A dessin on the base: a description of mutually non-local 7-branes without using branch cuts

We consider the special roles of the zero loci of the Weierstrass invariants $g_2(\tau(z))$, $g_3(\tau(z))$ in F-theory on an elliptic fibration over $P^1$ or a further fibration thereof. They are defined as the zero loci of the coefficient functions $f(z)$ and $g(z)$ of a Weierstrass equation. They are thought of as complex co-dimension one objects and correspond to the two kinds of critical points of a dessin d'enfant of Grothendieck. The $P^1$ base is divided into several cell regions bounded by some domain walls extending from these planes and D-branes, on which the imaginary part of the $J$-function vanishes. This amounts to drawing a dessin with a canonical triangulation. We show that the dessin provides a new way of keeping track of mutual non-localness among 7-branes without employing unphysical branch cuts or their base point. With the dessin we can see that weak- and strong-coupling regions coexist and are located across an $S$-wall from each other. We also present a simple method for computing a monodromy matrix for an arbitrary path by tracing the walls it goes through.


I. INTRODUCTION
The importance of F-theory [1][2][3] in modern particle physics model building cannot be too much emphasized. The SU (5) GUT, which can naturally explain the apparently complicated assignment of hypercharges to quarks and leptons, is readily achieved in F-theory. Another virtue of F-theory is that it can yield matter in the spinor representation of SO (10), into which all the quarks and leptons of a single generation are successfully incorporated, and which cannot be achieved in pure D-brane models. These features are shared by E 8 × E 8 heterotic models, but F-theory models have an advantage in that they may evade the issue of the relation between the GUT and Planck scales in heterotic string theory first addressed in [4]. Also, the Yukawa couplings perturbatively forbidden in D-brane models [5,6] can be successfully generated in F-theory.
Almost ten years after the first development in F-theory, there was much progress in the studies of local models of F-theory (See [7][8][9][10][11][12][13][14][15][16] for an incomplete list.). In this class of theories, one basically considers a supersymmetric gauge theory 1 on a stack of 7-branes in F-theory, whose coalescence is supposed to give rise to a gauge symmetry depending on the fiber type in the Kodaira classification. In particular, if the fiber type is either IV * , III * or II * , the gauge symmetry will be E 6 , E 7 or E 8 , respectively, and then the brane was called an exceptional brane [8]. 2 The fiber type of such a codimension-one singularity can be labeled by the (conjugacy class of the) SL(2, Z) monodromy around the fiber. It was shown that all the types of Kodaira fibers can be represented by some product of monodromies of a basic set of 7branes: A=D-brane, B=(1, 1)-brane and C=(1,−1)-brane [51][52][53], as shown in Table in Appendix 3 . The relation between the resolution of the singularity and the gauge symmetry on a coalescence of 7-branes has been clearly explained by using string junctions. String junctions are also useful to describe chiral matter [55], non-simply-laced Lie algebras [56], the 1 More precisely, the compact part of the theory is "twisted" so that the Casimirs of the gauge fields correctly transform as sections of Looijenga's weighted projective space bundle [17]. 2 More recently, after the LHC run in particular, global F-theory models have been attracting much interest.
For recent works on global F-theory models, see e.g. . 3 In this paper, we identify these 7-branes as the monodromy matrices M p,q defined in [51] with the sign of q reversed (as we have adopted Schwarz's convention for the tension [54]), which are the inverse of K [p,q] in [52,53]; this is consistent as the orderings of the branes and K [p,q] 's are in reverse to each other.
From the table one can see that the singular fibers of the exceptional type consist of a Bbrane and two C-branes in addition to the ordinary D(=A)-branes. Thus, in this algebraic approach, the exceptional branes are seen to emerge due to the coalescence of these Band C-branes which are distinct from D-branes. From a geometrical point of view, however, these branes are just the zero loci of the discriminant of a Weierstrass equation and there are no a priori differences from each other; they all are locally D-branes.
In this paper, we consider the special roles of the zero loci of the Weierstrass invariants g 2 (τ (z)), g 3 (τ (z)) in F-theory on an elliptic fibration over P 1 , or a further fibration thereof.
They are defined as the zero loci of the coefficient functions f (z) and g(z) of a Weierstrass equation. They are thought of as complex co-dimension one objects, and we call them "elliptic point planes".
