − i Re ¯ βz ψ α ,α ( z, τ ) , (114) e A ( z ) = A ( z ) − d [Re ¯ βz ] = πM Im τ Im (cid:18)(cid:18) ¯ z − i Im τπM ¯ β (cid:19) dz (cid:19) , (115)where β is a complex number, ψ α ,α satisfies Eqs. (42) and (43), A ( z ) is in Eq. (11). We canregard i Im τπM β ≡ e a w as the WL. Accordingly, χ ( z ) and χ ( z ), defined in Eqs. (12) and (13), aredeformed as e χ ( z ) = πM Im τ Im (cid:18) z + i Im τπM β (cid:19) = χ ( z ) + Re β, (116) e χ ( z ) = πM Im τ Im¯ τ (cid:18) z + i Im τπM β (cid:19) = χ ( z ) + Re¯ τ β. (117)Therefore, the boundary conditions of the gauge transformed wavefunction e ψ α ,α are modifiedfrom Eqs. (42) and (43) as e ψ α ,α ( z + 1 , τ ) = e πiα − i Re β e i e χ ( z ) e ψ α ,α ( z, τ ) , (118) e ψ α ,α ( z + τ, τ ) = e πiα − i Re¯ τβ e i e χ ( z ) e ψ α ,α ( z, τ ) . (119)22hen we chose β = − iπ α τ − α Im τ , the gauge transformed wavefunction, e ψ α ,α ( z, τ ) = e πi Im( α τ − α z Im τ ψ α ,α ( z, τ ) , (120)has the WL, M e a w = α τ − α , and the vanishing SS phases, ( e α , e α ) = (0 , α , α ), can be converted into the WL, M e a w = α τ − α , through gauge transformationin Eq. (120). Actually, the j -th wavefunction can be expressed as e ψ ( j + α ,α ) ,MT ( z, τ ) = e − πi α α M ψ ( j +0 , ,MT ( z + e a w , τ ) . (121)Next, let us consider the modular transformation. When M =even ( x = 0), the WLtransforms as T ( M e a w ) = α ( τ + 1) − ( α + α ) = M e a w , T = (cid:18) (cid:19) , (122) S ( M e a w ) = − α (cid:18) − τ (cid:19) − α = M e a w − τ , S = (cid:18) − (cid:19) , (123)that is, it transforms as γ ( M e a w ) = M e a w cτ + d , γ = (cid:18) a bc d (cid:19) . (124)In this case, as mentioned in Ref. [13], the modular transformation for the wavefunction in right-hand side of Eq. (121) is the same as Eq. (33). Furthermore, in this case, the gauge phase inEq. (120) is invariant under the modular transformation and then the modular transformationfor the gauge transformed wavefunction in left hand side of Eq. (120) or Eq. (121) is the sameas Eqs. (48) and (49). These are consistent. When M =odd ( x = 1), the T transformation forthe WL as T ( M e a w ) = α ( τ + 1) − (cid:18) α + α − M (cid:19) = M (cid:18)e a w + 12 (cid:19) . (125)Under the T transformation, the wavefunction with the WL in right-hand side of Eq. (121) istransformed as ψ ( j +0 , ,MT (cid:18) z + e a w + 12 , τ + 1 (cid:19) = e πij e πi j M e πi Mz + α τ − α τ ψ ( j +0 , ,MT ( z + e a w , τ ) . (126)On the other hand, in this case, the gauge phase in Eq. (120) is also transformed: e ψ α ,α ( z, τ + 1) = e πi M Im z Im τ e πi Im( α τ − α z Im τ ψ α ,α ( z, τ + 1) . (127)Considering this equation and Eq. (49), actually, the T transformation for the wavefunction inleft-hand side of Eq. (121) is consistent with Eq. (126).23 Z N Scherk-Schwarz phases and Z N shift modes Here, we also show that the wavefunctions on magnetized T ≃ C / Λ with the Z N SS phasesare related to the Z N -eigenmode wavefunctions on magnetized full Z N shifted orbifold of e T ≃ C / e Λ ( e Λ = N Λ) without the SS phases as follows.First, the lattice vectors e e k ( k = 1 ,
