Modular symmetry in magnetized/intersecting D-brane models

We study the modular symmetry in four-dimensional low-energy effective field theory, which is derived from type IIB magnetized D-brane models and type IIA intersecting D-brane models. We analyze modular symmetric behaviors of perturbative terms and non-perturbative terms induced by D-brane instanton effects. Anomalies are also investigated and such an analysis on anomalies suggests corrections in effective field theory.


I. INTRODUCTION
T-duality in string theory relates a theory with the compact space size R to another theory with the size 1/R. Thus, T-duality is a quite non-trivial symmetry in string theory.
Indeed, one type of superstring theory is related to different type of superstring theory by T-duality. (See for review [1].) T-duality has also a remnant in four-dimensional (4D) low energy effective field theory derived from superstring theory. In particular, 4D low energy effective field theory of heterotic string theory with certain compactification is invariant under the modular transformation of the moduli τ , with ad − bc = 1 and a, b, c, d ∈ Z, at least at the perturbative level. This is the symmetry inside a 4D effective field theory, but not between two theories. We refer this symmetry inside one effective field theory as the modular symmetry in order to distinguish this symmetry from the T-duality between two theories.
The modular symmetry plays an important role in studies on 4D low energy effective field theory of heterotic string theory. For example, moduli stabilization and supersymmetry breaking were studied with the assumption that non-perturbative effects are also modular invariant [2,3]. Moreover, anomalies of this symmetry were analyzed [4,5]. The anomaly structure in heterotic string theory has a definite structure. 1 Their phenomenological applications were also studied (see e.g. [7,8]). In addition, modular invariant potential of the modulus was studied for cosmic inflation [9]. Thus, the modular symmetry in 4D low energy effective field theory is important from several viewpoints such as theoretical one, particle physics and cosmology.
In this paper, we study the modular symmetry in 4D low-energy effective field theory derived from type II superstring theory. In particular, we consider the 4D low-energy effective field theory derived from type IIB magnetized D-brane models and type IIA intersecting Dbrane models. Their 4D low-energy effective field theories have been studied (see for review [10,11]). We study the modular symmetry at perturbative level in their low-energy effective field theories. The T-duality of Yukawa couplings between magnetized D-brane models and intersecting D-brane models was studied in [12]. That is very useful to our purpose. We extend such analysis to show modular transformation of 4D low-energy effective field theory including 3-point and higher order couplings. Also, their anomalies are examined and the anomaly structure could provide non-trivial information like those in heterotic string theory.
Furthermore, we discuss non-perturbative effects.
The paper is organized as follows. In section II, we study the modular symmetry of Yukawa couplings and higher order couplings at the perturbative level in the 4D low-energy effective field theory derived from type IIB magnetized D-brane models. In section III, we study supergravity theory derived from type IIA intersecting D-brane models. In particular, we investigate the anomaly structure of the modular symmetry. In section IV, we study the modular symmetry of non-perturbative terms induced by D-brane instanton effects. Section V is conclusion.

II. MODULAR SYMMETRY
Here, we study the modular symmetry in the 4D low-energy effective field theory derived from type IIB magnetized D-brane models.

A. Magnetized D-brane models
We start with magnetized D9-brane models in type IIB theory. We compactify six dimensional (6D) space to the 6D torus, e.g. three 2-tori. The metric of the r-th 2-torus for r = 1, 2, 3 is written by on the real basis (x r , y r ), where τ r denotes the complex structure modulus. We denote the volume of the r-th 2-torus by A r = R 2 r Imτ r . We use the complex coordinate z r = x r + τ r y r .