In fact, mathematically, our construction amounts to drawing a "dessin d'enfant" of Grothendieck on the P 1 base with a canonical triangulation 4 . We show that this drawing provides a new way of keeping track of mutual non-localness among 7-branes in place of the conventional ABC 7-brane description. In our approach, all the discriminant loci are treated democratically, and with this "dessin" we can see that weak-and strong-coupling regions coexist and are located across an S-wall from each other. We also present a simple method for computing a monodromy matrix for an arbitrary path by tracing the walls it goes through. The method for studying monodromies by tracing the contours on the J(τ )-plane was developed long time ago by Tani [55]. This paper is organized as follows: In section 2, we introduce the basic setup of this paper, including the motivations and definitions of the elliptic point planes, the domain walls extended from them, and the cell region decomposition of the P 1 base of the elliptic fibration. The various definitions of the new notions and objects are summarized as a miniglossary at the end of this section. In section 3, we briefly explain what is a "dessin d'enfant" and the relation to our present construction. In section 4, we discuss the basic properties of the two kinds of elliptic point planes, the f -plane and the g-plane. In section 5, we present a new method for computing the monodromy by drawing the dessin. In the final section we conclude with a summary of our findings. Appendix A contains a table of fiber types of the Kodaira classification. The plots presented in this paper have been generated with the aid of Mathematica.

II. WHAT IS AN ELLIPTIC POINT PLANE?
Consider a Weierstrass equation where y, x, f and g are sections of an O(3), an O(2), an O(4) and an O(6) bundle over the base P 1 . This is a rational elliptic surface, which we regard as one of the two rational elliptic surfaces arising in the stable degeneration limit of a K3 surface. It may also be thought of as the total space of a Seiberg-Witten curve (with the "u"-plane being the base) of an N = 2 SU (2) gauge theory or an E-string theory. In an affine patch of P 1 with the coordinate z, the coefficient functions f (z) and g(z) are a 4th and a 6th order polynomial in z. 5 As is well known, the modulus τ of the elliptic fiber of (1) is given by the implicit function: where J is the elliptic modular function. The denominator of the right hand side is called the discriminant. Near its zero locus z = z i , Imτ goes to ∞ (if one has chosen the "standard" fundamental region) for generic (that is, nonzero) f and g. Examining the behavior of J(τ ) around ∞, we find 5 Although we introduce and define various notions in this simple setup, most of them can be generalized to a lower-dimensional F-theory compactification on a higher-dimensional elliptic Calabi-Yau, whose base W is a P 1 fibration over some base manifold B, by simply taking y, x, f and g to be sections of K −3 W , K −2 W , K −4 W and K −6 W , respectively, where K W is the canonical class of W. The equation (1) then describes a K3 fibered Calabi-Yau over B. A configuration of the elliptic point planes, D-branes and various walls are then a "snapshot" of a P 1 fiber over some point on B with fixed coordinates. which implies the existence of a D7-brane at each discriminant locus. 6 On the other hand, since a locus of f (z) = 0 or g(z) = 0 alone does not mean ∆ = 0, it is not a D-brane. However, if the loci of f (z) = 0 and g(z) = 0 are present together with a D-brane, they play a significant role in generating a (p, q)-7-brane by acting SL(2, Z) conjugate transformations on a D-brane or as components of an orientifold plane, as we show below. In this paper, we will collectively call the loci of f (z) = 0 and g(z) = 0 "elliptic point planes". 7 Elliptic point planes consist of two types, the loci of f (z) = 0 and g(z) = 0, which have different properties. In this paper, we call the locus of f (z) = 0 an f= 0 locus plane, or an f -plane for short, and that of g(z) = 0 a g = 0 locus plane, or a g-plane for short. 8 At the location of an f -plane, the value of the J-function is which corresponds to τ = e 2πi 3 . On the other hand, at the position of a g-plane , so this implies τ = i. In their neighborhoods, J(τ ) is expanded as where K(k) is the complete elliptic integral of the first kind Thus τ = e 2πi 3 is a triple zero of J(τ ) and τ = i is a double zero of J(τ ) − 1.
6 Thus, henceforth in this paper, we refer to a locus of the discriminant as (a locus of) a "D-brane". As we will see, however, the monodromy around it is not always T (13) for a general choice of the reference point, due to the presence of the elliptic point planes. 7 In the standard fundamental region of the modular group of a two-torus, there are two elliptic points and i. They are fixed points of actions of some elliptic elements of SL(2, Z), hence the name. 8 Despite the name "plane", an elliptic point plane is no more a rigid object but a smooth submanifold when the elliptic fibration over P 1 is further fibered over another manifold, just like a D-brane.
