Four-dimensional $N=1$ theories, S-fold constraints on T-branes, and behaviors in IR and UV

We analyze four-dimensional (4d) $N=1$ superconformal field theories (SCFTs) obtained as deformations of 4d $N=2$ SCFTs on S-folds by tilting 7-branes. Geometric compatibility with the structures of S-folds constrains the forms of T-branes. As a result, brane monodromies are constrained. We also discuss two 4d $N=1$ theories on probe D3-branes, where the two theories behave identically in IR, but they originate from different theories in UV. Studying the global structure of their geometry is useful in constructing these two theories.

4d N = 2 superconformal field theories (SCFTs) have recently been studied by utilizing S-fold constructions [16,17]. A few years before these studies, the authors of [24] constructed 4d theories with N = 3 supersymmetry 1 by considering finite cyclic groups Z k acting on torus fibrations, wherein the Z k -actions also act on torus fibers. The 4d N = 2 SCFTs studied in [17] flow to 4d N = 3 theories, as constructed in [24], by mass deformations.
The subjects of study in this note are 4d N = 1 SCFTs on S-folds 2 obtained as in the context of deforming the 4d N = 3 theories constructed in [24]. It would be natural to consider deforming the 4d N = 2 SCFTs in [17] to yield such superconformal field theories, and we take this approach to study 4d N = 1 SCFTs.
There are several advantages to analyzing 4d N = 1 SCFTs in this manner. For example, the resulting 4d N = 1 SCFTs relate to 4d N = 2 and N = 3 theories by construction, and they are also relevant to studies of the structures of S-folds. Furthermore, as we discuss later in this note, they are also related to T-branes [35] 3 . Recent progress of T-branes can be found, e.g., in [37,38,39,40,41,42,43,44].
We utilize the operation of tilting 7-branes in 4d N = 2 SCFTs to yield 4d N = 1 SCFTs, as described in [45,46]. This operation corresponds to allowing the Higgs field φ on the 7-branes stack to have a non-zero position-dependent vacuum expectation value (vev) over the base space, and as a result a correction term is added to the superpotential [45]. Deformation generated by this operation provides an opportunity to consider T-branes when φ and φ † do not commute [35].
In the 4d F-theory background with a gauge field flux, φ and φ † do not need to commute [45]. When they do not commute, the N = 2 supersymmetry is broken to N = 1 [45,46]. We study the 4d N = 1 SCFT on the D3-brane probing a stack of 7-branes. A D3-brane probes 4 the infrared (IR) 4d N = 1 SCFT, which corresponds to seeing a local patch of a global geometry.
There are two themes of this note. The first is to analyze 4d N = 1 theories 5 obtained as deformations of 4d N = 3 theories, as aforementioned. In the situation where φ and φ † do not commute, there is room to consider the effect of T-branes. Because the geometry considered here is an S-fold built as the quotient of elliptic fibration with finite cyclic group Z k actions, the equations describing T-branes are constrained to be compatible with the Z k actions.
A T-brane is given by a spectral equation [57,58] that yields a multifold cover of a stack of 7-branes, where the 7-brane stack is given by z = 0. Because the parameter z transverse to the stack of 7-branes now has an equivalence relation under the Z k action, a consistency condition is imposed on the spectral equation of T-branes to make them compatible with the equivalence relation (1). A physical consequence of this compatibility argument is that brane monodromies [57,59,60,61] are constrained owing to the quotient action of the Z k groups. The other theme is to discuss the physical phenomenon where two 4d N = 1 SCFTs in the IR limit are indistinguishable but behave quite differently in the ultraviolet (UV) limit. We utilize a global perspective of geometry in F-theory to observe this phenomenon. This phenomenon occurs because the two globally distinct geometries have locally identical structures in a local patch that a D3-brane probes, making them appear the same in IR, but their difference becomes evident in UV.
The S-fold constructions discussed in [24,17] were obtained as Z k -actions on elliptic fibrations over an open base wherein the complex structures of the elliptic fibers are fixed over the entire base, so the complex structures of the total spaces are compatible with the Z k actions. However, elliptic fibrations can be constructed over a closed base space wherein the torus fibers have constant fixed complex structures over the base. Some nontrivial examples of such global geometries can be found, such as those in [62,63,64,65,66,67]. Here, global geometries represent elliptic fibrations over compact base spaces 6 . Such global geometries have some applications in 4d N = 1 theories.
We can observe a physical phenomenon that might be interesting: one can construct two global geometries on which physics appear completely identical in the IR limit, but in the UV limit, they describe different physical theories. This is owing to the fact that while the two geometries have different structures globally, their geometric structures appear locally identical. Physically, in the IR limit, they yield two identical 4d N = 1 SCFTs probed by D3-branes that flow to very different theories in UV.
