Defect (p,q) Five-branes

We study a local description of composite five-branes of codimension two. The formulation is constructed by virtue of $SL(2,{\mathbb Z}) \times SL(2,{\mathbb Z)}$ monodromy associated with two-torus. Applying conjugate monodromy transformations to the complex structures of the two-torus, we obtain a field configuration of a defect $(p,q)$ five-brane. This is a composite state of $p$ defect NS5-branes and $q$ exotic $5^2_2$-branes. We also obtain a new example of hyper-K\"{a}hler geometry. This is an ALG space, a generalization of an ALF space which asymptotically has a tri-holomorphic two-torus action. This geometry appears in the conjugate configuration of a single defect KK5-brane.


Introduction
A Neveu-Schwarz five-brane, called an NS5-brane for short, plays a significant role in string theory. This is a soliton coupled to the B-field magnetically in ten-dimensional spacetime, whereas the fundamental string is coupled to the B-field electrically [1,2]. The setup of two parallel NS5-branes with various D-branes attached with them is quite an important configuration to explore dualities among supersymmetric gauge theories [3,4]. The NS5-brane is uplifted to an M5-brane in Mtheory, which plays a central role in studying non-perturbative features of gauge theories in lower dimensions [5]. Applying T-duality to the NS5-brane, a Kaluza-Klein monopole [6], or referred to as a KK5-brane, emerges. If one performs another T-duality to the KK5-brane, one finds an exotic 5 2 2 -brane [7]. This is a strange object whose background geometry is no longer single-valued. However, this strange object does also contribute to quantum aspects of spacetime [8].
NS5-branes and KK5-branes have been investigated from various viewpoints [9]. In particular, in order to analyze quantum stringy corrections to five-branes, the worldsheet approach to fivebranes [2] has been developed in terms of two-dimensional supersymmetric gauge theory, called the gauged linear sigma model (GLSM) [10,11,12,13]. In the case of the exotic 5 2 2 -brane, the situation is different. The background geometry is written by multi-valued function because the exotic 5 2 2 -brane is codimension two. These days, branes of codimension two are referred to as defect branes [14]. The exotic 5 2 2 -brane is the typical example of defect five-branes. Indeed, it was difficult to construct both the worldsheet theory and worldvolume theory for the exotic 5 2 2 -brane. However, there was a breakthrough in this topic. The exotic 5 2 2 -brane in the GLSM framework was successfully obtained in [15]. This formulation enabled us to study the quantum aspects of the exotic 5 2 2 -brane [16] in the same way as the NS5-brane and the KK5-brane [11,12,13]. The worldvolume theory for the exotic 5 2 2 -brane was also constructed [17,18] by following the work [19]. In the analyses of five-branes, people often encounter configurations of many number of them. A typical example is the defect (p, q) five-branes. This is the bound state of p defect NS5-branes and q exotic 5 2 2 -branes [7,8]. This is one of the most significant situation to formulate the globally welldefined description of defect five-branes. This resembles to the (p, q) seven-branes in type IIB theory [20,21,22,23]. Unfortunately, the globally well-defined description of defect (p, q) five-branes is still lacking, though the global configuration of the (p, q) seven-branes has been established. So far, when people argue the defect (p, q) five-branes, they have applied string dualities to the description of the (p, q) seven-branes in type IIB theory [24].
It is quite an important task to construct the globally well-defined description of defect (p, q) five-branes. In order to complete this, we study the "local" description of such configurations as the first step. In this paper, we exhaustively utilize the monodromy structures of defect five-branes. Applying the aspects of the monodromy to the field configuration, we obtain the explicit form of the defect (p, q) five-branes. Even though the formulation tells us only the local structure of the five-branes, it would be a big step to find the globally well-defined form. In addition, we find a new hyper-Kähler geometry, as a bonus. This is called an ALG space [25,26], a generalization of an ALF space which asymptotically has a tri-holomorphic two-torus action. This is the conjugate geometry of the single defect KK5-brane.