Yukawa couplings
Here, we review analysis on Yukawa couplings in [12]. Our setup includes several stacks of D9-branes with magnetic fluxes. We assume that our setup preserves 4D N=1 supersymmetry. Among several D-branes, we consider two stacks of N a and N b D9-branes, which correspond to the U(N a )×U(N b ) gauge symmetry. We put magnetic fluxes, F a r (= F a zrzr ) and F b r (= F b zrzr ) on these D-branes along U(1) a and U(1) b directions of U(N a ) = U(1) a ×SU(N a ) and U(N b ) = U(1) b × SU(N b ). The magnetic fluxes must be quantized as F a r = πi Imτr m r a in the complex basis. For simplicity, here we do not include Wilson lines [12].
The open strings between these magnetized branes have massless modes. There appear I r ab zero-modes on the r-th 2-torus, where I r ab = m r a − m r b , and the total number of massless modes is given by their product, I ab = 3 r=1 I r ab . Their zero-mode profiles on the r-th 2-torus are written by [12] ψ j,N (τ r , z r ) = N r · e iπN zrImzr/Im τr · ϑ for N = I r ab > 0, where j denotes the zero-mode index for j = 1, · · · , N (mod N), and N r is the normalization factor given by The ϑ-function is defined as These zero-modes are also written by another basis, These bases are related as Note that the zero-mode profiles of bosonic and fermionic modes are the same in supersymmetric models. For N = I r ab < 0, the zero-mode profiles are obtained by ψ j,N (τ r , z r ) * . In addition to the above two stacks of D-branes, we consider another stack of N c D9branes. Then, there appear three types of massless modes, a − b, b − c, and c − a modes.
Their Yukawa couplings among cannonically normalized fields can be obtained by overlap integral of wavefunctions, y ijk = C abc e φ 10 /2 3 r=1 dz r dz r ψ i,I r ab (z r ) · ψ j,I r ca (z r ) · ψ k,I r cb (z r ) * , where C abc is the moduli-independent coefficient and φ 10 denotes the ten-dimensional dilaton.
Here, we set I r ab + I r ca = −I r bc = I r cb , because of gauge invariance. To be exact, we should replace the zero-mode indexes i, j, k by i r , j r , k r . However, we denote them as i, j, k to simplify the equations. Hereafter, we use a similar simplification. In this computation, the following relation of zero-mode profiles, is very useful. Then, the Yukawa coupling is written by [12] y ijk = C abc e φ 10 /2 where Similarly, the Yukawa couplings can be written in the basis χ, It would be convenient to use the 4D dilaton, and we defineĨ r = I r /A r . Then, we can write the Yukawa coupling

Modular symmetry
Now, let us study the modular transformation of the complex structure moduli τ r . Recall that we use the basis, that the fields are normalized cannonically. Thus, we just investigate the modular transformation of the Yukawa couplings. The modular transformation (1) is generated by the two generators, s and t, The modular function satisfies where n is called its modular weight. It is obvious that Imτ is invariant under t. Under s, The ϑ-function ϑ a b (0, τ ) is the modular function with the modular weight 1/2.
The ϑ-function part in the Yukawa coupling is transformed under s : Furthermore, using the Poisson resummation formula, we find Thus, the τ dependent part in the Yukawa coupling transforms under s This is nothing but the τ dependent part of the Yukawa coupling in the χ basis. Therefore, the Yukawa coupling terms in 4D low energy effective field theory are invariant under modular transformation including basis change.
The above results can be extended to the magnetic flux, by replacing I ab as I ab = n b m a − n a m b .

Higher order couplings
We can study higher order couplings in a similar way [13]. For example, the 4-point coupling can be obtained by computing integral of zero-mode profiles, We use the relation (9), and then we obtain [13] where M r = I r ab + I r bc and i + j + k + I r ab m + (I r ab + I r bc )n = ℓ + kI ad r with a certain integer n. Similarly, we can compute generic n-point couplings [13], whose τ dependence as well as φ 4 dependence appears in the form, for proper values of δ r i and α r i , because we use the relation (9). Note that the ϑ-function multiplied Im τ −1/4 is invariant under modular transformation. Thus, 4D low-energy effective field theory is invariant at perturbative level under modular transformation of the complex structure moduli, up to change of field basis.
Similarly, we can study the orientifold and orbifold compactifications. For example, the zero-mode profiles on the Z 2 orbifold can be written by linear combinations of zero-mode profiles on the torus [14], Thus, the Yukawa couplings on the orbifold as well as higher order couplings can be written by linear combinations of Yukawa couplings on the torus [14]. Then, the Yukawa couplings on the orbifold are also modular invariant in the same way as those on the torus. Furthermore, the modular symmetry in magnetized D5 and D7-brane models can be studied in a similar way.

III. SUPERGRAVITY AND ANOMALY
In this section, we study modular symmetry within the framework of string-derived supergravity and investigate its anomaly.

