Heterotic-Type IIA Duality and Degenerations of K3 Surfaces

We study the duality between four-dimensional N=2 compactifications of heterotic and type IIA string theories. Via adiabatic fibration of the duality in six dimensions, type IIA string theory compactified on a K3-fibred Calabi-Yau threefold has a potential heterotic dual compactification. This adiabatic picture fails whenever the K3 fibre degenerates into multiple components over points in the base of the fibration. Guided by monodromy, we identify such degenerate K3 fibres as solitons generalizing the NS5-brane in heterotic string theory. The theory of degenerations of K3 surfaces can then be used to find which solitons can be present on the heterotic side. Similar to small instanton transitions, these solitons escort singular transitions between different Calabi-Yau threefolds. Starting from well-known examples of heterotic--type IIA duality, such transitions can take us to type IIA compactifications with unknown heterotic duals.


Introduction
The duality between heterotic string theory and Type II string theories hints at non-trivial relations between seemingly totally unrelated mathematical objects. For example, heterotic string compactifications involve gauge field moduli, whereas only the compactification geometry must be specified for Type II compactifications. This article addresses some aspects of the duality dictionary of heterotic-type IIA duality in four dimensions.
Heterotic-type IIA duality in 4D [1] is not only a historical precursor to heterotic-Ftheory duality at 6D [2,3]. The former can also be regarded as a more general version of the latter, and moreover, we expect to formulate the duality in terms of world-sheet string theory.
The study of duality begins with finding a correspondence between discrete data of the compactifications on both sides, followed by an identification of the moduli. At the level of discrete data, however, we must say that the correspondence remains to be understood very poorly even today, apart from a few cases that have been studied in the context of heterotic-F-theory duality in 6D. This article intends to provide a survey on the status of understanding on this problem, and also to make a little progress.
Starting with the heterotic-type IIA duality in six dimensions, the key principle in understanding the correspondence of discrete data is the idea of adiabatically fibering the dual six dimensional theories over a base P 1 [4,5,6]. Armed with this principle, the problem of discrete data correspondence roughly splits into two fronts. One is to find out the variety in fibering the duality at higher dimensions without violating the adiabaticity; this is the sub-ject of Section 3 in this article. There are often multiple adiabatic fibrations of the duality for a given pair of lattices Λ S ⊕ Λ T ⊂ II 4,20 . We introduce an approach to use the hypermultiplet moduli information to distinguish multiple adiabatic fibrations from one another. This approach is used, for example, to indicate that the heterotic ST -model is dual to the type IIA compactification on (12) ⊂ W P 4 [1:1:2:2:6] , while three other candidate compactifications 1 of type IIA are excluded because of the hypermultiplet moduli information.
The other front is to study how the adiabaticity condition can be violated, and how to maintain the duality correspondence in the presence of such a violation. We address this question in Sections 4 and 5. Some background material from mathematics (Kulikov's theory of degenerations of K3 surfaces) is reviewed in Appendix B. A degeneration of a K3 fibre in a type IIA compactification should be regarded as a soliton in the heterotic dual. The variety in degenerations of a K3 fibre translates into the variety of generalizations of the NS5-brane in heterotic string theory. The classification theory of degenerations of latticepolarized K3 surfaces indicates which pairs of solitons can (co)exist in a BPS configuration. Furthermore, we can learn about the phase structure of the moduli space of solitons from the phase structure of the Kähler cone of the Calabi-Yau threefolds on the type IIA side.
Apart from the issues discussed in Appendix B, we do not try to make this article strictly self-contained. The review article [7] contains a lot of useful material about K3 surfaces. A review of those parts of lattice theory which are heavily used in the study of K3 surfaces can be found e.g. in [8]. We use the same notation as in [8], and mostly only offer brief explanations here. Similarly, we use the same notation as in [9] for toric geometry. We refer to [9] for definitions and explanations concerning the methods of toric geometry used in this article.

A Quick Review
Type IIA string theory compactified on certain Calabi-Yau threefolds M are known to be dual to certain compactifications of heterotic string theory preserving N = 2 supersymmetry in four dimensions [1]. This D=4 heterotic-IIA duality is best understood as an adiabatic fibration [4,5,6] of the D=6 heterotic-IIA duality, where a dual pair is formed of a Narain compactification of the heterotic string on T 4 and a K3 compactification of the type IIA string [10,11,12,13,14]. For type IIA compactification, we hence focus on non-singular Calabi-Yau threefolds M that admit a K3-fibration morphism, A generic fibre S t.A := π −1 M (t) for t ∈ P 1 A \∆ (∆ is a set containing a finite number of points in P 1 A ) is a non-singular K3 surface, the subscript A for the base P 1 and the fibre K3 S t is a mnemonic for their use in type IIA compactification.
To a Calabi-Yau threefold M with a K3-fibration morphism π M we can naturally associate two lattices: the Neron-Severi (NS) lattice NS(S t.A ) (rank ρ) of a generic K3 fibre, and the lattice polarization Λ S (rank r) of the K3 fibration. A K3-fibration morphism π M is said to be Λ S -polarized, when a set of divisors {D 1 , · · · , D r } of M are restricted to a generic fibre S t.A to generate a sublattice Λ S of NS(S t.A ).