In recent years, there has been progress in the appearance of discrete gauge groups in F-theory, e.g., in [68,69,70,71,72,73,74,75,76,77,63,64,65,78,79,80,81]. The compactification spaces in F-theory, wherein a discrete gauge group arises, are genus-one fibrations that "lack a global section" [69]. By tuning the coefficients as parameters of the moduli of such geometries without a global section, one can deform the geometry to an elliptic fibration with global sections. In the moduli space, physics on a genus-one fibration lacking a global section and physics on an elliptic fibration with global sections are very different; a discrete gauge symmetry arises in the former geometry, while a discrete gauge symmetry does not form and U(1) does form in the latter geometry. If we consider inserting a D3-brane probe into this geometric deformation, the characteristic physical phenomenon that we mentioned previously can be observed: the D3-brane is insensitive to the subtle deformation in geometry from genus-one fibration without a global section into an elliptic fibration with sections. When one moves to UV, the effect of the global structure of the geometry becomes explicit, and the corresponding physical theories before and after the deformation behave very differently. Our argument in section 3 suggested that a U(1) factor coming from the global sections of the elliptic fibration and a discrete gauge group arising from the geometry of the genus-one fibration lacking a global section are not reflected in the flavor symmetry group of the 4d theory on the probe D3-brane.
As a byproduct of the discussion of the second theme, we observe a puzzle in 4d SCFT concerning a subtlety in the structure of the flavor symmetry group. An elliptic fibration has the notion of a "global section." This can be seen as a copy of the base space embedded in the total elliptic fibration [82]. If an elliptic fibration admits a global section, global sections form an Abelian group known as a "Mordell-Weil group." This group decomposes into a direct sum of free part Z r and torsion part i Z i . In F-theory compactification [83,84,82] 7 , the structure of the Mordell-Weil group is reflected in the gauge group structure; the rank of the Mordell-Weil group 8 yields the number of U(1)s formed in F-theory [82], and the global structure of the gauge group is divided by the Mordell-Weil torsion [111,112,113] 9 . Is the structure of such Mordell-Weil torsion dividing the flavor symmetry group observed in "local" 4d SCFTs probed by D3-branes? Two opposing answers to this question seem possible, as we will demonstrate. Because the two views are opposite, one must choose one of the possible two viewpoints. We discuss this puzzle in section 4. 7 Local models of F-theory model constructions are discussed in [85,86,87,88]. Studies of the compactification geometry of F-theory from the global perspective can be found, e.g., in [89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110]. 8 The rank of the Mordell-Weil group is the number of copies of Z in the free part. 9 Computations of the Mordell-Weil torsions of elliptic fibrations in F-theory compactifications and the global structures of the gauge groups that form on the 7-branes are also discussed in [64,65,78,66,67].
By gauge group in sections 3 and 4, we mean the gauge group of spacetime when the theory is compactified on a Calabi-Yau 4-fold. In the D3-brane probe theory, this symmetry is perceived as (part of) the flavor symmetry of the 4d theory.
We would like to make a remark that, when we discuss UV theories in this study, they come from the D3-branes probing compact genus-one fibered Calabi-Yau 4-folds, and 4d gravity is not entirely decoupled. The flow from UV theory to IR theory is the decoupling limit. When we consider a limit at which the size of the Calabi-Yau space becomes large enough, however, the theory approaches the situation where the D3-brane probes a non-compact space and the gravity decouples. Considering this limit if necessary, we do not consider the effect of gravity in this note.
When gravity is not decoupled, because the 7-branes are wrapped on a compact space, the gauge field on the 7-branes is dynamical. Thus, the symmetries on the 7-branes yield the gauge groups in UV theories. Since the gauge field is decoupled in the decoupling limit, in the D3-brane probe theory in the IR the symmetry on the 7-branes is perceived as the flavor symmetry of the 4d theory.
In section 2.1, we briefly review the 4d N = 3 theories constructed in [24] and the 4d N = 2 SCFTs realized in F-theory on S-folds in [17] as deformations of 4d N = 3 theories. We construct and analyze 4d N = 1 SCFTs as deformations of these theories in section 2.2. Tilting 7-branes deforms 4d N = 2 SCFTs, which yields 4d N = 1 SCFTs in IR [46]. After we discuss this construction, we analyze brane monodromies in the resulting 4d N = 1 theories in section 2.2. The structures of T-branes are constrained by the symmetry of S-fold constructions. We analyze the constraints that brane monodromies receive as a consequence of this.
