SU(n)× Z 2 in F-theory on K3 surfaces without section as double covers of Halphen surfaces
KKEK-TH 2057 SU ( n ) × Z in F-theory on K3 surfaces without section as double covers ofHalphen surfaces 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 investigate F-theory models with a discrete Z gauge symmetry and SU ( n ) gaugesymmetries. We utilize a class of rational elliptic surfaces lacking a global section, knownas Halphen surfaces of index 2, to yield genus-one fibered K3 surfaces with a bisection,but lacking a global section. We consider F-theory compactifications on these K3 surfacestimes a K3 surface to build such models. We construct Halphen surfaces of index 2 withtype I n fibers, and we take double covers of these surfaces to obtain K3 surfaces withouta section with two type I n fibers, and K3 surfaces without a section with a type I n fiber. We study these models to advance the understanding of gauge groups that formin F-theory compactifications on the moduli of bisection geometries.Our results also show that the Halphen surfaces of index 2 can have type I n fibers upto I . We construct an example of such a surface and determine the complex structureof the Jacobian of this surface. This allows us to precisely determine the non-Abeliangauge groups that arise in F-theory compactifications on genus-one fibered K3 surfacesobtained as double covers of this Halphen surface of index 2, with a type I fiber timesa K3 surface. We also determine the U (1) gauge symmetries for compactifications whenK3 surfaces as double covers of Halphen surfaces with type I fiber are ramified over asmooth fiber. a r X i v : . [ h e p - t h ] J u l ontents I , I , I , I fibers 5 P × P . . . 52.2 Construction of Halphen surfaces with type I , I , I fibers . . . . . . . . . . . 72.2.1 Halphen surface with I fiber . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Halphen surface with I fiber . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Halphen surface with I fiber . . . . . . . . . . . . . . . . . . . . . . . 102.3 Construction of Halphen surface with type I fiber . . . . . . . . . . . . . . . . 102.4 Upper bound on the degree of A n singularity of a Halphen surface of index 2,and the determination of the complex structure of the Jacobian fibration of aHalphen surface with I fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 The Tate-Shafarevich groups and the Weil-Châtelet groups of the Jacobians ofthe Halphen surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 I , I , I fibers . . . . . . . . . . . . 203.3 Gauge groups in F-theory compactifications on K3 surfaces without a sectionas double covers of Halphen surfaces with I fiber . . . . . . . . . . . . . . . . 21 U (1) gauge symmetries 22 I fiber, and U (1) gauge symmetries in F-theory compactifications . . . . 22 Introduction
Building models in particle physics using the F-theory [1, 2, 3] approach has several advan-tages; in this approach, the SU (5) GUT model is naturally realized with matter fields in SO (10) spinor representation. Additionally, the F-theory approach can avoid the problemof weakly coupled heterotic string theory, addressed in [4]. Furthermore, up-type Yukawacouplings can be generated in this approach without difficulty. Local models [5, 6, 7, 8] havebeen mainly considered in recent studies of F-theory. Gravity, however, decouples when thelocal models are considered. Therefore, global models need to be investigated to discuss theproblems related to gravity, such as the inflation. In this note, we analyze the global geometricstructures of the compactification spaces in F-theory.Compactification spaces in F-theory have the structure of a genus-one fibration. Thecomplex structure of a torus as a fiber of a genus-one fibration of a compactification spaceis identified with the axio-dilaton, enabling the axio-dilaton to possess SL (2 , Z ) monodromy.7-branes in F-theory are wrapped on the components of the locus in the base space of a genus-one fibration, over which the fiber degenerates and becomes singular, namely the discriminantlocus . The gauge symmetries and matter that arise are determined from the structure of thegenus-one fibration of the compactification space. [9, 10] classified the types of singular fibersof elliptic surfaces . The types of the singular fibers of a genus-one fibration correspond to thenon-Abelian gauge groups that arise on the 7-branes in F-theory compactification [3, 13]. Thisrelationship is summarized in Table 1 below as the correspondence of the singularity types ofthe compactification space and the types of the singular fibers of a genus-one fibration thatthe compactification space admits.Type ofsingular fiber Singularity type I n ( n ≥ ) A n − I ∗ m ( m ≥ ) D m III A IV A IV ∗ E III ∗ E II ∗ E I none. II none.Table 1: The types of the singular fibers and the corresponding singularity types of thecompactification space. [11, 12] discussed techniques to determine the type of singular fibers of elliptic surfaces. arises in these models. The mechanism which accountsfor the origin of discrete gauge symmetry in the moduli of F-theory on genus-one fibrationsis discussed in [28]. The Tate–Shafarevich group of (the Jacobian fibration of) a genus-onefibration and the discrete gauge symmetry that arises in the model on the genus-one fibrationare identified [56]. When a genus-one fibered Calabi–Yau manifold M is given, the Jacobianfibration J ( M ) of it is considered. A genus-one fibered Calabi–Yau manifold M and the Jaco-bian fibration J ( M ) have the identical τ functions. The Calabi–Yau genus-one fibrations, theJacobian fibrations of which are isomorphic to J ( M ) , form a group. This group is referredto as the Tate–Shafarevich group X ( J ( M )) . A discrete gauge group that arises in F-theorycompactification on the Calabi–Yau genus-one fibration M is given by the Tate–Shafarevichgroup X ( J ( M )) .In general, a discrete Z n symmetry arises in F-theory compactification on a genus-onefibration which possesses an n -section. F-theory models with discrete gauge symmetries arediscussed, for example, in [28, 30, 31, 32, 33, 52, 34, 35, 39, 40]. Discrete Z , Z , Z , Z symmetries are mainly studied in these constructions.The structure of the genus-one fibration, including a multisection that it contains, needsto be analyzed to deduce gauge groups and matters that arise in F-theory compactifications.The demonstration of the existence of a model in F-theory with a discrete gauge symmetrywith a specific gauge group is non-trivial. We show the existence of some models with adiscrete Z symmetry with specific gauge groups in the moduli of F-theory.In this note, we construct several models of F-theory compactifications with a discrete Z symmetry with type I n fibers. We advance the understanding of models with type I n fibers in the moduli of F-theory compactified on bisection geometries. Concretely, we firstconstruct surfaces that belong to a class of rational elliptic surfaces without a global section,known as Halphen surfaces of index 2. We explicitly construct examples of these surfaceswith type I , I , I , and I fibers. By utilizing these surfaces, we yield K3 surfaces thatlack a global section but have a bisection. F-theory compactification on the resulting K3surfaces times a K3 surface show the existence of models with SU ( n ) × Z and SU (2 n ) × Z gauge groups, n = 4 , , , . This result can advance the understanding of the non-Abeliangauge symmetries that arise in the moduli of F-theory with a discrete Z gauge symmetry onbisection geometries.Rational elliptic surfaces admit a genus-one fibration, but do not necessarily have a globalsection. Halphen surfaces are examples of rational elliptic surfaces that lack a global section. See, e.g., [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55] for recent studies of discrete gaugesymmetries. Discussions on the structure of Halphen surfaces can be found in [57, 58]. An application of Halphen P × P at 8 points of thespecific configuration: We consider a bi-degree (4,4) curve in P × P , k = 0 , with eight simplesingularities (the curve k = 0 may have more singularities, other than the eight singularities).We choose a smooth bi-degree (2,2) curve, l = 0 , which passes through these eight simplesingularities. The process of blowing up these eight singular points yields a Halphen surface.The projection onto P induced by taking the ratio [ k : l ] yields a genus-one fibration. Theexceptional divisors that arise when the eight singularities are blown up yield bisections tothe fibration. The structure of the Halphen surfaces of index 2 is reviewed in section 2.1. Itis explained there that Halphen surfaces do not have a global section.In this study, we construct several examples of Halphen surfaces of index 2 with type I n fibers. We mainly consider the blow-ups of P × P at 8 points to construct Halphen surfacesof index 2. These constructions yield Halphen surfaces of index 2 with I , I , and I fibers.We also consider a blow-up of P at 9 points to yield a Halphen surface of index 2 with an I fiber. We consider the case in which the polynomial k = 0 is reducible into lines or curves toconstruct these surfaces. Specific configurations of these lines and curves yield type I n fibers after blow-ups, n = 4 , , , .Taking double covers of the examples of the Halphen surfaces of index 2 that we constructin this study yields genus-one fibered K3 surfaces which lack a global section . The resultingK3 surfaces are bisection geometries. F-theory compactifications on these K3 surfaces times aK3 surface reveal the existence of models in which a discrete Z symmetry and SU ( N ) gaugegroup arise.We will show in section 2.4 that when a Halphen surface of index 2 possesses a type I n fiber, the upper bound on the degree n is 9. The Halphen surface of index 2 with a type I fiber that we construct in this study provides such an example. This fact imposes someconstraints on the non-Abelian gauge symmetries that arise in F-theory compactification ongenus-one fibered K3 surfaces constructed as double covers of Halphen surfaces with type I n fibers times a K3 surface.This study is structured as follows: In section 2, after we briefly review the structure ofHalphen surfaces of index 2, we construct Halphen surfaces of index 2 with type I n fibers, n = 4 , , , . The constructions of these surfaces are discussed in section 2.2 and section 2.3.We also show in section 2.4 that when a Halphen surface of index 2 has an A n singularity, thehighest is A . Thus, a Halphen surface of index 2 can have a type I m fiber up to an I fiber.The Tate–Shafarevich groups of the Jacobian fibrations of the Halphen surfaces of index 2are trivial. For the Halphen surfaces, the information of the multisections that they possessis contained in the Weil–Châtelet groups of their Jacobians. We discuss this in section 2.5.In section 3, we construct genus-one fibered K3 surfaces lacking a section using examplesof Halphen surfaces of index 2 as described in section 2. There are two types of constructions surfaces of index 2 to string theory is discussed in [40]. Halphen surfaces of index 2 with type I m fibers, m = 2 , , , , are constructed in [40]. The K3 surfaces with involution are considered in [59]. These K3 surfaces belong to this class. Similar constructions obtained by considering double covers of Halphen surfaces of index 2, realized asblow-ups of P at nine points can be found in [40].