A. Intersecting D-brane models
In the previous section, we have studied modular symmetry in 4D low energy effective field theory of magnetized D-brane models for cannonically normalized fields. Here, we study type IIA intersecting D-brane models, which are T-dual to magnetized D-brane models. In intersecting D-brane models, Kähler metric of matter fields was computed [10,[15][16][17][18]. In this section, we study the modular symmetry from the viewpoint of supergravity derived from intersecting D-brane models. In particular, we study intersecting D6-brane models, where two sets of D6-branes, e.g. D6 a and D6 b , intersect each other at the angle πθ r ab on the r-th 2-torus.
First, we write the supergravity fields in type IIB theory as where the imaginary parts of S and T r correspond to certain axion fields. Their Kähler potential is written by We take T-dual along x r direction on each 2-torus from magnetized D9-branes to intersecting D6-branes. Then, we replace We have seen that low-energy effective field theory of cannonically normalized fields is modular symmetric for τ r in type IIB magnetized D-brane models . Thus, the low-energy effective field theory of type IIA intersecting D-brane models must have the symmetry under the modular transformation, both in canonically normalized field basis and in supergravity basis.
We take the T-dual of the Yukawa coupling (13) of magnetized D9-brane models, and then we can write the Yukawa coupling of intersecting D-brane models, where Within the framework of supergravity, physical Yukawa couplings are written by where W ijk denotes the holomorphic Yukawa coupling in the superpotential, i.e., K is the Kähler potential, and K ab , K bc , K ca are the Kähler metric of the ab, bc, ca sectors, respectively. Then, the relation (31) requires that The Kähler metric of matter fields was computed [10,[15][16][17][18]. The Kähler metric of the ab sector would be written as For example, in Refs. [16][17][18], the following Ansatz, was discussed by comparing the holomorphic and physical gauge couplings and threshold corrections. They satisfy the above relation (33) when sign(I ab )θ r ab + sign(I bc )θ r bc + sign(I ca )θ r ca = 0 .
Similarly, the n-point couplings in magnetized D-brane models include the τ dependent factor (23). Then, its T-dual intersecting D-brane models include (2Re T r ) n−2 /4. That requires that the product of the Kähler metric satisfies K a 1 a 2 K a 2 a 3 · · · K ana 1 = r (T r +T r ) −n/2 .
We can take the T-dual of type IIA intersecting D-brane models along the y r direction, type IIB model X ⇐⇒ and then obtain type IIB magnetized D-brane models, which are different from one discussed in the previous section, The relation between these two type IIB models was studied in [12], in particular Yukawa couplings. Our results in the precious section can be understood as such two different theories through double T-duality such as [12], but in any rate we are interested in the modular symmetry in one 4D low-energy effective field theory as mentioned in Introduction.

B. Anomaly
In the previous section, the modular symmetry in the supergravity basis was studied. The chiral multiplet, Φ ab in the ab sector has the Kähler metric (34). Thus, the chiral multiplet, under the modular transformation (28). That is, the matter field has the modular weight ν(θ r ab ) under the modular transformation of the r-th 2-torus. Such a modular transformation may be anomalous. The supergravity Lagrangian includes the following couplings, where λ denotes the gaugino, K ii is the Kähler metric of Φ i with the bosonic and fermionic components, φ i and ψ i , These couplings induce the anomaly of modular symmetry. Its anomaly coefficient of mixed anomaly with the SU(N a ) gauge group is written by [4] A r a = −C 2 (G a ) + matter,b T (R a )(1 + 2ν(θ r ab )), where C 2 (G a ) is the quadratic Casimir and T (R a ) is the Dynkin index of the representation R a . For simplicity, we consider the intersecting D-brane models on torus. In this case, we can write This anomaly can be cancelled by two ways [4,5]. One is moduli dependent threshold corrections and another is generalized Green-Schwarz mechanism. The latter would lead to mixing of moduli, e.g. in Kähler potential. In order to see it, we first review briefly on anomalous U(1) and Green-Schwarz mechanism in the next subsection [10,11,19].

Anomalous U(1)
First, let us consider the D6 b -branes wrapping the 3-cycle [Π b ], whose wrapping numbers are (n r b , m r b ) along (x r , y r ). We introduce the basis of 3-cycles, [α 0 ] and [α k ] with k = 1, 2, 3, such that [α 0 ] is along (1, 0) for all of (x r , y r ), while [α k ] is along (1, 0) only for r = k and (0, 1) for the others. We also introduce their duals [β k ] such that [α i ] · [β k ] = δ ik . These D6-branes correspond to U(N b ) gauge group, and its gauge kinetic function f b is written by where where i = j = k = i. Now, we study the U(1) a − SU(N b ) 2 mixed anomaly. Its anomaly coefficient can be written by where This anomaly can be cancelled by the shift of moduli, in the gauge kinetic function f b under the U(1) transformation, This means that the Kähler potential is not invariant, but the following Kähler potential is invariant, The Green-Schwarz mechanism is the same in the toroidal, orientifold and orbifold compactifications.