We discuss in section 3 two 4d N = 1 theories that behave in an identical manner in IR but flow to very distinct theories in UV. In these constructions, considering the global aspects of the geometry is essential. When a similar argument is applied to the global structure of the gauge group formed on 7-branes, we observe a puzzle, which we discuss in section 4. We present our concluding remarks in section 5 alongside some open problems.
2 Brane monodromies in 4d N = 1 theories as deformations of 4d N = 2 SCFTs on S-folds 2.1 Reviews of 4d N = 3 theory and 4d N = 2 SCFTs on S-folds We first briefly review the 4d N = 3 theories constructed in [24]. We then review the 4d N = 2 SCFTs on S-folds constructed in [17]. We will discuss 4d N = 1 SCFTs obtained as deformations of these theories, and T-branes and brane monodromies in the resulting 4d N = 1 theories in section 2.2. We review the 4d N = 2 and N = 3 theories in this section, as we require them to construct 4d N = 1 theories. F-theory compactification on the product C 3 × T 2 yields a 4d N = 4 theory. The coordinates of C 3 are denoted by z 1 , z 2 , and z 3 , and the complex coordinate of the two-torus in the product is denoted by z 4 . The authors of [24] considered the following action of a finite cyclic group Z k on the product C 3 × T 2 : Here, the complex structure τ of the two-torus as an elliptic fiber is constant over the base C 3 , so the product C 3 × T 2 is a direct product as a complex manifold. Because the Z k -action acts on the two-torus as a fiber, it is necessary to confirm that the action is compatible with the elliptic fibration structure of C 3 × T 2 . It was shown in [24] that the compatibility conditions require the order k of the cyclic group Z k to take one of the values k = 2, 3, 4, or 6, and the complex structure of the elliptic curve as a fiber must take a specific value for each of the values of k [24]. We describe the complex structures of elliptic fibers of an elliptic fibration in detail here. An elliptic curve (two-torus seen as a complex curve) is described by a Weierstrass equation. A Weierstrass equation is an equation of the following form: Then, the complex structure of an elliptic curve as a fiber of the fibration is uniquely labeled by a j-invariant, which is an invariant under the isomorphisms of elliptic curves. A j-invariant j is given in terms of the Weierstrass coefficients f, g, as follows [114]: Modular parameter τ also specifies the complex structure of an elliptic curve. τ takes a value in C modulo SL(2, Z)-action.
When the order k of the group Z k takes the values k = 3, 4, 6, N = 4 supersymmetry breaks down to N = 3, and as a result of the orbifold action (2), a 4d N = 3 theory is obtained [24].
Before we review 4d N = 2 SCFTs on S-folds, we make a remark. Construction of a genus-one fibration whose elliptic fiber has a constant complex structure throughout the base, that is not a direct product of a two-torus as a fiber and the base space, is possible. Genus-one fibered K3 surfaces, whose fibers have constant complex structures τ = exp( πi 3 ) and τ = i over the base P 1 (equivalently, they have j-invariants 0 and 1728, respectively) can be found in [63,64] 10 . These global geometries play a role in the construction of 4d N = 1 theories in section 3.
The authors of [17] constructed 4d N = 2 SCFTs by considering F-theory compactifications on S-folds as deformations of the 4d N = 3 theories constructed in [24], and we briefly review this construction.
The S-folds in [17] were built by acting finite cyclic groups Z k on the space K3 • × C 2 . Here, K3 • denotes an open patch in an elliptic K3 surface such that the base of K3 • is C. K3 • is an elliptic fibration over C. The coordinate of this base C is denoted as z 1 , and the coordinates of C 2 in the product K3 • × C 2 are denoted as z 2 , z 3 . z 1 yields the coordinate transverse to the stacks of 7-branes, and z 2 , z 3 yield the coordinates parallel to the stacks of 7-branes. z 1 is identified with a chiral superfield, Z 1 , which parameterizes the Coulomb branch, and z 2 , z 3 are identified with a decoupled hypermultiplet, Z 2 , Z 3 in the 4d theory on the probe D3-brane. The Z k -action acting on K3 • × C 2 is relaxed from condition (2) to the following condition [17]: where φ 2 and φ 3 are required to satisfy This construction yields S-folds, and F-theory on the resulting S-folds with a probe D3-brane yields 4d N = 2 SCFTs [17]. Analogous to 4d N = 3 theories, compatibility conditions are imposed on the values of k and the complex structures of the elliptic fibers. The complex structure of the fibers of the elliptic fibration K3 • must be constant over the base C, i.e., the axio-dilaton must take a constant value over C 3 . ∆ 7 is used to represent the scaling dimension of the Coulomb branch (CB) operator for the 7-brane in K3 • × C 2 . When Z k -groups act on the product to yield S-folds, the authors of [17] argued that the theories on the resulting S-folds should have the scaling dimensions k∆ 7 and that these correspond to a phase rotation of supercharges. They deduced from these that k∆ 7 = 2, 3, 4, 6, and consequently, they found that the S-folds that are compatible with Z k -actions are as follows [17]: Z 2 S-folds with type H 2 , D 4 , E 6 7-branes 11 ; Z 3 S-folds with type H 1 , D 4 7branes; and Z 4 S-folds with type H 2 7-brane. The axio-dilaton has a constant value over the base space, and the value depends on the value of k∆ 7 [17].