The orthogonal complement of Λ S in H 2 (S t.A ; Z) ∼ = II 3,19 -denoted by Λ T -is well-defined (regardless of t ∈ P 1 A ) up to lattice isometry. Since the generators of H 0 (S t.A ) and H 4 (S t.A ) of the fibre remain well-defined over the base P 1 A , we can replace the lattice Λ S by Λ S := U [−1] ⊕ Λ S . The pair of lattices Λ S ⊕ Λ T ⊂ II 4,20 (2) can be used to classify Calabi-Yau threefolds fibred by a lattice-polarized K3 surface used for type IIA compactification. Heterotic duals of such type IIA vacua are obtained by adiabatically fibering the Narain compactification over the base P 1 Het . The central charge (k 8 + ik 9 ) : II 4,20 ⊃ Λ S −→ C remains non-trivial and invariant over the base P 1 Het , while (k 6 + ik 7 ) takes values in Λ T ⊗ C and is allowed to undergo monodromy transformations over the base P 1 Het . An intuitive heterotic description is available for such dual pairs of vacua whenever where W and R are some even negative-definite lattices. In particular, R is regarded as an overlattice of an ADE root lattice. 2 When the first condition above is satisfied, the threefold for heterotic string compactification can be regarded as T 2 × K3 Het . A supergravity description is available (for most of the moduli space), because we can take the volume of T 2 to be parametrically larger than α ; even a lift to Het-F duality at 6D is possible in this case. When the second condition above is satisfied, we can think of the value of (k 6 + ik 7 ) varying over the base P 1 Het as an instanton in the gauge group corresponding to the algebra R.
Even when those two conditions are not satisfied, however, it is common belief that such heterotic vacua do indeed exist. We can provide a zero-th order approximation of what those compactifications are by using the language of adiabatic fibrations of Narain moduli. When there are points in the base P 1 where the adiabatic argument fails, extra care needs to be taken. This is the subject of Section 4 in this article.

Examples of Algebraic K3 Surfaces
In this section we collect some results about algebraic K3 surfaces, impatient readers are recommended to directly proceed to section 3.
To an algebraic K3 surface, we can assign its Neron-Severi lattice NS K3 (signature (1, ρ − 1)) and its transcendental lattice T K3 . Conversely, to see which pair of lattices (N S K3 , T K3 ) can be realized for some algebraic K3 surface, Morrison's theorem [15] is useful.
• Any even lattice NS K3 with signature (1, ρ − 1) with ρ ≤ 10 can be realized as the Neron-Severi lattice of an algebraic K3 surface, and furthermore, the corresponding lattice T K3 is determined uniquely (modulo isometry) for a given such NS K3 .
• Any even lattice T K3 with signature (2, 20 − ρ) with 12 ≤ ρ can be realized as the transcendental lattice of an algebraic K3 surface, and furthermore, the lattice NS K3 is determined uniquely (modulo isometry) for a given such T K3 .
Thus, lattice polarizations with low ρ are best worked out by classifying even signature (1, ρ − 1) lattices Λ S modulo lattice isometry, and those with high ρ by classifying even signature (2, 20 − ρ) lattices Λ T modulo isometry. Clearly, only some algebraic K3 surfaces listed up in this way satisfy the conditions (4). Here, we list up a few choices of NS K3 ⊕ T K3 , some of which are used in the discussion later.
Picard number 1 cases are classified simply by the degree, The signature of the Neron-Severi lattice is (1, 0). The degree 2k is an arbitrary even positive integer, without an upper limit in the value. The theorem above guarantees that Only the case k = 1 satisfies the second condition of (4), because −2 = A 1 . The degree 2k = 2 case has a realization in the form of a double cover over P 2 , ramified over a sextic curve. The degree 2k = 4 case is realized by the quartic K3 in P 3 . The degree 2k = 6 and degree 2k = 8 cases are realized in the form of complete intersections, (2) ∩ (3) ⊂ P 4 and (2) ∩ (2) ∩ (2) ⊂ P 5 , respectively. For higher degrees, the construction becomes more involved, see [16] for further information.
This family of K3 surfaces is elliptically fibred with the elliptic fibre given in the form of a Weierstrass model, embedded in the ambient space W P 2 [1:2 :3] . There is a unique choice of fibration of W P 2 [1:2:3] over P 1 realizing an elliptic Weierstrass K3 surface as a hypersurface. There are infinitely many cases with Picard number 2. The intersection form of the Neron-Severi lattice of any ρ = 2 case can be written as once a basis is chosen. The determinant (4ac − b 2 ) is independent of the choice of basis. 3 There is no upper limit for the value of (b 2 − 4ac) > 0, although (b 2 − 4ac) ≡ 0, 1 mod 4. Among these infinitely many algebraic K3 surfaces with ρ = 2, just nine are realized as a generic hypersurface of a 3D toric variety. Only the (b 2 − 4ac) = 1 case corresponds to the E 8 -elliptic K3 surface; more generally, an algebraic K3 surface with ρ = 2 admits an elliptic fibration morphism to P 1 if and only if (b 2 − 4ac) = D 2 for some integer D ∈ N; this elliptic fibration has a D-section (cf [17]). E 7 -elliptic K3 surfaces come with a choice. For this class of elliptic fibrations, the elliptic fibre curve is embedded in W P 2 [1: 1:2] and there is a choice we can make in fibering this ambient space over P 1 . Let the toric vectors of the ambient space be (ν 1 , ν 2 , ν 3 , ν 4 , ν 5 3 The discriminant form (G, q) is a better invariant for the classification of lattices modulo isometry. Even the discriminant group G alone, taken in the standard form G ∼ = Z n1 × Z n2 with n 1 |n 2 for the ρ = 2 cases, provides more detailed information than just the value of their product n 1 n 2 = b 2 − 4ac. The ρ = 2 cases with an E 7 -elliptic K3 surface (n = 0, 1 in (10)), however, provide an example where the discriminant form q is necessary for distinction between them.
