Modular symmetry anomaly in magnetic flux compactification

We study modular symmetry anomalies in four-dimensional low-energy effective field theory derived from six-dimensional super $U(N)$ Yang-Mills theory with magnetic flux compactification. The gauge symmetry $U(N)$ breaks to $U(N_a) \times U(N_b)$ by magnetic fluxes. It is found that Abelian subgroup of the modular symmetry corresponding to discrete part of $U(1)$ can be anomalous, but other elements independent of $U(1)$ in the modular symmetry are always anomaly-free.


I. INTRODUCTION
The modular symmetry is a geometrical feature, which torus compactfication as well as orbifold compactification has. Furthermore, the modular symmetry plays an important role in four-dimensional (4D) low-energy effective field theory derived from higher dimensional field theory and superstring theory.
The modular symmetry in string-induced supergravity theory was studied in Ref. [1] and also its anomaly was studied in Ref. [2,3]. (See also for anomalies in explicit heterotic orbifold models Ref. [4].) Recently, these studies were extended to supergravity theory induced by magnetized and intersecting D-brane models [5]. Furthermore, their anomalies are also interesting from the phenomenological viewpoint [3,6,7].
Also it was studied how massless modes transform under modular symmetry in heterotic orbifold models [8][9][10]. Recently, modular transformation behavior of massless modes was studied in magnetized D-brane models as well as intersecting D-brane models [11][12][13][14]. Then, it was found that the modular symmetry transforms massless modes each other, and that is a sort of flavor symmetries. On the other hand, it was shown that non-Abelaian discrete flavor symmetries appear in heterotic orbifold models [15][16][17][18][19][20] and magnetized/intersecting D-brane models [21][22][23][24][25][26] through analysis independent of modular symmetry. Indeed, a relation between modular symmetry and non-Abelian discrete flavor symmetry was also studied [13].(See also Ref. [27].) Non-Abelian discrete flavor symmetries are interesting from the phenomenological viewpoints [28][29][30]. Various finite groups have been utilized such as S 3 , A 4 , S 4 , A 5 , etc. For 4D field-theoretical model building, many models have been proposed in order to realize quark and lepton masses and their mixing angles and CP phases. The modular group includes S 3 , A 4 , S 4 , A 5 as its finite subgroups [31]. This aspect in addition to the above string compactification inspired a new approach of 4D field-theoretical model building [32], where finite subgroups of the modular symmetry are used as non-Abelian discrete flavor symmetries and also couplings and masses are assumed to transform non-trivially under such finite subgroups. Such a new approach has been applied to models with S 3 , A 4 , S 4 , A 5 modular symmetries [33][34][35][36][37][38][39][40][41].
Thus, the modular symmetry is important from both theoretical and phenomenological viewpoints. In general, continuous and discrete symmetries can be anomalous. (See for anomalies of Abelian and non-Abelian discrete symmetries Refs. [42][43][44][45].) Anomalous symmetries can be broken by non-perturbative effects. That is, breaking terms induced by non-perturbative effects appear in Lagrangian. Such breaking terms may have important implications. The purpose of this paper is to study the anomaly structure of the modular symmetry in 4D low-energy effective field theory derived from magnetic flux compactification of higher dimensional super Yang-Mills theory, which is effective field theory of magnetized D-brane models.
This paper is organized as follows. In Sec. II, we present our setup and give a brief review on magnetic flux compactification and the modular transformation of zero-modes. In Sec. III, we study the anomaly structure of the modular symmetry. Sec. IV is our conclusion. Here, we present our setup and give a brief review on magnetic flux compactification.
We start with six-dimensional U(N) super Yang-Mills theory on two-dimensional torus T 2 , which can be derived from D-brane system. Similarly, we can study higher dimensional theory such as ten-dimensional super Yang-Mills theory on T 2 × T 2 × T 2 , which can also be derived from D-brane system.
We use the complex coordinate z = x 1 + τ x 2 on T 2 , where τ is the complex modular parameter, and x 1 and x 2 are real coordinates. The metric on T 2 is given by We identify z ∼ z + 1 and z ∼ z + τ on T 2 .
We introduce the following magnetic flux along the diagonal direction, where N a + N b = N, I N a,b ×N a,b denotes the (N a,b × N a,b ) identity matrix and M a,b must be integer. This form of magnetic flux corresponds to the vector potential, Because of this gauge background, the U(N) gauge symmetry breaks to U(N a ) × U(N b ). Now let us study the gaugino sector. On T 2 , the spinor has two components, λ ± . They Here λ aa and λ bb correspond to the gaugino fields of unbroken gauge groups, U(N a ) and The zero-mode equation with the above gauge background (3), has chiral solutions. When M = M a − M b is positive, λ ab + and λ ba − have M degenerate zero-modes, whose profiles are written by [11] with j = 0, 1, · · · , (M − 1), where ϑ denotes the Jacobi theta function, Here, N denotes the normalization factor given by with A = 4π 2 R 2 Im τ . On the other hand, for M is negative, λ ab − and λ ba + have |M| degenerate zero-modes, whose profiles are the same as ψ j,M (z) except replacing M by |M|. Hereafter, we set M to be positive.
Because of the chiral spectrum, U(1) a and U(1) b are anomalous in 4D low-energy effective field theory. For example, both the mixed anomalies, are proportional to M. When we embed this super Yang-Mills theory into D-brane system, such anomalies can be canceled by the Green-Schwarz mechanism. The Green-Schwarz mechanism cancels anomalies by the shift of axionsχ a,b , under U(1) a,b transformation, where α a,b are U(1) a,b gauge transformation parameters [46].
Those axions are eaten by U(1) a,b gauge bosons and then U(1) a,b gauge bosons become massive.
In the next section, we will study the T 2 /Z 2 orbifold background. The zero-mode wavefunctions on T 2 /Z 2 are obtained from the above wavefunctions [47]. The above wavefunctions have the following property: Thus, the T 2 wavefunction with j = 0 is still the Z 2 -even zero-mode on T 2 /Z 2 . Also, when That is, we obtain for j = 0, M/2. For the other, the Z 2 -even and odd zero-modes can be written by When M = even, totally the numbers of Z 2 -even and odd zero-modes are equal to (M/2 + 1) and (M/2 − 1), respectively. When M = odd, the numbers of Z 2 -even and odd zero-modes are equal to ((M − 1)/2 + 1) and ((M − 1)/2), respectively.
The anomalies of U(1) a and U(1) b on the T 2 /Z 2 orbifold, e.g. for the Z 2 -even modes , can be studied in the same way as on the torus. Those anomalies can also be canceled by the Green-Schwarz mechanism.
Under the modular transformation, the modulus τ transforms as This group includes two important generators, S and T , T : τ −→ τ + 1.
Under S, the zero-mode wavefunctions transform as On the other hand, the zero-mode wavefunctions transform as under T . Generically, the T -transformation satisfies [12] T on the zero-modes, ψ j,M . Furthermore, in Ref. [12] it is shown that

