Structure of stable degeneration of K3 surfaces into pairs of rational elliptic surfaces

Journal of High Energy Physics, Mar 2018

Yusuke Kimura

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP03%282018%29045.pdf

Structure of stable degeneration of K3 surfaces into pairs of rational elliptic surfaces

HJE Structure of stable degeneration of K3 surfaces into pairs of rational elliptic surfaces 0 1-1 Oho , Tsukuba, Ibaraki 305-0801 , Japan 1 KEK Theory Center, Institute of Particle and Nuclear Studies , KEK F-theory/heterotic duality is formulated in the stable degeneration limit of a bration on the F-theory side. In this note, we analyze the structure of the stable Di erential and Algebraic Geometry; F-Theory; Gauge Symmetry; Super- - K3 degeneration limit. We discuss whether stable degeneration exists for pairs of rational elliptic surfaces. We demonstrate that, when two rational elliptic surfaces have an identical complex structure, stable degeneration always exists. We provide an equation that systematically describes the stable degeneration of a K3 surface into a pair of isomorphic rational elliptic surfaces. When two rational elliptic surfaces have di erent complex structures, whether their sum glued along a smooth ber admits deformation to a K3 surface can be determined by studying the structure of the K3 lattice. We investigate the lattice theoretic condition to determine whether a deformation to a K3 surface exists for pairs of extremal rational elliptic surfaces. In addition, we discuss the con gurations of singular bers under stable degeneration. The sum of two isomorphic rational elliptic surfaces glued together admits a deformation to a K3 surface, the singular bers of which are twice that of the rational elliptic surface. For special situations, singular bers of the resulting K3 surface collide and they are enhanced to a ber of another type. Some K3 surfaces become attractive in these situations. We determine the complex structures and the Weierstrass forms of these attractive K3 surfaces. We also deduce the gauge groups in F-theory compacti cations on these attractive K3 surfaces times a K3. E6, E7, E8, SU(5), and SO(10) gauge groups arise in these compacti cations. string Vacua 2 Stable degeneration of K3 to a pair of isomorphic rational elliptic sur Equation for the degeneration of K3 to a pair of isomorphic rational elliptic surfaces Extremal rational elliptic surfaces Example of generic deformation using an extremal rational elliptic surface Attractive K3 surfaces as a special deformation of two extremal rational elliptic surfaces glued together 2.4.1 Complex structures of attractive K3 surfaces, and gauge groups in 2.4.2 2.4.3 F-theory compacti cations Weierstrass equation Anomaly cancellation condition 3 Stable degeneration of K3 to a pair of non-isomorphic rational elliptic 1 Introduction faces 2.1 2.2 2.3 2.4 surfaces 3.1 3.2 4 Conclusion Lattice condition for stable degeneration limit Pairs of extremal rational elliptic surfaces 1 Introduction The F-theory approach to particle physics model building has several advantages. It naturally realizes SU(5) grand uni ed theories with matter in spinor representations of SO(10). In contrast to D-brane models, there is no di culty in generating up-type Yukawa couplings. Furthermore, it can evade the problem of weakly coupled heterotic string theory addressed in [1]. Recent studies on F-theory model building [2{5] have emphasized the use of local models. However, in order to address the issue of gravity such as in ation, a global model of compacti cation need to be considered eventually. In particular, many insights can be gained by the duality between heterotic string and F-theory [6{10], which states the equivalence between the former compacti ed on an elliptically bered CY n-fold and the latter on a K3 bered CY (n + 1)-fold in the stable degeneration limit [10, 11].1 The aim of the present paper is to develop a systematic study of the process of stable degeneration of a K3 surface for a pair of rational elliptic surfaces. 1For recent discussion of the stable degeneration limit of F-theory and F-theory/heterotic duality, see, for example, [12{16]. { 1 { K3 surface stably degenerates into two rational elliptic surfaces in two distinct ways: i) K3 surface splits into two rational elliptic surfaces with an identical complex structure ii) K3 surface splits into two rational elliptic surfaces with di erent complex structures. We discuss these cases separately, in section 2 and section 3, respectively. We demonstrate in section 2.1 that, when a pair of rational elliptic surfaces are isomorphic, stable degeneration can be described by a systematic equation. We analyze the geometry of stable degeneration of the rst kind i) using this equation. We determine that given any pair of isomorphic rational elliptic surfaces, there is some K3 surface that stably degenerates into the pair. However, it is considerably di cult to describe stable degeneration by an equation when two rational elliptic surfaces have di erent complex structures. Moreover, a pair of non-isomorphic rational elliptic surfaces glued together along a smooth ber does not necessarily admit a deformation to a K3 surface. Complex structures and con gurations of singular bers are classi ed for a speci c class of rational elliptic surfaces, called extremal rational elliptic surfaces. We focus on the extremal rational elliptic surfaces to analyze the stable degeneration of the second kind ii). We determine whether stable degeneration exists for pairs of these rational elliptic surfaces. We use the lattice theoretic approach to analyze this process. We also study the con guration of singular bers under the stable degeneration limit. In F-theory, non-Abelian gauge symmetries on the 7-branes are in correspondence with the types of singular bers. Therefore, analyzing the con gurations of singular bers under the stable degeneration limit is of physical interest. The outline of this study is as follows. In section 2, we discuss the stable degeneration limit where a K3 surface degenerates into two isomorphic rational elliptic surfaces. We provide an equation that systematically describes this process. We determine that any