The structure of this paper is as follows. In section 2, we review standard five-branes and defect five-branes. First, we exhibit their local descriptions. Next, we discuss the O(2, 2; Z) monodromies of the defect five-branes and mention a nongeometric feature. In section 3, we further study the monodromies of the defect five-branes by virtue of the equivalence O(2, 2; Z) = SL(2, Z)×SL(2, Z). We introduce two complex structures associated with two SL(2, Z). They are the key ingredients to analyze composites of the defect five-branes. In section 4, we investigate conjugate monodromies and construct their corresponding configurations. We find the local descriptions of the conjugate configurations as composites of defect five-branes. In particular, we obtain the formulation of the defect (p, q) five-branes. This is the composite of p defect NS5-branes and q exotic 5 2 2 -branes. We also obtain the conjugate configuration of the defect KK5-branes. This provides a new hyper-Kähler geometry as an ALG space. Section 5 is devoted to summary and discussions. In appendix A, we prepare the T-duality transformation rules applied to the field configurations and the monodromy matrices. In appendix B, we discuss another defect KK5-brane which is different from the reduction of the standard KK5-brane, and analyze its conjugate configurations.

A review of defect five-branes 2.1 Standard five-branes
In this subsection we briefly mention the explicit description of an H-monopole and a KK-monopole [9]. The H-monopole is the smeared version of a single NS5-brane along the 9-th direction, while the KK-monopole is the T-dualized object of the H-monopole along the 9-th direction. These two objects have been well investigated in the framework of GLSM [11,12,13], and doubled formalism [27,28] (see also [8,29]).
We begin with the H-monopole. In ten-dimensional spacetime, we describe the background metric G M N , the B-field B M N and the dilaton φ as Here α ′ is the Regge parameter in string theory. The NS5-brane is expanded in the 012345-directions whose spacetime metric is flat, while the transverse space of the 6789-directions is R 3 × S 1 . The vector x lives in the transverse 678-directions R 3 . This five-brane is smeared along the transverse 9-th compact direction whose radius is R 9 . This configuration is governed by a harmonic function H. The B-field is given by a function V i which is subject to the monopole equation (2.1d), where the index i represents the spatial directions i = 6, 7, 8.
Next, we consider the KK-monopole, or referred to as the KK5-brane. This is obtained via the T-duality transformation (see appendix A) along the smeared direction of the H-monopole (2.1), Due to the T-duality transformation, the B-field in the H-monopole (2.1) is involved into the offdiagonal part of the metric as the KK-vector V . We also see that the dilaton becomes trivial. The transverse space of the 6789-directions becomes the Taub-NUT space, a non-compact hyper-Kähler geometry. In order to emphasize the T-duality transformation along the 9-th direction, we refer to this coordinate as y 9 .

Defect five-branes
In the previous subsection we mentioned two standard five-branes of codimension three. It is interesting to consider five-branes of codimension two, called the defect five-branes [14]. We can easily find defect five-branes from the H-monopole and the KK5-brane if one of the transverse directions is further smeared 1 . One of the most interesting defect five-branes is the exotic 5 2 2 -brane. This has been investigated in the various viewpoints [7,24,8,15].
We first discuss a defect NS5-brane smeared along the 8-th direction of the H-monopole (2.1). The configuration is given as 3c) Here R 8 is the radius of the compact circle along the smeared 8-th direction. We notice that the harmonic function H is reduced to a logarithmic function. Here µ is the renormalization scale and h is the bare quantity which diverges if we go infinitely away from the five-brane. In this sense the representation (2.3) is valid only close to the defect five-brane.
There exist two isometries along the 8-the and 9-th directions of the defect NS5-brane (2.3). Taking the T-duality transformation along the 9-th direction, we obtain a defect KK5-brane, This is also found if the KK5-brane of codimension three (2.2) is smeared along the 8-th direction.
Here the B-field and the dilaton are again trivial.
If we take the T-duality transformation along the 8-th direction instead of the 9-th direction of the defect NS5-brane (2.3), we also find the configuration of another defect KK5-brane of different type. This will be discussed in appendix B.