4f K3 surfaces. These two types of constructions consider taking double covers of Halphensurfaces that are obtained in section 2.2 and section 2.3, ramified over either a smooth fiber,or a singular fiber. Consequently, we obtain two types of K3 surfaces without a section.The two constructions yield K3 surfaces with type I n fibers, and K3 surfaces with type I n fibers. These are bisection geometries. The two constructions are presented in section 3.1. Insection 3.2 and section 3.3, we discuss F-theory compactifications on the genus-one fibered K3surfaces lacking a section as described in section 3.1 times a K3 surface. This yields a four-dimensional theory. We study the gauge symmetries that arise in these compactifications.Because the constructed K3 surfaces are bisection geometries, discrete Z gauge symmetriesarise in these models.We discuss the Jacobian fibrations of Halphen surfaces of index 2, and the constructedK3 surfaces as described in section 3 in section 4. These surfaces have bisection geometries,therefore, as discussed in [27], the Jacobian fibrations of these surfaces always exist. By takingthe Jacobian fibration, the double fiber of a Halphen surface of index 2 becomes a smoothfiber. In section 4.2, we determine the Weierstrass equation of the Jacobian fibration of a K3surface obtained as a double cover of the Halphen surface with a type I fiber ramified over asmooth fiber. We also determine the Mordell–Weil rank of this Jacobian fibration. Using thisresult, we deduce that there is no U (1) gauge symmetry in F-theory compactification on thedirect product of a K3 surface obtained as a double cover of the Halphen surface with a type I fiber, ramified over a smooth fiber, times a K3 surface. We state our concluding remarksin section 5. I , I , I , I fibers P × P We review the structure of Halphen surfaces of index 2 that are constructed as a blow-up of P × P at eight points. These surfaces are genus-one fibered and have a bisection, but theylack a global section.We take a curve, k = 0 , of bi-degree (4,4) in P × P that has 8 simple singularities. (Thecurve k = 0 can have more singularities, other than the 8 singularities.) We choose a smoothbi-degree (2,2) curve, l = 0 , which passes through the 8 singularities of the curve k = 0 . The(2,2) curve in P × P has 9 monomials, therefore, a smooth (2,2) curve that passes throughthe fixed 8 points always exists.The blow-up of P × P at the 8 simple singularities of the (4,4) curve k = 0 gives a rationalsurface. The ratio of the (4,4) curve k = 0 to the square of the (2,2) curve l = 0 , [ k : l ] ,gives a projection onto P . This map endows the rational surface obtained as the blow-upof P × P at the 8 simple singularities of the curve k = 0 with a fibration structure. Therational surface obtained as the blow-up of P × P at the 8 simple singularities together with Discussion of the Jacobian fibrations of elliptic curves can be found in [60]. [ k : l ] , is called a Halphen surface of index 2 .The curve k = 0 yields a divisor belonging to the following complete linear system: | H + 4 H − i =1 P i | , (1)where we have used H to denote the line class that the first P in the product P × P defines,and H to denote the line class that the second P in P × P defines. We used P i to denote theexceptional divisors that arise when the 8 simple singularities are blown up. The bi-degree(2,2) curve l = 0 defines a divisor which belongs to the following complete linear system: | H + 2 H − Σ i =1 P i | . (2)A fiber F of the projection [ k : l ] onto P is given by k + λ l = 0 , (3)where λ denotes a constant. This is a (4,4) curve with eight simple singularities. Therefore,by the genus formula, the genus g ( F ) of a fiber (3) is given by: g ( F ) = (4 − − − δ = 9 − . (4) δ = 8 is the number of singularities of a fiber (3). This shows that a generic fiber of theprojection [ k : l ] is a genus-one curve, namely, the projection [ k : l ] is a genus-one fibration.The following equation describes the fiber over the point [ a : b ] in the base P : a · l − b · k = 0 . (5)The class [ k = 0] represents the fiber at the origin [0 : 1] of the base P , and the class [ l = 0] ∼ · [ l = 0] represents the fiber at the infinity [1 : 0] . Fibers of a rational surface arelinearly equivalent. Thus we obtain the following linear equivalence relations: [ k = 0] ∼ · [ l = 0] ∼ F. (6) F in (6) denotes the fiber class. The linear equivalence relations (6) means that the intersec-tion number of any divisor with the fiber F is a multiple of 2. A global section and the fiber F have the intersection number 1. Thus, it follows that a Halphen surface of index 2 doesnot have a global section.Blowing up the 8 singularities of the curve k = 0 yields bisections to the genus-one fibrationof a Halphen surface of index 2. The fiber at the infinity [1 : 0] of the base P is a uniquedouble fiber.In section 2.2 and section 2.3, we particularly consider the cases in which the (4,4) curve k = 0 is reducible into curves of bi-degree (1,0), (0,1), or (1,1) to construct Halphen surfacesof index 2 with type I n fibers, n = 4 , , , . Halphen surfaces of index 2, constructed as blow-ups of P at nine singularities are reviewed in [40]. .2 Construction of Halphen surfaces with type I , I , I fibers We construct Halphen surfaces of index 2 with type I , I , and I fibers. We consider a blow-up of P × P at 8 points, and we choose specific curves k = 0 to realize these constructions. I fiber We choose the bi-degree (4,4) polynomial k as the product of four irreducible (1,1) curves k , k , k , and k : k = k k k k . (7)Each pair of distinct (1,1) curves, k i and k j , i (cid:54) = j , intersect at 2 points in P × P . Wehave (cid:0) (cid:1) = 6 pairs of bi-degree (1,1) curves. Therefore, we have 12 intersection points intotal. These are the simple singularities of the (4,4) curve k = 0 . We choose four pointsamong these 12 intersection points so that the bi-degree (1,1) curves passing through thesefour points form a quadrangle. We show the image of four chosen points and the bi-degree(1,1) curves k i = 0 , i = 1 , , , in Figure 1. The four chosen points are not blown up, and weconsider the blow-up of the remaining eight points. Because each bi-degree (1,1) curve k i = 0 , i = 1 , , , , has the genus (1 − −