pair of isomorphic rational elliptic surfaces glued together deforms to a K3 surface, which is a double cover of P2 rami ed over a sextic curve. Moreover, we discuss the con gurations of singular bers under the degeneration. Furthermore, we review the properties of the extremal rational elliptic surfaces. We discuss some examples of stable degeneration using an extremal rational elliptic surface. The sum of two isomorphic rational elliptic surfaces glued together admits a deformation to a K3 surface, the singular bers of which are twice that of the rational elliptic surface. For special situations, two bers of the same type of the resulting K3 surface collide, and they are enhanced to a ber of another type. These situations can be considered as special cases of stable degeneration. For such situations, some K3 surfaces become attractive. In section 2.4, we determine the complex structures of these attractive K3 surfaces and their Weierstrass forms. We also deduce the non Abelian gauge symmetries that form on the 7-branes in F-theory compacti cations on these attractive K3 surfaces times a K3. E6, E7, E8, SU(5), and SO(10) gauge groups arise in these models. In section 3, we investigate the stable degeneration limit where a K3 surface degenerates into two non-isomorphic rational elliptic surfaces. Whether such a degeneration exists can be determined by studying the lattice structure of the second integral cohomology { 2 { surfaces liptic surfaces group H2(S; Z) of a K3 surface S. We obtain the lattice condition under which two nonisomorphic rational elliptic surfaces glued together admits a deformation to a K3 surface. We determine whether a deformation to a K3 surface exists for pairs of extremal rational elliptic surfaces. We study the con guration of singular bers under stable degeneration using the lattice theoretic argument. We state the concluding remarks in section 4. 2 Stable degeneration of K3 to a pair of isomorphic rational elliptic 2.1 Equation for the degeneration of K3 to a pair of isomorphic rational elwith a section. The base space of a rational elliptic surface is isomorphic to P1 by Luroth's theorem, and it is known that every rational elliptic surface with a section is the blow-up of P2 in the nine base points of a cubic pencil [17]. The Picard number of a rational elliptic surface is 10, and the rank of the Mordell-Weil group ranges from 0 to 8: 0 rk MW 8: (2.1) A generic rational elliptic surface has Mordell-Weil rank 8. Rational elliptic surfaces with Mordell-Weil rank 0 are called extremal rational elliptic surfaces. For extremal rational elliptic surfaces, the rank of the singularity type is 8, which is the highest for a rational elliptic surface. We will review the properties of extremal rational elliptic surfaces in section 2.2. In section 3, we focus on extremal rational elliptic surfaces to discuss the stable degeneration of a K3 surface into a pair of non-isomorphic rational elliptic surfaces. When two rational elliptic surfaces have isomorphic smooth elliptic bers, we can glue two rational elliptic surfaces in the following fashion: we choose a point in base P1 of each rational elliptic surface over which the ber is smooth and isomorphic, and we glue two rational elliptic surfaces by identifying the isomorphic smooth bers over the chosen points. As described in section 3, when a certain lattice theoretic condition is satis ed, the sum of two (not necessarily isomorphic) rational elliptic surfaces glued together can be deformed to a K3 surface. In this section, we particularly consider the case wherein two rational elliptic surfaces that are glued together are isomorphic. For this particular case, we explicitly provide an equation that describes the stable degeneration of a K3 surface into the pair of isomorphic { 3 { rational elliptic surfaces glued together. As stated above, a rational elliptic surface X is the blow-up of P2 in the base points of a cubic pencil; we denote the cubic pencil of a rational elliptic surface X as f . The double cover of P2 rami ed along a degree 6 curve is, in general, a K3 surface. We particularly consider double covers of P2 rami ed over a degree 6 curve given by the following form of equations: where g is a polynomial of degree 3 and g is the cubic pencil of the same type as the pencil f , but the ratio of the coe cients of the pencil g is generally di erent from the ratio of the coe cients of the pencil f . For this situation, K3 surface (2.2) is elliptically bered. To HJEP03(218)45 be explicit, when the cubic pencil f is given by (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) Each of the above two equations in (2.5) describes a rational elliptic surface given by cubic pencil f ; therefore, when a cubic polynomial g goes to cubic pencil f , the K3 surface (2.2) splits into two isomorphic rational elliptic surfaces, each given by the cubic pencil f . This is the stable degeneration limit of the K3 surface (2.2) splitting into two copies of rational elliptic surfaces X. Thus, we conclude that two isomorphic rational elliptic surfaces glued along an isomorphic smooth ber always admit a deformation to a K3 surface. We only consider rational elliptic surfaces with a global section in this note; therefore, they admit transformation into the Weierstrass form. The coe cients of the Weierstrass form depend on the coordinate of the base P1. We denote the homogeneous coordinate of the base P 1 as [u : v]. In terms of the coordinate [u : v], the stable degeneration of a K3 surface (2.2) splitting into two isomorphic rational elliptic surfaces is described by the following equation: 2 = u2 + 2kuv + v2: k in equation (2.6) denotes a parameter of deformation. k varies along the deformation, and when k assumes the values equation (2.6) splits into linear factors. This occurs when a K3 surface splits into two isomorphic rational elliptic surfaces. where a; b are coe cients of the pencil f , we choose the cubic pencil g as follows: c; d are coe cients of the pencil g, and the ratio [c : d] is generally di erent from the ratio [a : b]. For the limit at which polynomial g goes to cubic pencil f (i.e. for the limit at which ratio [c : d] goes to the ratio [a : b]), equation (2.2) is split