Performing the T-duality transformation along the 8-th direction of the defect KK5-brane (2.4), we obtain the configuration of the exotic 5 2 2 -brane [7,24,8,15], Here the B-field and the dilaton are non-trivial as in the configuration of the defect NS5-brane (2.3). However, their features are quite different from the ones in (2.3). Indeed, not only the spacetime metric, but also the B-field and the dilaton are no longer single-valued. It is impossible to remove such features by the coordinate transformations or by the B-field gauge transformation. This is the reason why this configuration is the "exotic" five-brane. In the next subsection we capture the exotic structure by virtue of monodromy.

O(2, 2; Z) monodromy
When we go around a defect five-brane along the ϑ coordinate in the 67-plane, we can capture monodromy generated by the two-torus T 89 . The analysis of monodromy is important to investigate the exotic structure of defect five-branes. Now we package the 89-directions of the metric and B-field [31]. The numerator O(2, 2) is related to the T-duality symmetry O(2, 2; Z) on the two-torus T 89 , while the denominator O(2) × O(2) describes the local symmetry related to the coordinate transformations and the B-field gauge transformation. When we go around a five-brane along the coordinate ϑ from 0 to 2π, the matrix M is transformed as The transformation matrix Ω indicates the monodromy of the system. This monodromy takes valued in O(2, 2; Z). We discuss the monodromy matrix more in detail [31,32,33,34,29]. The matrix Ω is described as [34] where A, D, Θ and β are 2 × 2 block matrices. The blocks A and D govern the coordinate transformations, while Θ gives rise to the B-field gauge transformation. If the block β exists nontrivially, the T-duality is involved into the geometrical structure. It is called a T-fold [31]. Such a space is locally geometric but globally nongeometric.
Now we analyze the matrices M and Ω of the defect five-branes. The defect NS5-brane (2.3) has the following matrices, We note that the monodromy matrix Ω NS contains the Θ part, while it does not contain the β part. This is consistent with the configuration (2.3), where the 2π shift of the coordinate ϑ is removed by the B-field gauge transformation. When we move around the defect NS5-brane ϑ = 0 → 2πn with n ∈ Z, the monodromy is given by (Ω NS ) n . This is equal to the monodromy Ω NS whose components ±2πℓ are replaced to ±2πℓn. In the same way, we study the matrices of the defect KK5-brane (2.4), The monodromy matrix Ω KK does not contain the Θ part and the β part. This is also consistent with the configuration (2.4), where the 2π shift of the ϑ coordinate can be eliminated by the coordinate transformations. However, the matrices of the exotic 5 2 2 -brane (2.5) are different from the ones in the above two systems, The monodromy matrix Ω E contains the β part. Indeed, in the configuration (2.5), the 2π shift of the coordinate ϑ cannot be removed by the coordinate transformations and the B-field gauge transformation. This shift is generated by the T-duality symmetry along the two-torus T 89 . Hence we can interpret that the background geometry of the exotic 5 2 2 -brane is a typical example of T-folds [31].
Monodromy is quite useful to investigate the (non)geometric aspects. Furthermore, if we apply the equivalence O(2, 2; Z) = SL(2, Z) × SL(2, Z) to the analysis, we can explore the geometries of defect five-branes in a deeper level. In the next two sections we will carefully analyze the SL(2, Z) × SL(2, Z) monodromy and construct new configurations of defect five-branes.

Two complex structures
The O(2, 2; Z) monodromy is generated by the two-torus T 89 . Let us study the monodromy by the equivalent group SL(2, Z) × SL(2, Z). Each SL(2, Z) should also be governed by the structure of T 89 . Associated with these two SL(2, Z), we introduce two complex structures τ and ρ. τ is the complex structure of the two-torus T 89 , while ρ is defined in terms of the B-field and the metric on T 89 in such a way as [8], In terms of the two complex structures, we can represent the metric G mn and the B-field B mn on the two-torus T 89 , and the dilaton φ, where τ = τ 1 + iτ 2 and ρ = ρ 1 + iρ 2 . Then, instead of the analysis of the matrix M(̺, ϑ), we will investigate the monodromy structures of the two complex structures τ and ρ of the defect five-branes.