The intersection form of the NS lattice is −2n 2 2 (10) in the basis of {D ν 3 , D ν 4 } for n = 0, 1. A basis change D ν 3 → D ν 3 + D ν 4 changes the upperleft entry −2n by 4, so that the two cases n = 0, 1 cannot be the same. For these two cases, the K3 surface has the E 7 -elliptic curve as the fibre and there is a 2-section realized by the divisor D ν 3 . In the n = 2 case, however, the ν 3 vector is not a vertex of the polytope in Z ⊕3 ⊗ R, but an interior point of an edge ν 4 , ν 5 of the polytope. The dual face (an edge) has one interior point so that ρ = 3; the 2-section obtained as the divisor D ν 3 now consists of two irreducible pieces, each of which provides an ordinary section. The intersection form of the NS lattice is E 6 -elliptic K3 also comes with a choice. These are characterized by using W P 2 [1:1:1] = P 2 as the ambient space of the elliptic fibre. When we choose the toric ambient space to be given by the K3 surface will have ρ = 2, and the divisor D ν 3 in the K3 surface provides a 3-section over the base P 1 for most of the ten possible choices of (a, b). When we choose (a, b) = 2(1, 0), 2(0, 1) or 2(−1, −1), however, ν 1 (or ν 2 or ν 3 ) is an interior point of an edge of the 3D polytope in Z ⊕3 ⊗ R and its dual face (an edge) contains two interior points. We hence have a ρ = 4 family of K3 surfaces. The intersection form of the NS lattice is given by A series of choices of (NS K3 , T K3 ), which are discussed in [18,2,3] in the context of F-theory/heterotic duality, is where (R vis , R str ) are (none, E 8 ), (A 2 , E 6 ), (D 4 , D 4 ), (E 6 , A 2 ), (E 7 , A 1 ) and (E 8 , none) and the conditions (4) are satisfied. When this series of (NS K3 , T K3 ) is used for heterotic-type IIA duality in four-dimensions (as stated in section 2), the heterotic string description of the dual vacua is a T 2 × K3 Het compactification with R vis -valued Wilson lines on T 2 and (R str + E 8 )valued instantons on K3 Het . Note also that the (R vis , R str ) = (A 1 , E 7 ) and (A 2 , E 6 ) choices in this series are not the same as the E 7 -elliptic and E 6 -elliptic K3 cases above.

Choices of Lattice-Polarized KFibration and Duality
In this section, we focus our attention to K3-fibred Calabi-Yau threefolds where a) The lattice polarization of the fibration, Λ S , is equal to NS K3 (not a proper subset).
b) The fibre K3 surface remains irreducible everywhere over the base P 1 A . This is where the adiabatic argument has full strength. To get started, we use toric hypersurface constructions to illustrate how often these two conditions are satisfied. Once a pair of lattices NS K3 = Λ S and T K3 = Λ T is chosen, there are still discrete choices to be made in how to take the corresponding algebraic K3 surface into the fibre to form a threefold M for type IIA compactification. We find, towards the end of section 3.1, that there are multiple choices for many pairs of (Λ S , Λ T ). This general phenomenon motivates a case study in section 3.2.

Discrete Choices in K3 Fibrations
As a preparation for later in this article, let us first consider one of the best known cases: an E 8 -elliptic K3 surface (NS K3 = U ) as the generic fibre. An E 8 -elliptic K3 surface can be constructed as a hypersurface of a toric ambient space whose toric vectors are given by Let ∆ F be the polytope in N F ⊗ R spanned by the four vertices above. In order to obtain a K3-fibred Calabi-Yau threefold M with such a K3 surface in the fibre, we construct an appropriate toric ambient space as follows. Consider a toric variety given by the toric vectors (16) where ν 6 F is chosen so that This choice secures that the polytope ∆ which forms the convex hull of ν 1 · · · ν 6 is reflexive. A Calabi-Yau hypersurface M = M U of such a toric ambient space has the E 8 -elliptic K3 surface over generic points in the base P 1 A . In this article, we often use Λ S or NS K3 in the subscript, as in M U , to have the lattice polarization of the fibre manifest. 4 Not all the Calabi-Yau threefolds M U constructed in this way realize completely adiabatic fibrations over P 1 A , however. In the polytope ∆ F for the E 8 -elliptic K3 surface in the fibre, there are two facets-ν 1,3,4 F ∩ ∂ ∆ F and ν 2,3,4 F ∩ ∂ ∆ F -that contain interior points. The condition for the absence of extra vertical divisors (equivalent to the K3 fibre being irreducible everywhere) is equivalent to choosing ν 6 F such that those interior points in the facets of ∆ F remain interior points of facets of ∆, i.e.
There are five points of this kind. The corresponding Calabi-Yau threefolds are denoted by M n U , with −2 ≤ n ≤ 2. They are known to be the same as E 8 -elliptic fibrations over F n , with −2 ≤ n ≤ 2. Type IIA compactification on M n U is dual to heterotic string compactification on T 2 × K3, with instantons distributed by 12 + n and 12 − n among the two E 8 's.