Suppose that z = 0 is a locus of f = 0. Since Furthermore, each region on the z-plane corresponding to a definite fundamental region is divided by a domain wall (a dashed green line) into two regions ImJ(τ ) > 0 and ImJ(τ ) < 0.
Since the value of J is ∞ at a discriminant locus for generic (i.e. nonzero) values of f and g, D-branes can never, by definition, touch nor pass through (a non-end point of) the domain walls because ImJ(τ ) must vanish at the domain walls.
In this way, the z-space (= P 1 ) is divided into several "cell regions", which correspond to different fundamental regions in the preimage of the J-function, by the domain walls extended from the elliptic point planes (=f -planes and g-planes) and D-branes ( Fig.1). In in what has been called a "B-brane" or a "C-brane" in the discussions of string junctions.
That is, while the monodromy matrix is necessarily We would like to emphasize here that such a local S transformation never takes place without these "elliptic point planes" (=f -planes and g-planes). If it were not for elliptic point planes but there are only D-branes, the domain walls extended from them are only the ones (green lines) and (dashed green lines). So crossing through these walls only leads to a T transformation which commutes with the original monodromies of D-branes.
In the discussion below, we refer to the domain wall (14) (a green lines) as T -wall and the one (15) (a dashed green line) as T -wall, whereas we call the type of domain wall (12) (a blue line) S-wall.
To conclude this section we summarize the definitions of the new objects and notions introduced in this section as a mini-glossary.
Mini-glossary T -wall A (real) co-dimension-1 object (domain wall) corresponding to a zero locus of ImJ with ReJ < 0, extending from a D-brane and a f -plane. Represented by a green line.
T -wall A (real) co-dimension-1 object (domain wall) corresponding to a zero locus of ImJ with ReJ > 1, extending from a D-brane and a g-plane. Represented by a dashed green line.
S-wall A (real) co-dimension-1 object (domain wall) corresponding to a zero locus of ImJ with 0 < ReJ < 1, extending from a f -plane and a g-plane. Represented by a blue line.
cell region A closed region on the z-plane (P 1 base of the elliptic fibration) bounded by the T -, T -and S-walls. Each cell region corresponds to either half of the (closure of the) 9 fundamental region with ImJ > 0 or ImJ < 0 of the fiber modulus.
shaded cell region The cell region corresponding to the (closure of the) half fundamental region with ImJ > 0 ( Figure 1).

III. RELATION TO "DESSIN D'ENFANT" OF GROTHENDIECK
In fact, the construction in the previous section is nothing but drawing a "dessin d'enfant" of Grothendieck [60], known in mathematics, on the P 1 base with a canonical triangulation. 10 9 Below we abuse terminology and refer to a "fundamental region" as one modulo points on its boundary. 10 The contents of this section are triggered by a suggestion made by the anonymous referee of Phys. Rev. D.
white points and lines connecting these points, drawn according to a special rule. To demonstrate the rule, let us consider, for example, a function [61]: where x ∈ P 1 . F is a map from P 1 to P 1 . At almost everywhere on P 1 , F is a homeomorphism, sending a small disk to another in a one-to-one way. However, F maps a small disk centered at x = 1 to one centered at F = 0 in a three-to-one way. Similarly, F is a two-to-one map from a small disk centered at x = 3 ± 2 √ 3 to one centered at F = 1. The points The equation (16) induces a branched covering over P 1 . Treating this graph as a combinatorial object, one can reproduce the information of the branched covering as follows: One first adds a point ∞ to each region of the dessin. One then connects each ∞ with lines to the black or white points as many times as they appear on the boundary of the region. This yields a triangulation of the dessin. Assigning either the upper-or the lower-half plane to each triangle depending on the ordering of 0, 1, ∞, and glueing these half planes together, one obtains a branched covering equivalent to the original one [61].
In the present case, the equation (10)  What is special about (10) is that it induces a local homeomorphism between the P 1 base and the upper-half plane. Indeed, as we saw in the previous section, the correspondence is one-to-one everywhere, even in the vicinity of the elliptic orbits τ = e 2πi 3 and i. This is so because the J = 0 (f = 0) points are always critical points with ramification index three, and the J = 1 (g = 0) points are always with ramification index two. In this paper, we treat the dessin not as just a combinatorial graph, but draw the ∞ points and the triangulating lines (the T -and T -walls) also as preimages of the J-function, as shown in FIG.3(b). The special feature of (10) then allows us to use the (triangulated) dessin as a convenient tool to compute monodromies, as we see below. (or its SL(2, Z) equivalents).
