Four-dimensional N=1 theories, S-fold constraints on T-branes, and behaviors in IR and UV
KKEK-TH-2267
Four-dimensional N = 1 theories, S-fold constraints on T-branes,and behaviors in IR and UV Yusuke Kimura KEK Theory Center, Institute of Particle and Nuclear Studies, KEK,1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
E-mail: [email protected]
Abstract
We analyze four-dimensional (4d) N = 1 superconformal field theories (SCFTs) obtainedas deformations of 4d N = 2 SCFTs on S-folds by tilting 7-branes. Geometric compatibilitywith the structures of S-folds constrains the forms of T-branes. As a result, brane mon-odromies are constrained. We also discuss two 4d N = 1 theories on probe D3-branes, wherethe two theories behave identically in IR, but they flow to different theories in UV. Studyingthe global structure of their geometry is useful in constructing these two theories. a r X i v : . [ h e p - t h ] N ov ontents N = 1 theories as deformations of 4d N = 2 SCFTson S-folds 4 N = 3 theory and 4d N = 2 SCFTs on S-folds . . . . . . . . . . . . 42.2 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 7 N = 1 theories behaving identically in IR but behaving differently inUV 104 Puzzle in the global structure of the gauge group 125 Concluding remarks and open problem 14 There have been numerous advances in four-dimensional (4d) N = 2 supersymmetric gauge the-ories. Seiberg and Witten obtained the exact prepotential for low-energy effective theories [1, 2],which is among the major advances in this field. There have been vigorous efforts in the field sincethis progress; for example, [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].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 4dgauge theories with N = 3 supersymmetry by considering finite cyclic groups Z k acting on torusfibrations, wherein the Z k -actions also act on torus fibers. The 4d N = 2 SCFTs studied in [17]were obtained as deformations of 4d N = 3 theories, as constructed in [24].The subjects of study in this note are 4d N = 1 SCFTs on S-folds obtained as in the contextof deforming the 4d N = 3 theories constructed in [24]. It would be natural to consider deformingthe 4d N = 2 SCFTs in [17] to yield such superconformal field theories, and we take this approachto study 4d N = 1 SCFTs.There are several advantages to analyzing 4d N = 1 SCFTs in this manner. For example, theresulting 4d N = 1 SCFTs relate to 4d N = 2 and N = 3 gauge theories by construction, andthey are also relevant to studies of the structures of S-folds. Furthermore, as we discuss later inthis note, they are also related to T-branes [35] . Recent progress of T-branes can be found, e.g.,in [37, 38, 39, 40, 41, 42, 43, 44]. Properties of 4d N = 3 theories were discussed in [25]. Discussions of 4d N = 3 theories can also be found in[26, 27, 28, 29, 30, 31, 32, 33, 16, 34]. See also [16] for a discussion of N = 1 S-fold. Nilpotent Higgs fields on branes were discussed in [36].
1e 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-branesstack 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 bythis 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 the4d N = 1 SCFT on the D3-brane probing a stack of 7-branes. A D3-brane probes 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 obtainedas deformations of 4d N = 3 theories, as aforementioned. In the situation where φ and φ † donot commute, there is room to consider the effect of T-branes. Because the geometry consideredhere is an S-fold built as the quotient of elliptic fibration with finite cyclic group Z k actions, theequations 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 of7-branes, where the 7-brane stack is given by z = 0. Because the parameter z transverse to thestack of 7-branes now has an equivalence relation z ∼ e πik z (1)under the Z k action, a consistency condition is imposed on the spectral equation of T-branes tomake them compatible with the equivalence relation (1). A physical consequence of this compat-ibility argument is that brane monodromies [57, 59, 60, 61] are constrained owing to the quotientaction of the Z k groups.The other theme is to discuss the physical phenomenon where two 4d N = 1 SCFTs in the IRlimit are indistinguishable but behave quite differently in the ultraviolet (UV) limit. We utilize aglobal perspective of geometry in F-theory to observe this phenomenon. This phenomenon occursbecause the two globally distinct geometries have locally identical structures in a local patch thata D3-brane probes, making them appear the same in IR, but their difference becomes evident inUV.The S-fold constructions discussed in [24, 17] were obtained as Z k -actions on elliptic fibrationsover 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, ellipticfibrations can be constructed over a closed base space wherein the torus fibers have constant fixedcomplex structures over the base. Some nontrivial examples of such global geometries can befound, such as those in [62, 63, 64, 65, 66, 67]. Here, global geometries represent elliptic fibrationsover compact base spaces . Such global geometries have some applications in 4d N = 1 theories. Studies of D3-branes probing F-theory singularities can be found, for example, in [47, 48, 49, 50, 51]. For recent studies on 4d N = 1 theories, see, for example, [52, 53, 54, 55, 16, 56]. For example, an elliptic fibration over C is a “local” geometry because C is not closed; in contrast, as a base,an elliptic fibration over P describes a global geometry because P is closed.