into the following two equations: = f = f: 2 = f g; f = a h1 + b h2; g = c h1 + d h2: k = 1; { 4 { We observe from equation (2.6) that the K3 surface (2.2) is the quadratic base change of the rational elliptic surface (2.5) into which the K3 surface splits in the stable degeneration limit, when g is the cubic pencil of the same type as the pencil f . In other words, the Weierstrass equation of the K3 surface is obtained when some appropriate quadratic equations are substituted into variables u; v in the coe cients of the Weierstrass form of a rational elliptic surface. Thus, the generic K3 surface (2.2) that results from the deformation of two isomorphic rational elliptic surfaces (2.5) glued together has twice the number of singular bers as a rational elliptic surface. For special situations, singular bers of the same type of the K3 surface, that is obtained as the quadratic base change of a rational elliptic surface, collide and they are HJEP03(218)45 enhanced to a ber of another type. We discuss these situations in section 2.4. The aforementioned argument applies to the deformation of every pair of two isomorphic rational elliptic surfaces with a global section glued along smooth ber to a K3 surface. 2.2 Extremal rational elliptic surfaces We summarize the properties of extremal rational elliptic surfaces. In section 2.3, we discuss the quadratic base change of extremal rational elliptic surfaces that rami es only over smooth bers. In section 2.4, we discuss the limits at which singular bers of the same type collide in the quadratic base change of extremal rational elliptic surfaces. We discuss the structures of attractive K3 surfaces that result from the quadratic base change of extremal rational elliptic surfaces in section 2.4. Extremal rational elliptic surfaces have the singularity type of rank 8, and the MordellWeil groups only have torsion parts. Singular ber types of the extremal rational elliptic surfaces are classi ed, and the complex structure of an extremal rational elliptic surface is uniquely determined by the ber type, except for the surfaces with the ber type [I0 ; I0 ] [18]. J-invariant of bers of an extremal rational elliptic surface with the ber type [I0 ; I0 ] is constant over the base P1, and the complex structure of a rational elliptic surface with the ber type [I0 ; I0 ] depends on the value of j. The complex structure of an extremal rational elliptic surface with the ber type [I0 ; I0 ] is xed when the j-invariant of the ber is chosen. Provided these facts, in this note, we denote an extremal rational elliptic surface using its ber type as the subscript. For example, an extremal rational elliptic surface with the ber type [IV; IV ] is denoted as X[IV; IV ]. We simply use n to represent the In ber, and m to represent the Im ber. Therefore, an extremal rational elliptic surface with singular bers of type IV , I3, and I1 is denoted as X[IV ; 3;1]. We denote a surface with the ber type [I0 ; I0 ] as X[0 ; 0 ](j), because the complex structure of such a surface depends on the j-invariant of bers. We list the con gurations of singular ber types of the extremal rational elliptic surfaces in table 1. The Weierstrass forms of the extremal rational elliptic surfaces were also derived in [18]. We include the Weierstrass forms of the extremal rational elliptic surfaces in table 1. The cubic pencils for all the extremal rational elliptic surfaces, except the surfaces X[II; II ], X[III; III ], X[IV; IV ], X[0 ; 0 ](j), were obtained in [19]. The cubic pencil for the { 5 { Extremal rational elliptic surface III , III IV , IV I0 , I0 II I1 I1 III I2 I1 IV I3 I1 I4 I1 I1 I2 I2 I2 I1 I4 I1 I9 I1 I1 I1 I8 I2 I1 I1 I6 I3 I2 I1 I5 I5 I1 I1 3u4 uv3 2u5v v5(u v) v4(16u2 72uv + 54v2) uv3(2u2 9v2) (u v)3(u3 + v3) u(u 2v)3(2u2 9v2) 2(u6 + 36u3v3 + 216v6) 2u6 + 12u4v2 + 15u2v4 2v6 2u6 + 12u5v + 12u4v2 14u3v3 +3u2v4 6uv5 + 2v6 2(u6 18u5v + 75u4v2 +75u2v4 + 18uv5 + v6) 2u6 3u4v2 3u2v4 + 2v6 2(u6 + 20u3v3 8v6) 2 = x3 + a4x + a6. [u : v] is the homogeneous coordinate on the base P1. For surface 4s3 + 27t2 6= 0. j-invariant of bers of surface X[0 ; 0 ](j) is a function of s; t. X[0 ; 0 ](j), s; t in coe cients a4; a6 of the Weierstrass form are complex numbers, s; t 2 C, with extremal rational elliptic surface X[IV; IV ] is given by a yz(y + z) + b x3: [x : y : z] represents the homogeneous coordinates on P2, and [a : b] represents the homogeneous coordinate on P1. The cubic pencils of extremal rational elliptic surfaces are listed in table 2. 2.3 Example of generic deformation using an extremal rational elliptic surface We discuss an example of results in section 2.1 using an extremal rational elliptic surface. We consider the surface X[IV; IV ]. As stated in section 2.2, the cubic pencil of surface X[IV; IV ] is given by (2.8) (2.9) f = a yz(y + z) + b x3: { 6 { a f(2x + z)(xz + y2) + x3g + b x(xz + y2) a (x + y)(xy + z2) + b x2y a xf(x + y)z + y2g + b (x + y)z2 a (x2y + y2z + z2x 3xyz) + b xyz a fx(xz y2) + y2zg + b (y + 2z)(xz y2 + z2) a z(x2 + xy + xz + y2) + b ( x2z + xyz + y3) a yz(x + y + z) + b x(x + y)(x + z) a (x y)(xy z2) + b xy(x + y 2z) a (x3 + y3 + z3) + b xyz (2.10) (2.11) g to be the pencil of the same type as the cubic pencil f : g = c yz(y + z) + d x3: We x c; d to be non-zero constants. We choose c; d so that the ratio [c : d] is generally di erent from the ratio [a : b] of the coe cients a; b in the pencil f . For the cubic pencil f (2.9) and cubic pencil g (2.10), equation generically provides a K3 surface with two type IV bers and two type IV bers. For the limit at which ratio [c : d] goes to [a : b], the cubic pencil g coincides with the cubic pencil f . In this stable degeneration limit, equation (2.11) is split into linear factors; accordingly, the K3 surface (2.11) splits into two extremal rational elliptic surfaces X[IV; IV ]. We stated in section 2.1 that gluing together two isomorphic rational elliptic surfaces, each given by cubic pencil f , to obtain a K3 surface (2.2) is equivalent to the quadratic base change of a rational elliptic surface over P1, when the cubic pencil g is chosen to be the same type as the pencil f . We consider, as an example, the case wherein two extremal rational elliptic surfaces X[IV; IV ] are glued together to form a K3 surface, to demonstrate this explicitly. We discuss the generic situation in which the quadratic base change rami es 2 = f g { 7 { over smooth bers. As given in table 1, the Weierstrass form of X[IV; IV ] is given by Gluing together two X[IV; IV ]s and deforming the resulting surface to a K3 surface is equivalent to substituting the following quadratic equations into variables u; v: u = v = 1u~2 + 2u~v~ + 3v~2 = 4u~2 + 5u~v~ + 6v~2 = 1(u~ 4(u~ 1v~)(u~ 3v~)(u~ 2v~) 4v~): i, i = 1; ; 6, are some constants, and the quadratic terms are split into linear factors on the right extreme hand sides in equation (2.13). (In (2.13), we assume that 1 6= and 3 6= 4.) The resulting K3 surface has the following Weierstrass form: this kind, in which singular bers of the same type collide in the quadratic base change of an extremal rational elliptic surface, will be discussed in section 2.4. 