Monodromy matrices
Let us first analyze the SL(2, Z) τ × SL(2, Z) ρ monodromy of the defect NS5-brane. Plugging the configuration (2.3) into the formulation (3.1), we can read off the explicit forms of the two complex structures, where we defined the complex coordinate z ≡ ̺ e iϑ in the 67-plane. When we go around the defect NS5-brane z → z e 2πi , the complex structure ρ has the monodromy as ρ → ρ + 2πℓ, whilst τ is invariant. The SL(2, Z) descriptions of the monodromy are It turns out that the two-torus T 89 is not deformed under the monodromy, while the field configuration of B 89 is changed. However, this change can be removed by the B-field gauge transformation. Then the configuration (2.3) is invariant under the monodromy transformation. This is consistent with the previous analysis in terms of Ω NS (2.8).
Next, we discuss the SL(2, Z) τ × SL(2, Z) ρ monodromy of the defect KK5-brane. Substituting the configuration (2.4) into (3.1), the two complex structures are given as Now ρ becomes trivial. Here it is convenient to introduce λ = −1/τ = V ℓ + iH ℓ . Under the shift z → z e 2πi , the complex structure λ is transformed as λ → λ ′ = λ + 2πℓ, while ρ is invariant. Their SL(2, Z) representations are given as follows, This implies that the complex structure of the two-torus T 89 is changed under the monodromy, while the B-field and the determinant of the metric is invariant. However, we can remove the change of the complex structure by the coordinate transformations. This is also consistent with the previous discussion in terms of Ω KK (2.9).
Finally, we study the SL(2, Z) τ × SL(2, Z) ρ monodromy of the exotic 5 2 2 -brane. Applying the configuration (2.5) to the complex structures (3.1), we can read off the following forms, Again the complex structure of the two-torus T 89 becomes trivial. For convenience, we introduce ω ≡ −1/ρ = V ℓ + iH ℓ . Under the shift z → z e 2πi , we see that ω has the monodromy ω → ω ′ = ω + 2πℓ, while τ is invariant. The SL(2, Z) matrix forms of the monodromy are , This behavior implies that the monodromy transformation does not change the complex structure of the two-torus, while the field configuration is changed. Furthermore, caused by the form ρ ′ = B ′ 89 + i det G ′ mn , this change cannot be eliminated completely in terms of the coordinate transformations and the B-field gauge transformation. As discussed in the analysis of Ω E , this is nothing but the aspect of T-fold.
In the next section we will investigate conjugates of the SL(2, Z) τ × SL(2, Z) ρ monodromy. We will find various new configurations of defect five-branes as their composite states.

Conjugate configurations
In type IIB theory, there exists a D7-brane which also has the SL(2, Z) monodromy generated by the combination of the dilaton and the axion [20]. Applying a generic SL(2, Z) transformation to the monodromy, we can find conjugate systems of the D7-brane [21,22]. In the same analogy, the conjugates of the exotic 5 2 2 -brane has been discussed in [24,8]. In this section, we develop the analyses to the conjugate configurations of the defect five-branes. To do this, we prepare a set of generic SL(2, Z) τ × SL(2, Z) ρ matrices, By using U τ,ρ , we construct a set of conjugate monodromy matrices Ω τ,ρ , Simultaneously, we transform the two complex structures τ and ρ in terms of U τ,ρ to new complex structures τ and ρ. Plugging them into (3.1), we can read of new field configurations G mn , B 89 and φ.

