Metaplectic Flavor Symmetries from Magnetized Tori
Yahya Almumin, Mu-Chun Chen, Victor Knapp-Perez, Saul Ramos-Sanchez, Michael Ratz, Shreya Shukla
UUCI–TR–2021–08
Metaplectic Flavor Symmetries from Magnetized Tori
Yahya Almumin a,α , Mu–Chun Chen a,β , V´ıctor Knapp–P´erez b,c,γ ,Sa´ul Ramos–S´anchez b,δ , Michael Ratz a,ε and Shreya Shukla a,ϕ a Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575 USA b Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, POB 20-364, Cd.Mx. 01000, M´exico c Address from September 2021: Department of Physics and Astronomy, University of California, Irvine, CA92697-4575 USA
Abstract
We revisit the flavor symmetries arising from compactifications on tori with magnetic back-ground fluxes. Using Euler’s Theorem, we derive closed form analytic expressions for the Yukawacouplings that are valid for arbitrary flux parameters. We discuss the modular transformations foreven and odd units of magnetic flux, M , and show that they give rise to finite metaplectic groupsthe order of which is determined by the least common multiple of the number of zero–mode flavorsinvolved. Unlike in models in which modular flavor symmetries are postulated, in this approachthey derive from an underlying torus. This allows us to retain control over parameters, such asthose governing the kinetic terms, that are free in the bottom–up approach, thus leading to anincreased predictivity. In addition, the geometric picture allows us to understand the relativesuppression of Yukawa couplings from their localization properties in the compact space. We alsocomment on the role supersymmetry plays in these constructions, and outline a path towardsnon–supersymmetric models with modular flavor symmetries. α [email protected] β [email protected] γ [email protected] δ ramos@fisica.unam.mx ε [email protected] ϕ [email protected] a r X i v : . [ h e p - t h ] F e b Introduction
The Standard Model (SM) of particle physics is believed to be an effective theory. One reason whythis is so is that it has many parameters that have to be adjusted by hand to fit data. The bulk ofthese parameters resides in the flavor sector, i.e. concerns the fermion masses, mixing angles and CP phases. An ultraviolet (UV) completion of the SM will have to explain these parameters. Turning thisaround, one may hope to get more insights on the UV completion by constructing a working theoryof flavor.Recently, a new approach to address the flavor problem has been put forward [1]: Yukawa couplingscould be modular forms. There are two main ways in which this proposal has been utilized:1. symmetry based (SB) , i.e. impose the modular flavor symmetry to construct the Lagrange density[2–17], and2. torus based (TB) , in which one derives the symmetries from an underlying torus or relatedsetup [18–30].Both strategies have strong points and challenges. In the SB approach, very good fits to data havebeen achieved. However, this is, in part, possible because one can postulate the symmetry and otherdata like modular weights and representations at will. Apart from the arbitrariness of the flavorgroup and modular weights, the kinetic terms of the fields are not very constrained by the modulartransformations [31]. The TB approach is much more restrictive, in particular when embedded intostring theory [22–25]. However, while these models have great promise and certainly fix the above–mentioned problems of arbitrariness, it is probably fair to say that they do not yet provide us withunequivocal predictions on flavor parameters that can be tested in the foreseeable future.The purpose of this paper is to explore the details of the relation between these approaches. Morespecifically, we derive metaplectic symmetries from magnetized tori. Earlier works on this subjectinclude [18–21, 26–28, 30]. To accomplish this, we work out closed–form expressions for the Yukawacouplings that are valid for arbitrary flux parameters, and thus generalize the results of the pioneeringwork by Cremades, Ib´a˜nez and Marchesano [32]. We also present consistent modular transformationlaws for both even and odd numbers of generations. Models derived from magnetized tori also allow usto understand to which extent supersymmetry is crucial for modular flavor symmetries, which we willargue to be less important than usually assumed. Additional motivation for looking at magnetized tori,with and without supersymmetry, comes from the fact that even without supersymmetry interactingscalar masses seem to be protected from quantum corrections [33–36].This paper is organized as follows. In section 2 we review the zero modes on magetized tori.Section 3 concerns the computation of the Yukawa couplings of these settings. We derive closedform expressions that are valid for arbitrary flux parameters. In section 4 we show how modulartransformations amount to flavor rotations. We will show that the torus compactifications give rise tofinite metaplectic groups, which have been studied using the SB approach in [15, 17]. In section 5 wecomment on the role that supersymmetry plays in the scheme of modular flavor symmetries. Section 6contains our conclusions. Various appendices contain some details of our derivations. Let us consider a gauge theory with two extra dimensions. The two extra dimensions are compactifiedon a 2–torus T , which is endowed with a magnetic flux. By the index theorem, the flux will giverise to chiral zero–modes. Throughout our discussion we will ignore questions on the vacuum energy,the stability and even anomaly cancellation. We think of this torus as a little local playground thatis embedded in a more complete setup. However, we will address some of the questions in section 5.The main goal of this section is to review some of the properties of the zero–modes. The wavefunctions of the zero modes of the Dirac operator on tori with magnetic flux have been worked out1 igure 2.1: Squares of the absolute values of the wave functions on a quadratic torus for M = 4. in [32]. They are given by ψ j,M ( z, τ, ζ ) = N e π i M ( z + ζ ) Im( z + ζ )Im τ ϑ (cid:104) jM (cid:105)(cid:0) M ( z + ζ ) , M τ (cid:1) . (2.1)Here, M ∈ N indicates the units of flux, 0 ≤ j ≤ M − z the coordinate in theextra dimensions, ζ a so–called Wilson line parameter, and τ the torus parameter or half-period ratio.The wave functions from equation (2.1) correspond to left–handed particles in 4D whereas there areno right–handed particles for positive M . On the other hand, for negative values of the integer M there are no solutions for left–handed particles, but there are | M | right–handed particles described by ψ j,M (¯ z, ¯ τ , ¯ ζ ), with 0 ≤ j ≤ | M | −
1. Furthermore, notice that despite what the notation may suggest,the ψ j,M are neither holomorphic functions of z , nor of τ . ϑ denotes the so–called Jacobi ϑ –function,cf. appendix A. The normalization is given by N = (cid:18) M Im τ A (cid:19) / , (2.2)where A = (2 πR ) Im τ is the area of the torus (cf. appendix B). In figure 2.1, we show the profiles ofsome zero–modes.We find it instructive to derive the quantization condition on M . Let us follow the discussionby [32]. Consider a U(1) gauge group in the torus with a magnetic flux given by the gauge potential A ( z + ζ ) = B τ Im (cid:0) (¯ z + ¯ ζ ) d z (cid:1) . (2.3)Then, if the wave function ψ j,M ( z, τ, ζ ) has charge q under this U(1), its transformation under torustranslations are ψ j,M ( z + 1 , τ, ζ ) = exp (cid:18) i qB τ Im( z + ζ ) (cid:19) ψ j,M ( z, τ, ζ ) , (2.4a) ψ j,M ( z + τ, τ, ζ ) = exp (cid:18) i qB τ Im( z + ζ )¯ τ (cid:19) ψ j,M ( z, τ, ζ ) . (2.4b)2n order to have consistency through a contractible loop in the torus, we must get the same wavefunction shifting z → z + τ + 1 as in the case where we shift by z → z + 1 + τ . Then, ψ j,M ( z + τ + 1 , τ, ζ ) = exp (cid:18) i qB τ Im (cid:0) ( z + ζ + 1 (cid:1) ¯ τ ) (cid:19) ψ j,M ( z + 1 , τ, ζ )= exp (cid:18) i qB τ Im (cid:0) ( z + ζ + 1 (cid:1) ¯ τ ) (cid:19) exp (cid:18) i qB τ Im( z + ζ ) (cid:19) ψ j,M ( z, τ, ζ )= exp (cid:18) i qB Im ¯ τ τ (cid:19) exp (cid:18) i qB τ Im (cid:0) ¯ τ ( z + ζ ) (cid:1)(cid:19) exp (cid:18) i qB τ Im( z + ζ ) (cid:19) ψ j,M ( z, τ, ζ ) , (2.5)where we used first equation (2.4b) and then equation (2.4a). On the other hand, ψ j,M ( z + τ + 1 , τ, ζ ) = exp (cid:18) i qB Im( z + ζ + τ )2 Im τ (cid:19) ψ j,M ( z + τ, τ, ζ )= exp (cid:18) i qB Im( z + ζ + τ )2 Im τ (cid:19) exp (cid:18) i qB τ Im ¯ τ ( z + ζ ) (cid:19) ψ j,M ( z, τ, ζ )= exp (cid:18) i qB Im τ τ (cid:19) exp (cid:18) i qB τ Im(¯ τ ( z + ζ )) (cid:19) exp (cid:18) i qB τ Im( z + ζ ) (cid:19) ψ j,M ( z, τ, ζ ) , (2.6)where we used first equation (2.4a) and then equation (2.4b). Imposing that equation (2.5) andequation (2.6) yield the same wave function leads to the flux quantization condition qB = 2 πM , (2.7)with M an arbitrary integer. Therefore, in what follows, we will not consider q and B individually,but only the integer M instead. Then, we have ψ j,M ( z + 1 , τ, ζ ) = exp (cid:18) i πM Im τ Im( z + ζ ) (cid:19) ψ j,M ( z, τ, ζ ) , (2.8a) ψ j,M ( z + τ, τ, ζ ) = exp (cid:18) i πM Im τ Im( z + ζ )¯ τ (cid:19) ψ j,M ( z, τ, ζ ) . (2.8b) One of the main rationales of working out the wave functions in section 2 is that the overlaps of wavefunctions yield the (Yukawa) couplings of the model. Let us consider a 4 + 2 dimensional theory whichis compactified on a torus T . There is a gauge group breaking U( N ) → U( N a ) × U( N b ) × U( N c ) with N = N a + N b + N c due to the introduction of a magnetic flux in the compact dimensions given by F z ¯ z = π iIm τ m a N a N a × N a m b N b N b × N b
00 0 m c N c N c × N c , (3.1)where we will assume that s α = m α N α is an integer for α ∈ { a, b, c } . Then, in [32, equation (5.7)] onefinds that Yukawa couplings of the 4D effective theory are given by Y ijk ( (cid:101) ζ, τ ) = g σ abc (cid:90) T d z ψ i, I ab ( z, τ, ζ ab ) ψ j, I ca ( z, τ, ζ ca ) (cid:0) ψ k, I cb ( z, τ, ζ cb ) (cid:1) ∗ . (3.2)3ere, ψ i, I ab ( z, τ, ζ ab ) are the wave functions of equation (2.1) that represent chiral fermions bifunda-mentals transforming as ( N a , N b ) under U( N a ) × U( N b ), and similarly for ψ j, I ca and ψ k, I cb . Themultiplicities of I αβ of these bifundamentals are given by I αβ = s α − s β , (3.3)which implies that I ab + I bc + I ca = 0 . (3.4)Furthermore, g is the (4 + 2)–dimensional gauge coupling, and σ abc = sign( I ab I bc I ca ) [32] is a signwhich is equal to − ζ αβ are given by ζ αβ = s α ζ α − s β ζ β s α − s β (3.5)for α, β ∈ { a, b, c } . Finally, ζ α are the Abelian Wilson lines associated to the group U( N α ) for α ∈ { a, b, c } . ζ cb and ζ ca are defined similarly. As one can see from equation (2.1), the ζ α representtranslation of the torus origin. However, as shown in [32] if all three wave functions are shifted by thesame Wilson line, then the values of the Yukawa couplings are unaffected. Let us now discuss how one can reduce the overlap integrals (3.2) to a linear combination of ϑ –functions. We follow the strategy of [32], but generalize the result to the cases I ab > I ab , I ca , I bc ) >
1, with I ab , I ca > I bc <