2e can observe a physical phenomenon that might be interesting: one can construct two globalgeometries on which physics appear completely identical in the IR limit, but in the UV limit, theydescribe different physical theories. This is owing to the fact that while the two geometries havedifferent structures globally, their geometric structures appear locally identical. Physically, in theIR limit, they yield two identical 4d N = 1 SCFTs probed by D3-branes that flow to very differenttheories 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 compactificationspaces in F-theory, wherein a discrete gauge group arises, are genus-one fibrations that “lack aglobal section” [69]. By tuning the coefficients as parameters of the moduli of such geometrieswithout 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 onan elliptic fibration with global sections are very different; a discrete gauge symmetry arises inthe former geometry, while a discrete gauge symmetry does not form and U(1) does form in thelatter geometry. If we consider inserting a D3-brane probe into this geometric deformation, thecharacteristic physical phenomenon that we mentioned previously can be observed: the D3-braneis insensitive to the subtle deformation in geometry from genus-one fibration without a globalsection into an elliptic fibration with sections. When one moves to UV, the effect of the globalstructure of the geometry becomes explicit, and the corresponding physical theories before andafter 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 arisingfrom the geometry of the genus-one fibration lacking a global section are not reflected in 4d gaugetheory on the probe D3-brane.As a byproduct of the discussion of the second theme, we observe a puzzle in 4d SCFT concern-ing a subtlety in the structure of the gauge group. An elliptic fibration has the notion of a “globalsection.” 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 knownas a “Mordell–Weil group.” This group decomposes into a direct sum of free part Z r and torsionpart (cid:81) i Z i . In F-theory compactification [83, 84, 82] , the structure of the Mordell–Weil groupis reflected in the gauge group structure; the rank of the Mordell–Weil group yields the numberof U(1)s formed in F-theory [82], and the global structure of the gauge group is divided by theMordell–Weil torsion [111, 112, 113] . Is the structure of such Mordell–Weil torsion dividing thegauge group observed in “local” 4d SCFTs probed by D3-branes? Two opposing answers to thisquestion seem possible, as we will demonstrate. Because the two views are opposite, one mustchoose one of the possible two viewpoints. We discuss this puzzle in section 4. Local models of F-theory model constructions are discussed in [85, 86, 87, 88]. Studies of the compactificationgeometry 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]. The rank of the Mordell–Weil group is the number of copies of Z in the free part. Computations of the Mordell–Weil torsions of elliptic fibrations in F-theory compactifications and the globalstructures of the gauge groups that form on the 7-branes are also discussed in [64, 65, 78, 66, 67].