Conjugate configuration of defect NS5-brane
First, we investigate conjugate configurations of the defect NS5-brane (2.3). Transforming the original monodromy matrices Ω NS τ and Ω NS ρ (3.3) in terms of the rule (4.1), the conjugate monodromy matrices can be obtained as Here Ω NS τ is again trivial because the original complex structure τ is trivial τ = i. Compared Ω NS ρ with Ω NS ρ (3.3) and Ω E ρ (3.7), it turns out that the conjugate system is a composite of p defect NS5-branes and q exotic 5 2 2 -branes [24,8]. Associated with the transformation rule (4.1), we also arrange the complex structure ρ by means of U −1 ρ , Since the monodromy of the original ρ is ρ → ρ ′ = ρ + 2πℓ under the shift z → z e 2πi , then the new complex structure ρ is transformed as This indicates that the new complex structure ρ reproduces the SL(2, Z) ρ conjugate monodromy Ω NS ρ (4.2). Then it turns out that ρ denotes the conjugate configuration of the defect NS5-brane (2.3).
We would like to construct the local expression of the conjugate configuration. Substituting the conjugate complex structures τ and ρ (4.3) into (3.1), we can read off the field configuration, This is a generic form of the composite system of p defect NS5-branes and q exotic 5 2 2 -branes under the constraint sp − qr = 1. The system of a single defect NS5-brane can be realized by setting (p, q, r, s) = (1, 0, 0, 1), while the system of a single exotic 5 2 2 -brane can be expressed by (p, q, r, s) = (0, 1, −1, 0). In a generic case of non-vanishing p and q, the expression (4.5) is rather lengthy.
In order to reduce the expression of the generic (p, q) configuration, we rewrite the conjugate complex structure ρ, . (4.6) Here we removed r by using sp − qr = 1. Now s is no longer constrained by (p, q), then we set s to zero without loss of generality, For convenience, we further introduce a new expression ω ≡ −1/( ρ + p q ). This is transformed as ω → ω ′ = ω + 2πℓq 2 under the shift z → z e 2πi . Then we can read off the monodromy of ρ = −1/ ω in such a way as It turns out that the reduced complex structure ρ again reproduce the conjugate monodromy Ω NS ρ (4.3). Throughout the above reduction, the other complex structure τ is unchanged. Finally, plugging τ = i and ρ (4.7) into the definition (3.1), we explicitly obtain the defect (p, q) five-branes, i.e, the configuration of p defect NS5-branes and q exotic 5 2 2 -branes, Of course, this configuration satisfies the equations of motion of supergravity theories. Actually, this configuration is desired in the framework of the GLSM [35].
There is a comment. The conjugate complex structure ρ (4.7) contains a constant term −p/q. Compared this with (3.1), we think that this constant might be eliminated by the B-field gauge symmetry. However, if the term −p/q in (4.7) is gauged away, the monodromy is reduced to (Ω E ρ ) q , which no longer represents the composite system of the defect NS5-branes and the exotic 5 2 2 -branes.

Conjugate configuration of defect KK5-brane
Next, we consider the conjugate of the defect KK5-brane (2.4). Applying the transformation rules (4.1) to the monodromies Ω KK τ and Ω KK ρ (3.5), the conjugate monodromy matrices are given as We note that the conjugate monodromy Ω KK ρ is identical with Ω KK ρ because the complex structure ρ is trivial (3.4). Compared Ω KK τ with the monodromy matrices Ω KK τ (3.5) and Ω AK τ (B.5), the conjugate monodromy Ω KK τ denotes that the system is a composite of −s ′ defect KK5-branes (2.4) and r ′ defect KK5-branes of another type (B.1). Let us focus on the complex structure τ . This is also changed in terms of U −1 τ in such a way as Since the original τ is transformed as in (3.5) under the shift z → z e 2πi , the new complex structure τ is transformed, This provides the same conjugate monodromy matrix Ω KK τ (4.10). Plugging τ (4.11) and ρ = i into the definition (3.1), we find Let us find the simple form of the generic (−s ′ , r ′ ), We introduce λ = −1/ τ . Applying the constraint s ′ p ′ − q ′ r ′ = 1 to this, we obtain . (4.14a) Now the parameter p ′ is no longer constrained by the other parameters (−s ′ , r ′ ). Then we can set p ′ = 0 without loss of generality, .