2.4 Attractive K3 surfaces as a special deformation of two extremal rational elliptic surfaces glued together 2.4.1 Complex structures of attractive K3 surfaces, and gauge groups in Ftheory compacti cations In section 2.1, we mainly discussed the quadratic base change whose rami cation occurs over smooth bers. A smooth ber remains smooth after the quadratic base change. When the quadratic base change is unrami ed over a singular ber, we obtain two copies of that ber. As we saw in section 2.1, the quadratic base change of a rational elliptic surface to obtain a K3 surface is the reverse of a process in which a K3 surface splits into two isomorphic rational elliptic surfaces under stable degeneration. A quadratic base change generically rami es only over smooth bers; we obtain two copies of the singular ber for each singular ber after the quadratic base change. Thus, the resulting K3 surface has twice as many singular bers as the rational elliptic surface for such a generic situation. We consider the special situation of the quadratic base change, where singular bers of the same type collide and they are enhanced to a singular ber of another type. We particularly consider the examples in which the resulting K3 surfaces after the quadratic base change are enhanced to attractive K3 surfaces.2 We show in table 3 the resulting ber type after the collision of two singular bers of the same type [20]. 2Following the convention of the term in [21], we refer to a K3 surface with the Picard number 20 as an attractive K3 surface in this study. { 8 { Original ber type of a pair of identical singular bers Resulting ber type In II III IV I2n IV I 0 IV We consider stable degeneration in which two isomorphic extremal rational elliptic surfaces X[II; II ] are glued together. When two type II bers collide in the quadratic base change, we nd from table 3 that the resulting ber has type IV . The resulting K3 surface has singular bers of types II , II , and IV . The corresponding ADE type is E2 8 A2. This is an extremal K3 surface. An extremal K3 surface is an attractive elliptic K3 surface with a global section, with the Mordell-Weil group of rank 0. The extremal K3 surface with ADE type E2 8 A2 is discussed in [22]. We next consider the gluing of two isomorphic extremal rational elliptic surfaces X[III; III ], in which two type III bers collide. The resulting K3 surface has type III , III , and I0 bers. The corresponding ADE type is D4. This is also an extremal K3 surface. This K3 surface is discussed in [23] as the Jacobian bration of some K3 genus-one bration without a section. For the gluing of two isomorphic extremal rational elliptic surfaces X[IV; IV ], when two type IV bers collide, the resulting ber has type IV . Therefore, the resulting K3 surface has three type IV singular bers. The corresponding ADE type is E3. This is an extremal K3 surface, and 6 this K3 surface is discussed in [24] as the Jacobian bration of an attractive K3 genus-one bration without a section. For the gluing of two isomorphic extremal rational elliptic surfaces X[II ; 1;1], we consider the situation in which two pairs of type I1 bers collide in the quadratic base change. The resulting II , and two bers have type I2; the resulting K3 surface has two singular bers of type bers of type I2. When we consider the limit of the quadratic base change of extremal rational elliptic surface X[III ; 2;1] at which two type I2 bers and two type I1 bers collide, the resulting K3 surface has bers of types III , III , I4, and I2. The limit of quadratic base change of extremal rational elliptic surface X[IV ; 3;1], at which two bers of type I3 and two bers of type I1 collide, gives K3 surface with singular bers of types IV , IV , I6, and I2. For the gluing of two extremal rational elliptic surfaces X[4 ; 1;1], when we consider the limit of the quadratic base change at which two pairs of type I1 bers collide, the resulting K3 surface has two singular bers of type I4 and two singular bers of type I2. For extremal rational elliptic surface X[2 ; 2;2], when we consider the limit of the quadratic base change at which two pairs of type I2 bers collide, the resulting K3 surface has two type I2 bers and two type I4 bers. For the gluing of two extremal rational elliptic surface X[1 ; 4;1], when we consider the limit of the quadratic base change at which two type I4 bers and two type I1 bers collide, the resulting K3 surface has type I1 , I1 , I8, and I2 bers. { 9 { HJEP03(218)45 For extremal rational elliptic surface X[5;5;1;1], when we consider the limit of the quadratic base change at which two bers of type I5 and two bers of type I1 collide, the resulting K3 surface has 1 type I10 ber, 2 type I5 bers, 1 type I2 ber, and 2 type I1 bers. From ADE types of the 10 K3 surfaces that we obtained above as the quadratic base change of extremal rational elliptic surfaces, we conclude that they are extremal K3 surfaces. ADE types of the extremal K3 surfaces and the corresponding complex structures were classi ed in [25]. Mordell-Weil groups of extremal K3 surfaces were also derived in [25]. Using table 2 in [25], we can deduce the complex structures of the 10 extremal K3 surfaces from their ADE types. The complex structure of an attractive K3 surface S is speci ed by the transcendental lattice T (S), which