2) of the lattice e Λ = N Λ are written by ones of the latticeΛ, e k ( k = 1 , e e k = N e k . Then, the coordinate and the modulus of e T ≃ C / e Λ, ( e z, e τ ) ≡ ( u/ e e , e e / e e ) are related to ones of T ≃ C / Λ, ( z, τ ) ≡ ( u/e , e /e ), as ( e z, e τ ) = ( z/N, τ ), where u is the coordinate of C . Note that e z + 1 ∼ e z and e z + e τ ∼ e z are satisfied on e T .The e T / Z N full shifted orbifold [13], on which the full modular symmetry remains, can beobtained by furthre identifying any Z N shifted points e z + ( r + s e τ ) /N ( ∀ r, s ∈ Z N ) with e z .(See also Ref. [39].) Then, the boundary conditions of the wavefunction on the e T / Z N fullshifted orbifold with the magnetic flux f M and the vanishing SS phases are just the followingtwo conditions, ψ e T / Z ( ℓ ,ℓ N (cid:18)e z + 1 N , e τ (cid:19) = e πi ℓ N e πi f M Im e zN Im e τ ψ e T / Z ( ℓ ,ℓ N ( e z, e τ ) , (128) ψ e T / Z ( ℓ ,ℓ N (cid:18)e z + e τN , e τ (cid:19) = e πi ℓ N e πi f M Im ¯ e τN e z Im e τ ψ e T / Z ( ℓ ,ℓ N ( e z, e τ ) , (129)where ℓ , ℓ ∈ Z N are the Z N -eigenvalues. From the above boundary conditions, f M /N ≡ M ∈ Z should be satisfied. The above wavefunction on the magnetized e T / Z N full shifted orbifoldwithout the SS phases, ψ j,M e T / Z ( ℓ ,ℓ N , can be expanded by the wavefunction on the magnetized e T without the SS phases as ψ j,M e T / Z ( ℓ ,ℓ N ( e z, e τ ) = 1 √ N N − X k =0 e − πik ℓ N ψ ( Nj + ℓ )+ kNM,N M e T ( e z, e τ ) . (130)Furthermore, by considering the relation, ( e z, e τ ) = ( z/N, τ ), the boundary conditions inEqs. (128) and (129) correspond to ones in Eqs. (42) and (43) with the Z N SS phases, ( α , α ) =( ℓ /N, ℓ /N ) ( ℓ , ℓ ∈ Z N ). Actually, the above wavefunction with the Z N -eigenvalue, ( ℓ , ℓ ),on the e T / Z N full shifted orbifold with the magnetic flux f M and the vanishing SS phases isrelated to the wavefunction on T with the magnetic flux M and the Z N SS phases, ( α , α ) =( ℓ /N, ℓ /N ), as ψ j,M e T / Z ( ℓ ,ℓ N (cid:16) zN , τ (cid:17) = 1 √ N N − X k =0 e − πik ℓ N ψ ( Nj + ℓ )+ kNM,N M e T (cid:16) zN , τ (cid:17) = e πi ( j + ℓ N ) ℓ N /M ψ ( j + ℓ N , ℓ N ) ,MT ( z, τ ) . (131)The analyses of the modular transformation are also consistent.Similarly, the wavefunction on the magnetized e T / Z twisted and full shifted orbifold with-out the SS phases is related to one on the magnetized T with the Z SS phases. Their behaviorof the modular transformation are consistent each other.24 e ∆(6 M ) as subgroup of e Γ M Here, we prove the generators in Eq. (66), in particular for M ∈ Z , satisfy the algebraicrelations of e ∆(6 M ) in Eq. (67), where the algebraic relations of e Γ M in Eqs. (27)-(31) with N = 2 M and also the additional relation in Eq. (64) are satisfied. Note that when we have k/ k =integer [even] and N = 2 M , which correspondto the algebraic relations of Γ ′ M [Γ M ] in Eqs. (8) and (9) with N = 2 M , we can find that thegenerators in Eq. (66) corresponds to ones in Eq. (93) [Eq. (65)] and they satisfy the algebraicrelations of ∆ ′ (6 M ) [∆(6 M )] in Eq. (95) [Eq. (63)].First, by using Eqs. (27)-(30), Eq. (64) can be rewritten as( S T ) = ( S − T ) = . (132)By using Eqs. (27)-(30), and (132), the generator a ′ in Eq. (66) can be rewritten as a ′ = ST S − T − = ST T S − T T − = ST T − ST − T − ST − T − = S − T − S − T − S T − ST − = T ST − ST − = T S − T − S T − S − T − = T ST SS ST ST − = T ST S − T − = T ( ST S − T − ) T − ⇔ a ′ = T − ( ST ST − ) T = T − ST S − . (133)Then, we can obtain ST p S − T q = ( ST S − ) p T q = T q ST p S − , p, q ∈ Z , (134)in general. Similarly, by using this relation, the generator a in Eq. (66) can be rewritten as a = ST S T = T ST S . (135)Thus, we can obtain a M = S − M T M ST M S − = 1 , a ′ M = T − M ST M S − = 1 , (136) aa ′ = ST S T = a ′ a, (137) When we consider Eqs. (8) and (9) with N = 2 M, M = 1 ,
2, we can check that Eq. (132) is already satisfied.