It is straightforward to generalized this observation. Suppose that an algebraic K3 surface with a Neron-Severi lattice (NS K3 , T K3 ) can be constructed as a generic toric hypersurface. There are 4319 toric hypersurface families of K3 surfaces realized via pairs of reflexive threedimensional polytopes [19]. Let ∆ F be the polytope in N F ⊗ R = R 3 . A toric ambient space for a threefold M is obtained by fibering the toric ambient space for K3 over P 1 A . When we choose a toric vector ν 5 = ( 0, −1) ∈ Z ⊕4 , we can take ν 6 = (ν 6 F , +1) T ∈ Z ⊕4 , with any one of to construct a reflexive four-dimensional polytope.
Different choices of ν 6 will, in general, result in different geometries, in particular, the Hodge numbers for the resulting threefolds can be different. As reviewed shortly, for some choices of the fibre K3 polytope, ∆ F , it depends on the choice of ν 6 whether conditions a) and b) are satisfied.
The condition b) is at stake whenever we consider a fibre K3 polytope ∆ F with a facet Θ [2] with an interior lattice point. To keep the condition b) in a threefold M , we need to choose ν 6 so that the polytope ∆ has a facet Θ [3] that contains Θ [2] and its interior lattice points altogether (see section 4.1 and Appendix A for more explanations). The restriction (18) in the example of E 8 -elliptic K3 surface came about precisely for this purpose. To take a few other examples, consider degree-2 and degree-4 K3 surfaces (NS K3 = +2 and +4 , respectively); they are both realized as toric hypersurfaces. None of the facets of the polytope ∆ F for the degree-4 (quartic) K3 surface contains an interior point, while just one facet of the polytope ∆ F for the degree-2 K3 surface has an interior point. In constructing a Calabi-Yau threefold M +4 that has a quartic K3 surface in the fibre over P 1 A , one can therefore use any one of the lattice points in 2 ∆ F for ν 6 F . In the case of degree-2 K3 fibred Calabi-Yau threefolds, however, only lattice points in one facet of 2 ∆ F are permitted if we want to satisfy condition b). Such different choices of taking a given polarized K3 surface as a fibre have been a subject of study, for example, in [20].
The condition a) is at stake when a fibre K3 polytope ∆ F and its dual polytope ∆ F have a dual pair of 1-dimensional faces, Θ [1] and Θ [1] such that * ( Θ [1] ) > 0 and * (Θ [1] ) > 0. A divisor D ν of the generic fibre S t,A corresponding to an interior point ν of Θ [1] is reducible and each one of the irreducible components of D ν is an independent generator of Pic(S t,A ) and contributes to ρ. When we choose ν 6 F ∈ 2 ∆ F ∩ M to construct a Calabi-Yau threefold M , however, the divisor D remains to have * (Θ [1] ) + 1 irreducible components only when the choice of ν 6 F is such that the point ν remains to be an interior point of some twodimensional face of ∆. In this case, the contributions of this divisors to NS K3 and Λ S agree. If ν 6 F is chosen such that Θ [1] becomes a one-dimensional face 5 of ∆, on the other hand, the * (Θ [1] ) + 1 irreducible components of the divisor D ν in a generic K3 fibre undergo monodromy transformations over the base P 1 A and do not define separate independent divisors of the threefold M . Correspondingly, this lattice point ν leads to only one irreducible divisor in h 1,1 (M ) and contributes only by a single class to the lattice polarization of fibration, Λ S .
Among the 4319 three-dimensional reflexive polytopes ∆ F to be use for the fibre K3 polytope [19], 131 do not allow a single choice of ν 6 where condition b) is satisfied; for the remaining 4188 polytopes, there is at least one choice of ν 6 such that condition b) is satisfied. Among these 4188 fibre K3 polytopes ∆ F , 1071 do not allow a choice of ν 6 where condition a) is satisfied as well. For the remaining 3117 fibre K3 polytopes, however, there is at least one choice of ν 6 such that both the conditions a) and b) are satisfied. For each one of these 3117 fibre K3 polytopes ∆ F , we must be able to use the adiabatic argument to discuss heterotic-type IIA duality. By a computer scan we have found that for 1134 of the polytopes among the 3117 options the choice of ν 6 for which the conditions a) and b) are satisfied is unique. The remaining 1983 fibre K3 polytopes ∆ F allow multiple choices 6 of ν 6 satisfying both conditions. Multiple choices available in (18), even after fixing (NS K3 , T K3 ) = (Λ S , Λ T ), can be regarded as an example of this more general phenomenon.
Focussing on the 1983 polytopes ∆ F for which there are multiple choices of ν 6 satisfying both conditions a) and b), it turns out that the value of h 2,1 (M ) depends on the choice of ν 6 for some polytopes ∆ F , but remains invariant for others. The example discussed above and the example presented in the nect section are among those ∆ F for which there are different fibration options satisfying conditions a) and b) which all have the same h 2,1 (M ).

Duality Dictionary in a Case Study: Degree-2 K3 in the Fibre
The moduli space of type IIA compactification on a K3-fibred Calabi-Yau threefold is therefore classified by the choice of (Λ S , Λ T ), and further by discrete choices of the fibration. The moduli space of heterotic string compactifications should also have the same structure, and there should be a duality map that translates discrete as well as continuous data of the two moduli spaces. The dictionary on the (Λ S , Λ T ) part simply descends from the dictionary of the heterotic-type IIA duality at higher dimensions (reviewed in section 2). We can then ask the question how the discrete choices of fibration are mapped to heterotic string language.