1) = 0 , bi-degree (1,1) curve k i = 0 is rational, i.e,it is isomorphic to P . Therefore, this construction yields a Halphen surface of index 2 withtype I fiber at the origin of the base P under the projection [Π i =1 k i : l ] . I fiber We choose the bi-degree (4,4) curve k as the product of three bi-degree (1,0) curves, k , k , k ,three bi-degree (0,1) curves, k , k , k , and a bi-degree (1,1) curve k : k = Π i =1 k i . (8)We assume that these seven curves, k , · · · , k , are in a general position. A distinct pair oftwo bi-degree (1,0) curves do not intersect. Similarly, a distinct pair of two bi-degree (0,1)curves do not meet. A pair consisting of a (1,0) curve and a (0,1) curve meet at 1 point. Thereare 9 pairs of a (1,0) curve, k i , i = 1 , , , and a (0,1) curve, k j , j = 4 , , . The bi-degree(1,1) curve k and a bi-degree (1,0) curve or (0,1) curve, k i , i = 1 , · · · , , intersect at 1 point.Therefore, we have intersection points in total. We choose seven points amongthese 15 intersection points, so that the curves k i passing through the seven chosen pointsform a 7-gon. The chosen seven points are not blown up, and we blow up the remaining 8points. The image of the seven points and the seven irreducible curves, k , · · · , k , are shownin Figure 2. Each of the seven irreducible curves, k , · · · , k , has the genus 0. Therefore, theyare each isomorphic to P . This construction yields a Halphen surface of index 2, with a type I fiber at the origin under the projection [Π i =1 k i : l ] . Curves of bi-degrees ( a, b ) and ( c, d ) in P × P have the number of intersections as ad + bc . A bi-degree ( a, b ) smooth curve in P × P has the genus ( a − b − . k K K Figure 1: Configuration of the four (1,1) curves, k , k , k , k , and twelve intersection points.The blue dots represent the four points which are left un-blown up. The four (1,1) curvespassing through the four points form a quadrangle. The remaining eight intersections areblown up. 8 K K K K K K Figure 2: Configuration of the lines and the (1,1) curve, k , · · · , k , and their fifteen inter-sections. The seven blue dots represent the seven points which are chosen to not undergoblowning up. The lines and the (1,1) curve passing through these chosen points form a 7-gonas indicated by the blue lines and the blue curve in the image.9 .2.3 Halphen surface with I fiber There are at least two ways to construct a Halphen surface of index 2 with a type I fiber. Oneconstruction is given as follows: we choose the bi-degree (4,4) curve k as the product of fourbi-degree (1,0) curves, k , · · · , k , and four bi-degree (0,1) curves, k , · · · , k . These curveshave × intersection points. We choose eight points among these 16 intersection points,so that the irreducible curves k i , i = 1 , · · · , , passing through the chosen eight points forman octagon. The chosen eight points are not blown up, and we blow up the remaining eightpoints. The image of the chosen eight points that were not blown up, and the eight irreduciblecurves k , · · · , k , are shown in Figure 3. This construction yields a Halphen surface of index2, with an I fiber at the origin of the base P under the projection [Π i =1 k i : l ] .Another construction of a Halphen surface of index 2 with a type I fiber arises from theconsideration of a special configuration of the seven irreducible curves k , · · · , k , as describedpreviously in section 2.2.2, to construct a Halphen surface of index 2 with a type I fiber. Weconsidered a general configuration of the seven curves k , · · · , k , so that the bi-degree (1,1)curve meet with the six curves k , · · · , k at six points in the previous section 2.2.2. Here, weconsider a special configuration in which the (1,1) curve k passes through the intersectionpoint of the line k and the line k . We refer to this intersection point of the curves k , k ,and k as p . We then consider the blow-up of the intersection point p . This operation yieldsan exceptional divisor E p ∼ = P at the point p , and this separates the three curves k , k ,and k . Each of these curves intersects the exceptional divisor E p at 1 point. These threepoints together with the other 12 intersections of the curves give 15 intersection points intotal. Eight specific points among these are chosen so that the curves passing through thechosen eight points form an octagon. The exceptional divisor E p was utilized as an edge ofthis octagon. The chosen eight points were not blown up, and we blew up the remainingseven points . The image of the configuration of the curves forming an octagon is shown inFigure 4. This yields a Halphen surface of index 2 with a type I fiber at the origin of thebase P under the projection [Π i =1 k i : l ] . I fiber As described in [40], a Halphen surface of index 2 can also be constructed by taking a sexticcurve k in P with 9 simple singularities , and by blowing up P at these 9 simple singularities.We choose a smooth cubic curve l that passes through the 9 simple singularities of the curve k , and a genus-one fibration of the constructed Halphen surface is given by the projection [ k : l ] onto P . Using this type of construction of a Halphen surface of index 2, we yield aHalphen surface of index 2 with a type I fiber . We once blew up the point p . Therefore, the remaining number of blow-ups required to yield a Halphensurface is eight minus one, i.e., 7. As explained in [40], the sextic curve k can have more singularities, other than the 9 singularities. By considering the construction of a Halphen surface as the blow-up of P at nine simple singularities ofa sextic curve, Halphen surfaces of index 2 with type I , I , I , I , I fibers are constructed in [40]. K K K K K K K Figure 3: Configuration of the eight lines, k , · · · , k . These lines have sixteen points ofintersection in total. The eight blue dots represent the eight points which are chosen toremain un-blown up. As the blue lines in the image show, the lines passing through thechosen eight points form an octagon. 11 K K K K K K K P Figure 4: The blue circle in the image numbered 3 represents the exceptional divisors E p .We use the exceptional divisors E p as an edge to form an octagon. The curves used as theedges of the octagon are numbered with the blue numbers. We choose the two points, theintersections of E p with the curves k and k , to not undergo blowning up. The six blue dotsin the image represent the remaining six points that are chosen to remain un-blown up.12e consider the situation in which the sextic curve k is reducible into six lines k i , i =1 , · · · , : k = Π i =1 k i . (9)We consider a specific configuration of the six lines k i , i = 1 , · · · , , as shown in Figure 5.At each of the three points P, Q, R in Figure 5, three lines intersect at 1 point. Blowing upeach of these points,