0. Note that the analogous discussion applies tothe case in which I ab and I ca are negative [32, cf. the discussion around equation (5.6)].In order to find closed–form expressions for the Yukawa couplings, one uses two important facts [32]:1. products of ϑ –functions can be expanded in terms of ϑ –functions, see [32, equation (5.8)], andthat2. the ϑ –functions fulfill certain orthogonality and completeness relations.These facts allow one to find analytic expressions for the Yukawa couplings (3.2) that do no longerinvolve integrals [32]. In more detail, starting from (3.2), one obtains (cf. [32, equation (5.15)]) Y ijk ( (cid:101) ζ, τ ) = N abc e H ( (cid:101) ζ,τ )2 (cid:88) m ∈ Z I bc δ k,i + j + I ab m ϑ (cid:34) I ca i −I ab j + I ab I ca m −I ab I bc I ca (cid:35)(cid:0)(cid:101) ζ, τ |I ab I bc I ca | (cid:1) , (3.6)where N abc = g σ abc (cid:18) τ A (cid:19) / (cid:12)(cid:12)(cid:12)(cid:12) I ab I ca I bc (cid:12)(cid:12)(cid:12)(cid:12) / (3.7)is a normalization constant and the Wilson line dependence is encoded in the quantities (cid:101) ζ := −I ab I ca ( ζ ca − ζ ab ) = d αβγ s α ζ α I βγ (3.8)and H ( (cid:101) ζ, τ )2 := π iIm τ ( I ab ζ ab Im ζ ab + I bc ζ bc Im ζ bc + I ca ζ ca Im ζ ca )= π iIm τ |I ab I bc I ab | − (cid:101) ζ Im (cid:101) ζ Im τ . (3.9)4ith d αβγ = , if { α, β, γ } is an even permutation of { , , } , , otherwise , (3.10)where we have used [32, equation (5.28)]. Cremades et al. obtain then [32, equation (5.15)] Y ijk ( (cid:101) ζ, τ ) = N abc e H ( (cid:101) ζ,τ )2 ϑ (cid:34) − (cid:16) j I ca + k I bc (cid:17) / I ab (cid:35)(cid:0)(cid:101) ζ, τ |I ab I bc I ca | (cid:1) for i = k − j mod I ab . (3.11)This expression yields the correct couplings only if I ab = 1, which implies that d = 1, where d := gcd (cid:0) |I ab | , |I ca | , |I bc | (cid:1) . (3.12)To see that we need to demand that d = 1 for (3.11) to hold, notice that in (3.6) the integers i , j and k are only defined modulo I ab , I bc and I ca , respectively. This is evident from the overlap integral (3.2),where e.g. ψ i, I ab ( z, τ, ζ ab ) = ψ i + I ab , I ab ( z, τ, ζ ab ). However, if gcd( |I ab | , |I ca | ) > |I ab | >
1, shifting i (or j ) by |I ab | (or ( |I ca | ), which leaves the wave functions invariant and hence has to produce thesame overlap integral, leads to different results for the Yukawa couplings when using (3.11).To obtain the general expression, let us look at [37, Proposition II.6.4. on p. 221] ϑ (cid:104) j I ab (cid:105) ( z , I ab τ ) · ϑ (cid:104) j I ca (cid:105) ( z , I ca τ ) = (cid:88) m ∈ Z I ab + I ca ϑ (cid:34) i + j + I ab m I ab + I ca (cid:35)(cid:0) z + z , ( I ab + I ca ) τ (cid:1) ϑ (cid:34) I ca i −I ab j + I ab I ca m I ab I ca ( I ab + I ca ) (cid:35) ( I ca z − I ab z , I ab I ca ( I ab + I ca ) τ ) , (3.13)which was used in [32]. In our wave functions, z = I ab ( z + ζ ab ) and z = I ca ( z + ζ ca ), so that in theoverlap integral z + z = I cb ( z + ζ cb ) and I ca z −I ab z = (cid:101) ζ . One thus obtains (cf. [32, equation (5.12)]) ψ i, I ab ( z, τ, ζ ab ) · ψ j, I ca ( z, τ, ζ ca ) = A − / (2 Im τ ) / (cid:12)(cid:12)(cid:12)(cid:12) I ab I ca I bc (cid:12)(cid:12)(cid:12)(cid:12) / e H ( (cid:101) ζ,τ )2 (cid:88) m ∈ Z |I bc | ψ i + j + I ab m, I cb ( z, τ, I cb ) ϑ (cid:34) I ca i −I ab j + I ab I ca m I ab I ca ( I ab + I ca ) (cid:35)(cid:16)(cid:101) ζ, I ab I ca ( I ab + I ca ) τ (cid:17) . (3.14)The product (3.14) gets projected on a third wave function ψ k, I ab + I ca via the overlap integral (3.2).This means that k has to “match”, i.e. m has to be a solution of the congruence equation I ab m + i + j = k mod I bc . (3.15)Now observe that, since I ab + I bc + I ca = 0, gcd (cid:0) |I ab | , |I cb | (cid:1) = d with d from equation (3.12).Equation (3.15) is a linear congruence equation for the variable m . It is known that (cf. e.g. [38, Lemma3 on p. 37]) if k − i − j = 0 mod d , (3.16)the linear congruence of equation (3.15) has d solutions. Otherwise there is no solution. Note that thecondition (3.16) provides us with a selection rule for the Yukawa couplings, which can be interpretedas a Z d flavor symmetry (cf. [39]). We thus know that the Yukawa couplings will be proportional to∆ ( d ) i + j,k := (cid:40) , if i + j = k mod d , , otherwise . (3.17)5onsider now combinations of i , j and k satisfying the selection rule (3.16). This means that k − i − j = m (cid:48) d (3.18)with some integer m (cid:48) = ( k − i − j ) /d . Define now I (cid:48) ab = I ab /d , I (cid:48) ca = I ca /d and I (cid:48) bc = I bc /d , whichare integers because of equation (3.12). We can thus divide equation (3.15) by d to get |I (cid:48) ab | m = m (cid:48) mod |I (cid:48) bc | , (3.19)where gcd (cid:0) |I (cid:48) ab | , |I (cid:48) bc | (cid:1) = 1. Equation (3.19) can be solved with e.g. the Mathematica command
FindInstance . However, as we shall discuss now, one can find a closed–form expression for thesolution. The linear congruence (3.19) has one (inequivalent) solution m = m , which is given by (cid:2) |I (cid:48) ab | (cid:3) ( |I (cid:48) bc | ) m (cid:48) where (cid:2) |I (cid:48) ab | (cid:3) ( |I (cid:48) bc | ) is the multiplicative inverse of |I (cid:48) ab | modulo |I (cid:48) bc | . According toEuler’s theorem (cf. e.g. [38, Theorem 1 on p. 64]), the multiplicative inverse can be expressed via theEuler φ –function, (cid:2) |I (cid:48) ab | (cid:3) ( |I (cid:48) bc | ) = ( I (cid:48) ab ) φ (cid:0) |I (cid:48) bc | (cid:1) − . This means that m = ( I (cid:48) ab ) φ (cid:0) |I (cid:48) bc | (cid:1) − k − i − jd mod |I (cid:48) bc | . (3.20)Note that the Euler φ –function is implemented in Mathematica as EulerPhi . Relation (3.20) impliesthat one particular solution m of equation (3.15) satisfies I ab m = ( I (cid:48) ab ) φ (cid:0) |I (cid:48) bc | (cid:1) ( k − i − j ) mod |I bc | . (3.21)Given the solution m in equation (3.20), the d solutions of equation (3.15) are given by m = m − |I (cid:48) bc | t for t = 0 , . . . , ( d − . (3.22)Thus, using equation (3.22) in (3.6), we see that the Yukawa couplings are given by Y ijk ( (cid:101) ζ, τ ) = N abc e H ( (cid:101) ζ,τ )2 ∆ ( d ) i + j,k d − (cid:88) t =0 ϑ (cid:34) I ca i −I ab j + I ab I ca m |I ab I bc I ca | + td (cid:35)(cid:16)(cid:101) ζ, |I ab I ca I bc | τ (cid:17) , (3.23)Equation (3.23) can be simplified further. Let us define P := |I ab I ca I bc | , (3.24a) λ := lcm (cid:0) |I ab | , |I ca | , |I bc | (cid:1) . (3.24b)Next we note that P = λ d . (3.25)Then equation (3.23) can be recast as Y ijk ( (cid:101) ζ, τ ) = N abc e H ( (cid:101) ζ,τ )2 d − (cid:88) t =0 ϑ (cid:34) d (cid:16) (cid:98) α ijk λ + t (cid:17) (cid:35)(cid:16)(cid:101) ζ, P τ (cid:17) , (3.26) To see this, consider two positive integers a and b , and define c = gcd( a, b ) = gcd (cid:0) a, b, ( a + b ) (cid:1) such that a = a (cid:48) c and b = b (cid:48) c with integers a (cid:48) and b (cid:48) . Then lcm (cid:0) a, b, ( a + b ) (cid:1) = c lcm (cid:0) a (cid:48) , b (cid:48) , ( a (cid:48) + b (cid:48) ) (cid:1) . Since a (cid:48) , b (cid:48) and ( a (cid:48) + b (cid:48) ) do nothave a nontrivial common divisor,lcm (cid:0) a, b, ( a + b ) (cid:1) = c a (cid:48) b (cid:48) ( a (cid:48) + b (cid:48) ) , so that a b ( a + b ) = [gcd (cid:0) a, b, ( a + b ) (cid:1) ] lcm (cid:0) a, b, ( a + b ) (cid:1) . (cid:98) α ijk = I (cid:48) ca i − I (cid:48) ab j + I (cid:48) ca I ab m (3.27)is an integer. Using equation (3.20), (cid:98) α ijk becomes (cid:98) α ijk = I (cid:48) ca i − I (cid:48) ab j + I (cid:48) ca ( I (cid:48) ab ) φ (cid:0) |I (cid:48) bc | (cid:1) ( k − i − j ) mod λ d . (3.28)Now we can use equation (A.4) to express the sum (3.26) as Y ijk ( (cid:101) ζ, τ ) = N abc e H ( (cid:101) ζ,τ )2 d − (cid:88) t =0 ∞ (cid:88) (cid:96) = −∞ exp (cid:34) i π (cid:18) d (cid:98) α ijk λ + 1 d t + (cid:96) (cid:19) P τ (cid:35) exp (cid:20) π i (cid:18) (cid:98) α ijk λ d + td + (cid:96) (cid:19) (cid:101) ζ (cid:21) = N abc e H ( (cid:101) ζ,τ )2 ∞ (cid:88) (cid:96) = −∞ d − (cid:88) t =0 exp (cid:34) i π (cid:18) (cid:98) α ijk λ + t + d (cid:96) (cid:19) λ τ (cid:35) exp (cid:34) π i (cid:18) (cid:98) α ijk λ + t + (cid:96)d (cid:19) (cid:101) ζd (cid:35) = N abc e H ( (cid:101) ζ,τ )2 ∞ (cid:88) (cid:96) (cid:48) = −∞ exp (cid:34) i π (cid:18) (cid:98) α ijk λ + (cid:96) (cid:48) (cid:19) λ τ (cid:35) exp (cid:34) π i (cid:18) (cid:98) α ijk λ d + (cid:96) (cid:48) (cid:19) (cid:101) ζd (cid:35) = N abc e H ( (cid:101) ζ,τ )2 ϑ (cid:34) (cid:98) α ijk λ (cid:35)(cid:32) (cid:101) ζd , λ τ (cid:33) . (3.29)Here, (cid:96) (cid:48) = d (cid:96) + t . When (cid:96) runs over all integers, and t runs from 0 to d − (cid:96) (cid:48) runs over all integers.The (cid:98) α ijk are integers. Therefore, the physical Yukawa couplings are given by Y ijk ( (cid:101) ζ, τ ) = N abc e H ( (cid:101) ζ,τ )2 ∆ ( d ) i + j,k ϑ (cid:34) I (cid:48) ca i −I (cid:48) ab j + I (cid:48) ca ( I (cid:48) ab ) φ ( |I(cid:48) bc | ) ( k − i − j ) λ (cid:35)(cid:32) (cid:101) ζd , λ τ (cid:33) (3.30)with d from equation (3.12), ∆ ( d ) i + j,k from equation (3.17), λ from equation (3.24b) and assuming I ab , I ca > I bc <
0. Note that if d = 1 and I ab = 1, this formula reproduces equation (3.11).Further, a priori this expression does not rely on supersymmetry, it is simply derived from the overlapof wave functions. However, one may expect the scalar wave function to be subject to substantialcorrections in non–supersymmetric theories. In section 5 we will argue that magnetized tori may notcomply with these expectations, and that this formula may even be a good leading–order result in anon–supersymmetric theory. The normalization factors in equation (3.30) are N abc = g σ abc (cid:18) τ A (cid:19) / λ / (cid:12)(cid:12)(cid:12)(cid:12) I (cid:48) bc (cid:12)(cid:12)(cid:12)(cid:12) / (3.31)with g being the gauge coupling. In equation (4.39) we will express the normalization in terms ofK¨ahler potential terms. Notice that if there are nontrivial relative Wilson lines, the normalizationof the fields changes compared to the case without Wilson lines [32, equation (7.37)]. This has tobe taken into account when computing physical Yukawa couplings. In what follows, we will set theWilson lines to zero, leaving the detailed study of their impact on the modular flavor symmetries forfuture work. As mentioned above, the selection rule (3.16) entails a Z d symmetry. As we discuss inmore detail in appendix D, out of a priori P = λ d entries, at most λ / + 1 are distinct. The modular group Γ = SL(2 , Z ) can be defined by the presentation relations S = ( S T ) = and S T = T S , (4.1)7here the generators S and T are usually chosen as S = (cid:32) − (cid:33) and T = (cid:32) (cid:33) . (4.2)These generators act on the modulus τ according to τ S (cid:55)−−→ − τ and τ T (cid:55)−−→ τ + 1 . (4.3)Hence, a general modular transformation acts on the modulus τ as τ γ (cid:55)−−→ a τ + bc τ + d =: γ τ , where γ = (cid:32) a bc d (cid:33) ∈ Γ , (4.4)such that ad − bc = 1 and a, b, c, d ∈ Z . Consequently, functions of τ also transform under γ . Thisis particularly true for modular forms, which are holomorphic functions of τ (also at τ → i ∞ ) withIm τ > f (cid:98) α ( τ ) of modular weight k ∈ N and level N = 2 , , , . . . build finitevector spaces and transform under a modular transformation γ ∈ Γ as [9] f (cid:98) α ( τ ) γ (cid:55)−−→ f (cid:98) α ( γ τ ) := ( c τ + d ) k ρ r ( γ ) (cid:98) α (cid:98) β f (cid:98) β ( τ ) , (4.5)where (cid:98) α, (cid:98) β are considered here just as (integer) counters, ( c τ + d ) k often gets referred to as automorphyfactor, and ρ r ( γ ) denotes an r -dimensional (irreducible) representation matrix of γ under the finitemodular group Γ (cid:48) N ∼ = SL(2 , Z N ). These finite groups are defined by the relations S = ( S T ) = , S T = T S , T N = (4.6)and an additional relation that ensures finiteness for N > φ i transform under a general modulartransformation γ ∈ Γ as φ i γ (cid:55)−−→ ( c τ + d ) k φ ρ s ( γ ) ij φ j . (4.7)Here ρ s ( γ ) is the s -dimensional (reducible or irreducible) Γ (cid:48) N