We check the monodromy of the new complex structure τ . It is convenient to introduce ζ ≡ −1/( λ − s ′ r ′ ). Since this is transformed as ζ → ζ ′ = ζ + 2πℓr ′2 under the shift z → z e 2πi , we can immediately read off the transformation of τ = −1/ λ in such a way as This guarantees that the new complex structure τ also generates the conjugate monodromy (4.10).
Plugging τ (4.15) and ρ = i into the definition (3.1), we find the local expression of the metric, the B-field and the dilaton for the composite of −s ′ defect KK5-branes (2.4) and r ′ defect KK5-branes of another type (B.1), This configuration also satisfies the equations of motion of supergravity theories. We note that the transverse space of 6789-directions in (4.17) is Ricci-flat. Since this configuration preserves a half of supersymmetry, the transverse space is also hyper-Kähler. Indeed this belongs to a class of ALG spaces [25,26] 3 .
The conjugate system can also be obtained from the conjugate configuration of the defect NS5-brane (4.9) via the T-duality transformation along the 9-th direction with relabeling (p, q) = (−s ′ , r ′ ). The exchange of the conjugate complex structures (τ, ρ) in (4.9) and (ρ, λ) in (4.17) also occurs.
We notice that the constant term s ′ /r ′ in the conjugate complex structure τ (4.15) should not be eliminated in terms of the coordinate transformations. If we remove s ′ /r ′ , the complex structure produces the monodromy (Ω AK τ ) r ′ (B.5) rather than the conjugate monodromy Ω KK τ (4.10).

Conjugate configuration of exotic 5 2 2 -brane
Finally, we investigate the conjugate system of the exotic 5 2 2 -brane. The SL(2, Z) τ × SL(2, Z) ρ monodromy matrices (3.7) are transformed by using (4.1), Since the complex structure τ (3.6) is trivial, the conjugate monodromy Ω E τ coincides with the original monodromy Ω E τ . On the other hand, the conjugate monodromy Ω E ρ implies that the conjugate system consists of r defect NS5-branes and −s exotic 5 2 2 -branes. In order to obtain the explicit field configuration of the conjugate system, we also transform the complex structure ρ, Recall that the original ρ is transformed as in (3.7) under z → z e 2πi . Then the new complex structure ρ is also transformed as This reproduces the conjugate monodromy Ω E ρ (4.18). The complex structures τ and ρ gives rise to the configuration of the metric, the B-field and the dilaton, This contains the case of the single exotic 5 2 2 -brane (2.5) by setting (p, q, r, s) = (−1, 0, 0, −1) and the case of the single defect NS5-brane by (p, q, r, s) = (0, −1, 1, 0). However, in the case of a generic (r, −s) = (0, 0), the expression (4.21) is cumbersome. Fortunately, as in the previous configurations, we can reduce (4.21). For convenience, let us introduce ω = −1/ ρ, where we used sp − qr = 1 to remove q. Since the parameter p is now arbitrary without any constraints, we can set the following form without loss of generality, . (4.23) We check the monodromy of the new complex structure ρ. For convenience, we define ζ ≡ −1/( ω − s r ). This is transformed as ζ → ζ ′ = ζ + 2πℓr 2 under the shift z → z e 2πi . Then we can read off the monodromy of ρ = −1/ ω in such a way as Thus we confirm that the new complex structure ρ is subject to the conjugate monodromy (4.18). Applying τ = i and ρ (4.23) to the definition (3.1), it turns out that the conjugate configuration is described as ds 2 = ds 2 012345 + H ℓ (d̺) 2 + ̺ 2 (dϑ) 2 + ω 2 | ω| 2 (dy 8 ) 2 + (dy 9 ) 2 , (4.25a) (4.25b) This is the local expression of the defect (r, −s) five-branes, i.e., the composite of r defect NS5branes (2.3) and −s exotic 5 2 2 -branes. This is similar to the previous form (4.9), while the roles of conjugate complex structures are different. Indeed, the configuration (4.25) is generated via the T-duality transformations along the 8-th and 9-th direction of (4.9), with relabeling (p, q) to (−s, r). Simultaneously, the complex structures (τ, ρ) in (4.9) are changed to (τ, ω) in (4.25).