Parameterize a small circle around z = 0 by z = e iθ ( > 0), then if one goes around along it once, so does τ once around e and an S-wall (blue) again, the transformation to the final SL(2, Z) equivalent point is T −1 S generates a Z 3 group, which is the isotropy group of the elliptic point τ = e

B. Basic properties of g-planes
Likewise, the expansion of J(τ ) around τ = i is given by (8). Let a g-plane be at z = 0 this time. f (z) and g(z) are expanded as Since τ (z) approaches with a very simple method to compute the monodromy matrices along an arbitrary path around branes on the complex plane (= an affine patch of the P 1 or the "u-plane" of a

A. The method
To illustrate the method, let us consider the Seiberg-Witten curve of N = 2 pure (N f = 0) SU (2) supersymmetric gauge theory [62]. The equation is Taking u as the coordinate z, we obtain a Weierstrass equation with whose dessin is shown in the upper panel of Figure 4. Let us compute the monodromy around each discriminant locus. Choosing a starting point near the left locus (shown as a cross), the left path crosses the walls as where G denotes the T -wall, B the S-wall and dG the T -wall. 12 The monodromy matrices for various patterns of crossings are where the first wall of each row is the crossing from a shaded cell region (ImJ > 0) to an unshaded one (ImJ < 0), and the second is from an unshaded to a shaded one. 13 The 12 G, B and dG are respectively the first letters of Green, Blue and dashed Green. We have avoided using T , S or T here as the monodromy matrices for the crossing do not coincide with the names of the walls. 13 Therefore, these rules only apply when one computes a monodromy for a path that starts from and ends in a shaded cell region (ImJ > 0). The rules for computing a monodromy for a path from an unshaded monodromy matrices are defined as as usual, where we say that the monodromy matrix is They are defined only in P SL(2, Z), i.e. up to a multiplication of −1.
By using the rule (29), we can immediately find the monodromy matrix for the path (28) as where ∼ denotes the equality in P SL(2, Z).
Similarly, the crossed walls for the right path are Using rule (29) again, we find that the monodromy is A confusing but important point of the rule is that, in the first example, the monodromy matrix T −1 which corresponds to the crossings → G → dG → taking place after the cell region (ImJ < 0) to another are similar but different: in which M is multiplied from the left.
the walls where W i (i = 1, . . . , k) are either of G, B or dG, and let M γ denote the associated monodromy matrix of γ. k is an even positive integer. (If it is odd, a shaded cell region is mapped to an unshaded cell region or vice versa, and the transformation cannot be an SL(2, Z) transformation). Let γ 1 , γ 2 be paths specified by the series of the walls crossed by them and let γ 1 + > γ 2 be the jointed path where we use the new symbol + > to denote the operation of jointing two paths. 14 Then Proposition.
Remark. As we noted above, the monodromy matrix corresponding to a later crossing comes to the right, unlike (36) in which the matrix for the later transformation is multiplied from the left.
Proof. By induction with respect to the total number of crossed walls, it is enough to show the statement for the cases when γ 2 is any of the crossing patterns (29). Suppose that γ 1 starts from a cell region C 0 and ends in another C 1 , and that γ 2 goes from the cell region C 1 to another C 2 , where γ 2 is taken to be any of the crossing patterns (29), say, γ 2 =→ dG → G → and M γ 2 = T . Let P γ i (i = 1, 2) be the associated maps which send points in the cell region C i−1 to those in the cell region C i , respectively, such that the torus modulus over the point is SL(2, Z) equivalent. We say two points on P 1 are SL(2, Z) equivalent if the torus fiber moduli over them are SL(2, Z) equivalent. Using this terminology, we can say that P γ i (i = 1, 2) are the maps which send the points in C i−1 to their SL(2, Z) equivalent points in C i , respectively. Since τ (z) is holomorphic in z and J(τ ) is holomorphic in τ , the domain of the map P γ 1 is not necessarily restricted to only C 0 but can be extended to outside C 0 as far as it is in a small neighborhood of z 0 .