Anomaly cancellation of modular symmetry
As mentioned above, the modular anomaly can be canceled by two ways [4,5]. One is moduli dependent threshold corrections and another is generalized Green-Schwarz mechanism. In general, the gauge kinetic function has one-loop threshold corrections due to massive modes as where the first term in the RHS corresponds to Eq.(45). The threshold corrections are computed explicitly [17,18,20,21] and its typical form is whereb is beta-function coefficient due to massive modes, and η(iT ) is the Dedekind eta function. The Dedekind eta function has the modular weight 1/2. This threshold correction can cancel the anomaly partially. The other part of anomaly can be canceled by the generalized Green-Schwarz mechanism, where we impose the following transformation under the modular transformation (28). That is, the generalized Green-Schwarz mechanism could cancel the anomaly proportional to By comparison with the total anomaly as well as the U(1) anomaly, a plausible Ansatz would be, where c is constant. In this case, the coefficientb may be obtained to cancel the modular anomaly. Indeed, the threshold correction, was discussed in [17,18].
The transformation (54) implies that Kähler potential is not invariant under the modular transformation. The Kähler potential must be modified as That is, the moduli mix, and instead of S and U i , the linear combinations, must appear in 4D low-energy effective field theory. Similar linear combinations were discussed in [18], although linear combinations in [18] include mixture of all the moduli. 2 Here, we return back to the type IIB model studied in section II . Similar to the above, we may need to replace, in 4D low-energy effective field theory. For example, the 4D dilaton factor in the Yukawa coupling would be modified as

IV. D-BRANE INSTANTON EFFECTS
In section II, we studied the modular symmetry of perturbative terms in Lagrangian. In this section, we study terms due to non-perturbative effects, in particular terms induced by D-brane instanton effects. First, we study an illustrating example, and then we will discuss generic aspects.

A. Example
In this subsection, we study a Majorana mass term induced by a E5-brane in Type IIB magnetized orientifold models with O9-planes compactified on Z 2 × Z ′ 2 torus . In these models, the non-perturbative corrections to superpotential are written as [22,23] In (63), α i denotes a fermionic zero-mode of the E5-brane and S denotes the classical action of E5-brane. S int denotes interaction terms including fermionic zero-modes as where y i 1 ···in,j 1 ···jm is a (n + m)-point coupling and Φ j is the chiral superfield of the models.
Then, we can obtain a Majorana mass term if there are two fermionic zero-modes and 3-point couplings like y ijk α i β j Φ k . The Majorana mass is generated as In this subsection, we concentrate on the rth 2-dimensional torus with two D-branes wrapping whole compact space for simplicity. We put the magnetic fluxes Im τ πi F a r = 2 on one D-brane and Im τ πi F b r = −2 on the other D-brane. For simplicity, all Wilson lines are set to zero in this subsection too. Then, there are three chiral fermions between these two branes.
These modes are given by the linear combinations of the wave functions on the covering torus ψ i , where i ∈ {0, 1, 2, 3}. The three zero-modes on the orbifold are given by Eq.(24) [14]. That is, two of them, Φ 0 and Φ 2 correspond to ψ 0 and ψ 2 , respectively, while Φ 1 is given by In addition, a E5-brane with no magnetic flux induces two zero-modes between the E-brane and the D-branes. These zero-modes are given by Then, Yukawa couplings are written by

B. Generic discussion
The example in the previous subsection shows the modular symmetry of non-perturbative terms induced by D-brane instanton effects for the complex structure moduli in type IIB magnetized D-brane models. Moreover, this example suggests a generic aspect. The D-brane instantons induce the non-perturbative terms such as where C is a moduli-independent coefficient 4 . Here, V ol(E5) denotes the volume of D-brane instanton in the compact space, and it depends only on A r , but not τ . Furthermore, y (n) denote the couplings among zero-modes and 4D fields Φ i , and these are computed in the same way as perturbative couplings shown in section II. The τ dependence appears only through these couplings y (n) . Therefore, terms induced by D-brane instanton effects are also modular symmetric.
In this section, we have not taken into account the moduli mixing so far. However, the discussion in section III would lead to modification such as (61).

V. CONCLUSION
We have studied the 4D low-energy effective field theory, which is derived from type IIB magnetized D-brane models and type IIA intersecting D-brane models. We have studied modular symmetric behavior of perturbative terms. Also, such analysis has been extended to non-perturbative terms induced by D-brane instanton effects. We have also investigated the anomaly of the modular symmetry. Its cancellation would require moduli mixing correction terms in low-energy effective field theory. Thus, the modular symmetry is important to understand the 4D low-energy effective field theory of superstring theory.