In the case that the Calabi-Yau threefold M for type IIA compactification has an E 8elliptic K3 surface in the fibre, the dual background for heterotic string theory is wellknown [1,2,3]. Different choices of fibering an E 8 -elliptic K3 surface over P 1 A -M n U with −2 ≤ n ≤ +2-correspond to the (12 + n, 12 − n) distribution of 24 instantons into the two E 8 's in heterotic string theory. This is a rare example, however, where the heterotic string interpretation of the discrete data is known. Different choices of instanton number distribu-tion in the heterotic string often result in different unbroken symmetries, which correspond to different choices of (Λ S , Λ T ), not the different choices of fibration with the same (Λ S , Λ T ).
Here, we address this question for the Λ S = +2 case, where a degree-2 K3 surface is the fibre for a K3 fibred Calabi-Yau threefold used for type IIA compactification. This is not as easy as the Λ S = U (E 8 -elliptic) case, but still remains relatively tractable. We start off by listing up discrete choices of degree-2 K3 fibred Calabi-Yau threefold M +2 (see also [20]).
The toric vectors in N F ∼ = Z ⊕3 for the degree-2 K3 surface (NS K3 = +2 , ρ = 1) can be chosen as follows 7 The polytope ∆ F ⊂ N F ⊗ R has four facets, only one of which, where X 1,2,3,4 are the homogeneous coordinates associated with the ν i F . This provides the picture of a double cover over P 2 (homogeneous coordinates [X 2 : X 3 : X 4 ]) ramified over a sextic curve {F (6) = 0} ⊂ P 2 .

Four branches with
A toric hypersurface Calabi-Yau threefold M with a Λ S = +2 -polarized K3 surface in the fibre can be constructed by using a toric ambient space constructed as in (16). Any one of the choices of ν 6 F satisfying (19) can be used to construct a non-singular threefold M +2 for type IIA compactification. In this section, we focus on the choices where the condition b) is satisfied (condition a) is automatic), such that h 1,1 (M +2 ) = ρ + 1 = 2 and the resulting D=4 N = 2 effective theory has h 1,1 (M ) = 2 vector multiplets. This narrows down the choice of ν 6 F to 2ν 2,3,4 There are ten choices for the integers (k 2 , k 3 , k 4 ) in as shown in Figure 1. The ambient space for M +2 is Figure 1: The ten lattice points 2ν 2,3,4 F ∩ ∂(2 ∆ F ) ∩ N F , which can be used for a toric vector ν 6 F in constructing a Calabi-Yau threefold with degree-2 K3 fibration satisfying conditions a) and b). Lattice points parametrized by (k 2 , k 3 , k 4 ) in (22) are labelled [n a ] when we can choose k a = n and two other k's zero. Due to the S 3 symmetry acting on this graph, at most four of them (maybe only three as discussed in the main text) define mutually non-isomorphic complex geometries, however.
There is an S 3 symmetry transformation acting on the lattice N which keeps ∆ F invariant. It acts as a permutation on the toric vectors ν 2,3,4 ] can be regarded as (a resolution of) the weighted projective space W P 4 [1:1:2:2:6] , while the three others (n = 1, 0, −1) cannot be regarded as such.
Wall's theorem states that the diffeomorphism class of a threefold M is characterized up to a finite number of possibilities by H 3 (M, Z), H 2 (M, Z), the intersection ring and the second Chern class. Furthermore, if H 3 (M, Z) is torsion free, the diffeomorphism class is characterized uniquely. The intersection rings of M n +2 with n = 2, 1, 0 do not agree for any identification of integral cohomology groups H 2 (M n +2 ; Z), so that these manifolds cannot be diffeomorphic. This means that type IIA compactifications on M n +2 with n = 2, 1, 0 each have their own separate moduli spaces [20]. There is such an identification between H 2 (M n=2 +2 ; Z) and H 2 (M n=−1 +2 ; Z), on the other hand. This indicates that M n=2 +2 and M n=−1

+2
are the same as real manifolds. It is not known, however, whether the complex structure moduli space of this real manifold has just one connected component. 8 We therefore have not ruled out the possibility that type IIA compactifications on M n +2 with n = 2 and n = −1 describe physically different vacua, and we treat them separately in the rest of this article. 9 At the very beginning of the study of heterotic-type IIA duality [1], type IIA compactification on M n=2 +2 = [(12) ⊂ W P 4 [1:1:2:2:6] ] was pointed out as the dual of a heterotic compactification, the ST -model. 10 Primary evidence for this duality claim is comprised of i) the agreement of the pair of lattices ( Λ S , Λ T ) and ii) the agreement of the number of vector and hypermultiplets in the D=4 N = 2 effective theory, on both sides of heterotic and type IIA descriptions. It turns out, however, that all of the Calabi-Yau threefolds M n +2 with n = 2, 1, 0, −1-sharing the lattices ( Λ S , Λ T )-have [20] Hence from observations i) and ii) not just type IIA compactification on M n=2 +2 , but on any one of the M n +2 's must be regarded as an eligible candidate for the dual of the heterotic ST -model.