Summary and discussions
In this paper, we studied the SL(2, Z) τ × SL(2, Z) ρ monodromy structures of various defect fivebranes. We also investigated the conjugate configurations of them by virtue of the conjugate monodromy matrices and the corresponding complex structures. Once we found the explicit forms of the complex structures which reproduce the conjugate monodromies, we immediately constructed the field configurations of the conjugate system. In this process we constructed the metric, the Bfield and the dilaton for the defect (p, q) five-branes, i.e., the composite of p defect NS5-branes and q exotic 5 2 2 -branes, in a concrete manner. Since the configuration of the single defect five-brane is not globally well-defined, the expression of the composite system would be quite helpful to find the globally well-defined formulation of defect five-branes, as in the case of seven-branes in type IIB theory. If we find the global description for defect five-branes, we will understand the importance of the exotic 5 2 2 -brane in a deeper level. In this work we also obtained the new hyper-Kähler geometry (4.17) as the conjugate system of the defect KK5-branes. This is an ALG space, a generalization of an ALF space which asymptotically has a tri-holomorphic two-torus T 89 action [25,26]. Since this is originated from the Taub-NUT space via the smearing and the conjugating, we can interpret this as the conjugated defect Taub-NUT space.
There are several discussions. First, all of the conjugated configurations represent the composites of coincident defect five-branes. In order to find a further general configuration where each defect five-brane is located at arbitrary point in the 67-plane, we should introduce new parameters into the system. In the case of multi-centered five-branes of codimension three, We have already known the harmonic function, where q k is the position of the k-th five-brane in the 678-directions. In the case of defect five-branes, however, it is difficult to recognize the sum of the harmonic functions for the single defect five-brane as the one for multiple defect five-branes. This is because the individual harmonic function involves the renormalization scale µ. Then we have to control many number of scale parameters in order that the harmonic function is well-defined in the whole region of the 67-plane. This estimate is too naive to describe a number of separated defect five-branes. In order to acquire the consistent form for such a configuration, we have to find the globally well-defined function as the modular invariant function for seven-branes in type IIB theory. Nevertheless, the local descriptions of composite defect five-branes would be helpful to construct the globally well-defined function.
Apart from the above current difficulty, we can still study other topics. (i) In the previous work [35], we tried to construct the GLSM for two defect five-branes. Since we obtained the explicit configuration for them in the current work, it can be possible to find the improved version of [35]. In particular, it would be quite interesting to construct string worldsheet theories for ALG spaces related to the previous work [15], by virtue of the T-duality transformation rules on the GLSM [36,37]. (ii) In the system of the defect (p, q) five-branes (4.9), the 9-th direction is compactified and smeared. Let us consider the string worldsheet instanton corrections along the 9-th direction. From the viewpoint of the defect NS5-branes, the worldsheet instanton corrections to the 9-th direction can be interpreted as the KK momentum corrections [11,12,13]. On the other hand, from the viewpoint of the exotic 5 2 2 -branes, the worldsheet instanton corrections can be understood as the winding corrections to the configuration [16]. Then, how should we interpret the worldsheet instanton corrections to the defect (p, q) five-branes? This is quite a fascinating question. (iii) There are various hyper-Kähler geometries. If some of them coincide with the conjugate configurations discussed in this paper, we would be able to find a novel relation among various five-branes from the (non)geometrical viewpoint [18,38].
The second expression is a part of the monodromy transformations (2.6). The T-duality transformations along the 8-th and 9-th directions are represented in terms of 4 × 4 matrices U 8,9 in such a way as In terms of these matrices, we can see the T-duality relations among the monodromies Ω NS , Ω KK and Ω E , The matrix description of the T-duality transformations can be also seen in [40,24], and so forth.