3n section 2.1, we briefly review the 4d N = 3 theories constructed in [24] and the 4d N = 2SCFTs realized in F-theory on S-folds in [17] as deformations of 4d N = 3 theories. We constructand analyze 4d N = 1 SCFTs as deformations of these theories in section 2.2. Tilting 7-branesdeforms 4d N = 2 SCFTs, which yields 4d N = 1 SCFTs in IR [46]. After we discuss thisconstruction, 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 analyzethe 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 butflow to very distinct theories in UV. In these constructions, considering the global aspects of thegeometry is essential. When a similar argument is applied to the global structure of the gaugegroup formed on 7-branes, we observe a puzzle, which we discuss in section 4. We present ourconcluding remarks in section 5 alongside some open problems. N = 1 theories as deforma-tions of 4d N = 2 SCFTs on S-folds 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 = 2SCFTs on S-folds constructed in [17]. We will discuss 4d N = 1 SCFTs obtained as deformationsof these theories, and T-branes and brane monodromies in the resulting 4d N = 1 theories insection 2.2. We review the 4d N = 2 and N = 3 theories in this section, as we require them toconstruct 4d N = 1 theories.F-theory compactification on the product C × T yields a 4d N = 4 theory. The coordinatesof C are denoted by z , z , and z , and the complex coordinate of the two-torus in the product isdenoted by z . The authors of [24] considered the following action of a finite cyclic group Z k onthe product C × T : z → e πik z (2) z → e − πik z z → e πik z z → e − πik z . Here, the complex structure τ of the two-torus as an elliptic fiber is constant over the base C ,so the product C × T is a direct product as a complex manifold. Because the Z k -action acts onthe two-torus as a fiber, it is necessary to confirm that the action is compatible with the ellipticfibration structure of C × T . It was shown in [24] that the compatibility conditions require theorder k of the cyclic group Z k to take one of the values k = 2 , ,
4, or 6, and the complex structureof 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. Anelliptic curve (two-torus seen as a complex curve) is described by a Weierstrass equation. A4eierstrass equation is an equation of the following form: y = x + f x + g. (3)Then, the complex structure of an elliptic curve as a fiber of the fibration is uniquely labeled bya j-invariant, which is an invariant under the isomorphisms of elliptic curves. A j-invariant j isgiven in terms of the Weierstrass coefficients f, g , as follows [114]: j = 1728 · f f + 27 g . (4)Modular parameter τ also specifies the complex structure of an elliptic curve. τ takes a valuein C modulo SL (2 , Z )-action.When the order k of the group Z k takes the values k = 3 , , N = 4 supersymmetry breaksdown 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-onefibration 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 K3surfaces, whose fibers have constant complex structures τ = exp ( πi ) and τ = i over the base P (equivalently, they have j-invariants 0 and 1728, respectively) can be found in [63, 64] . Theseglobal 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 compactificationson S-folds as deformations of the 4d N = 3 theories constructed in [24], and we briefly review thisconstruction.The S-folds in [17] were built by acting finite cyclic groups Z k on the space K ◦ × C . Here, K ◦ denotes an open patch in an elliptic K3 surface such that the base of K ◦ is C . K ◦ is anelliptic fibration over C . The coordinate of this base C is denoted as z , and the coordinates of C in the product K ◦ × C are denoted as z , z . z yields the coordinate transverse to the stacks of7-branes, and z , z yield the coordinates parallel to the stacks of 7-branes. z is identified witha chiral superfield, Z , which parameterizes the Coulomb branch, and z , z are identified with adecoupled hypermultiplet, Z , Z in the 4d theory on the probe D3-brane. The Z k -action actingon K ◦ × C is relaxed from condition (2) to the following condition [17]: z → e πik z (5) z → e i φ z z → e i φ z , where φ and φ are required to satisfy φ + φ = 0 mod 2 πn. (6) Higher dimensional generalizations to genus-one fibered Calabi–Yau 3-folds and genus-one fibered Calabi–Yau4-folds can be found in [65, 66, 67]. N = 2 SCFTs [17].Analogous to 4d N = 3 theories, compatibility conditions are imposed on the values of k andthe complex structures of the elliptic fibers. The complex structure of the fibers of the ellipticfibration K ◦ must be constant over the base C , i.e., the axio-dilaton must take a constant valueover C . ∆ is used to represent the scaling dimension of the Coulomb branch (CB) operatorfor the 7-brane in K ◦ × C . When Z k -groups act on the product to yield S-folds, the authorsof [17] argued that the theories on the resulting S-folds should have the scaling dimensions k ∆ and that these correspond to a phase rotation of supercharges. They deduced from these that k ∆ = 2 , , ,