Let D i be the divisors of the toric ambient space corresponding to the toric vector ν i (i = 1, · · · , 6), andD i := D i | M n +2 . Let us focus on M n +2 with n = 2, 1, 0. Then we can choose two curves C 2 and C 5 in M n

+2
to generate the cone of effective curves (Mori cone) of M n +2 ; here, C 2 ·D 2 = C 5 ·D 5 = 1 and C 2 ·D 5 = C 5 ·D 2 = 0. Complexified Kähler parameters (t 2 , t 5 ) are introduced (Im(t 2 ) > 0 and Im(t 5 ) > 0), and the complexified Kähler form is given by (B + iJ) = t = t 2D2 + t 5D5 . The Gromov-Witten invariants of vertical curve classes, namely, β = n 2 C 2 + n 5 C 5 with n 5 = 0, remain independent of the discrete choices (n = 2, 1, 0) of the fibration [20]. This type IIA information corresponds to the 1-loop threshold correction in heterotic computations. Thus, the experimental evidence so far allows an interpretation that all of type IIA compactifications with M n=2,1,0 +2 are dual to the heterotic string ST -model defined at the perturbative level (in the g s expansion of the heterotic string) and the presence of multiple choices of M n +2 in type IIA (n = 2, 1, 0) is an indication that multiple non-perturbative completions are possible in heterotic string theory [20]. 8 An example of this phenomenon is discussed e.g. in [21]. 9 The Kähler cones of M n=2 +2 and M n=−1

+2
are mapped to each other under the identification φ. The genus zero Gromov-Witten invariants of M n +2 with n = 2 and n = −1 also seem to agree under the identification φ. The result in section 3.2.3, however, makes us hesitate from saying that M 2 +2 and M −1 +2 are the same. 10 The ST -model is a heterotic string compactification on "T 2 × K3" with one S 1 ⊂ T 2 at the self-dual radius. The S 1 at the self-dual radius gives rise to an extra gauge group A 1 and the 24 instantons are distributed as 4, 10 and 10 between A 1 , E 8 and E 8 , respectively. This is an attractive interpretation, but we must say that it still sounds odd. Certainly the third term in the D = 4 N = 2 prepotential come from 1-loop threshold correction in a heterotic string computation, but the second term-computed from the intersection ring of M n +2 in type IIA compactifications-is also supposed to come from heterotic string 1-loop threshold correction. A given heterotic string compactification (say, the ST -model) cannot take multiple values. If type IIA compactification on one of the Calabi-Yau threefolds M n +2 is dual to the ST -model, the IIA compactifications on the other M n +2 cannot be dual to the ST -model. Reference [22] indicates how to extract the coefficients of such tri-linear term in the prepotential, for some examples of heterotic string compactifications to four-dimensions. 11 Hence, it is possible to pursue this approach, which exploits information on the vector multiplet moduli space on both sides of the duality. In this article, we provide an alternative method to study the duality dictionary of the choices of fibration, which uses the hypermultiplet moduli space. It is better to have more tools than less!
To get started, let us go back to the case where we choose the generic fibre to be the E 8 -elliptic K3 surface, because a lot more is known in the physics literature. In this case, NS K3 = U and T K3 = E ⊕2 8 ⊕ U ⊕2 . Our discussion in the following is valid in the weak coupling regime of heterotic vacua, or equivalently, the large volume region of P 1 A of type IIA vacua. The Narain moduli of heterotic string theory and the period integral of the fibre K3 surface for type IIA can be compared fibrewise in this situation. In order to study the duality map of discrete data (such as the instanton number distributions and the choice of fibration of a given lattice polarized K3 surface), it is enough to use any corner of moduli space that is continuously connected.
The hypersurface equation of an E 8 -elliptic K3 surface is written down as in For a specific fibre, f k and g m are complex numbers. This K3 surface is to be used for type IIA compactification. On the heterotic side, we have the Narain moduli (ρ, τ, a I=1,··· ,16 ) corresponding to the volumeρ and complex structure τ of T 2 , as well as Wilson lines a I=1,··· , 16 .
To establish a dictionary between (f k , g m ) and (ρ, τ, a I ) is simple, at least conceptually. One merely needs to compute period integrals for transcendental cycles, and express them in terms of (f k , g m ).
In practice, it is not a simple task to determine period integrals depending on 18 complex variables, but even knowing their qualitative behaviour goes a long way. 12 At the qualitative level, there are well-known constraints on the (f k , g m ) for the E 8 ⊕ E 8 gauge symmetry of the heterotic string to remain unbroken, and furthermore, it is known how to scale (f k , g m ) such that the T 2 volume of the heterotic description is large (i.e.,ρ → i∞) [3]. In this way, we learn which part of the complex coefficients of the hypersurface equation corresponds to the moduli controlling geometric aspects of the heterotic compactification. This dictionary has been extended to some extent to include the moduli controlling the breaking of the E 8 ⊕ E 8 symmetry. Such a map has been discussed in the context of local mirror symmetry for any ABCDE group [24,25]. For a compact K3 surface, the duality map has been discussed for the case the symmetry breaking stays within SU(5) × SU(5) ⊂ E 8 × E 8 [26,27,28,29]. See [30] for symmetry breaking in SU(6) ⊂ E 8 , and [31] for more general cases. For other aspects of hypermultiplet moduli map, see e.g. [32,33,34,35] and references therein.
