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.
T-duality in string theory relates a theory with the compact space size to another theory with the size . Thus, T-duality is a quite non-trivial symmetry in string theory. Indeed, one type of superstring theory is related to a different type of superstring theory by T-duality. (For a review, see Ref. .)
T-duality also has 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 a certain compactification is invariant under the modular transformation of the moduli ,
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., Refs. [7,8]). In addition, the modular invariant potential of the modulus was studied for cosmic inflation . Thus, the modular symmetry in 4D low-energy effective field theory is important from several viewpoints: theoretical, 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 D-brane models. Their 4D low-energy effective field theories have been studied before (for a review, see Refs. [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 Ref. . This is very useful for our purpose. We extend such analysis to show modular transformation of 4D low-energy effective field theory including three-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 Sect. 2, 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 Sect. 3, we study supergravity theory derived from type IIA intersecting D-brane models. In particular, we investigate the anomaly structure of the modular symmetry. In Sect. 4, we study the modular symmetry of non-perturbative terms induced by D-brane instanton effects. Section 5 provides the conclusions.
2. Modular symmetry
Here, we study the modular symmetry in the 4D low-energy effective field theory derived from type IIB magnetized D-brane models.
2.1. Magnetized D-brane models
2.1.1. Yukawa couplings
Here, we review the analysis of Yukawa couplings in Ref. . Our setup includes several stacks of D9-branes with magnetic fluxes. We assume that our setup preserves 4D supersymmetry. Among several D-branes, we consider two stacks of and D9-branes, which correspond to the gauge symmetry. We put magnetic fluxes, and on these D-branes along and directions of and . The magnetic fluxes must be quantized as in the complex basis. For simplicity, we do not include Wilson lines here .
The open strings between these magnetized branes have massless modes. There appear zero-modes on the th 2-torus, where , and the total number of massless modes is given by their product, . Their zero-mode profiles on the th 2-torus are written by 
Note that the zero-mode profiles of bosonic and fermionic modes are the same in supersymmetric models. For , the zero-mode profiles are obtained by .
In addition to the above two stacks of D-branes, we consider another stack of D9-branes. Then, there appear three types of massless modes, , , and modes. Their Yukawa couplings among canonically normalized fields can be obtained by overlap integral of wavefunctions,
2.1.2. Modular symmetry
Now, let us study the modular transformation of the complex structure moduli . Recall that we use the basis, so that the fields are normalized canonically. Thus, we just investigate the modular transformation of the Yukawa couplings. The modular transformation (1) is generated by the two generators, and ,
2.1.3. Higher-order couplings
We can study higher-order couplings in a similar way . For example, the four-point coupling can be obtained by computing the integral of zero-mode profiles,
Similarly, we can compute generic -point couplings , whose dependence as well as dependence appears in the form
Similarly, we can study the orientifold and orbifold compactifications. For example, the zero-mode profiles on the orbifold can be written by linear combinations of zero-mode profiles on the torus ,
3. Supergravity and anomaly
In this section, we study modular symmetry within the framework of string-derived supergravity and investigate its anomaly.
3.1. Intersecting D-brane models
In the previous section, we studied modular symmetry in 4D low-energy effective field theory of magnetized D-brane models for canonically 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, the Kähler metric of matter fields was computed [10,15–18]. 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. and , intersect each other at an angle on the th 2-torus.
We take the T-dual of the Yukawa coupling (13) of the magnetized D9-brane models, and then we can write the Yukawa coupling of intersecting D-brane models:
In the previous section, the modular symmetry in the supergravity basis was studied. The chiral multiplet in the sector has the Kähler metric (34). Thus, the chiral multiplet, , transforms as
These couplings induce the anomaly of modular symmetry. Its anomaly coefficient of mixed anomaly with the gauge group is written by 
This anomaly can be canceled in two ways [4,5]. One is moduli-dependent threshold corrections and the other is the generalized Green–Schwarz mechanism. The latter would lead to mixing of moduli, e.g. in the Kähler potential. In order to see this, we first review briefly anomalous and the Green–Schwarz mechanism in the next subsection [10,11,19].
First, let us consider the -branes wrapping the 3-cycle , whose wrapping numbers are along . We introduce the basis of 3-cycles, and with , such that is along for all of , while is along only for and for the others. We also introduce their duals such that . These D6-branes correspond to the gauge group, and its gauge kinetic function is written by
The Green–Schwarz mechanism is the same in the toroidal, orientifold, and orbifold compactifications.
3.2.2. Anomaly cancelation of modular symmetry
As mentioned above, the modular anomaly can be canceled in two ways [4,5]. One is moduli-dependent threshold corrections and the other is the generalized Green–Schwarz mechanism. In general, the gauge kinetic function has one-loop threshold corrections due to massive modes as
The transformation (54) implies that the Kähler potential is not invariant under the modular transformation. The Kähler potential must be modified as
Here, we return to the type IIB model studied in Sect. 2. Similar to the above, we may need to replace
4. D-brane instanton effects
In Sect. 2, we studied the modular symmetry of perturbative terms in the 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 illustrative example, and then we will discuss generic aspects.
In this subsection, we study a Majorana mass term induced by an E5-brane in type IIB magnetized orientifold models with O9-planes compactified on a torus. In these models, the non-perturbative corrections to the superpotential are written as [22,23]3
4.2. 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\break D-brane models. Moreover, this example suggests a generic aspect. The D-brane instantons induce the non-perturbative terms such as
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 the 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 cancelation would require moduli mixing correction terms in low-energy effective field theory. Thus, the modular symmetry is important in understanding the 4D low-energy effective field theory of superstring theory.
T. K. and S. U. are supported in part by Grants-in-Aid for Scientific Research No. 26247042 and No. 15J02107 from the Ministry of Education, Culture, Sports, Science, and Technology in Japan.
Open Access funding: SCOAP3.