P, Q, R , yields the exceptional divisors E P , E Q , and E R , each of whichis isomorphic to P . This operation separates three lines that meet at 1 point. Each of thesethree lines meets an exceptional divisor at 1 point. We show the configuration of the threelines and the exceptional divisor E P when the blow-up is performed at the point P in Figure6. The situations are identical for the points Q, R . After the three points
P, Q, R are blownup, there are 9 points for the intersections of the separated lines and the three exceptionaldivisors, E P , E Q , and E R . Together with the other six intersections of the lines, there are 15intersection points in total after the three points P, Q, R are blown up. We chose nine specificpoints among the 15 intersection points so that the lines and the exceptional divisors passingthrough these chosen points form a 9-gon. The exceptional divisors E P , E Q , and E R wereused as three edges among the nine edges forming this 9-gon. We show the configuration ofthe nine chosen points and lines forming the 9-gon in Figure 7. The chosen nine points werenot blown up, and we blew up the remaining six points. This yields a Halphen surface ofindex 2 with a type I fiber at the origin of the base P under the projection [Π i =1 k i : l ] . A n singularity of a Halphen sur-face of index 2, and the determination of the complex structureof the Jacobian fibration of a Halphen surface with I fiber We constructed Halphen surfaces of index 2 with A n type singularities. By considering theJacobian fibration, we show that a Halphen surface of index 2 with the singularity type A or with A n type singularity of higher degree does not exist. Halphen surfaces of index 2 arebisection geometries. Therefore, as discussed in [27], the Jacobian fibration of a Halphensurface of index 2 always exists. The Jacobian fibration of a Halphen surface is a rationalelliptic surface with a global section. By the Shioda–Tate formula [61, 62, 63], the followingequality holds for the rank of the Mordell–Weil group and the rank of the singularitytype of the Jacobian fibration of a Halphen surface: rk MW + rk ADE = 8 , (10) We blew up P at the three points, P, Q, R , therefore, the number of blow-ups remaining to yield aHalphen surface is nine minus three, i.e., six. This equality is used to yield several families of rational elliptic surfaces with a section with variousMordell–Weil ranks in [25]. Identical pairs of these surfaces are glued to obtain elliptic K3 surfaces, on which U (1) gauge symmetries of various ranks arise in F-theory compactifications, in [25]. [64] classified the Mordell–Weil groups of rational elliptic surfaces that admit a global section. Q R
Figure 5: Configuration of the six lines, k , · · · , k . Three lines meet at one point at each ofthe three points P, Q, R . 14 P Figure 6: Blow-up at P separates the three lines that meet at one point at P . The verticalline represents the exceptional divisor E P that arises after blow-up. The horizontal threelines are the separated three lines. Each of these lines intersects the exceptional divisor atone point. 15 Q R
12 3 4 5678 9
Figure 7: The blue lines in the image are the edges of the 9-gon. Each of these edges arenumbered, using the blue numbers. Blue circles numbered 2, 4, and 9 in the image representthe exceptional divisors E P , E Q , and E R , respectively. These exceptional divisors are used asthree edges, a part of the nine edges, to form the 9-gon.16here rk ADE denotes the rank of the singularity type of the Jacobian fibration of a Halphensurface. Particularly, the following inequality holds rk ADE ≤ . (11)The singularity types of the original Halphen surface and the Jacobian fibration are identical.Therefore, the inequality (11) shows that a Halphen surface of index 2 can have an A n typesingularity up to A . This proves that a Halphen surface of index 2 can have a type I n fiberup to type I .A Halphen surface with a type I fiber has the singularity type A , thus the Jacobianfibration of this Halphen surface is an extremal rational elliptic surface . The complexstructures of the extremal rational elliptic surfaces were classified in [69]. The complex struc-ture of an extremal rational elliptic surface with A singularity is uniquely determined [69].The extremal rational elliptic surface with A singularity has 1 type I fiber, and 3 type I fibers [69]. This extremal rational elliptic surface is denoted as X [9 , , , in [68]. From theaforementioned argument, we deduce that the Jacobian fibration of the Halphen surface ofindex 2 with a type I fiber constructed in section 2.3 is isomorphic to the extremal rationalelliptic surface X [9 , , , . Utilizing this result, we precisely determine the non-Abelian gaugesymmetry that arises in F-theory compactification on K3 surface obtained as a double coverof the Halphen surface of index 2 with a type I fiber in section 3.3. The Jacobian fibrations of the Halphen surfaces generally have trivial Tate–Shafarevich groups.The Weil–Châtelet group instead contains the information of the multisections for the Halphensurfaces. We discuss the structures of these groups of the Jacobian fibrations of the Halphensurfaces.The Tate–Shafarevich group is a subgroup of the Weil–Châtelet group. Among genus-onefibrations as elements of the Weil–Châtelet group, those which locally admit a section form asubgroup, and this subgroup is the Tate–Shafarevich group [60].A Halphen surface X of index n does not have a global section, but it admits an n -section, therefore, the element X generates a Z n group in the Weil–Châtelet group of theJacobian fibration J ( X ) , W C ( J ( X )) : < X > ∼ = Z n ⊂ W C ( J ( X )) . (12) Extremal rational elliptic surfaces are the rational elliptic surfaces with a section with the Mordell–Weilrank 0. They have the singularity types of rank 8. Applications of extremal rational elliptic surfaces that appear in the stable degeneration [65, 66] of F-theory/heterotic duality [1, 2, 3, 67, 65] to string theory are discussed in [68]. Halphen surface of index n is obtained by blowing up P at nine singularities of multiplicities n of a degree n curve. The exceptional divisors yield n -sections.