representation matrix. As for modu-lar forms, the powers k φ are also known as modular weights and are identical for the fields in thetransformation. Thus, matter fields build a representation of the finite modular group Γ (cid:48) N , which canbe adopted as a symmetry of the underlying (quantum) field theory. In this scenario, Γ (cid:48) N can beconsidered a “modular flavor symmetry”.In string–derived models, it is known that matter fields are subject to modular transformationssimilar to equation (4.7). Moreover, Yukawa couplings also transform as in equation (4.5). However,as we shall see in this section, the modular weights can be fractional and, hence, the emerging modularflavor symmetry is not necessarily one of the Γ (cid:48) N . Yet, to obtain fractional modular weights it is notnecessary to go all the way to strings, they already emerge from simpler settings such as magnetizedtori (see e.g. [18–21, 26–28, 30]). As we discuss in detail in detail in section 4.2, this follows alreadyfrom the τ –dependence of the normalization of the wave functions [32].As a first step, let us review the modular symmetries associated with modular forms with half–integral modular weights [15]. In this case, one must consider instead of SL(2 , Z ) its double cover,the so–called metaplectic group (cid:101) Γ = Mp(2 , Z ). The generators (cid:101) S and (cid:101) T of (cid:101) Γ satisfy the presentation (cid:101) S = ( (cid:101) S (cid:101) T ) = and (cid:101) S (cid:101) T = (cid:101) T (cid:101) S , (4.8)8hich are represented by the choice (cid:101) S = ( S, −√− τ ) and (cid:101) T = ( T, +1) , S, T ∈ Γ . (4.9)In terms of these, the elements of the metaplectic group are given by (cid:101) Γ = (cid:110)(cid:101) γ = ( γ, ϕ ( γ, τ )) | γ ∈ Γ , ϕ ( γ, τ ) = ± ( c τ + d ) / (cid:111) , (4.10)subject to the multiplication rule( γ , ϕ ( γ , τ ))( γ , ϕ ( γ , τ )) = ( γ γ , ϕ ( γ , γ τ ) ϕ ( γ , τ )) . (4.11)To determine the sign of ϕ ( γ, τ ) for an arbitrary element (cid:101) γ ∈ (cid:101) Γ, one has to express (cid:101) γ as a product ofthe metaplectic generators (4.9) and then use the multiplication rule (4.11).The modular transformations (cid:101) γ act on the modulus still just as γ , according to equation (4.4). Incontrast, modular forms of modular weight k / and level 4 N , where k, N ∈ N , transform as f (cid:98) α ( τ ) (cid:101) γ (cid:55)−−→ f (cid:98) α ( (cid:101) γ τ ) := ϕ ( γ, τ ) k ρ r ( (cid:101) γ ) (cid:98) α (cid:98) β f (cid:98) β ( τ ) . (4.12)Here ϕ ( γ, τ ) k is now the automorphy factor, and ρ r ( (cid:101) γ ) is an (irreducible) representation matrix of (cid:101) γ in the finite metaplectic modular group (cid:101) Γ N . The generators (cid:101) S and (cid:101) T of this discrete group satisfy (cid:101) S = ( (cid:101) S (cid:101) T ) = , (cid:101) S (cid:101) T = (cid:101) T (cid:101) S , (cid:101) T N = (4.13)and, for N >
1, a relation to ensure the finiteness of the group. This amounts to finding appropriatecombinations of (cid:101) S and (cid:101) T that yield mod 4 N , where the modulo condition is to be understoodcomponentwise, and then demand that this combination yields identity in the finite group. For N = 2we adopt the choice by [15, equation (21)] (cid:101) S (cid:101) T (cid:101) S (cid:101) T (cid:101) S (cid:101) T (cid:101) S (cid:101) T = , (4.14)and for N = 3 we choose (cid:101) S (cid:101) T (cid:101) S (cid:101) T − (cid:101) S − (cid:101) T (cid:101) S (cid:101) T − (cid:101) S − (cid:101) T (cid:101) S − (cid:101) T − = . (4.15)Note that (cid:101) Γ N is the double cover of Γ (cid:48) N . It is known that (cid:101) Γ ∼ = [96 , (cid:101) Γ ∼ = [768 , (cid:101) Γ is a group of order 2304. We use here the unique identifiers assigned by the computer program GAP [41]. For example, [96 ,
67] denotes a finite discrete group of order 96, where 67 labels the group.Finally, in field theories endowed with (cid:101) Γ N symmetries, the modular transformations of matterfields are given by φ i (cid:101) γ (cid:55)−−→ ϕ ( γ, τ ) k φ ρ s ( (cid:101) γ ) ij φ j , (4.16)where ρ s ( (cid:101) γ ) is now a (reducible or irreducible) (cid:101) Γ N representation. As we shall see, this behavior isnatural in toroidal compactifications with magnetic fluxes. The wave functions in equation (2.1) satisfy (cid:90) T d z | ψ j,M ( z, τ, ζ ) | = A (cid:90) d x (cid:90) d y | ψ j,M ( x + τ y, τ, ζ ) | = 1 , (4.17)9bject ψ j,M φ j,M Ω j,M Y ijk W modular weight k / − / / − Table 4.1: Modular weights of the T wave functions ψ j,M , 4D fields φ j,M , 6D fields Ω j,M , Yukawa couplings Y ijk , and superpotential W . where T denotes the fundamental domain of the torus, cf. appendix B. The normalization constant N ∝ (Im τ ) − / in equation (2.2) is chosen in such a way that the normalization condition (4.17) holds.This implies, in particular, that the K¨ahler metric is proportional to (Im τ ) − / , i.e. K i ¯ ı ∝ τ ) / , (4.18)i.e. the modular weight of the 4D fields φ j,M describing the zero modes is k φ = − / . We survey themodular weights of the fields, coupling and superpotential in table 4.1. The modular weights k ψ of thewave functions can be inferred from their normalization factor N in equation (2.2) to be k ψ = + / ,as we shall also confirm through their explicit modular transformations, equation (4.37). Therefore,the 6D fields,Ω j,M = φ j,M ( x µ ) ⊗ ψ j,M ( z, τ ) , (4.19)have trivial modular weights, as they should. The modular weights of the Yukawa couplings, k Y = / ,can be explicitly determined from their modular transformations, equations (E.8) and (E.10). Sincethe superpotential terms describing the Yukawa couplings involve three 4D fields and one coupling“constant”, the superpotential W has modular weight k W = 3 k φ + k Y = −
1. This means that undera modular transformation the superpotential picks up an automorphy factor W γ (cid:55)−−→ ( c τ + d ) − W . (4.20)The automorphy factor ( c τ + d ) − can in general be “undone” by so–called K¨ahler transformations [42],under which W (cid:55)→ e − F (Φ) W (Φ) , (4.21a) K (Φ , Φ) (cid:55)→ K (Φ , Φ) + F (Φ) + F (Φ) , (4.21b)where Φ denotes the collection of 4D superfields, and F a holomorphic function. In our case, theK¨ahler potential is, after setting the “matter” fields to zero and at the classical level, given by (cf.e.g. [32, equation (5.50)]) (cid:98) K = − ln( S + S ) − ln( T + T ) − ln( U + U ) ⊂ K , (4.22)in terms of the axio–dilaton S , the K¨ahler modulus T and the complex structure modulus U . Thesechiral fields are related to the gauge coupling g , the torus volume A and τ according to Re S ∝ /g ,Re T ∝ A and Re U = Im τ . Consequently, τ appears in the K¨ahler potential as − ln( U + U ) = − ln( − i τ + i ¯ τ ) . (4.23)Given that τ − ¯ τ γ (cid:55)−−→ | c τ + d | − ( τ − ¯ τ ) , (4.24)it is easy to see that K under a modular transformation of τ becomes K γ (cid:55)−−→ K + ln( c τ + d ) + ln( c ¯ τ + d ) . (4.25)10 K¨ahler transformation (4.21) with F = − ln( c τ + d ) then absorbs simultaneously the modulartransformation of K and W , see equation (4.20), yielding a modular invariant supersymmetric theory.That is, the supergravity K¨ahler function G (Φ , Φ) = K (Φ , Φ) + ln | W (Φ) | (4.26)is automatically invariant under the simultaneous transformation (4.20) and (4.25). In other words, wecannot dial the modular weight of the superpotential at will, it is already determined by the (classical)K¨ahler potential of the torus (4.22). In particular, setting the modular weight of the superpotential tozero is not an option in this approach, in which we derive modular flavor symmetries from an explicittorus. It has been stated in the literature [26, 27] that the wave functions given by equation (2.1) do notsatisfy the boundary conditions given by the lattice periodicity when transformed under equation (4.3)for odd units of flux, M . If true, this would mean that a physical wave function gets mapped to anunphysical one just by looking at an equivalent torus, which would indicate that either the expressionsfor the wave functions were incorrect, or there is something fundamentally wrong with odd M . Inthis case, simple explanations of three generations would be at stake.However, as we shall see, the transformed wave functions do obey the correct boundary conditions,both for even and odd M . The important point is that, if our original wave function ψ j,M ( z, τ, τ , after a modular transformation τ (cid:55)→ τ (cid:48) the transformed wave function ψ j,M ( z, τ (cid:48) ,
0) needs to fulfill the conditions for τ (cid:48) , and not for τ .For the modular S transformations, the boundary conditions, given by equations (2.8a) and (2.8b),are now ψ j,M (cid:18) − zτ + 1 , − τ , (cid:19) = exp (cid:18) i πM Im( − z / τ )Im( − / τ ) (cid:19) ψ j,M (cid:18) − zτ , − τ , (cid:19) = exp (cid:18) − i πM Im z ¯ τ Im τ (cid:19) ψ j,M (cid:18) − zτ , − τ , (cid:19) , (4.27a) ψ j,M (cid:18) − zτ − τ , − τ , (cid:19) = exp (cid:18) i πM Im( − z / τ )( − / ¯ τ )Im( − / τ ) (cid:19) ψ j,M (cid:18) − zτ , − τ , (cid:19) = exp (cid:18) i πM Im z Im τ (cid:19) ψ j,M (cid:18) − zτ , − τ , (cid:19) . (4.27b)The fact that the transformed wave functions follow the boundary condition is a consequence of thefact that the wave functions are functions of z and τ , which we can just replace by their image under S . Nonetheless we verify this explicitly in appendix C.1.Next, under the modular T transformation given by equation (4.3) the transformed boundaryconditions, equations (2.8a) and (2.8b), are ψ j,M ( z + 1 , τ + 1 ,
0) = exp (cid:18) i πM Im τ Im z (cid:19) ψ j,M ( z, τ + 1 , , (4.28a) ψ j,M ( z + τ + 1 , τ + 1 ,
0) = exp (cid:18) i πM Im τ Im (cid:0) (¯ τ + 1) z (cid:1)(cid:19) ψ j,M ( z, τ + 1 , . (4.28b)We can make the same argument as above but also verify the statement explicitly in appendix C.2.However, the transformed wave function equation (C.6), i.e. the wave functions “living” on a toruswith torus parameter τ (cid:48) = τ + 1 do not follow the original boundary conditions of equations (2.8a)11nd (2.8b) with τ . Indeed, from equation (C.6) we get ψ j,M ( z + τ, τ + 1 ,
0) = (cid:101) N e i πM Im τ [ z Im z + z Im τ + τ Im z + τ Im τ ] ϑ (cid:20) jMM (cid:21) ( M z + M τ, M τ )= (cid:101) N e i πM Im τ [ z Im z + z Im τ + τ Im z + τ Im τ ] e − i πMτ − π i ( Mz + M ) ϑ (cid:20) jMM (cid:21) ( M z, M τ )= e − π i M e i πM Im τ ( z Im τ + τ Im z + τ Im τ − τ Im τ − z Im τ ) (cid:101) N e iπMz Im z Im τ ϑ (cid:20) jMM (cid:21) ( M z, M τ )= e − π i M exp (cid:18) i M π Im τ Im ¯ τ z (cid:19) ψ j,M ( z, τ + 1 , , (4.29)where (cid:101) N := e − i π j (cid:0) − jM (cid:1) N and we have used equation (A.5b) in the second line. Thus, we find that ψ j,M ( z + τ, τ + 1 ,
0) = e − π i M exp (cid:18) i M π Im τ Im ¯ τ z (cid:19) ψ j,M ( z, τ + 1 , . (4.30)Therefore, for odd M equation (4.30) differs from equation (2.8b) by a phase. However, there isalso no reason why the transformed wave functions should obey boundary conditions for τ insteadof τ (cid:48) = τ + 1. Nevertheless, this fact will have important implications for the explicit form of the T –transformation, as we shall see in section 4.4.1. ψ j,M Crucially, physics should not depend on how we choose to parametrize the underlying torus. That is,if we subject the half–period ratio τ of the torus to a modular transformation, the physical predictionsof the theory have to stay the same. This means that there should be a dictionary between theorieswith seemingly different but equivalent values of τ , which are related by modular transformations.Let us now study the action of T , under which z (cid:55)→ z and τ (cid:55)→ τ + 1. We wish to establisha dictionary between the wave functions on a torus with parameter τ and an equivalent torus withparameter τ + 1. Let us now consider [27, equation (37)], ψ j,M ( z, τ, T (cid:55)−−→ ψ j,M ( z, τ + 1 ,