Consider taking the coefficients (f k , g m ) in (26) to be with |g m |, |f k | ∼ O(1) and η 1 and K 1. This is a generalization of the scaling in [3,36]. Under such a choice of complex coefficients, 10 out of the 24 discriminant points of this E 8 -elliptic K3 surface are found in the region z ∼ ( η 6 K ), 2 are in the region z ∼ η , 2 more are in the region z ∼ −1 η and the remaining 10 are found in the region z ∼ ( η 6 K ) −1 . The K -scaling power of individual coefficients above is determined such that the hypersurface equation (26) around (x, y, z) = (0, 0, 0) is well approximated by a deformation of an E 8 singularity. Indeed, we only need to rewrite (26) by using a set of local coordinates (ξ, η, ζ) in and drop all the terms with positive powers in η or K . It is also possible (though not necessary) to consider the (reducible) K3 surface associated with the stable degeneration limit corresponding to η → 0 and focus on one of the two irreducible components. We then have a rational elliptic surface [37] The f k and g m here correspond to β k and α m of dP 8 in [37]. This description (parametrization) of E 8 Wilson lines in T 2 is redundant. This is due to the fact that we have not fixed the automorphisms acting on the base P 1 of the E 8 -elliptic K3 surface. For two constants c 1 and c 2 , it is By allowing this redundancy in the parametrization, however, the collection of η scaling powers, {1, 2, 3, 4, 5, 6, 1, 2, 3, 4}, contains the full list of Dynkin labels of the extended Dynkin diagram of E 8 . The collection of K scaling powers, {0, 6, 12, 18, 24, 30, 2, 8, 14, 20}, contains degrees of all of the independent deformation parameters of the E 8 singularity in [38]. With this preparation, let us now consider a Calabi-Yau threefold M that has an E 8elliptic K3 surface in the fibre. Here, the coefficients f ±k and g ±m are promoted to sections of line bundles over the base P 1 A . When the fibration corresponds to a choice of ν 6 F in (18) with −2 ≤ n ≤ 2, g m ∈ Γ(P 1 A ; O(12 + n · m)) = Γ(P 1 A ; O(|m|η ± + (6|m| − 6)K P 1 )). (31) where η + = 12 + n for m > 0 and k > 0, and η − = 12 − n for m < 0 and k < 0. Therefore, we see that any one of (f k , g m ) which is required to have a scaling A η B K for the Het-sugra and near-symmetry-restoration takes its value in a line bundle O P 1 A (Aη + BK P 1 ). It is also known, based on the study of chains of Higgs cascades and singular transitions among various branches of the moduli spaces of F-theory (or type IIA) compactifications, that M is dual to heterotic string theory compactified on K3 (K3 x T 2 ) with the instantons distributed as η + and η − to For a given branch of the moduli space of type IIA compactifications on a Calabi-Yau manifold M with a lattice-polarized K3 fibration π M : M −→ B, we are hence motivated to ask if there is an assignment of scalings A η B K of the complex coefficients that leads to a decoupling of gravity and symmetry restoration. When there is such an assignment of the scalings, one can then further ask if there is a divisor η on the base B such that a section f under the scaling A η B K is a section of O B (Aη + BK B ). If such a divisor η is found, then we can take it to be the instanton number (more generally second Chern character ch 2 ) defined purely in terms of (hypermultiplet moduli of) the type IIA compactification. In cases where the heterotic-IIA duality dictionary is not understood well enough, it is important that we can extract such information intrinsically.
This is a generalization of the same idea that has been known to hold 13 in the case of symmetry breaking with the structure group SU(N ). When sections a r ∈ Γ(B; O B (η +rK B )) (r = 0, 2, · · · , N ) for some divisor η on B in a hypersurface equation has the scaling η r K for symmetry restoration of SU(N ), the heterotic dual involves a vector bundle with the instanton number (second Chern character ch (2)) specified by η.

E 8 ⊕ E 8 degeneration of degree-2 K3 surfaces
In order to argue what is the distribution of instanton numbers in the heterotic dual of type IIA compactifications on M n +2 (n = 2, 1, 0, −1), where the fibre is a degree-2 K3 surface, we first need to find a scaling behaviour of the complex coefficients of M n +2 that leads to symmetry restoration.
The E 8 ⊕E 8 ⊕A 1 part of two-cycles remains in the transcendental lattice T K3 for a generic fibre. Since transcendental cycles of a K3 surface correspond to divisors of its mirror K3 surface (up to a sublattice U ), lattice points of the dual polytope ∆ F can be used to capture those two-cycles. The dual polytope ∆ F of degree-2 K3 surfaces is shown in Figure 2 (b) along with that of E 8 -elliptic K3 surface. To each lattice point marked in Figure 2 (b), there is a corresponding transcendental two-cycle. They are all isomorphic to S 2 , except for the cycle corresponding to the lattice point at the top of the polytope, which is isomorphic to a T 2 . There are three linear (topological) relations among them, and after a computation of the intersection form, a rank-18 lattice of transcendental two-cycles, is obtained. Two more transcendental cycles are missing here, because they correspond to H 0 and H 4 of the mirror of S t.A . The entire transcendental lattice is T K3 (S t.A ) ⊕ U . There are three different ways in identifying the E 8 ⊕ E 8 lattice among those two-cycles, which can be traced back to the S 3 symmetry on the moduli space of degree-2 K3 surface. One of the three identifications is already shown in Figure 2 (b). With this picture in mind, it is now easy to figure out how to assign the scaling behaviour for the approximate restoration of the E 8 ⊕ E 8 symmetry in the moduli space of degree-2 K3 surfaces. Let us write down the hypersurface equation of a degree-2 K3 surface in a set of affine coordinates, (y, x 4 , x 3 , x 6 ) = (X 1 /X 3 2 , X 4 /X 2 , X 3 /X 2 , X 6 /X 5 ): Table 1: Each column corresponds to a Calabi-Yau threefold hypersurface M in the toric ambient space corresponding to one of the choices of ν 6 F shown in Figure 1. a r (r = 1, · · · , 6), b r (r = 1, · · · , 4), c r=1,2 , and a r , b r , c r are sections of line bundles on P 1 A whose degrees are indicated in the 1st-6th rows in this table.