is de ned to be the orthogonal complement of the Neron-Severi lattice in the K3 lattice H2(S; Z) [ 26 ]. Therefore, we represent the complex structure of an attractive K3 surface by the intersection matrix of the transcendental lattice in this study. For an attractive K3 surface, the transcendental lattice is a 2 2 integral, symmetric, positive de nite even lattice. See section 4 in [24] for a review of the correspondence of the complex structures of attractive K3 surfaces and the transcendental lattice. We list the ber types, corresponding ADE types, the complex structures and the Mordell-Weil groups of the 10 extremal K3 surfaces, that we obtained above as deformation of two isomorphic extremal rational elliptic surfaces glued together, in table 4. The global structures of the non-Abelian gauge symmetries3 that arise on the 7-branes in Ftheory compacti cations on the 10 extremal K3 surfaces times a K3 surface are also shown in table 4. Extremal K3 surface has the Mordell-Weil rank 0, therefore these F-theory compacti cations do not have U(1) gauge eld. 2.4.2 Weierstrass equation We obtained 10 extremal K3 surfaces in section 2.4.1 as the limit of the quadratic base change4 of extremal rational elliptic surfaces, at which singular bers of the same type collide. Therefore, by substituting appropriate quadratic polynomials into u; v in the Weierstrass forms of extremal rational elliptic surfaces, listed in table 1 in section 2.2, we can deduce the Weierstrass forms of the 10 extremal K3 surfaces, obtained in section 2.4.1. As an example, extremal K3 surface whose transcendental lattice has the intersection matrix 60 02 , with ADE type E2 6 A5 A1, is obtained via the quadratic base change of extremal rational elliptic surface X[IV ; 3;1], in which two bers of type I3 and two bers of type I1 collide. The Weierstrass form of extremal rational elliptic surface X[IV ; 3;1] is given by y2 = x3 + v3(24u 27v)x + v4(16u2 72uv + 54v2): The discriminant of the Weierstrass form (2.15) is [18] on the 7-branes. 3See [ 8, 27 ] for the correspondence of the types of the singular bers and the non-Abelian gauge groups 4The relationship of K3 surfaces and rational elliptic surfaces via base change is also discussed in [28]. (2.15) (2.16) Singular bers Complex Str. Mordell-Weil group II II IV III III I0 D4 E3 6 E2 A2 8 1 III III I4 I2 E2 A3 A1 7 groups that form on the 7-branes. Type IV ber is at [u : v] = [1 : 0], type I3 ber is at [u : v] = [0 : 1] and type I1 ber is at [u : v] = [1 : 1]. We consider the following substitutions for u; v in the Weierstrass form of extremal rational elliptic surface X[IV ; 3;1]: u = u~ 2 v = 2u~v~ 2 v~ : This gives the limit of the quadratic base change at which two bers of type I3 collide at [u : v] = [0 : 1] and two bers of type I1 collide at [u : v] = [1 : 1]. The resulting equation y2 = x3 + v3(2u v)3(24u2 + 27v2 54uv)x + v4(2u v)4(16u4 144u3v + 288u2v2 gives the Weierstrass form of extremal K3 surface with transcendental lattice 60 02 , with A1. The discriminant of the Weierstrass form (2.18) is given by We con rm from equations (2.18) and (2.19) that extremal K3 surface (2.18) has two type IV bers, at [u : v] = [1 : 0]; [1 : 2], type I6 ber at [u : v] = [0 : 1] and type I2 ber at [u : v] = [1 : 1]. (2.18) (2.19) E2 8 D82 D62 A3 A5 A1 A1 A1 f 0 0 12u6 11+25p5 (2uv v2) 18u10 11+25p5 (2uv v2) We show the Weierstrass forms of the 10 extremal K3 surfaces, which we obtained in section 2.4.1, in table 5. The discriminants of the Weierstrass forms in table 5 are listed in table 6. The Weierstrass forms of extremal K3 surfaces with ADE types E2 8 A21 are discussed in [22]. The Weierstrass forms of extremal K3 surfaces with ADE A2 and types E63 and E2 7 D4 are discussed in [24] and [23], respectively. 2.4.3 Anomaly cancellation condition We consider F-theory compacti cations on the extremal K3 surfaces obtained in section 2.4.1 times a K3 surface. The resulting theory is a four-dimensional theory with N = 2, without a four-form ux. The anomaly cancellation condition determines the form of the discriminant locus to be 24 K3 surfaces, counted with multiplicity; there are 24 7-branes wrapped on the K3 surfaces.5 We show the correspondence of the numbers of 7-branes and the ber types in table 7. The Euler numbers of the singular bers can be found in [29]. The Euler number of ber type can be considered as the number of the associated 7-branes. We con rm from tables 4 and 7 that there are in fact 24 7-branes in 5See, for example, [24] for discussion. HJEP03(218)45 E2 8 E2 7 E2 8 D82 D62 A2 D4 A2 1 A2 1 A2 3 of the Weierstrass forms in table 5 are shown. We suppressed the irrelevant constant factors of the discriminants. Fiber type # of 7-branes (Euler number) F-theory compacti cations on the 10 extremal K3 surfaces times a K3 surface. Therefore, we conclude that the anomaly cancellation condition is satis ed for these compacti cations. By turning on four-form ux [30{34], F-theory compacti cation on K3 times K3 gives four-dimensional theory with N = 1 supersymmetry. We con rm from table 1 in [35] and table 2 in [13] that for F-theory compacti cations on the 10 extremal K3 surfaces, obtained in section 2.4.1, times some appropriate attractive K3 surface, the tadpole [31] can be cancelled. See [23, 24] for the details. surfaces In this section, we discuss the stable degeneration where a K3 surface degenerates into two non-isomorphic rational elliptic surfaces. It is considerably di cult to provide a general equation to describe this kind of stable degeneration. Therefore, instead of providing an equation to describe the process, we use a lattice theoretic approach to determine whether stable degeneration exists for pairs of non-isomorphic rational elliptic surfaces. For the sake of brevity, in this section, we simply say \a pair of rational elliptic surfaces" to indicate a pair of non-isomorphic rational elliptic surfaces. We also discuss the con gurations of singular bers under stable degeneration. In section 3.1, we will discuss a lattice theoretic condition for the existence of stable degeneration for pairs of rational elliptic surfaces. Applying the lattice condition, in section 3.2, we demonstrate that stable degeneration exists for pairs of extremal rational elliptic surfaces. Elliptic ber of a rational elliptic surface generally has the moduli of dimension 1 over the base P1. Therefore, given a pair of rational elliptic surfaces, there is a pair of isomorphic smooth elliptic bers, and the pair of rational elliptic surfaces can be glued along the isomorphic smooth bers. However, when elliptic ber of a rational elliptic surface has the constant moduli over the base P1, such gluing is not necessarily possible. For the three extremal rational elliptic surfaces X[II; II ], X[III; III ], X[IV; IV ], the complex structure of elliptic bers is constant over the base P . 