0) = e i π j | M | ψ j,M ( z, τ, . (4.31)As shown in [27], this relation holds for even units of magnetic flux M . However, for odd M a relationof the form ψ j,M ( z, τ + 1 ,
0) = M − (cid:88) j (cid:48) =0 [ ρ ( T )] jj (cid:48) ψ j (cid:48) ( z, τ,
0) (4.32) cannot be true because according to equation (4.30) both sides have different periodicities. That is,on the left–hand side of the equality (4.31) we see a function that is supposed to be “periodic” under z (cid:55)→ z + τ (cid:48) whereas on the right–hand side the function is supposed to be “periodic” under z (cid:55)→ z + τ .According to equation (4.30), for odd M only one of these “periodicities” can hold.At first sight, this statement may appear odd. One might think that the zero modes ψ j,M form abasis of eigenmodes of the Dirac operator with eigenvalue 0. So one may expect that the transformedwave functions can be expanded in terms of the original ones as in equation (4.32). However, thisargument is incorrect. When we write down our wave functions we make a choice for the origin ofthe torus. A priori there are arbitrarily many choices possible, which may be parametrized by ∆ z in ψ j,M ( z + ∆ z, τ, ψ j,M ( z, τ + 1 ,
0) = M − (cid:88) j (cid:48) =0 [ ρ ( T )] jj (cid:48) ψ j (cid:48) ( z + ∆ z, τ,
0) (4.33)12or some appropriate real constant ∆ z . As we shall see, an appropriate choice of ∆ z will allow us toexpress the transformed wave functions in terms of the original one also for odd M . More concretely,we will impose that z (cid:55)→ z + ∆ z , with some real constant ∆ z that we are going to find. Inserting thisansatz leads to (cf. equation (C.9)) ψ j,M ( z + ∆ z, τ + 1 ,
0) = (cid:101) N e i πM ∆ z Im z Im τ e i πMz Im z Im τ ϑ (cid:104) jM (cid:105) ( M ( z + ∆ z + / ) , M τ ) (4.34)Thus, if N := M (∆ z + / ) is an integer, we might use equation (A.5a), which we recast here in aslightly different form ϑ (cid:104) jM (cid:105) ( M z + N, τ ) = e π i N α ϑ (cid:104) jM (cid:105) ( M z, M τ ) , (4.35)to get rid of the extra factor in the z coordinate of the ϑ function. Finally, after the redefinition z (cid:55)→ z − ∆ z , we obtain ψ j,M ( z, τ, T (cid:55)−−→ e i πM ∆ z Im( z )Im τ e i π j | M | +2i πj ∆ z ψ j,M ( z − ∆ z, τ, . (4.36)Note that in order to get an integer N , it is sufficient to demand an integer or half–integer ∆ z foreven M . For ∆ z = 0 equation (4.36) reproduces equation (4.31). However, for odd M we need ahalf–integer ∆ z . Specifically, for ∆ z = / we find that (see appendix C for details) ψ j,M ( z, τ, S (cid:55)−−→ e i π √ M (cid:18) − τ | τ | (cid:19) / M − (cid:88) k =0 e π i jk/M ψ k,M ( z, τ, − (cid:18) − τ | τ | (cid:19) / (cid:104) ρ ( S ) ψM (cid:105) jk ψ k,M ( z, τ, , (4.37a) ψ j,M ( z, τ, T (cid:55)−−→ e i πM Im z τ e i πj ( j/M +1) ψ j,M ( z − / , τ, i πM Im z τ (cid:104) ρ ( T ) ψM (cid:105) jk ψ k,M ( z − / , τ, , (4.37b)where (cid:104) ρ ( S ) ψM (cid:105) jk := − e i π/ √ M exp (cid:18) π i j kM (cid:19) , (4.38a) (cid:104) ρ ( T ) ψM (cid:105) jk := exp (cid:20) i π j (cid:18) jM + 1 (cid:19)(cid:21) δ jk . (4.38b)As we shall confirm shortly in equation (4.48), the matrices (4.38) equal, up to a phase in equa-tion (4.38a), representation matrices of the generators of finite metaplectic modular groups. Theyare compatible with [27–30], but (4.38a) differs from [26] by the e i π/ phase. For even M , the T transformation can rather be represented as in equations (4.37) and (4.38) or equation (4.31) due tothe freedom of choosing half–integer or integer ∆ z . However, since the Yukawa integral involves wavefunctions with both odd and even fluxes M , we need to be consistent in our choice of ∆ z to cancel the z –dependent phase appearing in equation (4.37b) (cf. equation (E.3)). Specifically, we need ∆ z = / for the T transformation also for even M , in which case our results differ from [26–29] by phase factorswhich are absent in equation (4.31). Nevertheless, the modular T transformation of the 2D compactwave functions for odd M was excluded in [26–29]. In [30] they were introduced through the so–calledScherk–Schwarz phases. In particular, our equations (4.37) and (4.38) are consistent with their dis-cussion in [30, equation (126)]. However, as discussed in section 4.3, we disagree with the statementmade in [26–30] that the modular transformed wave functions do not follow the appropriate boundaryconditions. As we have shown, the T transformation can generally not be represented by a matrixmultiplication of the set of wave functions, but necessarily goes beyond this. However, as we discussin appendix E, the extra exponential factors in equation (4.37b) get canceled in the overlap integral(3.2), thus allowing us to define a matrix representation for the transformation of the 4D fields, whichderive from equation (4.38). 13 .4.2 Modular flavor symmetries in the effective 4D theory Let us now define proper “modular flavor transformation” for the 4D fields. The first thing to noticeis that these transformations cannot be unique , at least not in models of this type. The reason is thatthere are additional symmetries at play, such as the remnant gauge factors, and we can always add anextra transformation to our transformation law. That is to say that the details of the representationmatrices of a modular flavor symmetries acting on the fields are somewhat ambiguous. Let us startwith something unambiguous: the transformation of the Yukawa couplings. As we have seen inequation (3.30), there are a priori λ Yukawa couplings, out of which at most λ/ Y ijk and “holomorphicYukawa couplings” Y ijk [43], which are related by (cf. [32, equation (5.41)]) Y ijk ( τ ) = e (cid:98) K/ Y ijk ( τ )( K i ¯ ı K j ¯ K k ¯ k ) / . (4.39)Here, (cid:98) K stands for the K¨ahler potential of the moduli, which is, in our truncated setup, at tree levelgiven by equation (4.22). The formula for the Yukawa couplings (3.30), which we obtained from theoverlap integral (3.2), contains the normalization factor (3.31), which is not holomorphic. In our case,the matter field K¨ahler metric is proportional to (Im τ ) − / (cf. equation (4.18)), so (cf. [32, section 5.3])e (cid:98) K/ ( K i ¯ ı K j ¯ K k ¯ k ) / = N abc ∝ g (cid:18) Im τ A (cid:19) / . (4.40)While Y ijk ( τ ) is normalized and thus “physical”, it is not holomorphic. On the other hand, thesuperpotential coupling Y ijk ( τ ) = ϑ (cid:34)(cid:98) α ijk /λ (cid:35) (0 , λ τ ) (4.41) is a proper modular form. Here, we have made use of the fact that the upper characteristic isof the form (cid:98) α ijk /λ with some integer (cid:98) α ijk , cf. the discussion below equation (3.30), and we set,as done throughout this section, the Wilson lines to zero. Further, all additional non–zero factorsappearing in equation (3.30) must be included in the K¨ahler potential, so that they are canceled inthe holomorphic couplings through the redefinition (4.39). The holomorphic coupling Y ijk ( τ ) differsfrom the physical coupling between canonically normalized fields by a non–holomorphic factor. Themodular transformations are seemingly non–unitary because of the automorphy factor has generallynot modulus 1. However, the automorphy factors get canceled, cf. our discussion below equation (4.46).As shown in appendix E, the λ –plet of Yukawa couplings transforms with the simple transformationlaw Y (cid:98) α ( τ ) (cid:101) γ (cid:55)−−→ Y (cid:98) α ( (cid:101) γ τ ) = ± ( c τ + d ) / ρ λ ( (cid:101) γ ) (cid:98) α (cid:98) β Y (cid:98) β ( τ ) , (4.42)where (cid:98) α and (cid:98) β are integers that label the distinct Yukawa couplings, and we use the metaplecticelement (cid:101) γ ∈ (cid:101) Γ instead of γ ∈ Γ because the Yukawa couplings have weight k Y = / . The transformationmatrices of the modular generators are given by ρ λ ( (cid:101) S ) (cid:98) α (cid:98) β = − e i π/ √ λ exp (cid:32) π i (cid:98) α (cid:98) βλ (cid:33) , (4.43a) ρ λ ( (cid:101) T ) (cid:98) α (cid:98) β = exp (cid:18) i π (cid:98) α λ (cid:19) δ (cid:98) α (cid:98) β . (4.43b) It has been suggested that the modular flavor symmetries can be defined by the requirement that the 6D fieldsremain invariant [26]. However, apart from the fact that this prescription fails for odd M since the 2D coordinates getsshifted (cf. equation (4.37b)), it is not clear to us why one should impose this very requirement. ρ λ ( (cid:101) S ) (cid:98) α (cid:98) β ] − = [ ρ λ ( (cid:101) S ) (cid:98) α (cid:98) β ] ∗ and [ ρ λ ( (cid:101) T ) (cid:98) α (cid:98) β ] − = [ ρ λ ( (cid:101) T ) (cid:98) α (cid:98) β ] ∗ . (4.44)Since there can be relations between the Yukawa couplings, this may not be an irreducible representa-tion. The relations between the Yukawa couplings depend on the choice of fluxes. We will specify theirreducible representations of the Yukawa couplings in our survey of models in section 4.5. The modu-lar transformations of the Yukawa couplings given by equation (3.2) were also studied in [26]. Althoughan explicit general formula for any combination of I αβ was not given in their work, our results fromequation (4.43) match their result up to the phase e i π in the models described in sections 4.5.2 to 4.5.4.This phase is crucial to have the transformation matrices (4.43) satisfy the presentation (4.13) and,thus, give rise to representations of a finite metaplectic modular group, as was noted in [29]. Note alsothat there is an extra minus in our equation (4.43) compared to [26, equations (64) and (108)] and [29].However, this sign comes only from our convention that the automorphy factor is ϕ ( S, τ ) = −√− τ inequation (4.12).Next, we discuss modular flavor symmetries. They are, by definition, symmetry transformationsof the 4D Lagrange density. In our present discussion, we are thus seeking transformations of the 4Dfields, φ j,M , which are such that superpotential couplings W ⊃ Y ijk ( τ ) φ i, I ab φ j, I ca φ k, I cb (4.45)are invariant up to K¨ahler transformations, cf. the discussion around (4.20). Here, I cb = −I bc = I ab + I ca >
0. That is, our modular flavor transformations are given by φ j,M (cid:101) γ (cid:55)−→ ± ( c τ + d ) − / (cid:104) ρ φM ( (cid:101) γ ) (cid:105) − jk φ k,M . (4.46)Notice that, due to equations (4.42) and (4.46), the superpotential acquires modular weight k W = − τ inthe K¨ahler potential followed by a K¨ahler transformation, see our discussion around equation (4.24).Therefore, the requirement that the modular transformations be a symmetry amounts to demandingthat Y ijk ( (cid:101) γ τ ) (cid:104) ρ φ I ab ( (cid:101) γ ) (cid:105) − ii (cid:48) φ i (cid:48) , I ab (cid:104) ρ φ I ca ( (cid:101) γ ) (cid:105) − jj (cid:48) φ j (cid:48) , I ca (cid:104) ρ φ I cb ( (cid:101) γ ) (cid:105) − kk (cid:48) φ k (cid:48) , I cb ! = Y ijk ( τ ) φ i, I ab φ j, I ca φ k, I cb . (4.47)As already mentioned, this condition does not fix the transformation laws of the 4D fields uniquely.However, we can use the transformation properties of the T wave functions, (cf. equations (E.3)and (E.4)), to infer the matrix structure of the transformations. One way in which we may infer thetransformations of the 4D fields is by using the quasi –inverse transformations of the compact wavefunctions, that is, the inverse transformations of equation (4.38). However, a more convenient choiceis ρ φ M ( (cid:101) S ) jk = − e i π (3 M +1) / √ M exp (cid:18) π i j kM (cid:19) , (4.48a) ρ φ M ( (cid:101) T ) jk = exp (cid:20) i π j (cid:18) jM + 1 (cid:19)(cid:21) δ jk , (4.48b)where we have chosen the transformation (4.38a) multiplied by a phase e π M in the S matrix represen-tation. These matrices fulfill equation (4.44), too. This choice has the virtue that ρ M ( (cid:101) γ ) = [ ρ M ( (cid:101) γ )] ∗ and that, as we will demonstrate in section 4.5, it yields the correct representation matrices for thegroup (cid:101) Γ λ . 15e also note that, as far as the Yukawa couplings are concerned, there is a U(1) symmetry due tothe condition of equation (3.4), which acts as φ j, I αβ U(1) (cid:55)−−−−→ e i qα I αβ φ j, I αβ , (4.49)where α, β ∈ { a, b, c } as in equation (3.3). Here, φ I ab , φ I ca have a charge +1 and φ I cb a charge −