17n this note, we constructed several examples of Halphen surfaces of index 2. These surfacesare bisection geometries, therefore, it follows from the aforementioned argument that theWeil–Châtelet groups of the Jacobian fibrations of these surfaces contain Z groups.We determined in section 2.4 that the Jacobian fibration of the Halphen surface of index2 with a type I fiber as we constructed in section 2.3 is isomorphic to the extremal rationalelliptic surface X [9 , , , . Thus, we find that the Weil–Châtelet group W C ( X [9 , , , ) of theextremal rational elliptic surface X [9 , , , contains a Z group, and the Tate–Shafarevichgroup X ( X [9 , , , ) is trivial: W C ( X [9 , , , ) ⊃ Z , X ( X [9 , , , ) ∼ = 0 . (13) We utilize the Halphen surfaces of index 2 with type I n fibers that we constructed in section2.2 and section 2.3 to yield a genus-one fibered K3 surface without a global section. Aspreviously seen, the constructed Halphen surfaces have type I , I , I , I fibers at the originof the base P . We consider two constructions of a genus-one fibered K3 surface without asection: double covers of Halphen surfaces ramified over a smooth fiber, and double covers ofHalphen surfaces ramified over the singular fiber at the origin of the base P . We considerF-theory compactifications on the resulting K3 surfaces without a global section times a K3surface, and deduce gauge groups that arise. We also precisely determine the non-Abeliangauge symmetries for K3 surfaces obtained as double covers of Halphen surfaces of index 2with type I fibers. A fiber of a Halphen surface of index 2 constructed as a blow-up of P × P at eight points isgiven by the following equation k + a l = 0 , (14)where k is a bi-degree (4,4) polynomial, so that the curve k = 0 has eight simple singularities,and l = 0 is a bi-degree (2,2) smooth curve that passes through these eight singularities, asdescribed in section 2.1. a is a constant.We consider a double cover of a Halphen surface of index 2 ramified over a fiber τ = k + a l . (15)18he equation (15) describes a double cover of a Halphen surface constructed as a blow-up of P × P at eight points, ramified over a bi-degree (4,4) curve. Thus, it gives a K3 surface.Blowing up the simple singularities of the curve k = 0 yields bisections to a Halphensurface of index 2. Using an argument similar to that presented in [40], we deduce that theresulting K3 surface (15) does not have a global section, and the pullback of a bisection to anoriginal Halphen surface of index 2 yields a bisection to the resulting K3 surface (15).We assume that a (cid:54) = 0 (16)in equation (15). For generic values of a (cid:54) = 0 , the equation (14) yields a smooth fiber. When a takes the value 0, a fiber becomes singular, and the equation (14) describes a type I n fiberat the origin of the base P . The case a = 0 will be discussed in section 3.1.2.We find that the K3 surface (15) that results as a double cover of a Halphen surface ofindex 2 is identical to the quadratic base change of the Halphen surface using an argumentsimilar to that given in [40]. Therefore, the singular fibers that the K3 surface (15) has aretwice the number of the original Halphen surface, when the ramification locus of the doublecover is a smooth fiber, namely a (cid:54) = 0 . We discuss the case where the ramification locus ofthe double cover becomes a singular fiber, which occurs when a = 0 , in section 3.1.2.When a = ∞ , the equation (15) can be expressed as follows: τ = l . (17)This equation splits into the following two equations: τ = l (18) τ = − l. This is the situation where a K3 surface degenerates into two rational elliptic surfaces asdiscussed in [68]. We assume that a (cid:54) = ∞ in this study.A double cover of a Halphen surface of index 2 constructed as a blow-up of P at ninepoints, given by an equation of the same form as (15), also gives a K3 surface without asection. This construction of a genus-one fibered K3 surface without a section is discussed in[40].The aforementioned argument shows that the K3 surface (15) obtained as a double coverof a Halphen surface ramified along a smooth fiber has two type I n fibers when an originalHalphen surface has a type I n fiber, n = 4 , , , , as constructed in section 2.2 and section2.3. We discuss the case where a in the equation (15) takes the value 0, and a K3 surface as adouble cover of a Halphen surface of index 2 is given by the following equation: τ = k. (19)19or this case, the ramification locus of a K3 surface as a double cover (19) occurs along thesingular fiber k = 0 . (20)The singular fiber (20) describes the type I n fiber of a Halphen surface at the origin of thebase P .An argument similar to that given in [40] proves that the K3 surface (19) generically lacksa global section, but it has a bisection.The quadratic base change that corresponds to the double cover (19) ramifies over type I n fiber at the origin of the base P , and the resulting K3 surface as a double cover (19) hasa type I n fiber , instead of two I n fibers as described in section 3.1.1.When the simple singularities of the bi-degree (4,4) polynomial k include a cusp, thepullback of the bisection which arises when the cusp is blown up splits into two sections .This is because a bisection that arises as an exceptional divisor when a cusp is blown up istangent to the branching locus. The K3 surface (19) admits a global section for this specialsituation. We do not consider this situation in this study, and we assume that the singularitiesof the polynomial k do not have a cusp.The aforementioned argument shows that the K3 surface (19) obtained as a double coverof a Halphen surface of index 2 ramified over a singular fiber k = 0 has a type I n fiber, n = 4 , , , , when an original Halphen surface has a type I n fiber as constructed in section2.2 and section 2.3. I , I , I fibers We discuss the gauge symmetries that arise in F-theory compactifications on the K3 surfaceswithout a global section that were constructed in section 3.1 as double covers of Halphensurfaces of index 2 with type I , I , I fibers times a K3 surface. We discuss F-theory com-pactifications on the K3 surfaces obtained as double covers of Halphen surfaces with a type I fiber in section 3.3 separately.As described in section 3.1.1, K3 surfaces constructed as double covers of Halphen surfaceswith a type I n fiber ramified over a smooth fiber has two type I n fibers, n = 4 , , , andthe K3 surfaces have a bisection. Therefore, the gauge symmetries that arise in F-theorycompactifications on the K3 surfaces times a K3 surface include a factor as follows: SU ( n ) × Z , (21) n = 4 , , . When the quadratic base change is ramified over a type I n fiber, two type I n fibers collide, instead ofsimply yielding two copies of type I n fiber, and they are enhanced to a singular fiber of type I n [70]. The curve k + a l does not generally have a cusp for nonzero a .