We remark here that the types of 7-branes for S-folds mentioned previously are the types before the quotient actions of Z k . The authors of [17] computed the Z k -quotients of the corresponding singular fiber types by analyzing the Z k -actions on the Weierstrass equations directly.
However, there is another method to compute the quotient fiber types, which we discuss here. The complex structure of a singular fiber is invariant under the action of the finite cyclic group Z k . Alternatively, the j-invariant of a singular fiber is unchanged under the quotient action of Z k . Utilizing this mathematical fact, one can deduce the singular fiber types as a result of the quotient action of Z k groups. The types of singular fibers of elliptic surfaces, including their j-invariants, were classified by Kodaira in [115,116]. Kodaira's classification of the types of singular fibers 12 for elliptic surfaces is presented in Table 1. brane type J-invariant Monodromy Order of Monodromy # of 7-branes singularity Table 1: Types of the singular fibers of elliptic surfaces and their properties [115,116] and wellknown brane interpretations. "Regular" for j-invariant of type I * 0 indicates the fact that j-invariant of type I * 0 fiber can take any finite value in C.
Because there are Z 2 and Z 4 S-folds with type H 2 7-branes according to the results of [17], Z 2and Z 4 -quotients of type IV fibers can be considered. Type IV fibers have j-invariant 0 according to Table 1; therefore, Z 2 -and Z 4 -quotients of this fiber type must have j-invariant 0. The order of the A 2 singularity corresponding to type IV fibers is 4, and the Z 2 -action doubles this order, so the resulting fiber type has a corresponding singularity type of order 8. Thus, we uniquely determine that the resulting fiber type is IV * . This is consistent with the result given in [17]. A similar reasoning applied to the Z 4 -quotient appears to show that the resulting fiber type has order 16, and this appears to be a bad singularity. In fact, this is not the case. To see this, consider the following reparameterization of the Weierstrass coefficients: Because the discriminant is given by ∆ ∼ 4f 3 + 27g 2 , this reparameterization reduces the order of the singularity by six, and the actual order of singularity is 10. Therefore, the Z 4 -quotient of a type IV fiber has a j-invariant whose corresponding singularity has order 10, and this is uniquely determined to be a type II * fiber. This is also consistent with the result in [17].
A similar reasoning applies to the Z k -quotient of other fiber types. The results of the quotient fiber types obtained in [17] are perfectly consistent with the results deduced via the method described herein.
S-fold quotient constrains the deformation parameters of the Weierstrass equation of an elliptic fibration [17,22], and the mass deformation parameters form the Casimir invariants of the resulting flavor symmetry. The flavor symmetries of 4d N = 2 SCFTs on S-folds were discussed in [17,21,22]. We list the rank-one 4d N = 2 SCFTs on S-folds with discrete torsion and their flavor symmetries deduced in [17] 13 in Table 2.
S-fold quotient of singular fiber rank-one 4d N = 2 SCFTs Table 2: Rank-one 4d N = 2 SCFTs on S-folds with discrete torsion corresponding to quotients of Kodaira fibers [17] are listed. Items in the left column represent Z n S-fold quotients of Kodaira fibers. For example, IV * , Z 2 indicates Z 2 quotient of type IV * Kodaira fiber. The first entries in the square brackets in the right column represent the Kodaira fiber types corresponding to the Coulomb-branch singularities. The groups appearing as the second entries in the square brackets in the right column represent the flavor symmetries of the probe D3-branes in the resulting rank-one 4d N = 2 SCFTs. We follow the notational convention utilized in [14].