1. This U(1) factor allows one to install “extra” phases of the above type. Note that while the T –transformed wave functions, for odd M , cannot be expanded in terms of untransformed wavefunctions, the additional factor in our dictionary (4.37b) cancels in the overlap integrals (3.2) so thatthere is a meaningful, well–defined modular flavor transformation of the 4D fields also for odd M . Ourproposal in equation (4.48) for the transformations of 4D fields φ j,M for even values of M differs fromthe results in [26]. While [26] assumes that the modular transformations of the 4D fields coincide withthose of the 2D wave functions, we assume the 4D fields transform quasi–inversely to the 2D wavefunctions. Furthermore, we have an extra phase e π M , which is useful to achieve metaplectic grouprepresentations. Note that we specify the T transformation, rather than just the T representation asin [26]. In this subsection, we survey a couple of toy models. These models are far from realistic but highlighthow modular flavor symmetries derive from some simple magnetized tori with even and odd numbersof repetitions of matter fields. In all of the next models we will use the representation matrices statedin equation (4.48) for the I cb –plet of φ k
4D fields, while the I ab –plet of φ i and I ca –plet of φ j
4D fieldswill transform in the conjugate representation. On the other hand, the λ –plet of Yukawa couplingswill follow the representation matrices found in equation (4.43). We will show that the modularflavor symmetries in these models are given by (cid:101) Γ λ with λ being the least common multiple of matterrepetition numbers (3.24b). Furthermore, using equation (3.25) one can see that for a fixed totalnumber of Yukawa couplings P , the largest number of independent Yukawa couplings, that is thelargest λ , is obtained by having the least possible d . Although we have proposed the representationmatrices for the 4D fields in equation (4.48), the ones for the Yukawa couplings equation (4.43)are unambiguous. In fact, in all models we discuss here we will find that the representations ρ λ satisfy equation (4.13) together with the finiteness conditions (4.14)–(4.15) for N = 2 ,
3, with λ =2 N . Thus, the modular transformations of the Yukawa couplings build representations of the finitemetaplectic group (cid:101) Γ λ . In [29] it was also noted that, for even numbers of flavors, the Yukawa couplingstransform as a λ –plet under the metaplectic group. However, in [29] it does not get mentioned that for λ > ρ φ I αβ build representations of the same group, so that (cid:101) Γ λ can be regarded as the modular flavor symmetry of the models. We are hence led to conjecturethat, with λ from equation (3.24b),magnetized tori with λ = lcm( (cid:101) Γ λ modular flavor symmetry . (4.50) I ab = I ca = 1 and I bc = − Let us consider a model based on a U(3) gauge symmetry and fluxes F = π iIm τ − . (4.51)16he fluxes break U(3) → U(1) a × U(1) b × U(1) c . Since the N α = 1 for α ∈ { a, b, c } , we thus have I ab = I ca = 1 and I bc = − . (4.52)According to (3.3) this means that we have one repetition of ψ i, I ab =1 and ψ j, I ca =1 each, and twocopies of ψ k, I bc = − . We can now compute the holomorphic Yukawa couplings of this model using(3.30), Y ijk (0 , τ ) = ϑ (cid:104) k (cid:105) (0 , τ ) , (4.53)which gives a doublet Y (cid:98) = (cid:32) Y Y (cid:33) := (cid:32) Y Y (cid:33) , (4.54)which transforms under the representation matrices given by ρ (cid:98) ( (cid:101) S ) = − e i π √ (cid:32) − (cid:33) and ρ (cid:98) ( (cid:101) T ) = (cid:32) (cid:33) . (4.55)Note that in this case we could have used [32, equation (5.17)] since I ab = d = 1. This Yukawa couplingcoincides (up to an irrelevant similarity transformation with diag(1 , − (cid:98) representation of (cid:101) Γ λ =4 = (cid:101) S ∼ = [96 ,
67] [15, cf. equation (41)]. This representation can be thought of as the fundamentalrepresentation of (cid:101) S in that all other nontrivial representations can be obtained by reducing tensorproducts of a suitable number of (cid:98) representations. The fields with multiplicity 2 transform with theinverses (or conjugates, see equation (4.44)) of the representation matrices (4.55). That is, these fieldstransform under (cid:98) . Altogether, we have a (cid:101) S theory with (holomorphic) Yukawa couplings given by W ⊃ φ ab φ ca (cid:16) ϑ (cid:2) (cid:3) (0 , τ ) φ cb + ϑ (cid:104) / (cid:105) (0 , τ ) φ cb (cid:17) , (4.56)where we suppress the trivial generation indices of the fields coming with repetition 1. Notice thatthe physical Yukawa coupling comes with extra normalization factors, see equation (4.39). I ab = I ca = 3 and I bc = − Let us consider a three generation toy model, based on a super–Yang–Mills theory in six dimensionswith gauge group U(4) [26]. The two extra dimensions are compactified on T , and the U(4) gaugesymmetry gets broken to SU(2) × U(1) a × U(1) b × U(1) c by the fluxes F = π iIm τ × − , (4.57)where we used equation (3.1). The chiral matter content of the supersymmetric model is given intable 4.2. They decompose into three generations of L particles, six generations of R particles andthree generations of H particles. The superpotential of this model is given by W ⊃ Y ijk L i H j R k , (4.58)where the (holomorphic) Yukawa couplings are given by equation (3.30), Y ijk ( τ ) = ϑ (cid:104) k − j (cid:105) (0 , τ ) . (4.59) Here, we use the notation (cid:98) from [15] to refer to the two–dimensional irreducible representation of (cid:101) Γ . × U(1) a × U(1) b × U(1) c quantum numbers L (1 , − , I ab = 2 − ( −
1) = 3 R (0 , +1 , − I bc = − − (5) = − H ( − , , I ca = 5 − (2) = 3 Table 4.2: Matter content of the 336 model.
Here we used the values from table 4.2 and assumed zero Wilson lines. The explicit transformationmatrices for the L i and H j are given by (4.48) for M = 3, and are the conjugates of ρ φ ( (cid:101) S ) = − √ i i ii e − π e − i π i e − i π e − π and ρ φ ( (cid:101) T ) = − π
00 0 e − π . (4.60)The explicit transformation matrices for the R k fields are given by equation (4.48) for M = 6 ρ φ ( (cid:101) S ) = − i e i π √ π i3 e π i3 − − π i3 e − π i3 π i3 e − π i3 π i3 e − π i3 − − −
11 e − π i3 e π i3 − π i3 e π i3 − π i3 e − π i3 − π i3 e π i3 , (4.61a) ρ φ ( (cid:101) T ) = diag (cid:0) , e − π i6 , e π i3 , i , e π i3 , e − π i6 (cid:1) . (4.61b)As discussed at the end of section 3, there are only λ / + 1 = 4 independent Yukawa couplings, Y := Y i = j,j,k =2 j , Y := Y i = j +1 ,j,k =2 j +1 = Y := Y i = j +2 ,j,k =2 j +5 , (4.62a) Y := Y i = j,j,k =2 j +3 , Y := Y i = j +2 ,j,k =2 j +2 = Y := Y i = j +1 ,j,k =2 j +4 , (4.62b)where i and j are understood to be modulo 3, and k modulo 6. The six–plet of holomorphic Yukawacoupling coefficients Y = ( Y , Y , Y , Y , Y , Y ) T obeys the transformation law equation (4.42) undermodular transformations, with the matrix representations ρ ( (cid:101) S ) = − i ρ φ ( (cid:101) S ) and ρ ( (cid:101) T ) = diag (cid:0) , e π i6 , e π i3 , − i , e π i3 , e π i6 (cid:1) . (4.63)However, the 6 × P → = √ √
00 0 0 10 0 √ √ , (4.64) The relations given in equation (4.62) are valid for both holomorphic and non–holomorphic Yukawa couplings.
18e can define the –plet of independent Yukawa couplings through Y = P T → Y , which transformas modular forms with the representation matrices given by ρ ( (cid:101) S ) = P T → ρ ( (cid:101) S ) P → = − e i π √ √ √ √ − −√ √ − − √ −√ √ − , and (4.65a) ρ ( (cid:101) T ) = P T → ρ ( (cid:101) T ) P → = π i6 π i3
00 0 0 − i . (4.65b)The representation matrices in equations (4.60), (4.61), (4.63) and (4.65) fulfill the conditions (4.13)for N = 3 and (4.15), which implies that this model exhibits a (cid:101) Γ λ =12 finite modular symmetry oforder 2304. The fact that there are only four distinct Yukawa entries implies that the 6–dimensionalrepresentation of the Yukawa couplings decomposes into (cid:101) Γ irreducible representations according to = ⊕ , as we have confirmed, where the doublet vanishes. As we shall discuss below, this can be alsoattributed to the existence of an outer automorphism. Using the character tables (cf. [44, section 3.4]),we find that the matter triplets and six–plets are reducible as well, = (cid:48)(cid:48) ⊕ (cid:48) and (cid:48) = (cid:48) ⊕ (cid:48) ,where we added primes to indicate that these are different representation matrices, and that thesinglet is nontrivial. We have verified that the reducible representation (cid:48) provides us with a faithfulrepresentation content of (cid:101) Γ and its tensor products yield all other representations of the group.The six Y (cid:98) α have been identified in [26], where they have been represented as sums of three different ϑ –functions each, and the relations Y = Y and Y = Y have been missed. The latter relations areactually quite interesting as they can be thought of as i ↔ j exchange symmetries, Y ijk ( τ ) = Y jik ( τ ) . (4.66)However, the wave functions labeled i and j , i.e. the L i and H j , have different quantum numbers in4D (and in the upstairs theory). This means that this symmetry is not an “ordinary” flavor symmetrybut an outer automorphism of the low–energy gauge symmetry. Note that the existence of this veryouter automorphism depends on the specifics of the model, i.e. while both the current model and theone presented in section 4.5.4 have a (cid:101) Γ λ = (cid:101) Γ metaplectic flavor symmetry, the form of the outerautomorphism is specific to the current model. Examples for such outer automorphisms include theso–called left–right parity [45]. It is known that such symmetries can originate as discrete remnants ofgauge symmetries either by dialing appropriate VEVs [46, 47] or by orbifolding [48]. As the exchangeof the U(1) factors is part of the original U(4) gauge symmetry of the model, we have identified yetanother way in which these outer automorphism can emerge from an explicit gauge symmetry. A geometric interpretation of Yukawa couplings.
It is instructive to discuss the geometricalinterpretation of these results. We have derived the couplings by computing the overlaps of wavefunctions, see (3.2). The result is that, up to a normalization factor, the Yukawa couplings are givenby Y (cid:98) α ∝ (Im τ ) − / ϑ (cid:34) (cid:98) α / λ (cid:35) (0 , λ τ ) = (Im τ ) − / ∞ (cid:88) (cid:96) = −∞ e − π λ (Im τ − i Re τ )( (cid:98) α / λ + (cid:96) ) , (4.67)where we have used equation (A.4). Here we choose to highlight the fact that the terms are expo-nentially suppressed by e − π λ Im τ ξ with some ξ > a Figure 4.1: Overlap of two Gaussians on a torus. The overlap of a given, say red, curve is not just the overlapwith one blue curve but with infinitely many of them, thus leading to an expression of the form (4.67). - - - - - - Im τ l og | Y α | Figure 4.2: Dependence of the magnitude of the Yukawa couplings Y (cid:98) α for Re τ = 0 .