6, and consequently, they found that the S-folds that are compatible with Z k -actionsare as follows [17]: Z S-folds with type H , D , E ; Z S-folds with type H , D Z S-folds with type H k ∆ [17].We remark here that the types of 7-branes for S-folds mentioned previously are the types beforethe quotient actions of Z k . The authors of [17] computed the Z k -quotients of the correspondingsingular 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 quotientaction 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 for elliptic surfaces is presented in Table 1.Because there are Z and Z S-folds with type H Z -and Z -quotients of type IV fibers can be considered. Type IV fibers have j-invariant 0 accordingto Table 1; therefore, Z - and Z -quotients of this fiber type must have j-invariant 0. The orderof the A singularity corresponding to type IV fibers is 4, and the Z -action doubles this order,so the resulting fiber type has a corresponding singularity type of order 8. Thus, we uniquelydetermine that the resulting fiber type is IV ∗ . This is consistent with the result given in [17].A similar reasoning applied to the Z -quotient appears to show that the resulting fiber type hasorder 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: f → z · f, g → z · g. (7)Because the discriminant is given by ∆ ∼ f + 27 g , this reparameterization reduces the orderof the singularity by six, and the actual order of singularity is 10. Therefore, the Z -quotient of atype IV fiber has a j-invariant whose corresponding singularity has order 10, and this is uniquelydetermined to be a type II ∗ fiber. This is also consistent with the result in [17]. Here H l , l = 1 ,
2, represents the Argyres–Douglas theories [4, 5, 6] that arise from an SU (2) theory possessing l + 1 flavors. [117, 118] discussed techniques to determine the types of the singular fibers of an elliptic fibration. I ∗ regular − (cid:18) (cid:19) D I b ∞ (cid:18) b (cid:19) infinite b A b − I ∗ b ∞ − (cid:18) b (cid:19) infinite b +6 D b +4 II (cid:18) − (cid:19) II ∗ (cid:18) −
11 1 (cid:19) E III (cid:18) − (cid:19) A III ∗ (cid:18) −
11 0 (cid:19) E IV (cid:18) − − (cid:19) A IV ∗ (cid:18) − −
11 0 (cid:19) E Table 1: Types of the singular fibers of elliptic surfaces and their properties [115, 116] and well-known brane interpretations. “Regular” for j-invariant of type I ∗ indicates the fact that j-invariantof type I ∗ fiber can take any finite value in C .A similar reasoning applies to the Z k -quotient of other fiber types. The results of the quotientfiber types obtained in [17] are perfectly consistent with the results deduced via the methoddescribed herein. 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 ofthe 4d N = 2 SCFTs on S-folds [17] that we reviewed previously. We require an operation thatbreaks N = 2 supersymmetry down to N = 1; we utilize an operation of tilting a stack of 7-branes, as described in [45, 46]. This operation corresponds to the Higgs field φ on the 7-branesstack acquiring a non-zero position-dependent vev [45]. Then, tilting the stack of 7-branes adds acorrection to the superpotential as T r G ( φ ( Z , Z ) · O ) [45]. When φ and φ † do not commute, the N = 2 supersymmetry is broken down to N = 1 [45].The adjoint-valued vev that φ takes, in the situation [ φ, φ † ] (cid:54) = 0, yields room to consider the7-brane structure [35]. When the stack of n z = 0, the T-brane is generallygiven by the following spectral equation [35]: z n + b z n − + b z n − + . . . + b n = 0 , (8)where z denotes the spectral parameter, and b , . . . , b n are polynomials in the variables of z and z . 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 equationcannot capture [39]. This physical degree of freedom in string theory is not fully determined bythe defining equation of the geometry and still receives constraints from the geometry when anS-fold is considered. In other words, the structure of the T-brane must be compatible with thequotient action of the Z k groups. We now explain this in more detail.The Z k quotient of K ◦ × C , k = 2 , ,
4, was used to construct S-folds in [17]. K ◦ denotes anopen 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 constructedin [17].The base of K ◦ as an elliptic fibration is isomorphic to C , and the coordinate of the base isdenoted by z . The coordinates of C in the product K ◦ × C are denoted by z , z . Z k actionare given by (5) [17], as mentioned in section 2.1, where phases φ , φ satisfy the relation (6).The product z z is invariant under Z k action (5). z is identified with e πik z under the quotientaction.Note here that when S-fold construction is considered, geometric conditions are imposed onthe equation (8). For a Z k S-fold, the coordinate z satisfies the equivalence relation z ∼ e πik z . (9)For simplicity, we focus on the situations where the coefficients b i , i = 2 , . . . , n , are polynomialsthat only depend on the product z z as a variable. Namely, we only