2.2 4d N = 1 SCFTs as deformations of 4d N = 2 SCFTs on S-folds, T-branes in 4d N = 1 S-fold and brane monodromy One of the aims of this note is to analyze the 4d N = 1 SCFTs obtained as deformations of the 4d N = 2 SCFTs on S-folds [17] that we reviewed previously. We require an operation that breaks N = 2 supersymmetry down to N = 1; we utilize an operation of tilting a stack of 7branes, as described in [45,46]. This operation corresponds to the Higgs field φ on the 7-branes stack acquiring a non-zero position-dependent vev [45]. Then, tilting the stack of 7-branes adds a correction to the superpotential as T r G (φ(Z 1 , Z 2 ) · O) [45]. O denotes dimension-two operators in the D3-brane probe theory transforming in the adjoint representation of G, where G represents the flavor symmetry group [45]. When φ and φ † do not commute, the N = 2 supersymmetry is broken down to N = 1 [45]. N = 1 deformation by tilting the stack of 7-branes corresponds to the superpotential deformation δW = T r G (φ(Z 1 , Z 2 ) · O) as discussed in [45,46].
The adjoint-valued vev that φ takes, in the situation [φ, φ † ] = 0, yields room to consider the T-brane structure [35]. When the stack of n 7-branes is located at z 1 = 0, the T-brane is generally given by the following spectral equation [35]: where z 1 denotes the spectral parameter, and b 2 , . . . , b n are polynomials in the variables of z 2 and z 3 . Equation (8) can be seen as the spectral equation for φ.
A T-brane is a structure associated with the coincident 7-branes that the Weierstrass equation cannot capture [39]. This physical degree of freedom in string theory is not fully determined by the defining equation of the geometry and still receives constraints from the geometry when an S-fold is considered. In other words, the structure of the T-brane must be compatible with the quotient action of the Z k groups. We now explain this in more detail.
The Z k quotient of K3 • × C 2 , k = 2, 3, 4, was used to construct S-folds in [17]. K3 • denotes an open patch on the K3 surface, as previously mentioned. We focus on these S-folds because the 4d N = 1 theories analyzed here are obtained as deformations of the 4d N = 2 SCFTs constructed in [17].
The base of K3 • as an elliptic fibration is isomorphic to C, and the coordinate of the base is denoted by z 1 . The coordinates of C 2 in the product K3 • × C 2 are denoted by z 2 , z 3 . Z k action are given by (5) [17], as mentioned in section 2.1, where phases φ 2 , φ 3 satisfy the relation (6). The product z 2 z 3 is invariant under Z k action (5). z 1 is identified with e 2πi k z 1 under the quotient action.
Note here that when S-fold construction is considered, geometric conditions are imposed on the equation (8). For a Z k S-fold, the coordinate z satisfies the equivalence relation For simplicity, we focus on the situations where the coefficients b i , i = 2, . . . , n, are polynomials that only depend on the product z 2 z 3 as a variable. Namely, we only consider the cases in which they are given as As stated previously, the product z 2 z 3 is invariant under Z k action (5). Under this assumption, conditions are imposed only on powers of z 1 in the spectral equation (8) to make it compatible with the Z k action. For simplicity of the argument, we focus on this situation and proceed.
The T-brane is then given by the spectral equation of the following form: where b i (z 2 z 3 ), i = 2, . . . , n, are polynomials in z 2 z 3 . The form of the T-brane must be invariant under Z k action (5). Because the coefficients b i (z 2 z 3 ), i = 2, . . . , n are invariant under Z k action (5), this is equivalent to the requirement that F (z) satisfies the following relation: Here, c denotes a non-zero constant number. When this condition is satisfied, under the quotient action of Z k , the equation F (z) = 0 is invariant; the form of the T-brane is unchanged under the quotient action. The condition (12) clearly constrains the form of the polynomial F (z) in the spectral equation. The physical consequence of this 14 is as follows. Because the Galois group of the polynomial F (z) yields brane monodromy [46,35], the S-fold compatibility condition (12) constrains the brane monodromy. When conditions are imposed on the coefficients of the polynomial F (z), the associated Galois group generally becomes smaller than the symmetric group S n . As a result, the brane monodromy group becomes smaller owing to the compatibility condition (12). Although the T-brane is a structure associated with the coincident 7-branes that the Weierstrass equation cannot fully capture, its structure still receives restrictions from the geometry, as we have shown. Compatibility with the Z k orbifold actions physically constrains the structures of T-branes and brane monodromies.
As an example, we consider the T-brane structure in 4d theory on an S-fold built from an open patch in K3 surface with an A 2 singularity corresponding to type IV fiber.
For this case, 7-branes in the open patch of the K3 surface have type H 2 , and the type-H 2 stack consists of four 7-branes. Z 2 action acted on the open patch K3 • times C 2 yields Z 2 S-fold. As we mentioned previously, the theory on probe D3-brane inserted in the resulting S-fold yields a 4d N = 2 SCFT 15 , and considering T-brane structure breaks half of N = 2 supersymmetry which yields 4d N = 1 theory. The spectral equation of the T-brane in the resulting 4d N = 1 theory is given by degree-four characteristic polynomial. The spectral equation must satisfy the relation (12) under the action of Z 2 group.