1. The black solid, orangedashed, green dotted and red dash–dotted curves represent (cid:98) α = 0, 1, 2 and 3, respectively. There is anexponential suppression with Im τ that depends on the “distance” between the wave functions (cid:98) α , i.e. the Im τ dependence is more pronounced for larger (cid:98) α . couplings with a simple overlap of Gaussians. For simplicity, we just consider two Gaussians, andconsider y ( a, b , b ) = ∞ (cid:90) −∞ d x N b e − x /b N b e − ( x − a ) /b = e − a / ( b + b ) √ π √ b + b (4.68)with Gaussian normalization factors N b = 1 / √ bπ . In order to compute the overlap on the torus, onedoes not only have to compute the overlap of a given Gaussian of width b , say, with one Gaussianof width b , but with all images of the second Gaussian under torus translations. This leads to anexpression which is qualitatively similar to the sum on the right–hand side of equation (4.67).Turning this around, the upper characteristics (cid:98) α in equations (3.30) and (4.67), or, more precisely,min (cid:0) | (cid:98) α/λ | , | − (cid:98) α/λ | (cid:1) with 0 ≤ | (cid:98) α/λ | <
1, has the interpretation of a “distance between the lociof the states”, i.e. a in figure 4.1. We illustrate this by plotting some sample Yukawa couplingsin figure 4.2. This geometric intuition may conceivably provide us with an understanding of theobserved hierarchies of fermion masses. Apart from the fact that the kinetic terms are under control,the geometric interpretation may be one of the strongest motivations for deriving the modular flavorsymmetries from explicit tori. 20 .5.3 Model with I ab = I ca = 2 and I bc = − Although we have stressed that our results are valid for odd repetitions of matter I αβ , they of coursealso apply to settings in which all I αβ are even. Let us consider a toy model with the same superpo-tential and gauge group breaking as in section 4.5.2 by the fluxes F = π iIm τ × − , (4.69)where we used equation (3.1). This means that we have two repetitions of L i and H j each, and fourcopies of R k . Similarly to what we have found in section 4.5.2, there are λ/ Y := Y i = j,j,k =2 j , Y := Y i = j +1 ,j,k =2 j +1 = Y := Y i = j +1 ,j,k =2 j +3 , (4.70a) Y := Y i = j,j,k =2 j +2 , (4.70b)where i and j are understood to be modulo 2, and k modulo 4. The equality Y = Y is alsoa consequence of the exchange symmetry given in equation (4.66). The four–plet of holomorphicYukawa couplings Y (cid:98) α follow the modular transformations (4.43), with the matrices ρ ( (cid:101) S ) = − e i π √ − − i1 − − − i − and ρ ( (cid:101) T ) = π i4 − π i4 . (4.71)Analogously to what we have done in section 4.5.2, due to the relations of the Yukawa couplings inequation (4.70), we can reduce the representation matrices through the projection matrix P → = √
00 0 10 √ . (4.72a)We can define the triplet of independent Yukawa couplings through Y = P T → Y , which transformsunder the representation matrices given by ρ ( (cid:101) S ) = P T → ρ ( (cid:101) S ) P → = − e i π √ √ √ −√ −√ , (4.73a) ρ ( (cid:101) T ) = P T → ρ ( (cid:101) T ) P → = π i4
00 0 − . (4.73b)The transformation matrices of the doublets L i and H j are obtained by setting M = 2 in equa-tion (4.48) and taking its conjugate. The resulting matrices are given by equation (4.55). Thefour–plets R k transform through equation (4.48) for M = 4, that is ρ φ (cid:48) ( (cid:101) S ) = e i π √ − − i1 − − − i − and ρ φ (cid:48) ( (cid:101) T ) = − π i4 − − π i4 . (4.74)21he representation matrices in equations (4.55), (4.71), (4.73) and (4.74) fulfill the conditions (4.13)for N = 2 and (4.14) which implies that we have a theory endowed with a (cid:101) Γ λ =8 ∼ = [768 , = ⊕ (cid:48) , where the singlet vanishes. On the other hand, the matter four–plets decompose as (cid:48) = (cid:48) ⊕ (cid:48)(cid:48) whereas the matter doublets are irreducible. While all these representations areunfaithful, the combination (cid:48) ⊕ provides us with a faithful representation content of (cid:101) Γ . The tensorproducts of (cid:48) and produce all (cid:101) Γ representations. I ab = 1, I ca = 2 and I bc = − It is important to show that our conjecture (4.50) that we have a (cid:101) Γ λ invariant superpotential is notonly valid for repeated values of I αβ . To this end we consider a toy model with the Yukawa couplingsas in equation (4.58) and with gauge group breaking U(3) → U(1) a × U(1) b × U(1) c by the fluxes F = π iIm τ − − . (4.75)Out of 3 · · a priori possible Yukawa couplings, λ/ Y := Y , Y := Y = Y := Y , (4.76a) Y := Y , Y := Y = Y := Y , (4.76b)where i can only be 0, and j and k are understood to be modulo 2 and 3 respectively. As d = 1,this setting has a comparatively large number of distinct couplings, i.e. 4 out of 6 entries are distinctwhereas e.g. in the model of section 4.5.2 only 4 out of 54 a priori contractions have nontrivialdistinct coefficients. In this case, I ab (cid:54) = I ca , so there is no exchange symmetry of the type (4.66),yet the number of independent Yukawa couplings gets reduced due to the symmetry Y jk = Y j, − k .Unlike the transformation that ensured the equality of Yukawa entries in the model discussed insection 4.5.2, this symmetry is not a nontrivial outer automorphism of the 4D continuous gaugesymmetries. The six–plet of holomorphic Yukawa couplings Y (cid:98) α transform with the matrices fromequation (4.63) that can be reduced by using equation (4.64) to a four–plet, which then transformsaccording to equation (4.65). Furthermore, using equation (4.48), we see that the singlet φ , I ab =1 is invariant under modular transformations, the doublet φ j, I ca =2 transforms with the representationmatrices from equation (4.55), and the triplet φ k, I cb =3 transforms using equation (4.60). It can beshown that all these matrices satisfy the conditions (4.13) for N = 3 and equation (4.15). Therefore,the superpotential is invariant under the finite metaplectic flavor symmetry (cid:101) Γ λ =12 of order 2304. As we have seen, the models derived from explicit tori give rise to the finite metaplectic groups, whichhave been discussed e.g. in [15] in the context of bottom–up model building. The models presentedhere do not attempt to make an immediate connection to particle phenomenology. At first sight, itseems to be hard to derive the models of [15] from tori for at least two reasons:1. our fields all have modular weight − / while in [15] they come with a variety of weights, and2. we have additional symmetries like the outer automorphism symmetry (4.66) and residual gaugefactors.On the other hand, deriving the modular transformations from an explicit higher–dimensional modelhas the virtue that normalization of the fields is known at tree level, and that otherwise free parametersget fixed. Of course, the K¨ahler potential is not exact, apart from the usual 4D corrections there are22dditional terms contributing (cf. [49]), yet the point that there is a zeroth order classical form pluscorrections, which are under control. On the other hand, in the SB approach every invariant K¨ahlerpotential is as good as others [31], and there are thus large uncertainties. An additional benefit ofderiving the modular flavor symmetries from explicit tori is the geometrical intuition one can developfor the Yukawa couplings, cf. our discussion at the end of section 4.5.2.One may now wonder if the price that one has to pay for all these benefits is the inability toconstruct semi–realistic models. In what follows, we will argue that this is not the case. First of all,the T model is just a building block of a more complete story. As explained in [32], these modelsare dual to some intersecting D –brane constructions. Moreover, the couplings of the latter are closelyrelated to heterotic string compactifications [50], which provide us with a large number of potentiallyrealistic models [51]. These more complete settings come with a variety of modular weights [52].Second, even if one is not adding more dimensions to the construction, fields with higher modularweights can emerge as composites of fields with modular weight − / . That is, if “quarks” of a modelwith an SU( N c ) have modular weights − / , then the “baryons” will have weights − N c / . Let us comment on the role supersymmetry (SUSY) plays in the discussion. While Cremades etal. work in a supersymmetric theory, they mention [32, see the beginning of section 5.3] that theirderivation “in principle is valid for toroidal compactifications where supersymmetry might be brokenexplicitly”. Of course, if one wants to claim that the couplings that one has computed are Yukawacouplings, one needs to make sure that one computes the overlap between two fermionic and onebosonic zero–modes. In supersymmetric models there is no problem because the superpartners aredescribed by the same wave functions.In a model without low–energy SUSY one may be worried that quantum corrections lead touncontrollable corrections to the wave function of the scalar. This is generally a very valid concern,yet is as recently been observed that in the magnetized tori there is an interesting cancellation ofcorrections to the scalar mass [33–36]. While this has not yet led to a complete solution of thehierarchy problem in the SM, it does suggest that in the context of the very models that we were ledto consider for the sake of deriving modular flavor symmetries the situation is “better” than in othernonsupersymmetric completions of the SM with a high UV scale. In fact, similar cancellations havebeen reported in [53], where they were attributed to modular symmetries.In a bit more detail, one could imagine a torus compactification in which the Yukawa couplingsemerge as outlined in [32], namely as the overlap of three wave functions. These wave functionsdescribe two fermions and one scalar, such as the SM Higgs. If the scalars remain light, they will stillbe approximate zero–modes, and thus the profile is approximately given by equation (2.1). So theYukawa couplings will, to some good approximation, be the ones of equation (3.2). So supersymmetryis not instrumental for having models with modular flavor symmetries.
We discussed how modular flavor symmetries derive from explicit tori which are endowed with mag-netic fluxes. Using Euler’s Theorem, we have derived a closed–form expression for the Yukawa cou-plings between zero modes. This expression generalizes the results of the literature to arbitrary fluxparameters I ab and I ca , which fix I bc , and is not restricted to the special case in which one fluxparameter equals 1. Each entry of the Yukawa tensor is a single ϑ –function, i.e. the holomorphic We adopt the convention to call models with realistic and unrealistic features “semi–realistic” while “potentiallyrealistic” models are constructions that have no obviously unrealistic features, but have not yet been analyzed in enoughdetail to be called realistic. Y ijk ( τ ) = ϑ (cid:34)(cid:98) α ijk /λ (cid:35) (0 , λ τ ) , (6.1)where (cid:98) α ijk = I (cid:48) ca i − I (cid:48) ab j + I (cid:48) ca ( I (cid:48) ab ) φ ( |I (cid:48) bc | ) ( k − i − j ) mod λ . (6.2)Here, φ denotes the Euler φ –function, I (cid:48) αβ = I αβ /d for α, β ∈ { a, b, c } , d = gcd (cid:0) |I ab | , |I ca | , |I bc | (cid:1) , λ = lcm (cid:0) |I ab | , |I ca | , |I bc | (cid:1) , and τ denotes the half–period ratio of the torus. The condensed formfor holomorphic Yukawa couplings as single ϑ –functions is instrumental for deriving the symmetriesbetween Yukawa couplings. As we have seen, these symmetries include outer automorphisms of thelow–energy gauge symmetry. These couplings are modular forms of weight / of level 2 λ that buildrepresentations under the metaplectic modular flavor symmetry (cid:101) Γ λ . There are at most 1+ λ/ λ of (cid:101) Γ λ . This meansthat e.g. a model with flux parameters ( I ab , I ca , I bc ) = (1 , , −