consider the cases in whichthey are given as b i = (cid:88) j c ij ( z z ) j . (10)As stated previously, the product z z is invariant under Z k action (5). Under this assumption,conditions are imposed only on powers of z in the spectral equation (8) to make it compatiblewith 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: F ( z ) = z n + b ( z z ) z n − + b ( z z ) z n − + . . . + b n ( z z ) = 0 , (11)where b i ( z z ), i = 2 , . . . , n , are polynomials in z z . The form of the T-brane must be invariantunder Z k action (5). Because the coefficients b i ( z z ), i = 2 , . . . , n are invariant under Z k action(5), this is equivalent to the requirement that F ( z ) satisfies the following relation: F ( e πik · z ) = c F ( z ) . (12)8ere, c denotes a non-zero constant number. When this condition is satisfied, under the quotientaction of Z k , the equation F ( z ) = 0 is invariant; the form of the T-brane is unchanged under thequotient action.The condition (12) clearly constrains the form of the polynomial F ( z ) in the spectral equation.The physical consequence of this 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 branemonodromy. When conditions are imposed on the coefficients of the polynomial F ( z ), the as-sociated Galois group generally becomes smaller than the symmetric group S n . As a result, thebrane monodromy group becomes smaller owing to the compatibility condition (12). Althoughthe T-brane is a structure associated with the coincident 7-branes that the Weierstrass equationcannot 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 andbrane monodromies.As an example, we consider the T-brane structure in 4d gauge theory on an S-fold built froman open patch in K3 surface with an A singularity corresponding to type IV fiber.For this case, 7-branes in the open patch of the K3 surface have type H , and the type- H stack consists of four 7-branes. Z action acted on the open patch K3 ◦ times C yields Z S-fold.As we mentioned previously, the theory on probe D3-brane inserted in the resulting S-fold yieldsa 4d N = 2 SCFT , and considering T-brane structure breaks half of N = 2 supersymmetrywhich yields 4d N = 1 theory. The spectral equation of the T-brane in the resulting 4d N = 1theory is given by degree-four characteristic polynomial. The spectral equation must satisfy therelation (12) under the action of Z group.A T-brane given by the spectral equation of the following form is invariant under the Z -action: F ( z ) = z + a z + b = 0 . (13) a, b are polynomials in z z 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 ,the dihedral group of order 8. Therefore, the 7-brane monodromy group for the 4d N = 1 theoryobtained from the Z S-fold is D .Utilizing an open patch in K3 surface with an A singularity corresponding to a type IV fiber,one can also construct a Z S-fold. For this construction, a T-brane described by the followingequation is invariant under the Z action: F ( z ) = z + a = 0 . (14)The Galois group of this polynomial is isomorphic to Z , and the 7-brane monodromy group forthe 4d N = 1 theory obtained from the constructed Z S-fold is Z . This theory flows to a 4d theory with N = 4 supersymmetry, instead of N = 3 theory [24, 17]. Two 4d N = 1 theories behaving identically in IR butbehaving differently in UV We discussed structures of some 4d N = 1 SCFTs probed by D3-branes in section 2. The D3-brane only probes a local small patch of the geometry, and theories behaving identically in IRdo not necessarily behave identically in UV. In this section, we aim to provide two 4d N = 1theories that behave identically in IR but differently in UV by analyzing the geometry from aglobal perspective. When the global structures of the geometries are distinct, as theories flow tothe 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 fibrationswith 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, asspecial cases of such geometries. As constructed in [63, 64], there are genus-one fibrations lackinga global section whose fibers have constant j-invariants 0 and 1728 throughout the base spaces (orequivalently, having complex structures τ = exp ( πi ) and τ = i ). One can construct “closed” Z k S-folds by acting Z k groups on these genus-one fibrations. Because the D3-brane only probesa local open patch, theories on such closed S-folds in the IR limit reduce to 4d N = 1 SCFTs onS-folds, as discussed in section 2.We have stated the capacity to which our argument in this section applies. Before discussingthe central point of our argument, we explain the physical meanings of multisection geometriesand elliptic fibrations with multiple sections. It is known that a discrete gauge group arises onmultisection geometry in F-theory [69]. A multisection is a multifold cover of the base space ofa 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]. Thespecial 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 multisectionto special values, a multisection splits into multiple global sections. The number of independentglobal sections that an elliptic fibration possesses minus one yields the number of U(1)s formed inF-theory on that elliptic fibration, as discussed in [82]. Therefore, when an n -section splits into n global sections, a U(1) n − forms in F-theory. Tuning an n -section geometry to an elliptic fibrationwith n sheets of global sections can be viewed as the reverse of the Higgsing process, wherein amodel with U(1) n − 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 ofthe geometry, it is insensitive to tuning of a genus-one fibration with an n -section into an ellipticfibration 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 locallyindistinguishable. 