III. MODULAR SYMMETRY ANOMALY
Here, we study the modular symmetry anomaly. Anomalies of non-Abelian discrete symmetries were studied in Ref. [45]. Each element of a non-Abelian discrete group, g, generates Abelian discrete symmetry, Z K i.e. g K = 1. Thus, basically anomalies of non-Abelian discrete group are studied by analyzing Abelian discrete anomalies of each element, g. However, states correspond to a multiplet under a non-Abelian discrete symmetry. That is, g is represented by a matrix. Suppose that zero-modes correspond to the (anti-)fundamental representation of SU(N b ). Then, if det g = 1, the mixed Z K −SU(N b ) 2 anomaly vanishes. Otherwise, the Z K symmetry generated by g can be anomalous. Furthermore, suppose that zero-modes correspond to the bi-fundamental representation (N a ,N b ) under SU(N a ) × SU(N b ). Then, if det g Na = 1, the mixed Z K − SU(N b ) 2 anomaly vanishes. Otherwise, the Z K symmetry generated by g is anomalous. Hence, the quantity det g is important to examine whether or not the corresponding discrete symmetry can be anomalous.
A. T 2 /Z 2 orbifold As mentioned above, the orbifold basis is more fundamental. Thus, we first study anomalies due to the Z 2 -even modes on the T 2 /Z 2 orbifold. Here, we study anomalies by examining det g for smaller M concretely.