3) has as many independent Yukawacouplings as a model with fluxes (3 , , − (cid:98) α ijk in equation (6.1) corresponds to a separation of the states, andlead to an exponential suppression with an exponent min( (cid:98) α ijk , − (cid:98) α ijk ) Im τ for unsuppressed Im τ .The 4D fields have a well–defined transformation behavior under (cid:101) Γ λ , regardless of whether the fluxis even or odd, and have weight k φ = − / . Our analysis is restricted to a magnetized torus T and itshalf–period ratio τ , which is contained in the so–called complex structure modulus. We also set theWilson lines to zero, and largely disregarded the K¨ahler modulus. While this is sufficient to derivemeaningful modular flavor symmetries, this analysis may be thought of as a building block of morecomplete, perhaps stringy models. It will be interesting to derive a generalization of equation (6.1)for such constructions.We have also commented on the role that supersymmetry plays in these constructions. As alreadypointed out in [32], supersymmetry is not instrumental as long as the profiles of the zero–modes donot get distorted too much. More recent analyses [33–36] indicate that magnetized tori have certainunusual properties in that scalar masses seem to be immune to quantum corrections even withoutsupersymmetry. This means that one can plausibly disentangle modular flavor symmetries from thequestion of low–energy supersymmetry. Acknowledgments
The work of Y.A. was supported by Kuwait University. The work of M.-C.C., M.R. and S.S. wassupported by the National Science Foundation, under Grant No. PHY-1915005. The work of S.R.-S.was partly supported by CONACyT grant F-252167. This work is also supported by UC-MEXUS-CONACyT grant No. CN-20-38. A ϑ –functions In this appendix, we collect some relevant facts on the ϑ –functions. Our conventions are based on [37]and [54]. One defines ϑ (cid:2) αβ (cid:3) ( z, τ ) := S β T α ϑ ( z, τ ) = e π i α β T α S β ϑ ( z, τ ) , (A.1)where [37, p. 4] ϑ ( z, τ ) := (cid:88) (cid:96) ∈ Z exp(i π (cid:96) τ ) exp(2 π i (cid:96) z ) (A.2)24ith Im τ >
0, and [37, p. 6]( S β f )( z ) := f ( z + β ) , (A.3a)( T α f )( z ) := e i π α τ +2 π i α z f ( z + α τ ) . (A.3b)This immediately gives us (cf. [54, p. 214 f.]) ϑ (cid:2) αβ (cid:3) ( z, τ ) = ∞ (cid:88) (cid:96) = −∞ e i π ( α + (cid:96) ) τ e π i ( α + (cid:96) ) ( z + β ) . (A.4)For an integer n ∈ Z one has torus periodicities ϑ (cid:2) αβ (cid:3) ( z + n, τ ) = e π i n α ϑ (cid:2) αβ (cid:3) ( z, τ ) , (A.5a) ϑ (cid:2) αβ (cid:3) ( z + n τ, τ ) = e − i π n τ − π i n ( z + β ) ϑ (cid:2) αβ (cid:3) ( z, τ ) . (A.5b)Further, the ϑ –function have several periodicities in the characteristics α and β , ϑ (cid:104) α +1 β (cid:105) ( z, τ ) = ϑ (cid:2) αβ (cid:3) ( z, τ ) , (A.6a) ϑ (cid:2) αβ +1 (cid:3) ( z, τ ) = e π i α ϑ (cid:2) αβ (cid:3) ( z, τ ) . (A.6b)The behavior under modular transformation is ϑ (cid:2) αβ (cid:3) ( z, τ + 1) = e − i π α ( α +1) ϑ (cid:104) αβ + α + (cid:105) ( z, τ ) , (A.7a) ϑ (cid:2) αβ (cid:3)(cid:18) − zτ , − τ (cid:19) = √− i τ e π i (cid:16) z τ + α β (cid:17) ϑ (cid:2) − βα (cid:3) ( z, τ ) . (A.7b)Another useful formula is ϑ (cid:104) j / M (cid:105)(cid:16) z, τM (cid:17) = M − (cid:88) k =0 e π i jk/M ϑ (cid:34) kM (cid:35) ( M z, M τ ) . (A.8) B Torus integration
The torus is defined by two lattice vectors, which can be chosen as e = 2 πR and e = 2 πR τ , wherethe real, dimensionful quantity R sets the length of one lattice vector, and τ with Im τ > G = (2 πR ) (cid:32) τ Re τ | τ | (cid:33) . (B.1)We can define the fundamental domain of the torus as T = { z ∈ C ; z = x e + y e with 0 ≤ x, y ≤ } . (B.2)It is straightforward to verify that the Jacobian of the transformation (Re z, Im z ) (cid:55)→ ( x, y ) is givenby (2 πR ) Im τ . Therefore, the integrals of an arbitrary function f ( z ) over the fundamental domainare given by (cid:90) T d z f ( z ) = (2 πR ) Im τ (cid:90) d x (cid:90) d y f ( xe + ye ) . (B.3)Let us now look at constant modes on the torus, or, equivalently integrate over the torus T todetermine its volume. We then havevol(torus) = (cid:90) T d z πR ) Im τ =: A . (B.4)25 Explicit verification of the boundary conditions for trans-formed wave functions
C.1 S transformation We now compute the S transformation, τ (cid:55)→ − /τ (cf. equation (4.3)), of equation (2.1). We have ψ j,M (cid:18) − zτ , − τ , (cid:19) = (cid:18) M Im − τ A (cid:19) / exp (cid:18) i πM zτ Im zτ Im − τ (cid:19) ϑ (cid:104) jM (cid:105)(cid:18) − M zτ , − Mτ (cid:19) = N (cid:112) | τ | exp (cid:18) i πM zτ Im z ¯ τ Im τ (cid:19)(cid:114) − i τM e πM i z τ ϑ (cid:104) jM (cid:105)(cid:18) z, τM (cid:19) = N (cid:112) | τ | exp (cid:18) i πM zτ Im z ¯ τ Im τ (cid:19)(cid:114) − i τM e i πM z τ M − (cid:88) k =0 e π i jk/M ϑ (cid:104) kM (cid:105) ( M z, M τ )= e i π √ M (cid:18) − τ | τ | (cid:19) / M − (cid:88) k =0 e π i jkM (cid:20) N exp (cid:18) i πM zτ Im z ¯ τ Im τ + i M π z τ (cid:19) ϑ (cid:104) kM (cid:105) ( M z, M τ ) (cid:21) = e i π √ M (cid:18) − τ | τ | (cid:19) / M − (cid:88) k =0 e π i jkM N e iπMz Im z Im τ ϑ (cid:104) jM (cid:105) ( M z, M τ ) , (C.1)where we used (2.2), (A.7b) and (A.8) in the first, second and third lines respectively. We thus arriveat (C.2). Therefore, the S modular transformation of the zero modes is ψ j,M (cid:18) − zτ , − τ , (cid:19) = e i π √ M (cid:18) − τ | τ | (cid:19) / M − (cid:88) k =0 e π i jk/M ψ k,M ( z, τ, . (C.2)It is straightforward to see that the wave function of equation (C.2) satisfies the boundary conditionsof equations (4.27a) and (4.27b). Note that ψ j,M (cid:18) − zτ + 1 , − τ , (cid:19) = ψ j,M (cid:18) − ( z − τ ) τ , − τ , (cid:19) = e i π √ M (cid:18) − τ | τ | (cid:19) / M − (cid:88) k =0 e π i jkM ψ k,M ( z − τ, τ,
0) = exp (cid:18) − i πM Im z ¯ τ Im τ (cid:19) ψ j,M (cid:18) − zτ , − τ , (cid:19) (C.3)and ψ j,M (cid:18) − zτ − τ , − τ , (cid:19) = ψ j,M (cid:18) − ( z + 1) τ , − τ , (cid:19) = e i π √ M (cid:18) − τ | τ | (cid:19) / M − (cid:88) k =0 e π i jkM ψ k,M ( z + 1 , τ,
0) = exp (cid:18) i πM Im z Im τ (cid:19) ψ j,M (cid:18) − zτ , − τ , (cid:19) . (C.4)Thus, from equations (C.3) and (C.4) we can see that the S transformed zero mode follows theboundary conditions of equations (4.27a) and (4.27b) for both odd and even M . C.2 T transformation Now, we compute the transformed wave function ψ j,M ( z, τ + 1 ,
0) and check that it satisfies bothequation (4.28a) and equation (4.28b). Applying the T modular transformation in equation (2.1)26ives ψ j,M ( z, τ + 1 ,
0) = N e i π Mz Im z Im τ ϑ (cid:104) jM (cid:105)(cid:0) M z, M ( τ + 1) , (cid:1) = e − i πj (cid:0) jM +1 (cid:1) N e i πMz Im z Im τ ϑ (cid:20) jM j + M (cid:21) ( M z, M τ )= e − i πj (cid:0) jM +1 (cid:1) N e i πMz Im z Im τ e π i jM j ϑ (cid:20) jMM (cid:21) ( M z, M τ )= e − i πj (cid:0) − jM (cid:1) N e i πMz Im z Im τ ϑ (cid:20) jMM (cid:21) ( M z, M τ ) , (C.5)where we used (A.7a) and (A.6b) in the second and third line, respectively. Defining (cid:101) N := e − i πj (cid:0) − jM (cid:1) N we can thus write ψ j,M ( z, τ + 1 ,
0) = (cid:101) N e i πMz Im z Im τ ϑ (cid:20) jMM (cid:21) ( M z, M τ ) . (C.6)Now we check that equation (C.6) satisfies boundary conditions given by equations (4.28a) and (4.28b).The first boundary condition is satisfied as shifting z → z + 1 in equation (C.6) gives ψ j,M ( z + 1 , τ + 1 ,
0) = (cid:101) N e i πM ( z +1) Im z Im τ ϑ (cid:20) jMM (cid:21) ( M ( z + 1) , M τ )= exp (cid:18) i πM Im z Im τ (cid:19) (cid:101) N e i πM ( z +1) Im z Im τ e π i M jM ϑ (cid:20) jMM (cid:21) ( M z, M τ )= exp (cid:18) i πM Im z Im τ (cid:19) ψ j,M ( z, τ + 1 , , (C.7)where we used equation (A.5a) in the second line. For equation (4.28b) we have ψ j,M ( z + τ + 1 , τ + 1 ,
0) = (cid:101) N e i πM ( z + τ +1)Im τ Im( z + τ +1) ϑ (cid:20) jMM (cid:21) ( M ( z + τ + 1) , M τ )= (cid:101) N e i πM Im τ [ z Im z +( τ +1) Im z + z Im( τ +1)+( τ +1) Im( τ +1) e π i jM M ϑ (cid:20) jMM (cid:21) ( M ( z + τ ) , M τ )= (cid:101) N e i πM Im τ [ z Im z +( τ +1) Im z + z Im( τ +1)+( τ +1) Im( τ +1) e − π i τM − π i ( Mz + M ) ϑ (cid:20) jMM (cid:21) ( M z, M τ )= e i πM Im τ [( τ +1) Im z + z Im( τ +1)+( τ +1) Im( τ +1) − τ Im τ − z Im τ − Im τ ] (cid:101) N e i πM Im τ z Im z ϑ (cid:20) jMM (cid:21) ( M z, M τ )= exp (cid:18) i πM Im τ Im(¯ τ + 1) z (cid:19) ψ j,M ( z, τ + 1 , , (C.8)where we have used equations (A.5a) and (A.5b) in the second and third line respectively. Therefore,the transformed modular wave function given by equation (C.6) follows the transformed boundaryconditions of equations (4.28a) and (4.28b) for even and odd M .Let us now tackle the problem of expressing the T transformed wave functions in terms of theoriginal ones. As noted in section (4.4.1), for odd values of M it is not possible to express the T transformed wave functions in terms of the original ones because of (4.30). At the level of the wavefunctions, one can refer to equation (C.6) and see that if M is even, then using (A.6b) confirms (4.31).However, if M is odd, in order to make use of (A.6b) we need to shift the z coordinate as z (cid:55)→ z + ∆ z with real ∆ z . Using equation (C.6) we have ψ j,M ( z + ∆ z, τ + 1 ,
0) = (cid:101) N e i πM ( z +∆ z ) Im z Im τ ϑ (cid:20) jMM (cid:21) ( M ( z + ∆ z ) , M τ )= (cid:101) N e i πM ∆ z Im z Im τ e i πMz Im z Im τ ϑ (cid:104) jM (cid:105) ( M ( z + ∆ z + ) , M τ ) , (C.9)27here we have used (A.4) to rewrite the lower characteristic of the ϑ –function as a shift in the z coordinate. Therefore, if we assume that ∆ z is a half–integer number, we can use (A.5a) which gives ψ j,M ( z + ∆ z, τ + 1 ,
0) = (cid:101) N e i πM ∆ z Im z Im τ e i πMz Im z Im τ e π i j ( +∆ z ) ϑ (cid:104) jM (cid:105) ( M z, M τ )= e i πM ∆ z Im z Im τ e π i j ( jM +2∆ z ) ψ j,M ( z, τ, . (C.10)Note that in order to use (A.5a), M (cid:0) + ∆ z (cid:1) needs to be an integer. Therefore, for odd M , ∆ z needsto be half–integer, whereas for even M both integer and half–integer ∆ z are valid choices. After aredefinition of z → z − ∆ z with some half–integer ∆ z , ψ j,M ( z, τ, T (cid:55)−−→ e i πM ∆ z Im z Im τ e i π j | M | +2i πj ∆ z ψ j,M ( z − ∆ z, τ, , (C.11)and this is valid for both even and odd values of M . D Symmetries between the Yukawa couplings
In this appendix we identify additional relations between the Yukawa couplings given in equation (3.30).Yukawa entries with different i , j and/or k are equal if the upper characteristic, I (cid:48) ca i − I (cid:48) ab j + I (cid:48) ca ( I (cid:48) ab ) φ ( |I (cid:48) bc | ) ( k − i − j ) λ =: u ijk , (D.1)with I (cid:48) αβ = I αβ /d , is the same. For instance, suppose i (cid:48) = i + r , j (cid:48) = j + s and k (cid:48) = k + r + s , so that i (cid:48) , j (cid:48) and k (cid:48) also satisfy the selection rule (cf. equation (3.16)). Then, for values of r and s satisfying I ca r − I ab s = 0 , (D.2)we find that u ijk = u i (cid:48) j (cid:48) k (cid:48) , thus implying that Y ijk = Y i (cid:48) j (cid:48) k (cid:48) . We know that I ca and I ab are divisibleby d = gcd (cid:0) |I ab | , |I ca | , |I bc | (cid:1) (cf. equation (3.12)), u ijk has the form u ijk = (cid:98) α ijk λ (D.3)with some integer (cid:98) α ijk given by equation (6.2). Further, as a shift of u ijk by 1 leaves the ϑ –functionin the Yukawa entry invariant (cf. equation (A.6a)), there are at most λ distinct entries, i.e. u ijk ∈ { , /λ, . . . ( λ − /λ } . (D.4)Additionally, for vanishing Wilson lines, the ϑ –function takes the simple form (cf. equation (A.4)) ϑ (cid:34) (cid:98) α ijk λ (cid:35) (0 , λ τ ) = ∞ (cid:88) (cid:96) = −∞ e i π ( (cid:98) α ijk /λ + (cid:96) ) λτ , (D.5)which shows that ϑ (cid:34) − (cid:98) α ijk λ (cid:35) (0 , λ τ ) = ϑ (cid:34) (cid:98) α ijk λ (cid:35) (0 , λ τ ) . (D.6)Therefore, there are λ/ − λ/ I ab = I ca , the overlap integral (3.2) becomes Y ijk = g σ abc (cid:90) T d z ψ i, I ab ( z, τ, ζ ) ψ j, I ab ( z, τ, ζ ) (cid:0) ψ k, I cb ( z, τ, ζ ) (cid:1) ∗ . (D.7)28his equation is symmetric under i ↔ j , which implies that Y ijk = Y jik . (D.8)As we discuss around equation (4.66) in the main text, the i ↔ j flip can entail an outer automorphismof the low–energy gauge symmetry. E Modular transformations of Yukawa couplings
In this appendix we will show the different ways in which the Yukawa couplings obtained from theoverlap integrals (3.2) transform under modular transformations and how that they indeed are modularforms according to equation (4.12).