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 locidiminish and eventually disappear, the n -section splits into n separate global sections. When one “Closed” means that they have compact base spaces. 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 doesnot 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 separateglobal sections while maintaining the type of the 7-branes stack at which the probe D3-brane isplaced. The probe D3-brane is thus insensitive to this deformation in the geometry, as we havedemonstrated.In particular, this means that in the IR limit, the D3-brane does not probe the Higgsingprocess of U(1) n − breaking down into a Z n gauge group occurring through the global geometricdeformation in the moduli. U(1) or Z n gauge groups arising from the global feature of the geometryin 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 onan elliptic fibration with n global sections, the two theories on the D3-brane probes flow to theidentical theory in IR. In UV, differences in the global geometric structures are probed and flowto distinct theories, reflecting the difference between U(1) n − and Z n gauge groups.We would like to demonstrate that it is possible to deform a multisection geometry to anelliptic fibration with multiple global sections while maintaining the type of stack of the 7-branesat which the D3-brane probes. We show this for the case of a bisection geometry [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 studiedin [64]: τ = (cid:89) i =1 ( t − α i ) x + (cid:89) j =5 ( t − α j ) , (15)where t and x represent inhomogeneous coordinates of P , and each α i , i = 1 , . . . ,
8, is a point in P . Thus, the equation (15) yields a double cover of P × P branched over a (4,4) curve, whichis a K3 surface [64]. A projection onto P in the product P × P yields a genus-one fibration, asdiscussed in [64]. We regard t as the coordinate of the base P of the genus-one fibration. TheK3 surface (15) is a bisection geometry [64]. The elliptic fibers of this fibration have a particularsymmetry, which forces the complex structure of the fibers to be constant with τ = i throughoutthe base P [64]. This constrains the types of 7-branes so that the fiber types over the 7-branescan only be III , I ∗ , or III ∗ [64].We focus on the case of α = α = 0 here. Then, the equation of the K3 surface (15) becomes τ = (cid:89) i =1 ( t − α i ) x + t ( t − α )( t − α ) . (16) Bisection refers to an n -section with n = 2. Equivalently, they have j-invariant 1728.
11e only consider the cases where α i , i = 1 , , ,
4, are mutually distinct and α i , i = 1 , , , , , ∼ (cid:89) i =1 ( t − α i ) · t ( t − α ) ( t − α ) . (17)7-branes at the origin t = 0 have type D , and each stack of 7-branes at t = α i , i = 1 , , ,
4, hastype H [64]. 7-brane stacks at t = α , α have type H when α and α are not equal. At thelimit at which α and α coincide, the 7-brane type at t = α is enhanced to D .For the reader’s convenience, we present the Jacobian fibration of the double cover (16). TheJacobian fibration admits transformation to the following Weierstrass form [119]: y = 14 x − (cid:89) i =1 ( t − α i ) t ( t − α )( t − α ) x. (18)We consider F-theory on a K3 surface (16) times a K3 surface. We consider the situationwhere the probe D3-brane is placed at the origin t = 0. We consider tilting the stack of 7-branesat the origin t = 0 to yield a 4d N = 1 SCFT, as discussed in section 2. For general parameters α i , i = 1 , , , , ,
8, the K3 surface (16) has a bisection, but it does not have a global section,and a Z gauge group forms in F-theory. As we discussed previously, this is not probed by theD3-brane in IR.One can tune the parameters so a bisection splits into two global sections. A bisection splitsinto global sections when either the coefficient of x or constant term in the quartic polynomialbecomes a perfect square [69]. This occurs when α and α become coincident in (16). Therefore,at the limit at which α and α coincide, the bisection splits into two global sections. Then,U(1), instead of the Z gauge group, forms in F-theory, but the theory on the D3-brane probe isinsensitive to this change.One can choose α and α far away from the origin so they do not lie in the local patch thatthe D3-brane probes. Tuning α such that it approaches α does not affect the singularity