M = 2
Here, we study the modular symmetry for M = 2. Note that the zero-modes on T 2 are the same as the Z 2 -even zero-modes on T 2 /Z 2 . First, we study diagonal elements, T and (ST ) 3 . Their explicit forms are written as where we have omitted vanishing off-diagonal entries. That is the Z 4 × Z 8 symmetry, and they satisfy det T (2) = 1 and det(S (2) T (2) ) 3 = 1. Thus, both symmetries can be anomalous.
However, their combination, has det T ′ (2) = 1 and is always anomaly-free. This is the Z 8 symmetry. Hence, the Z 4 × Z 8 symmetry can be broken to Z 8 by anomalies. The generator of the could-be anomalous symmetry is A (2) = (S (2) T (2) ) 3 . Note that (A (2) ) 4 = (T ′ (2) ) 4 . It is obvious that A (2) is commutable with any element. Therefore, at least the elements (A (2) ) k g (k = 1, 2, 3) with det g = 1 has det((A (2) ) k g) = 1 and can be anomalous among all of the elements, which are generated by S (2) and T (2) . Indeed, by explicit calculation it is found that the order of the full group generated by S (2) and T (2) is equal to 192, and among them the number of elements with det g = 1 is equal to 48. Thus, all of the elements with det h = 1 can be written by h = (A (2) ) k g (k = 1, 2, 3) with det g = 1. That is, only the element A (2) is important for anomalies.
The element A (2) can be anomalous. For example, it can lead to the mixing anomalies with SU(N a ) and SU(N b ). However, it is remarkable that the element A (2) corresponds to a subgroup of U(1) a as well as U(1) b . Thus, when we embed this system to D-brane models, the discrete anomaly corresponding to A (2) can also be canceled by the same Green-Schwarz mechanism as one for U(1) a and U(1) b . The other discrete parts independent of A (2) are always anomaly-free.
As mentioned in the previous section, in the Green-Schwarz mechanism the axion χ shifts under the U(1) transformation to cancel anomalies. Such an axion is the pure imaginary part of a complex field, the so-called modulus field U, where anxionic shift (9) leads to U → U + iα under the U(1) gauge transformation with the transformation parameter α. It implies that e −cU transforms linearly and it behaves to have the U(1) "charge" −c. Nonperturbative effects such as D-brane instanton effects induce new terms e −cU φ 1 φ 2 · · · in 4D low-energy effective field theory. Such terms are invariant under the anomalous U(1) and discrete symmetry with taking into account the transformation of e −cU . However, when we replace U by its vacuum expectation value, such terms correspond to breaking terms. Thus, breaking terms for anomalous symmetries appear. Similar breaking terms would also appear by field-theoretical instanton effects even if we do not take string non-perturbative effects into account.

M = 4
Similarly, we study the orbifold model with M = 4, in particular the Z 2 -even modes.
They correspond to the Z 8 ×Z 8 symmetry. We find that det T (4)+ = 1 and det(S (4)+ T (4)+ ) 3 = 1. They can be anomalous. However, their combination, has det T ′ (4)+ = 1, and is always anomaly-free. This is the Z 8 symmetry. The Z 8 ×Z 8 symmetry can be broken to Z 8 by anomalies. The generators of the could-be anomalous symmetry is A (4) = (S (4)+ T (4)+ ) 3 = e πi/4 I 3×3 , again, and this is commutable with any element. At least the elements (A (4)+ ) k g (k = 1, · · · , 7) with det g = 1 has det((A (4)+ ) k g) = 1 and can be anomalous among all of the elements, which are generated by S (4)+ and T (4)+ . Indeed, by explicit calculation we find that the order of the full group generated by S (4)+ and T (4)+ is equal to 768, and among them the number of elements with det g = 1 is equal to 96. Thus, all of the elements with det h = 1 can be written by h = (A (4)+ ) k g (k = 1, · · · , 7) with det g = 1.
The could-be anomalous element A (4)+ is a sub-element of U(1) a as well as U(1) b . Thus, anomalies originated from A (4)+ can be canceled by the Green-Schwarz mechanism when we embed this system in D-brane models.

M = 6
Similarly, we study the orbifold model with M = 6, in particular the Z 2 -even modes. The diagonal elements, T and (ST ) 3 , are explicitly written by where det(S (6)+ T (6)+ ) 3 = −1. They correspond to the Z 12 × Z 8 symmetry. They can be anomalous. By their combinations, we can construct the diagonal elements with det g = 1 such as etc. They include T k (6)+ only for k = 3k ′ with k ′ = integer, but the elements g including T k (6)+ for k = 3k ′ + 1 and k = 3k ′ + 2 have det g = 1 and can be anomalous. The order of the above group with det g = 1 in the Z 12 × Z 8 symmetry is equal to 16. Thus, its order reduces by the factor 1/6. Indeed, the order of the full group generated by S (6)+ and T (6)+ is equal to 2304, and among them the number of elements with det g = 1 is equal to 384. That is, the order reduces by the factor 1/6. Here, it seems that the group elements including T k Similarly, we can discuss T 2 models. The zero-modes of T 2 are combinations of Z 2 -even and odd modes on T 2 /Z 2 orbifold. ForM = 2, all of the zero-modes on T 2 are the Z 2 -even zero-modes. Thus, S and T are represented by S (2) and T (2) .

IV. CONCLUSION
We have studied the modular symmetry anomalies in magnetic flux compactifiction. Our model is six-dimensional super U(N) Yang-Mills theory, where U(N) gauge symmetry is broken down to U(N a ) × U(N b ) by magnetic fluxes in the compact space. Discrete subsymmetries of U(1) a,b in the modular symmetry can be anomalous, but other discrete elements independent of U(1) a,b are always anomaly-free. Anomalies of such discrete symmetries can be canceled by the Green-Schwarz mechanism when we embed our theory into D-brane system. As a result, breaking terms can be induced only for continuous and discrete U(1) a,b symmetries.
Here we have studied super U(N) Yang-Mills theory, which can be derived from D-brane models. Similar representations of S and T were derived in heterotic orbifold models [8][9][10].
It is interesting to carry out a similar analysis on heterotic orbifold models.