25y also using Eq. (31) with N = 2 M and M ∈ Z . Furthermore, from Eq. (28), we also have( S T ) = . (138)Then, we can prove ( S n +3 T n − ) = , n ∈ N , (139)on the mathematical induction. Thus, we can obtain the other relations in Eq. (67), b = T − M ( T M +3 S M − ) T M = 1 (140) c = ST M − ST M − ST M − ST M − = ST M − S − T − S − T M − ST M − S = ST M − ST M − ST M − = ( S M +3 T M − ) S − M − = S M +2 (141) c = S (142) c = 1 (143) cbc − = ST M − ST S M − T − M S − T − M S − = T ST M − S M T S − T − M S − T − M − = T ST M +3 S M − T − M S T ST − M − = T ST M +3 S M − T M +3 S T S − M +1 T − M − = T S M − T M +3 S M − T M +3 S − M − T S − M +1 T − M − = T T − M − S − M +1 S − M − T S − M +1 T − M − = T − M S − M +1 T − M − = b − (144) It is because S − M = 1 is satisfied only if M ∈ Z . However, when we consider the case that Eqs. (8)and (9) with N = 2 M are satisfied instead of Eqs. (27)-(31) with N = 2 M , S − M = 1 is satisfied even if M = 2(2 s −
1) ( s ∈ Z ). ab − = T M +3 S M − T M ST S T − M S − M +1 T − M − = T M +3 S M − T S − M − T − M − = T S − T S − T = T − S − T − ST ST = T − S − T − S − = a − a ′− (145) ba ′ b − = T M +3 S M − T M ST S − T − − M S − M +1 T − M − = T M +3 S M − T − S − M +1 T − M − = T − S − T − S − T − S T = ST SS ST ST = ST S T = a (146) cac − = ST M − ST M − ST S T − M S − T − M S − = ST M − S − T − S − T S T ST − M S = ST M − S T S T ST − M S = S − T M − T S T S T T ST − M S − = S − T M − ST ST − M S − = T ST − S − = a ′− (147) ca ′ c − = ST M − ST M − ST S − T − − M S − T − M S − = ST M − S − T − ST S − T − S − T − M S = ST M − S − T S T − M S = ST ST S T S = T − S T S T S = T − S T S T − = T − S − T − S − S S − T − S − T − = T − S − T − S − T − = T − S − T − S − = a − . (148)Therefore, when the relation in Eq. (64) is also satisfied in addition to the algebraic relationsof e Γ M , in particular for M ∈ Z , Eq. (66) can be the generators of e ∆(6 M ). Similarly, whenthe algebraic relations of Γ ′ M [Γ M ] and also Eq. (64) are satisfied, we can find that Eq. (93)[Eq. (65)] as well as Eq. (94) [the generators in footnote 8] can be the generators of ∆ ′ (6 M )[∆(6 M )]. 27 Three-dimensional modular forms
Here, we express three-dimensional modular forms obtained from the wavefunctions on mag-netized orbifolds at z = 0, which means the modular forms can be obtained from Z -even( m = n = 0) modes.We can obtain two three-dimensional modular forms of weight 1 / M, m ) = (4 ,
0) and (7 ,