1 6 We do not consider the three extremal rational elliptic surfaces X[II; II ], X[III; III ], X[IV; IV ] in this section. 3.1 Lattice condition for stable degeneration limit We use the Torelli theorem for K3 surfaces [36] to deduce a lattice theoretic condition that determines whether pairs of rational elliptic surfaces admit stable degeneration. The Torelli theorem for K3 surfaces states that the geometry of a K3 surface is determined by the structure of the K3 lattice K3. The K3 lattice K3 of a K3 surface S is the second integral cohomology group H2(S; Z). The K3 lattice K3 is the inde nite even unimodular lattice of signature (3,19), and it is the direct sum of three copies of the hyperbolic plane U and two copies of the E8 lattice: K3 = U 3 E82: (3.1) In this note, we assume that a rational elliptic surface has a global section; thus, we presume that the K3 surface obtained as a deformation of the sum of two rational elliptic surfaces also admits a global section.7 An elliptic K3 surface having a section is equivalent 6The complex structure of elliptic bers of extremal rational elliptic surface X[0 ; 0 ](j) is also constant over the base P1, but there is a degree of freedom in choosing j-invariant of an elliptic ber. Therefore, when j-invariant is appropriately chosen, X[0 ; 0 ](j) can be glued with another rational elliptic surface. For this reason, we include extremal rational elliptic surface X[0 ; 0 ](j). 7In general, genus-one bered K3 surfaces need not have a global section. For discussion of the geometry of genus-one bered K3 surfaces without a section and string compacti cations on such spaces, see, for example, [23, 24, 37]. For recent progress in F-theory compacti cations on genus-one brations without a section, see, for example, also [38{48]. to the primitive embedding of the hyperbolic plane U into the K3 lattice K3 [36, 49]. We denote the ADE types of two rational elliptic surfaces, X1 and X2, as R1 and R2, respectively. Applying the argument in [50] to the boundary of the closure of K3 moduli, from the Torelli theorem for K3 surfaces, we deduce that a K3 surface exists and it admits stable degeneration into two rational elliptic surfaces X1 and X2, exactly when there is a primitive embedding of the lattice U R2 into the K3 lattice K3: R1 R1 U R2 K3: (3.2) Whether lattice U R1 R2 primitively embeds into the K3 lattice K3 can be determined by the criterion given in [ 51 ]. Some lattice theoretic terms are necessary to state the criterion; we introduce some lattice theoretic terms rst. By lattice, we indicate a nite rank free Z-module with a non-degenerate integral symmetric bilinear form. The lattice L is said to be even, when for every element x of L, x2 = x x is even. The discriminant of lattice L, disc L, is the determinant of an intersection matrix (ei ej )ij for a basis feig of the lattice L. The lattice L is said to be unimodular when its discriminant is 1. U denotes the hyperbolic plane. The hyperbolic plane U is the even unimodular lattice of signature (1,1). E8 denotes the even unimodular lattice of signature (0,8). E8 and U are unique up to the isometries of lattice. When the lattice L1 embeds into the lattice L2, the embedding L1 L2 is said to be primitive when the quotient L2=L1 is free as a Z-module. The dual lattice of lattice L is the lattice Hom(L; Z), and is denoted as L . The quotient GL := L =L is a nite Abelian group, and this group is called the discriminant group. When L is an even lattice, the map qL : GL ! Q=2Z, qL(x) = x 2 mod 2Z (3.3) de nes a non-degenerate quadratic form of the discriminant group GL; form qL is called the discriminant form. When A is a nite Abelian group, its length l(A) is de ned to be the minimum number of elements required to generate group A. Further, we state the criterion of primitive lattice embedding. Criterion (C) [51]. M is an even lattice of signature (m+; m ), G is the discriminant group of M and q is the discriminant form of M . Then, M primitively embeds into some even unimodular lattice of signature (l+; l ) when all of the three conditions i){iii) are satis ed: i) l+ + l rk M > l(G). ii) l+ iii) l+ m+ l 0, l m 0 (mod 8). 3.2 Pairs of extremal rational elliptic surfaces Applying criterion (C), we discuss whether the lattice U R1 R2 of pairs of extremal rational elliptic surfaces, X1 q X2, primitively embeds into the K3 lattice K3. This Discriminant group Z=2Z Z=2Z 0 Z=2Z Z=3Z Z=2Z Z=4Z Z=3Z Z=4Z Z=6Z Z=5Z Z=2Z Z=2Z determines whether stable degeneration exists for pairs of extremal rational elliptic surfaces. An even unimodular lattice of signature (3,19) is unique up to the isometries of lattice [52]. Therefore, when the lattice U R1 R2 admits a primitive embedding into some even unimodular lattice of signature (3,19), it primitively embeds into the K3 lattice K3. As stated in section 2.2, the complex structures and singular ber types of the extremal rational elliptic surfaces were classi ed. For an extremal rational elliptic surface, the discriminant group and the Mordell-Weil group are identical [53]. The Mordell-Weil groups of the rational elliptic surfaces were computed in [ 54 ]. We list the discriminant groups and ADE types of extremal rational elliptic surfaces in table 8. Let G denote the product group of the discriminant groups of two extremal rational elliptic surfaces X1 and X2. We determine the pairs of extremal rational elliptic surfaces that satisfy the condition i) rk K3 rk U R1 R2 > l(G): in criterion (C). From table 8, we observe that the length of the discriminant group of an extremal rational elliptic surface