E.1 Transformation of the overlap integrals
Let us start by discussing how our dictionary (4.37b) between the wave functions with torus parameter τ and an equivalent torus with parameter τ +1 allows us to infer how the three index Yukawa couplings Y ijk transform. We start with the T transformation where we use (4.37b) for the 2D wave functions.As we have discussed around (4.34), our dictionary involves a shift of the z –coordinate, ∆ z . Fordefiniteness we use ∆ z = . Thus, from equation (3.2) we have Y ijk ( τ + 1) = (cid:90) T d z (cid:16) ρ ( T ) ψi,i (cid:48) e i π I ab Im z Im τ ψ i (cid:48) , I ab (cid:0) z − , τ, (cid:1)(cid:17)(cid:16) ρ ( T ) ψj,j (cid:48) e i π I ca Im z Im τ ψ j (cid:48) , I ca (cid:0) z − , τ, (cid:1)(cid:17) · (cid:16) ρ ( T ) ψk,k (cid:48) e i π I cb Im z Im τ ψ k (cid:48) , I cb (cid:0) z − , τ, (cid:1)(cid:17) ∗ = (cid:90) T d z e i π ( I ab + I ca + I bc ) Im z τ ρ ( T ) ψi,i (cid:48) ρ ( T ) ψj,j (cid:48) (cid:16) ρ ( T ) ψk,k (cid:48) (cid:17) ∗ · ψ i (cid:48) , I ab ( z − , τ, ψ j (cid:48) , I ca ( z − , τ, (cid:16) ψ k (cid:48) , I cb ( z − , τ, (cid:17) ∗ . (E.1)Using equation (3.4), we find that, Y ijk ( τ + 1) = ρ ( T ) ψi,i (cid:48) ρ ( T ) ψj,j (cid:48) (cid:16) ρ ( T ) ψk,k (cid:48) (cid:17) ∗ · (cid:90) T d z ψ i (cid:48) , I ab ( z − , τ, ψ j (cid:48) , I ca ( z − , τ, (cid:16) ψ k (cid:48) , I cb ( z − , τ, (cid:17) ∗ . (E.2)We can now define w := z − . Then d z = d w , i.e. the integration measure for torus coordinatesand the domain of integration remains invariant. Thus we find that Y ijk ( τ + 1) = ρ ( T ) ψi,i (cid:48) ρ ( T ) ψj,j (cid:48) (cid:16) ρ ( T ) ψk,k (cid:48) (cid:17) ∗ (cid:90) T d w ψ i (cid:48) , I ab ( w, τ, ψ j (cid:48) , I ca ( w, τ, (cid:16) ψ k (cid:48) , I cb ( w, τ, (cid:17) ∗ = ρ ( T ) ψi,i (cid:48) ρ ( T ) ψj,j (cid:48) (cid:16) ρ ( T ) ψk,k (cid:48) (cid:17) ∗ Y ijk ( τ )= e i π ( i / I ab + j / I ca − k / I cb + i + j − k ) Y ijk ( τ ) . (E.3)Thus the z –dependent phase appearing in our dictionary for T transformation (4.36) cancels out dueto the condition (3.4). 29or the S transformation of Y ijk , we use equation (4.37a), which gives Y ijk ( − /τ ) = (cid:90) T d z (cid:32) − (cid:18) − τ | τ | (cid:19) / ρ ( S ) ψi,i (cid:48) ψ i (cid:48) , I ab ( z, τ, (cid:33)(cid:32) − (cid:18) − τ | τ | (cid:19) / ρ ( S ) ψj,j (cid:48) ψ j (cid:48) , I ca ( z, τ, (cid:33) · (cid:32) − (cid:18) − τ | τ | (cid:19) / ρ ( S ) ψk,k (cid:48) ψ k (cid:48) , I cb ( z, τ, (cid:33) ∗ = − (cid:18) − τ | τ | (cid:19) / ρ ( S ) ψi,i (cid:48) ρ ( S ) ψj,j (cid:48) (cid:104) ρ ( S ) ψk,k (cid:48) (cid:105) ∗ Y i (cid:48) j (cid:48) k (cid:48) = − (cid:18) − τ | τ | (cid:19) / − e i π (cid:112) |I ab I bc I bc | I ab − (cid:88) i (cid:48) =0 I ca − (cid:88) j (cid:48) =0 I cb − (cid:88) k (cid:48) =0 e π i (cid:16) ii (cid:48)I ab + jj (cid:48)I ca + kk (cid:48)I bc (cid:17) Y i (cid:48) j (cid:48) k (cid:48) , (E.4)where have used the fact that the automorphy factor and the ρ ( S ) ψ matrix do not depend in the z coordinate, and then, can be taken out of the integral.Equations (E.3) and (E.4) give the modular transformations of Y ijk . They can be used to inferthe possible modular transformations of the 4D fields. E.2 Modular transformation of the λ –plet of Yukawa couplings The λ –plet of holomorphic Yukawa couplings (4.41), Y ijk ( τ ) = ϑ (cid:104) (cid:98) α ijk /λ (cid:105) (0 , λ τ ), transforms as amodular form of weight / . To see this, let us first investigate how Y (cid:98) α ( τ ), where (cid:98) α := (cid:98) α ijk ∈ Z λ ,behaves under T . Obviously, Y (cid:98) α ( τ ) T (cid:55)−−→ Y (cid:98) α ( τ ) = ∞ (cid:88) (cid:96) = −∞ exp (cid:34) i π (cid:18) (cid:98) αλ + (cid:96) (cid:19) λ ( τ + 1) (cid:35) . (E.5)The phase can be manipulated to givei π (cid:18) (cid:98) αλ + (cid:96) (cid:19) λ ( τ + 1) = i π (cid:18) (cid:98) αλ + (cid:96) (cid:19) λ τ + i π ( (cid:98) α + λ (cid:96) ) λ . (E.6)The second term can be rewritten asi π ( (cid:98) α + λ (cid:96) ) λ = i π (cid:98) α λ + 2 π i (cid:96) + i π λ (cid:96) . (E.7)Only the first term on the right–hand side yields a nontrivial phase. The two others are integermultiples of 2 π i because λ is even. Therefore, Y (cid:98) α ( τ + 1) = e i π (cid:98) α λ Y (cid:98) α ( τ ) . (E.8)Likewise, under S Y (cid:98) α ( τ ) S (cid:55)−−→ Y (cid:98) α ( − /τ ) = ϑ (cid:104) (cid:98) α/λ (cid:105)(cid:0) , − λ/τ (cid:1) =: ϑ (cid:104) (cid:98) α/λ (cid:105)(cid:0) , − /t (cid:1) (E.9)where t := τ /λ . Then Y (cid:98) α ( − /τ ) = √− i t ϑ (cid:104) (cid:98) α/λ (cid:105)(cid:0) , t (cid:1) = √− i τ √ λ λ − (cid:88) (cid:98) β =0 e π i (cid:98) α (cid:98) βλ ϑ (cid:104) (cid:98) β/λ (cid:105)(cid:0) , λ τ (cid:1) = ( − τ ) / λ − (cid:88) (cid:98) β =0 e π i4 √ λ e π i (cid:98) α (cid:98) βλ ϑ (cid:104) (cid:98) β/λ (cid:105)(cid:0) , λ τ (cid:1) = − ( − τ ) / λ − (cid:88) (cid:98) β =0 (cid:32) − e π i4 √ λ (cid:33) e π i (cid:98) α (cid:98) βλ Y (cid:98) β ( τ ) . (E.10)30ere we used equations (A.8) and (A.7b). This shows that the λ –plet of Y (cid:98) α ( τ ) picks up the correctautomorphy factors to be a modular form of weight / . Note that we choose the minus sign in equa-tion (E.10), anticipating that these transformations comply with equation (4.12), for ϕ ( S, τ ) = −√− τ ,and thus with equation (4.13). Therefore, from equations (E.8) and (E.10) we get the representationsof the λ –plet of Yukawa couplings (4.43b), which we recast here ρ λ ( (cid:101) S ) (cid:98) α (cid:98) β = − e i π/ √ λ exp (cid:32) π i (cid:98) α (cid:98) βλ (cid:33) , (E.11a) ρ λ ( (cid:101) T ) (cid:98) α (cid:98) β = exp (cid:18) i π (cid:98) α λ (cid:19) δ (cid:98) α (cid:98) β . (E.11b)Finally, although these matrices may be not be irreducible for some choice of I αβ , in section 4.5 weget the irreducible representation matrix in each case (cf. e.g. equations (4.63) and (4.65)). Therefore,equation (4.12) is satisfied and the Yukawa couplings given by equation (3.2) are modular forms ofweight k Y = / . Furthermore, as discussed in section 4.5, the representation matrix will correspondto a representation of the metaplectic group (cid:101) Γ λ , which implies that the Yukawa couplings have level2 λ . References [1] F. Feruglio,
Are neutrino masses modular forms? , From My Vast Repertoire ...: Guido Altarelli’sLegacy (A. Levy, S. Forte, and G. Ridolfi, eds.), 2019, pp. 227–266.[2] T. Kobayashi, K. Tanaka, and T. H. Tatsuishi, Phys. Rev. D (2018), no. 1, 016004, arXiv:1803.10391 [hep-ph].[3] J. Penedo and S. Petcov, Nucl. Phys. B (2019), 292, arXiv:1806.11040 [hep-ph].[4] J. C. Criado and F. Feruglio, SciPost Phys. (2018), no. 5, 042, arXiv:1807.01125 [hep-ph].[5] F. J. de Anda, S. F. King, and E. Perdomo, Phys. Rev. D (2020), no. 1, 015028, arXiv:1812.05620 [hep-ph].[6] H. Okada and M. Tanimoto, Phys. Lett. B (2019), 54, arXiv:1812.09677 [hep-ph].[7] G.-J. Ding, S. F. King, and X.-G. Liu, Phys. Rev. D (2019), no. 11, 115005, arXiv:1903.12588 [hep-ph].[8] P. Novichkov, J. Penedo, S. Petcov, and A. Titov, JHEP (2019), 165, arXiv:1905.11970 [hep-ph].[9] X.-G. Liu and G.-J. Ding, JHEP (2019), 134, arXiv:1907.01488 [hep-ph].[10] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, Phys. Rev. D (2019), no. 11, 115045, arXiv:1909.05139 [hep-ph], [Erratum: Phys.Rev.D 101, 039904 (2020)].[11] T. Asaka, Y. Heo, T. H. Tatsuishi, and T. Yoshida, JHEP (2020), 144, arXiv:1909.06520 [hep-ph].[12] G.-J. Ding, S. F. King, X.-G. Liu, and J.-N. Lu, JHEP (2019), 030, arXiv:1910.03460 [hep-ph].[13] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. Tatsuishi, and H. Uchida, Phys. Rev.D (2020), no. 5, 055046, arXiv:1910.11553 [hep-ph].3114] G.-J. Ding and F. Feruglio, JHEP (2020), 134, arXiv:2003.13448 [hep-ph].[15] X.-G. Liu, C.-Y. Yao, B.-Y. Qu, and G.-J. Ding, Phys. Rev. D (2020), no. 11, 115035, arXiv:2007.13706 [hep-ph].[16] G.-J. Ding, F. Feruglio, and X.-G. Liu, JHEP (2021), 037, arXiv:2010.07952 [hep-th].[17] C.-Y. Yao, X.-G. Liu, and G.-J. Ding, arXiv:2011.03501 [hep-ph].[18] T. Kobayashi, S. Nagamoto, and S. Uemura, PTEP (2017), no. 2, 023B02, arXiv:1608.06129 [hep-th].[19] T. Kobayashi, S. Nagamoto, S. Takada, S. Tamba, and T. H. Tatsuishi, Phys. Rev. D (2018),no. 11, 116002, arXiv:1804.06644 [hep-th].[20] T. Kobayashi and S. Tamba, Phys. Rev. D (2019), no. 4, 046001, arXiv:1811.11384 [hep-th].[21] Y. Kariyazono, T. Kobayashi, S. Takada, S. Tamba, and H. Uchida, Phys. Rev. D (2019),no. 4, 045014, arXiv:1904.07546 [hep-th].[22] A. Baur, H. P. Nilles, A. Trautner, and P. K. Vaudrevange, Phys. Lett. B (2019), 7, arXiv:1901.03251 [hep-th].[23] H. P. Nilles, S. Ramos-S´anchez, and P. K. S. Vaudrevange, Phys. Lett. B (2020), 135615, arXiv:2006.03059 [hep-th].[24] A. Baur, M. Kade, H. P. Nilles, S. Ramos-S´anchez, and P. K. S. Vaudrevange, JHEP (2021),018, arXiv:2008.07534 [hep-th].[25] A. Baur, M. Kade, H. P. Nilles, S. Ramos-S´anchez, and P. K. S. Vaudrevange, arXiv:2012.09586 [hep-th].[26] H. Ohki, S. Uemura, and R. Watanabe, Phys. Rev. D (2020), no. 8, 085008, arXiv:2003.04174 [hep-th].[27] S. Kikuchi, T. Kobayashi, S. Takada, T. H. Tatsuishi, and H. Uchida, arXiv:2005.12642 [hep-th].[28] S. Kikuchi, T. Kobayashi, H. Otsuka, S. Takada, and H. Uchida, JHEP (2020), 101, arXiv:2007.06188 [hep-th].[29] K. Hoshiya, S. Kikuchi, T. Kobayashi, Y. Ogawa, and H. Uchida, arXiv:2012.00751 [hep-th].[30] S. Kikuchi, T. Kobayashi, and H. Uchida, arXiv:2101.00826 [hep-th].[31] M.-C. Chen, S. Ramos-S´anchez, and M. Ratz, Phys. Lett. B (2020), 135153, arXiv:1909.06910 [hep-ph].[32] D. Cremades, L. Ib´a˜nez, and F. Marchesano, JHEP (2004), 079, hep-th/0404229 .[33] W. Buchm¨uller, M. Dierigl, E. Dudas, and J. Schweizer, JHEP (2017), 052, arXiv:1611.03798 [hep-th].[34] D. Ghilencea and H. M. Lee, JHEP (2017), 039, arXiv:1703.10418 [hep-th].[35] W. Buchm¨uller, M. Dierigl, and E. Dudas, JHEP (2018), 151, arXiv:1804.07497 [hep-th].[36] T. Hirose and N. Maru, JHEP (2019), 054, arXiv:1904.06028 [hep-th].[37] D. Mumford, Tata lectures on theta I , 1. ed., Birkh¨auser, 1983.3238] U. Dudley,
Elementary number theory: Second edition , Dover Books on Mathematics, Dover Pub-lications, 2012.[39] H. Abe, K.-S. Choi, T. Kobayashi, and H. Ohki, Phys. Rev. D (2009), 126006, arXiv:0907.5274 [hep-th].[40] J. Bruinier, G. van der Geer, G. Harder, and D. Zagier, , Springer, Berlin,2008.[41] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.0 , 2020.[42] J. Wess and J. Bagger,
Supersymmetry and supergravity , 1992, Princeton, USA.[43] V. S. Kaplunovsky and J. Louis, Phys. Lett. B (1993), 269, hep-th/9303040 .[44] P. Ramond,
Group theory: A physicist’s survey , 2010.[45] R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. (1980), 912.[46] T. W. B. Kibble, G. Lazarides, and Q. Shafi, Phys. Rev. D26 (1982), 435.[47] D. Chang, R. N. Mohapatra, and M. K. Parida, Phys. Rev. Lett. (1984), 1072.[48] S. Biermann, A. M¨utter, E. Parr, M. Ratz, and P. K. S. Vaudrevange, Phys. Rev. D (2019),no. 6, 066030, arXiv:1906.10276 [hep-ph].[49] P. Di Vecchia, A. Liccardo, R. Marotta, and F. Pezzella, JHEP (2009), 029, arXiv:0810.5509 [hep-th].[50] S. A. Abel and A. W. Owen, Nucl. Phys. B682 (2004), 183, hep-th/0310257 .[51] L. E. Ib´a˜nez and A. M. Uranga,
String theory and particle physics: An introduction to stringphenomenology , Cambridge University Press, 2 2012.[52] L. E. Ib´a˜nez and D. L¨ust, Nucl. Phys.
B382 (1992), 305, arXiv:hep-th/9202046 [hep-th].[53] K. R. Dienes, Nucl. Phys. B (1994), 533, hep-th/9402006hep-th/9402006