20s described in section 3.1.2, K3 surfaces without a global section as double covers ofHalphen surfaces with a type I n fiber, n = 4 , , , ramified over a singular fiber at the originof the base has a type I n fiber, and the K3 surfaces have a bisection. Therefore, the gaugesymmetries arising in F-theory compactifications on the resulting K3 surfaces times a K3surface have a factor: SU ( m ) × Z , (22) m = 8 , , . I fiber We saw in section 2.4 that the Jacobian fibration of the Halphen surface of index 2 with atype I fiber as constructed in section 2.3 has 1 type I fiber and 3 type I fibers. Therefore,the Halphen surface with a type I fiber as constructed in section 2.3 has 1 type I fiber, 3type I fibers, as well as a double fiber at infinity . Thus, a K3 surface obtained as a doublecover of the Halphen surface with type I fiber has 2 type I fibers and 6 type I fibers. Wededuce that the non-Abelian gauge group that arises in F-theory compactification on this K3surface times a K3 surface is precisely SU (9) . (23)As discussed in [36], F-theory compactification on a space constructed as the direct productof K3 surfaces yields a four-dimensional theory with N = 2 supersymmetry, and the anomalycancellation condition requires that 24 7-branes should be present. Type I n fiber correspondsto n . The number of 7-branes associated with 2 type I fibers and 6 type I fibersis 24. Therefore, we confirm that the anomaly cancellation condition is satisfied for F-theorycompactification on the K3 surface obtained as a double cover of the Halphen surface with atype I fiber ramified along a smooth fiber times a K3 surface. A K3 surface obtained as adouble cover of the Halphen surface with a type I fiber ramified along a smooth fiber has abisection, therefore the gauge group in F-theory compactification includes the following factor SU (9) × Z . (24)Next, we discuss the K3 surface obtained as a double cover of the Halphen surface witha type I fiber ramified along a singular fiber. The corresponding quadratic base changeramifies over the type I fiber; therefore, the resulting K3 surface has 1 type I fiber and6 type I fibers. We confirm that this agrees with the anomaly cancellation condition. The As explained in section 4.1, the types of the singular fibers of a Halphen surface of index 2 and the typesof the singular fibers of the Jacobian fibration are identical, except the double fiber that a Halphen surface ofindex 2 possesses. The number of 7-branes wrapped on a discriminant component is given by the Euler number of the fibertype over that component. The Euler numbers of the types of the singular fibers of an elliptic surface aregiven in [10]. SU (18) . (25)The resulting K3 surface has a bisection, thus the arising gauge group contains a factor asfollows: SU (18) × Z . (26) U (1) gauge symmetries Halphen surface of index 2 always has the Jacobian fibration. The types of the singular fibersand their locations over the base of a Halphen surface and those of the Jacobian fibration areidentical, except the multiple fiber of a Halphen surface. By taking the Jacobian fibration,the double fiber of a Halphen surface becomes a smooth fiber, as mentioned in [40].Halphen surfaces of index 2 and genus-one fibered K3 surfaces without a section as con-structed in section 3.1 are bisection geometries. Therefore, as discussed in [27], double coversof quartic polynomials describe these surfaces. Taking the resolvent cubic of quartic polynomi-als and equating with the term y yield the Weierstrass equations of their Jacobian fibrations[27]. I fiber, and U (1) gauge symmetries in F-theory compactifications Taking double covers of the Halphen surface with a type I fiber as constructed in section 2.3yields genus-one fibered K3 surfaces without a section as obtained in section 3.1. We deducethe Weierstrass equations of the Jacobians of these K3 surfaces when the ramifications occuralong a smooth fiber. We also determine the Mordell–Weil rank of the Jacobians. Usingthis, we find that F-theory compactifications on generic members of the genus-one fibered K3surfaces constructed as double covers of the Halphen surface with a type I fiber ramified overa smooth fiber do not have a U (1) gauge symmetry.First, we briefly review the quadratic base change . The quadratic base change of anelliptic fibration is an operation in which the coordinate of the base curve P is replaced byhomogeneous quadratic polynomial. We denote the coordinate of the base P by [ u : v ] . The A mathematical discussion of the quadratic base change can be found in [70]. y = x + f ( u, v ) x + g ( u, v ) , (27)we consider the following replacements of the coordinates: u → α u + α uv + α v (28) v → α u + α uv + α v . Constants α i , i = 1 , · · · , , in (28) give the parameters. These replacements transform thehomogeneous polynomial f of degree 4 and the homogeneous polynomial g of degree 6. Wedenote the resulting homogeneous polynomials by ˜ f and ˜ g which have degrees 8 and 12,respectively. The Weierstrass equation y = x + ˜ f x + ˜ g (29)gives an elliptic K3 surface that the quadratic base change of the original rational ellipticsurface (27) yields.As discussed previously in section 3.1.1, a K3 surface constructed as a double cover ofthe Halphen surface with