A T-brane given by the spectral equation of the following form is invariant under the Z 2 -action: a, b are polynomials in z 2 z 3 as we previously mentioned. The equation (13) satisfies the relation (12). A computation shows that the Galois group of the polynomial (13) is isomorphic to D 8 , the dihedral group of order 8. Therefore, the 7-brane monodromy group for the 4d N = 1 theory obtained from the Z 2 S-fold is D 8 .
Utilizing an open patch in K3 surface with an A 2 singularity corresponding to a type IV fiber, one can also construct a Z 4 S-fold. For this construction, a T-brane described by the following equation is invariant under the Z 4 action: The Galois group of this polynomial is isomorphic to Z 4 , and the 7-brane monodromy group for the 4d N = 1 theory obtained from the constructed Z 4 S-fold is Z 4 .
We make a remark on the flavor symmetries of the probe D3-branes in 4d N = 1 SCFTs obtained as deformations of the 4d N = 2 SCFTs on S-folds by T-branes. Because the effect of considering T-branes corresponds to tilting the stack of 7-branes, we expect that the resulting flavor symmetry algebra of the probe D3-brane in 4d N = 1 SCFT obtained as a deformation of a 4d N = 2 SCFT on an S-fold by T-branes is a subalgebra of the flavor symmetry algebra of that 4d N = 2 SCFT. This argument suggests that the flavor symmetry algebras of the 4d N = 1 SCFTs obtained as deformations of the 4d N = 2 SCFTs on S-folds by T-branes are given as subalgebras of the flavor symmetry algebras of the 4d N = 2 SCFTs deduced in [12,17,21,22].
3 Two 4d N = 1 theories behaving identically in IR but behaving differently in UV We discussed structures of some 4d N = 1 SCFTs probed by D3-branes in section 2. The D3brane only probes a local small patch of the geometry, and theories behaving identically in IR do not necessarily behave identically in UV. In this section, we aim to provide two 4d N = 1 theories that behave identically in IR but differently in UV by analyzing the geometry from a global perspective. When the global structures of the geometries are distinct, as theories flow to the UV regime, they become sensitive to the difference in the global structures of the geometries. Our argument in this section applies to general geometries constructed as genus-one fibrations with multisections and their deformations into elliptic fibrations with multiple global sections. Our argument applies to 4d N = 1 SCFTs constructed using S-folds, as discussed in section 2, as special cases of such geometries. As constructed in [63,64], there are genus-one fibrations lacking a global section whose fibers have constant j-invariants 0 and 1728 throughout the base spaces (or equivalently, having complex structures τ = exp( πi 3 ) and τ = i). One can construct "closed" Z k S-folds 16 by acting Z k groups on these genus-one fibrations. Because the D3-brane only probes a local open patch, theories on such closed S-folds in the IR limit reduce to 4d N = 1 SCFTs on S-folds, as discussed in section 2.
We have stated the capacity to which our argument in this section applies. Before discussing the central point of our argument, we explain the physical meanings of multisection geometries and elliptic fibrations with multiple sections. It is known that a discrete gauge group arises on multisection geometry in F-theory [69]. A multisection is a multifold cover of the base space of a genus-one fibration. The times a multisection wraps around over the base is referred to as the "degree" of the multisection. When a multisection has degree n, it is concisely referred to as an "n-section." A discrete Z n gauge group forms in F-theory on an n-section geometry [69]. The special case of n = 1 corresponds to a global section, yielding a copy of the base space.
By tuning the coefficients of the defining equation for a genus-one fibration with a multisection to special values, a multisection splits into multiple global sections. The number of independent global sections that an elliptic fibration possesses minus one yields the number of U(1)s formed in F-theory on that elliptic fibration, as discussed in [82]. Therefore, when an n-section splits into n global sections, a U(1) n−1 forms in F-theory. Tuning an n-section geometry to an elliptic fibration with n sheets of global sections can be viewed as the reverse of the Higgsing process, wherein a model with U(1) n−1 breaks down into a mode with a Z n gauge group [69].
It would be natural to expect that because the D3-brane only probes a small local patch of the geometry, it is insensitive to tuning of a genus-one fibration with an n-section into an elliptic fibration with n global sections. We demonstrate that this expectation is true.