typeat the origin t = 0.We particularly considered a bisection given by a specific form of an equation, but a similarreasoning applies to general bisection geometry. When an elliptic fibration has a section, the set of sections is known to form a group, and thisgroup is referred to as the “Mordell–Weil group.” When the Mordell–Weil group of an ellipticfibration has a torsion part, the torsion part divides the global structure of the gauge group formedin F-theory on that elliptic fibration [111, 112, 113].There are non-isomorphic elliptic K3 surfaces whose singular fiber types are identical and F-theory compactifications yield identical 7-brane types but their Mordell–Weil torsions are different.12e explicitly give an example of this. We utilize extremal K3 surfaces . A complex ellipticallyfibered K3 surface f : S → P with a global section is referred to as extremal when the Picardnumber of K3 surface S is 20 and the Mordell–Weil group, M W ( S, f ), is a finite group. ComplexK3 surfaces with the Picard number of 20 are referred to as attractive K3 surfaces .It is a mathematical fact that the complex structures of the attractive K3 surfaces are labelledby triplets of integers [122, 123]. The complex structures, singularity types, and Mordell–Weiltorsions of the K3 extremal fibrations were completely classified in [124]. A pair of non-isomorphicK3 surfaces whose singularity types are identical but Mordell–Weil torsions are different can befound in Table 2 in [124]; numbers are assigned to the singularity types of the K3 extremalfibrations in this table. No. 276 in this table corresponds to the singularity type E A A . Thereare two K3 surfaces possessing this singularity type according to the table, and they have differentcomplex structures, so they are non-isomorphic. F-theory on either of these two K3 surfaces timesa K3 surface yields a 4d N = 2 theory, and an E × SU (10) × SU (3) gauge group forms in bothcompactifications. However, one of the extremal K3 surfaces has no Mordell–Weil torsion, whilethe other K3 surface has Mordell–Weil torsion isomorphic to Z , so the gauge groups formed in thetwo theories are globally different. These theories flow to different theories in UV, possessingdistinct global structures of the gauge groups.A natural question is whether the Mordell–Weil torsional group dividing the gauge group isreflected in the theory on the D3-brane probe, which probes only a local patch. Let us take alocal patch of an elliptic fibration and place a probe D3-brane in the neighborhood of a stack of7-branes. Locally, one cannot see the structures of global sections; thus, the Mordell–Weil groupcannot be seen locally. Then, it is natural to consider that the Mordell–Weil torsions also cannotbe locally seen. A consequence of this reasoning seems to be that the effect of Mordell–Weil torsionis 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 F-theory, 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 thegauge group formed on the 7-branes that the D3-brane probes.Because these two viewpoints provide opposing interpretations, one must choose one of thetwo interpretations. The question of which one gives the correct physical interpretation is left forfuture study. Discussions of extremal elliptic K3 surfaces in the context of string theory can be found, e.g., in [63, 64, 65,66, 67, 120]. We follow the convention of the term used in [121]. We chose an example of extremal K3 fibrations, whose classification is known. However, the appearance oftwo non-isomorphic K3 surfaces whose singularity types are identical while having distinct Mordell–Weil torsionsis not limited to extremal K3 surfaces, and such cases typically arise in the whole moduli of algebraic K3 surfaces. Concluding remarks and open problem
In this note, we analyzed 4d N = 1 SCFTs obtained as deformations of 4d N = 2 SCFTs onS-folds by tilting 7-branes. We showed that T-branes receive constraints from the geometry whenS-fold constructions are considered. Consequently, brane monodromies are also constrained bythe geometric conditions, and we presented explicit examples of this.We also discussed two 4d N = 1 SCFTs behaving identically in IR but flowing to distincttheories in UV. To construct these theories, utilizing the global structure of genus-one fibrationslacking a global section was useful. Our argument also showed that U(1) and discrete Z n groupsarising from the global feature of the geometry in F-theory are not reflected in 4d N = 1 SCFTson the probe D3-brane.We encountered a dilemma when considering an effect of the global geometry on the theoryliving on the probe D3-brane, i.e., the action of the Mordell–Weil torsional group on the gaugegroup. Two opposing viewpoints seem possible for this problem, and which viewpoint yields acorrect interpretation is left to be determined in future work. Acknowledgments
We would like to thank Yosuke Imamura, Shun’ya Mizoguchi and Shigeru Mukai for discussions.
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