0) at z = 0 on the magnetized T / Z (t)2 twisted orbifold, as the followings, ϑ ( τ ) ϑ ( τ ) ϑ ( τ ) = ϑ (cid:20) (cid:21) (0 , τ ) √ (cid:18) ϑ (cid:20) (cid:21) (0 , τ ) + ϑ (cid:20) (cid:21) (0 , τ ) (cid:19) ϑ (cid:20) (cid:21) (0 , τ ) = ϑ (cid:20) (cid:21) (0 , τ ) √ ϑ (cid:20) (cid:21) (0 , τ ) ϑ (cid:20) (cid:21) (0 , τ ) ( M = 4) (149) ϑ ( τ ) ϑ ( τ ) ϑ ( τ ) = √ (cid:18) ϑ (cid:20) − (cid:21) (0 , τ ) − ϑ (cid:20) − (cid:21) (0 , τ ) (cid:19) √ (cid:18) ϑ (cid:20) − (cid:21) (0 , τ ) − ϑ (cid:20) − (cid:21) (0 , τ ) (cid:19) √ (cid:18) ϑ (cid:20) − (cid:21) (0 , τ ) − ϑ (cid:20) − (cid:21) (0 , τ ) (cid:19) = √ ϑ (cid:20) − (cid:21) (0 , τ ) √ ϑ (cid:20) − (cid:21) (0 , τ ) √ ϑ (cid:20) − (cid:21) (0 , τ ) ( M = 7) . (150)They are the modular forms of weight 1 / e Γ(8) and e Γ(56), respectively, and also they trans-form as the three-dimensional representations of e ∆(96) and P SL (2 , Z ) × Z , respectively .Similarly, we can obtain four three-dimensional modular forms of weight 1, two of which areobtained from the modes ( M (1) , m : M (2) , m ) = (4 , ,
0) and (7 , ,
0) at z = z = 0 onthe magnetized T / Z (t )2 × T / Z (t )2 orbifold, and the other two of which are obtained from themodes ( M, m, n ) = (2 , ,
0) and (5 , ,
0) at z = z = 0 on the magnetized ( T × T ) / ( Z (t)2 × Z (p)2 ) We can obtain N ( M )-dimensional modular forms of weight 1 / e Γ(2 M ) with M =even or e Γ(8 M ) with M =odd from the m = 0 modes at z = 0 on the magnetized T / Z (t)2 twisted orbifold, and then they transformas the N ( M )-dimensional representations of e Γ M or e Γ M , in general. ϑ (4 , ( τ ) ϑ (4 , ( τ ) ϑ (4 , ( τ ) = √ ϑ (cid:20) − (cid:21) (0 , τ ) ϑ (cid:20) (cid:21) (0 , τ )2 ϑ (cid:20) − (cid:21) (0 , τ ) ϑ (cid:20) (cid:21) (0 , τ ) √ ϑ (cid:20) − (cid:21) (0 , τ ) ϑ (cid:20) (cid:21) (0 , τ ) ( M (1) = 4 , M (2) = 3) (151) ϑ (7 , ( τ ) ϑ (7 , ( τ ) ϑ (7 , ( τ ) = ϑ (cid:20) − (cid:21) (0 , τ ) ϑ (cid:20) − (cid:21) (0 , τ )2 ϑ (cid:20) − (cid:21) (0 , τ ) ϑ (cid:20) − (cid:21) (0 , τ )2 ϑ (cid:20) − (cid:21) (0 , τ ) ϑ (cid:20) − (cid:21) (0 , τ ) ( M (1) = 7 , M (2) = 3) , (152) ϑ (2 , ( τ ) ϑ (2 , ( τ ) ϑ (2 , ( τ ) = ϑ (cid:20) (cid:21) (0 , τ ) ϑ (cid:20) (cid:21) (0 , τ ) √ ϑ (cid:20) (cid:21) (0 , τ ) ϑ (cid:20) (cid:21) (0 , τ ) ϑ (cid:20) (cid:21) (0 , τ ) ϑ (cid:20) (cid:21) (0 , τ ) ( M (1) = M (2) = M = 2) (153) ϑ (5 , ( τ ) ϑ (5 , ( τ ) ϑ (5 , ( τ ) = ϑ (cid:20) − (cid:21) (0 , τ ) ϑ (cid:20) − (cid:21) (0 , τ )2 √ ϑ (cid:20) − (cid:21) (0 , τ ) ϑ (cid:20) − (cid:21) (0 , τ )2 ϑ (cid:20) − (cid:21) (0 , τ ) ϑ (cid:20) − (cid:21) (0 , τ ) ( M (1) = M (2) = M = 5) . (154)They are the modular forms of weight 1 for Γ(24), Γ(84), Γ(4), and Γ(20) respectively, and alsothey transform as the three-dimensional representations of ∆ ′ (96) × Z , P SL (2 , Z ) × Z × Z , S ′ , and A × Z , respectively . References [1] G. Altarelli and F. Feruglio, Rev. Mod. Phys. (2010) 2701 [arXiv:1002.0211 [hep-ph]]. We can obtain N (0 , ( M )-dimensional modular forms of weight 1 for Γ(2 M ) with M =even or Γ(4 M ) with M =odd from the ( m, n ) = (0 ,
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