The key point to show this is that an n-section and n separate global sections are locally indistinguishable. An n-section is an n-fold branched cover of the base space. In other words, an n-section is n global sections combined along the branching loci. When the branching loci diminish and eventually disappear, the n-section splits into n separate global sections. When one takes a small open patch such that it avoids any point in the branching loci, that local patch does not distinguish the n-section and n separate global sections; the only difference is whether n copies of the base space are combined along the branching loci or not, but that local patch does not pass any branched point, so they locally have no difference. Now we tune a genus-one fibration with an n-section to an elliptic fibration with n separate global sections while maintaining the type of the 7-branes stack at which the probe D3-brane is placed. The probe D3-brane is thus insensitive to this deformation in the geometry, as we have demonstrated.
In particular, this means that in the IR limit, the D3-brane does not probe the Higgsing process of U(1) n−1 breaking down into a Z n group occurring through the global geometric deformation in the moduli. U(1) or Z n groups arising from the global feature of the geometry in F-theory are not reflected in the 4d N = 1 SCFT in IR.
For the two theories, realized as an F-theory on an n-section geometry and an F-theory on an elliptic fibration with n global sections, the two theories on the D3-brane probes flow to the identical theory in IR. In UV, differences in the global geometric structures are probed and flow to distinct theories, reflecting the difference between U(1) n−1 and Z n groups.
We would like to demonstrate that it is possible to deform a multisection geometry to an elliptic fibration with multiple global sections while maintaining the type of stack of the 7-branes at which the D3-brane probes. We show this for the case of a bisection geometry 17 [68,69,64]. A bisection geometry generally admits an expression as the double cover of a quartic polynomial, as discussed in [68,69].
We particularly discuss bisection geometries given by equations of the following form, as studied in [64]: where t and x represent inhomogeneous coordinates of P 1 , and each α i , i = 1, . . . , 8, is a point in P 1 . Thus, the equation (15) yields a double cover of P 1 × P 1 branched over a (4,4) curve, which is a K3 surface [64]. A projection onto P 1 in the product P 1 × P 1 yields a genus-one fibration, as discussed in [64]. We regard t as the coordinate of the base P 1 of the genus-one fibration. The K3 surface (15) is a bisection geometry [64]. The elliptic fibers of this fibration have a particular symmetry, which forces the complex structure of the fibers to be constant with τ = i 18 throughout the base P 1 [64]. This constrains the types of 7-branes so that the fiber types over the 7-branes can only be III, I * 0 , or III * [64]. We focus on the case of α 5 = α 6 = 0 here. Then, the equation of the K3 surface (15) becomes We only consider the cases where α i , i = 1, 2, 3, 4, are mutually distinct and α i , i = 1, 2, 3, 4, 7, 8 are non-zero. The discriminant of the K3 surface (16) is given by [64] ∆ 7-branes at the origin t = 0 have type D 4 , and each stack of 7-branes at t = α i , i = 1, 2, 3, 4, has type H 1 [64]. 7-brane stacks at t = α 7 , α 8 have type H 1 when α 7 and α 8 are not equal. At the limit at which α 7 and α 8 coincide, the 7-brane type at t = α 7 is enhanced to D 4 . For the reader's convenience, we present the Jacobian fibration of the double cover (16). The Jacobian fibration admits transformation to the following Weierstrass form [119]: We consider F-theory on a K3 surface (16) times a K3 surface. We consider the situation where the probe D3-brane is placed at the origin t = 0. We consider tilting the stack of 7-branes at the origin t = 0 to yield a 4d N = 1 SCFT, as discussed in section 2. For general parameters α i , i = 1, 2, 3, 4, 7, 8, the K3 surface (16) has a bisection, but it does not have a global section, and a Z 2 gauge group forms in F-theory. As we discussed previously, this is not probed by the D3-brane in IR.
One can tune the parameters so a bisection splits into two global sections. A bisection splits into global sections when either the coefficient of x 4 or constant term in the quartic polynomial becomes a perfect square [69]. This occurs when α 7 and α 8 become coincident in (16). Therefore, at the limit at which α 7 and α 8 coincide, the bisection splits into two global sections. Then, U(1), instead of the Z 2 gauge group, forms in F-theory, but the theory on the D3-brane probe is insensitive to this change.
One can choose α 7 and α 8 far away from the origin so they do not lie in the local patch that the D3-brane probes. Tuning α 8 such that it approaches α 7 does not affect the singularity type at the origin t = 0.
We particularly considered a bisection given by a specific form of an equation, but a similar reasoning applies to general bisection geometry.

Puzzle in the global structure of the gauge group
When an elliptic fibration has a section, the set of sections is known to form a group, and this group is referred to as the "Mordell-Weil group." When the Mordell-Weil group of an elliptic fibration has a torsion part, the torsion part divides the global structure of the gauge group formed in F-theory on that elliptic fibration [111,112,113].