a type I fiber is described by the quadratic base change of thatHalphen surface. Therefore, the Jacobian fibration of the K3 surface constructed as a doublecover of the Halphen surface with a type I fiber is given as the quadratic base change ofthe Jacobian fibration of the Halphen surface with a type I fiber, X [9 , , , . The Weierstrassequation of the Jacobian fibration of the Halphen surface with a type I fiber, X [9 , , , , isgiven as follows [69]: y = x − u ( u + 24 v ) x + 2( u + 36 u v + 216 v ) . (30)The type I fiber of the extremal rational elliptic surface X [9 , , , (30) is located over [ u : v ] =[1 : 0] in the base P [69]. Utilizing the quadratic base change (28), we find that the Jacobianfibration of a K3 surface constructed as a double cover of the Halphen surface with a type I fiber is described by the following Weierstrass equation: y = x − α u + α uv + α v ) · [( α u + α uv + α v ) + 24( α u + α uv + α v ) ] x + 2 [( α u + α uv + α v ) + 36( α u + α uv + α v ) ( α u + α uv + α v ) + 216( α u + α uv + α v ) ] . (31)The Jacobian fibration of a K3 surface constructed as a double cover of the Halphen surfacewith a type I fiber (31) can be seen as obtained by gluing a pair of identical rational elliptic23urfaces (30) [68]. Thus, as shown in [25], the Mordell–Weil rank of the Jacobian K3 (31) isequal to the Mordell–Weil rank of the original rational elliptic surface (30) for generic valuesof the parameters α i , i = 1 , · · · , . The rational elliptic surface (30) is an extremal rationalelliptic surface with the singularity type A . Therefore, it has the Mordell–Weil rank 0. Thus,it follows that the resulting Jacobian K3 (31) has the Mordell–Weil rank 0 for generic valuesof the parameters α i , i = 1 , · · · , .From this, we conclude that F-theory compactification on a K3 surface constructed as adouble cover of the Halphen surface with a type I fiber ramified along a smooth fiber timesa K3 surface, generically does not have a U (1) gauge symmetry.We also discuss the Jacobian fibration of a K3 surface constructed as a double cover ofthe Halphen surface with a type I fiber ramified along a singular fiber at the origin of thebase. We saw in section 3.3 that a K3 surface constructed as a double cover of the Halphensurface with a type I fiber ramified along a singular fiber has 1 type I fiber and 6 type I fibers. Thus, this K3 surface without a section has the singularity type A . (32)The Jacobian fibration has the identical singularity type A . The complex structure moduliof the elliptic K3 surfaces with a section with the singularity type A is constructed in [24]by considering a special limit of the quadratic base change of the extremal rational ellipticsurface X [9 , , , . The Jacobian fibration of the K3 surface obtained as a double cover of theHalphen surface with a type I fiber ramified along a singular fiber belongs to this moduli.The members of this moduli, namely elliptic K3 surfaces with a section with the singularitytype A , the Weierstrass equations of which are deduced in [24], correspond to special limitsof the base change (28) in which the parameters α i take special values as follows: α = α = 0 , α (cid:54) = 0 , α (cid:54) = 0 . (33) We constructed several Halphen surfaces of index 2 with type I n fibers ( n = 4 , , , ) in thisstudy. We obtained genus-one fibered K3 surfaces lacking a section by taking double coversof the constructed Halphen surfaces. Two types of K3 surfaces were obtained, depending onwhether the ramification locus of a double cover is a smooth fiber, or a singular fiber. Theseconstructions yielded K3 surfaces singular fibers of which include two type I n fibers, and K3surfaces singular fibers of which include a type I n fiber.We analyzed F-theory compactifications on these K3 surfaces lacking a section times aK3 surface. These K3 surfaces have bisection geometries. Gauge group that arises contains afactor SU ( n ) × Z in F-theory compactification on a K3 surface obtained as a double coverof a Halphen surface with a type I n fiber ramified along a smooth fiber times a K3 surface.Gauge group has a factor SU (2 n ) × Z in F-theory compactification on a K3 surface obtainedas a double cover of a Halphen surface with a type I n fiber ramified over a singular fiber timesa K3 surface. 24e showed that a Halphen surface of index 2 can have a type I n fiber up to I fiber. Weconstructed a Halphen surface of index 2 with a type I fiber, that saturates this upper bound.We determined the complex structure of the Jacobian fibration of this Halphen surface. Weprecisely obtained the non-Abelian gauge symmetries that arise in F-theory compactificationson K3 surfaces obtained as double covers of the constructed Halphen surface of index 2 witha type I fiber times a K3 surface.Fibering K3 surfaces obtained in this note over a base complex surface can yield genus-onefibered Calabi–Yau 4-folds. An investigation of this construction, and the effect of F-theorycompactification on the resulting Calabi–Yau geometries is a likely direction of future studies. Acknowledgments
We would like to thank Shun’ya Mizoguchi and Shigeru Mukai for discussions. This work ispartially supported by Grant-in-Aid for Scientific Research
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