is either 1 or 2. For two extremal rational elliptic surfaces, X1 and X2, the singular ber types R1 and R2 are rank 8 even lattices (3.4) (3.5) rk R1 = rk R2 = 8 of signature (0,8). Thus, the lattice U R2 has the signature (1,17). The di erence of the rank of the K3 lattice K3 and the rank of the lattice U R1 R2 is The length l(G) of the discriminant group G attains this bound only when the discriminant groups of two extremal rational elliptic surfaces both have the length 2. The product group G of the discriminant groups of extremal rational elliptic surfaces has the length 4 only for three pairs8 of extremal rational elliptic surfaces: X[4;4;2;2] q X[2 ; 2;2]; X[2 ; 2;2] q X[0 ; 0 ]; X[0 ; 0 ] q X[4;4;2;2]: For all other pairs of extremal rational elliptic surfaces, the criterion (C) applies. Both conditions ii) and iii) in criterion (C) are satis ed: HJEP03(218)45 (3.6) (3.7) (3.8) (3.9) and 3 rational elliptic surfaces. pairs9 of extremal rational elliptic surfaces. From lattice embedding Thus, we determine that the lattice U R2 primitively embeds into the K3 lattice K3 for all pairs of extremal rational elliptic surfaces, except the three pairs X[4;4;2;2] q X[2 ; 2;2], X[2 ; 2;2] q X[0 ; 0 ] and X[0 ; 0 ] q X[4;4;2;2]. We conclude that the stable degeneration of a K3 surface exists for all pairs of extremal rational elliptic surfaces except the three pairs. The remaining three pairs, X[4;4;2;2] q X[2 ; 2;2], X[2 ; 2;2] q X[0 ; 0 ] and X[0 ; 0 ] q X[4;4;2;2], have the ADE types D6 A4, D6 1 D42 A2, D42 1 A21, respectively. ADE types of the singular bers of elliptic K3 surfaces with a global section were classi ed in [ 55 ]. We conclude from table 1 in [ 55 ] that the lattice U R1 R2 primitively embeds into the K3 lattice K3 for the three pairs of extremal rational elliptic surfaces. (They correspond to No.2079, 2043, 2152 in table 1 in [ 55 ], respectively.) This demonstrates that the stable degeneration of a K3 surface exists for the remaining three pairs of extremal The aforementioned argument demonstrates that stable degeneration exists for all U R1 R2 K3; (3.10) we deduce that the ADE type of the singular bers of the resulting K3 surface is the sum of the ADE types of the singular bers of the two non-isomorphic extremal rational elliptic surfaces. 8The product of discriminant group Z=3Z Z=3Z and another discriminant group with length 2 has length 2. For example, the product of Z=3Z Z=3Z with Z=2Z Z=2Z is isomorphic to Z=6Z Z=6Z, which has the length 2. 9As we stated at the beginning of this section, the three extremal rational elliptic surfaces X[II; II ], X[III; III ], X[IV; IV ] are not considered in this section. degeneration limit. In this study, we analyzed the stable degeneration of a K3 surface into two rational elliptic surfaces. We also discussed the con gurations of singular bers under the stable We demonstrated that gluing together two isomorphic rational elliptic surfaces and deforming the resulting surface to a K3 surface is always possible. We provided an equation to describe this kind of stable degeneration. The sum of two isomorphic rational elliptic surfaces glued together admits a deformation to a K3 surface, the singular bers of which are twice the singular bers of the rational elliptic surface. For special cases, two bers of the same type of the resulting K3 surface collide, and they are enhanced to a ber of another type. Some K3 surfaces become attractive in these cases. We determined the complex structures and the Weierstrass forms of these attractive K3 surfaces. We also deduced the gauge groups in F-theory compacti cations on these attractive K3 surfaces times a K3. We also investigated the deformation of two non-isomorphic rational elliptic surfaces glued together to a K3 surface, using the Torelli theorem of K3 surfaces. We deduced the lattice theoretic condition that must be satis ed to ensure that a deformation to a K3 surface exists for pairs of non-isomorphic rational elliptic surfaces. We con rmed that the lattice condition is satis ed for all pairs of the extremal rational elliptic surfaces. Thus, all such pairs of extremal rational elliptic surfaces glued together admit a deformation to a K3 surface. This demonstrates that for any pair of extremal rational elliptic surfaces, except the three extremal rational elliptic surfaces X[II; II ], X[III; III ], X[IV; IV ], there is a K3 surface that stably degenerates into that pair. The ADE type of singular bers of the resulting K3 surface is the sum of those of the two non-isomorphic extremal rational elliptic surfaces glued together. The lattice condition discussed in this study can be extended to general pairs of rational elliptic surfaces. Acknowledgments We would like to thank Shun'ya Mizoguchi and Shigeru Mukai for discussions. This work is partially supported by Grant-in-Aid for Scienti c Research #16K05337 from The Ministry of Education, Culture, Sports, Science and Technology of Japan. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. (1996) 135 [hep-th/9602070] [INSPIRE]. (2011) 1237 [arXiv:0802.2969] [INSPIRE]. [1] E. Witten, Strong coupling expansion of Calabi-Yau compacti cation, Nucl. Phys. B 471 [2] R. Donagi and M. Wijnholt, Model building with F-theory, Adv. Theor. Math. Phys. 15 01 (2009) 058 [arXiv:0802.3391] [INSPIRE]. (2011) 1523 [arXiv:0808.2223] [INSPIRE]. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE]. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE]. HJEP03(218)45 [INSPIRE]. 187 (1997) 679 [hep-th/9701162] [INSPIRE]. (1997) 533 [hep-th/9705104] [INSPIRE]. [arXiv:1310.1931] [INSPIRE]. [10] R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. [arXiv:1401.5908] [INSPIRE]. [14] N. Cabo Bizet, A. Klemm and D. Vieira Lopes, Landscaping with uxes and the E8 Yukawa point in F-theory, arXiv:1404.7645 [INSPIRE]. [15] M. Cvetic, A. Grassi, D. Klevers, M. Poretschkin and P. Song, Origin of abelian gauge symmetries in heterotic/F-theory duality, JHEP 04 (2016) 041 [arXiv:1511.08208] [INSPIRE]. [arXiv:1607.07280] [INSPIRE]. Birkhauser, Germany (1989). 