There are non-isomorphic elliptic K3 surfaces whose singular fiber types are identical and Ftheory compactifications yield identical 7-brane types but their Mordell-Weil torsions are different. We explicitly give an example of this. We utilize extremal K3 surfaces 19 . A complex elliptically fibered K3 surface f : S → P 1 with a global section is referred to as extremal when the Picard number of K3 surface S is 20 and the Mordell-Weil group, M W (S, f ), is a finite group. Complex K3 surfaces with the Picard number of 20 are referred to as attractive K3 surfaces 20 .
It is a mathematical fact that the complex structures of the attractive K3 surfaces are labelled by triplets of integers [122,123]. The complex structures, singularity types, and Mordell-Weil torsions of the K3 extremal fibrations were completely classified in [124]. A pair of non-isomorphic K3 surfaces whose singularity types are identical but Mordell-Weil torsions are different can be found in Table 2 in [124]; numbers are assigned to the singularity types of the K3 extremal fibrations in this table. No. 276 in this table corresponds to the singularity type E 7 A 9 A 2 . There are two K3 surfaces possessing this singularity type according to the table, and they have different complex structures, so they are non-isomorphic. F-theory on either of these two K3 surfaces times a K3 surface yields a 4d N = 2 theory, and an E 7 × SU (10) × SU (3) gauge group forms in both compactifications. However, one of the extremal K3 surfaces has no Mordell-Weil torsion, while the other K3 surface has Mordell-Weil torsion isomorphic to Z 2 , so the gauge groups formed in the two theories are globally different. These theories 21 yield different theories in UV, possessing distinct global structures of the gauge groups.
A natural question is whether the Mordell-Weil torsional group dividing the gauge group is reflected in the theory on the D3-brane probe, which probes only a local patch. Let us take a local patch of an elliptic fibration and place a probe D3-brane in the neighborhood of a stack of 7-branes. Locally, one cannot see the structures of global sections; thus, the Mordell-Weil group cannot be seen locally. Then, it is natural to consider that the Mordell-Weil torsions also cannot be locally seen. A consequence of this reasoning seems to be that the effect of Mordell-Weil torsion is not reflected in the 4d N = 1 SCFT on the probe D3-brane in IR.
However, the 7-brane type is determined by the local information. In the language of Ftheory, the 7-brane type is determined by the type of a singular fiber over the 7-branes, which is "vertical" information not relevant to the global feature of the base space. Because the Mordell-Weil torsional group acts on the gauge groups, the effect of this group might be reflected in the flavor symmetry group formed on the 7-branes that the D3-brane probes.
Because these two viewpoints provide opposing interpretations, one must choose one of the two interpretations. The question of which one gives the correct physical interpretation is left for future study.

Concluding remarks and open problem
In this note, we analyzed 4d N = 1 SCFTs obtained as deformations of 4d N = 2 SCFTs on S-folds by tilting 7-branes. We showed that T-branes receive constraints from the geometry when S-fold constructions are considered. Consequently, brane monodromies are also constrained by the geometric conditions, and we presented explicit examples of this.
We also discussed two 4d N = 1 SCFTs behaving identically in IR but flowing to distinct theories in UV. To construct these theories, utilizing the global structure of genus-one fibrations lacking a global section was useful. Our argument also showed that U(1) and discrete Z n groups arising from the global feature of the geometry in F-theory are not reflected in 4d N = 1 SCFTs on the probe D3-brane.
We encountered a dilemma when considering an effect of the global geometry on the theory living on the probe D3-brane, i.e., the action of the Mordell-Weil torsional group on the flavor symmetry group. Two opposing viewpoints seem possible for this problem, and which viewpoint yields a correct interpretation is left to be determined in future work.
We utilized the property that the D3-brane only probes a local patch of the geometry to deduce several physical consequences in sections 3 and 4. Strictly speaking, however, when the space possesses an S-fold singularity, not only the local patch of the geometry and the modes coming from the interaction of the D3-branes and the S-fold but also the structure of the S-fold can contribute to the 4d physics, as explained in [24]. It might be interesting to investigate this subtle point, and this is left for future studies.
We remark that the conformal anomalies a IR and c IR for 4d N = 1 SCFTs analyzed in this study can be computed. The value of the trial central charge, a IR (t), is expressed in terms of a U V , c U V , k U V and a parameter of the Jordan block structure pertaining to the superpotential deformation δW = T r G (φ(Z 1 , Z 2 ) · O) utilizing 't Hooft anomaly matching [46]. The conformal anomalies a IR and c IR are obtained by applying the method in [46], or the method in [125] with the geometric interpretation given in [126].