Let z 0 be a point in C 0 , and let z 1 = P γ 1 (z 0 ) ∈ C 1 , z 2 = P γ 2 (z 1 ) ∈ C 2 . If we denote τ i (i = 0, 1, 2) be the modulus of the torus fiber over z i (i = 0, 1, 2), they satisfy where τ 1 and τ 2 are the values analytically continued from τ 0 along the paths γ 1 , and then γ 2 . Taking τ 0 in the standard fundamental region, the transformation from τ 0 to τ 1 is given by τ 1 = M γ 1 • τ 0 , but consecutive transformation from τ 1 to τ 2 is not M γ 2 • τ 1 , as τ 1 does not belong to the standard fundamental region in general. Rather, since P γ 1 is locally an isomorphism between a neighborhood around z 0 and that around z 1 , the final point z 2 can be written as the P γ 1 image of z 1 , where z 1 is the SL(2, Z) equivalent point in the cell region reached along the path γ 2 first from z 0 , if z 2 is close enough to z 1 ( Figure 5). If, on the other hand, z 2 is not close to z 1 , we can continuously deform the complex structure of the elliptic fibration so that z 2 may come close to z 1 . Since this is a continuous deformation, the monodromy transformation matrix does not change, as the entries of the matrix take discrete values. Thus we may assume that z 2 is close to z 1 .
Since τ 0 is taken in the standard fundamental region, τ 1 , the modulus of the torus fiber over z 1 , is given by Therefore, since τ 2 = M γ 1 • τ 1 , we find which is what the proposition claims.
In deriving (43), we did not use the fact that γ 2 was assumed to be a particular pattern among (29), but the relation (43)    The Weierstrass equation is (1) where However, one can also set the mass parameters of the same gauge theory so that the dessin of the Seiberg-Witten curve looks as shown in Figure 1. In this case, the S-walls spread into wide areas of the P 1 . There is not much difference among the six D-branes, and there is no obvious reason to distinguish particular two as B or C from the other four D-branes.
Remark. We have seen that a cluster of a D-brane and two elliptic point planes, in which the former is surrounded by the S-walls extended from the latter, may be effectively identified as a Bor a C-brane, if viewed from a distance of the size of the cluster. Thus one might think that an "exact" (p, q)-brane (whose monodromy is M p,q along arbitrary small loop) can be obtained by taking the f -and g-planes on top of each other so that the size of the cell region the S-walls surround becomes zero. This is not the case, however, since if the f -and g-planes collide, the order of the discriminant becomes two, implying that another D-brane also automatically comes on top of the D-brane, f -plane and g-plane. Since it contains two D-branes, it cannot be identified as a single (p, q)-brane in the ABC-brane description.

VI. CONCLUSIONS
The coexistence of D-branes and non-pure-D-7-branes is an essential feature of F-theory, as it enables us to achieve exceptional group gauge symmetries or matter in spinor representations by allowing string junctions to appear as extra objects ending on more than two different types of 7-branes, in addition to the open strings which can only connect two ordinary D-branes. These 7-branes are conventionally described algebraically in terms of ABC 7-branes. In this paper, noticing that all the discriminant loci are on equal footing and there is no a priori reason to distinguish one from the others, we have considered new complex co-dimension one objects consisting of the zero loci of the coefficient functions f and g of the Weierstrass equation, which we referred to as an "f -plane" and a "g-plane", collectively as "elliptic point planes". They are two kinds of critical points of a "dessin d'enfant" known in mathematics.
Although they do not carry D-brane charges, they play an essential role in achieving an exceptional gauge symmetry and/or a spinor representation by altering the monodromies around the branes. More precisely, if there are some elliptic point planes, the z-plane is divided into several cell regions, each of which corresponds to a (half of a) fundamental region in the preimage of the J-function. A cell region is bounded by several domain walls extending from these elliptic point planes and D-branes, on which the imaginary part of the J-function vanishes. In particular, the elliptic point planes extend a special kind of domain walls, which we call "S-walls", crossing through which implies that the type IIB complex string coupling is S-dualized. Consequently, on the z-plane coexist a theory in the perturbative regime and its nonperturbative S-dual simultaneously. The monodromy around several 7-branes is thus not just a product of monodromy around each 7-brane any more, but they get SL(2, Z) conjugated due to the difference of the corresponding fundamental regions the base points belong to.
In this sense one may say that the nonperturbative properties of F-theory -the realizations of exceptional group symmetry, matter in spinor representations, etc. -are the consequence of the coexisting "locally S-dualized regions" bounded by the S-walls extended from the elliptic point planes. In the orientifold limit [63], the D-branes and the elliptic point planes gather to form a I * 0 singular fiber, so that the S-walls extended from the elliptic point planes are contracted with each other and confined, so the S-walls are not seen from even a short distance.
We hope this new way of presenting the non-localness among 7-branes will be useful for understanding of the structure of higher-codimension singularities with higher-rank enhancement such as discussed in [2,3,11,[64][65][66][67].