8 (1986) 315. [16] S. Mizoguchi and T. Tani, Looijenga's weighted projective space, Tate's algorithm and Mordell-Weil Lattice in F-theory and heterotic string theory, JHEP 11 (2016) 053 [17] F.R. Cossec and I.V. Dolgachev, Enriques surfaces I, Progress in Mathematics volume 76, [18] R. Miranda and U. Persson, On extremal rational elliptic surfaces, Math. Z. 193 (1986) 537. [19] I. Naruki, Con gurations related to maximal rational elliptic surfaces, Adv. Stud. Pure Math. [20] M. Schutt and T. Shioda, Elliptic surfaces, Adv. Studies Pure Math. 60 (2010) 51 [21] G.W. Moore, Les Houches lectures on strings and arithmetic, hep-th/0401049 [INSPIRE]. [22] T. Shioda, K3 surfaces and sphere packings, J. Math. Soc. Japan 60 (2008) 1083. [23] Y. Kimura, Gauge symmetries and matter elds in F-theory models without sectioncompacti cations on double cover and Fermat quartic K3 constructions times K3, arXiv:1603.03212 [INSPIRE]. (1997) 1 [hep-th/9609122] [INSPIRE]. (2000) 69 [Erratum ibid. B 608 (2001) 477] [hep-th/9906070] [INSPIRE]. [24] Y. Kimura, Gauge groups and matter elds on some models of F-theory without section, JHEP 03 (2016) 042 [arXiv:1511.06912] [INSPIRE]. [25] I. Shimada and D.-Q. Zhang, Classi cation of extremal elliptic K3 surfaces and fundamental HJEP03(218)45 023 [hep-th/9908088] [INSPIRE]. 001 [hep-th/0506014] [INSPIRE]. arXiv:1404.1527 [INSPIRE]. [arXiv:1401.7844] [INSPIRE]. [INSPIRE]. [INSPIRE]. [36] I.I. Piatetski-Shapiro and I.R. Shafarevich, A Torelli theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971) 530. [35] P.S. Aspinwall and R. Kallosh, Fixing all moduli for M-theory on K3 K3, JHEP 10 (2005) [37] D.R. Morrison and W. Taylor, Sections, multisections and U(1) elds in F-theory, [38] V. Braun and D.R. Morrison, F-theory on genus-one brations, JHEP 08 (2014) 132 [39] L.B. Anderson, I. Garc a-Etxebarria, T.W. Grimm and J. Keitel, Physics of F-theory compacti cations without section, JHEP 12 (2014) 156 [arXiv:1406.5180] [INSPIRE]. [40] D. Klevers et al., F-Theory on all toric hypersurface brations and its Higgs branches, JHEP 01 (2015) 142 [arXiv:1408.4808] [INSPIRE]. [41] I. Garc a-Etxebarria, T.W. Grimm and J. Keitel, Yukawas and discrete symmetries in F-theory compacti cations without section, JHEP 11 (2014) 125 [arXiv:1408.6448] [42] C. Mayrhofer, E. Palti, O. Till and T. Weigand, Discrete gauge symmetries by Higgsing in four-dimensional F-theory compacti cations, JHEP 12 (2014) 068 [arXiv:1408.6831] [43] C. Mayrhofer, E. Palti, O. Till and T. Weigand, On discrete symmetries and torsion homology in F-theory, JHEP 06 (2015) 029 [arXiv:1410.7814] [INSPIRE]. [44] V. Braun, T.W. Grimm and J. Keitel, Complete intersection bers in F-theory, JHEP 03 (2015) 125 [arXiv:1411.2615] [INSPIRE]. [arXiv:1502.06953] [INSPIRE]. bered Calabi-Yau 4-folds without section | Hypersurface and double cover constructions, arXiv:1607.02978 [INSPIRE]. without section, JHEP 04 (2017) 168 [arXiv:1608.07219] [INSPIRE]. Izv. 14 (1980) 103. (1958). Topologia Algebraica (International Symposium on Algebraic Topology), Mexico City, Mexico [3] C. Beasley , J.J. Heckman and C. Vafa , GUTs and exceptional branes in F-theory | I, JHEP [4] C. Beasley , J.J. Heckman and C. Vafa , GUTs and exceptional branes in F-theory | II: experimental predictions , JHEP 01 ( 2009 ) 059 [arXiv: 0806 .0102] [INSPIRE]. [5] R. Donagi and M. Wijnholt , Breaking GUT groups in F-theory , Adv. Theor. Math. Phys . 15 [6] C. Vafa , Evidence for F-theory, Nucl . Phys. B 469 ( 1996 ) 403 [ hep -th/9602022] [INSPIRE]. [7] D.R. Morrison and C. Vafa , Compacti cations of F-theory on Calabi-Yau threefolds . 1, Nucl . [8] D.R. Morrison and C. Vafa , Compacti cations of F-theory on Calabi-Yau threefolds . 2, Nucl . [9] A. Sen , F theory and orientifolds , Nucl. Phys. B 475 ( 1996 ) 562 [ hep -th/9605150] [11] P.S. Aspinwall and D.R. Morrison , Point-like instantons on K3 orbifolds, Nucl . Phys. B 503 [12] L.B. Anderson , J.J. Heckman and S. Katz , T-branes and geometry , JHEP 05 ( 2014 ) 080 [13] A.P. Braun , Y. Kimura and T. Watari , The Noether-Lefschetz problem and gauge-group-resolved landscapes: F-theory on K3 K3 as a test case , JHEP 04 ( 2014 ) 050 [29] K. Kodaira , On compact analytic surfaces III, Ann . Math. 78 ( 1963 ) 1 . [30] K. Becker and M. Becker , M theory on eight manifolds , Nucl. Phys. B 477 ( 1996 ) 155 [31] S. Sethi , C. Vafa and E. Witten, Constraints on low dimensional string compacti cations , Nucl. Phys. B 480 ( 1996 ) 213 [ hep -th/9606122] [INSPIRE]. [32] E. Witten , On ux quantization in M-theory and the e ective action , J. Geom. Phys . 22 [33] S. Gukov , C. Vafa and E. Witten, CFT's from Calabi-Yau four folds , Nucl. Phys. B 584 [34] K. Dasgupta , G. Rajesh and S. Sethi , M theory, orientifolds and G- ux, JHEP 08 ( 1999 ) groups of open K3 surfaces , Nagoya Math. J. 161 ( 2001 ) 23 [math/0007171]. [26] T. Shioda and H. Inose , On singular K3 surfaces, in Complex analysis and algebraic geometry , in W.L. Jr. Baily and T. Shioda eds., Iwanami Shoten , Tokyo Japan ( 1977 ). [27] M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries , Nucl. Phys. B [28] M. Schu tt, Elliptic brations of some extremal K3 surfaces, Rocky Mount . J. Math. 37 [45] M. Cvetic et al., F-theory vacua with Z3 gauge symmetry , Nucl. Phys. B 898 ( 2015 ) 736 [46] L. Lin , C. Mayrhofer , O. Till and T. Weigand , Fluxes in F-theory compacti cations on genus-one brations , JHEP 01 ( 2016 ) 098 [arXiv: 1508 .00162] [INSPIRE]. [47] Y. Kimura , Gauge groups and matter spectra in F-theory compacti cations on genus-one [48] Y. Kimura , Discrete gauge groups in F-theory models on genus-one bered Calabi-Yau 4 -folds [50] S. Kondo , Type II degenerations of K3 surfaces , Nagoya Math J. 99 ( 1985 ) 11 . [51] V.V. Nikulin , Integral symmetric bilinear forms and some of their applications , Math. USSR [53] T. Shioda , On the Mordell-Weil lattices , Comment. Math. Univ. St. Pauli 39 ( 1990 ) 211 . [54] K. Oguiso and T. Shioda , The Mordell-Weil lattice of a rational elliptic surface , Comment. Math. Univ. St. Pauli 40 ( 1991 ) 83 . [55] I. Shimada , On elliptic K3 surfaces , Michigan Math. J. 47 ( 2000 ) 423 [math/0505140].


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP03%282018%29045.pdf

Yusuke Kimura. Structure of stable degeneration of K3 surfaces into pairs of rational elliptic surfaces, Journal of High Energy Physics, 2018, 45, DOI: 10.1007/JHEP03(2018)045