where we used an affine patch ( where η and K are taken to be small, while a r * , b r * and c r * are O(1). When the value of η is small, one set of E 8 transcendental cycles (visible sector) are found close to the point X 4 = X 3 = 0 in P 2 , while the other set of E 8 transcendental cycles (hidden sector) are located near the point X 4 = X 2 = 0. When K is set to zero, while η remains small but non-zero, we have an E 8 singularity, y 2 + a 1 η x 5 4 + d 0 x 3 3 = 0. The scaling behaviour assigned for a m and b k agrees with those for g m and f k in the case of an E 8 -elliptic K3.
When the degree-2 K3 surface is fibred over the base P 1 A , the coefficients a r , b r and c r for the visible E 8 and those for the hidden E 8 are promoted to sections of certain line bundles. We can work out the degree of those line bundles for any given choice of fibration in Figure  1. The results are shown in Table 1. If the complex structure moduli of a threefold M n +2 are to be interpreted as E 8 + E 8 instanton moduli in heterotic string, we expect that . For any other choice, the degrees of the relevant line bundles cannot have the right pattern to even define 14 the instanton numbers of E 8 + E 8 intrinsically in terms of the threefolds M n +2 (n = 1, 0, −1). We therefore conclude that only the type IIA compactification on M n=2 +2 is dual to the heterotic ST -model (where the instantons numbers are distributed by 4+10+10 in A 1 + E 8 + E 8 ). Type IIA compactifications on M n +2 with n = 1, 0, −1 are not, although the effective theories with D = 4 N = 2 supersymmetry have the same number of vector and hypermultiplets, and the special geometry passes highly non-trivial tests of duality (the third term of (25)). The hypermultiplets, however, do not seem to reproduce the instanton moduli expected in the heterotic ST -model and the heterotic dual of type IIA compactifications on M n +2 with n = 1, 0, −1 must be something other than the ST -model. A case study for Λ S = +2 was presented above, but this is a very small subset of all the O(2000) choices of Λ S , where there are multiple choices of fibering Λ S -polarized K3 surface over P 1 A . The method described above may be applied to cases where Λ T contains U ⊕ U ⊕ (⊕ a R a ) with an ADE root lattice R a ; after assigning the height A and degree B for a symmetry R a to monomials in the defining equation of M Λ S , one can ask whether an appropriate divisor η a for R a is found. In the case of Λ S = +4 (where we have a quartic K3 surface as the fibre in IIA language), for example, it is at least possible to talk about the instanton number assignment in the E 8 ⊕ E 8 part of Λ T , if not for the −4 part. It will be difficult, however, to apply this method to Calabi-Yau manifolds without a complete intersection construction, or to choices of Λ T without a U ⊕ U component, or a single factor of R a .

Degenerations of K3 Surfaces and Soliton Solutions
In the last section, we restricted our attention to K3-fibred Calabi-Yau threefolds where the K3 fibre remains irreducible everywhere over the base P 1 A and furthermore the entire NS K3 lattice of a generic fibre becomes the lattice polarization of the fibration, NS(S t.A ) = Λ S . The method of construction was limited to using a toric polytope ∆ spanned by just one toric vector ν 6 in addition to ν 1,2,3,4,5 . The spirit was to focus on situations where the adiabatic argument can be used.
In this section, we explore fibrations where the adiabatic argument does not hold at isolated points in the base P 1 A , and discuss their heterotic dual descriptions. In particular, we will relax the condition that the K3 fibre remains irreducible everywhere over the base P 1 A for a smooth threefold M . Examples in section 4.1 are such that only a single extra vertex ν 6 is introduced besides ν 1,2,3,4,5 in the toric polytope. In sections 4.2 and 4.3 we also relax this condition, and the construction in [40] (and its obvious generalization) is exploited. We will see a rich variety of branches of the type IIA compactification moduli space, even for a single choice of the lattice Λ S = NS K3 . After examining the reducible fibre geometries and how those branches are connected, we will study their heterotic string interpretation in section 4.4.

A Simple Fibration with Reducible Fibre(s)
Let us continue to work out discrete fibration choices of degree-2 K3 fibred Calabi-Yau threefolds M +2 realized as toric hypersurfaces corresponding to a polytope spanned by ν 1,2,3,4,5, and just one point ν 6 . Contrary to before, however, we do not require that the fibre K3 surface remains irreducible everywhere over the base P 1 A . This means that we can choose any one of (19), not just those in ν 2,3,4 For the choices of toric vectors ν 6 := (v 1 − 2, v 2 , v 3 , 1) T that are now allowed, there is one lattice point ν 7 = (ν 7 F , 0) T ∈ N placed in the interior of a two-dimensional face Θ [2] of ∆, but not interior to any one of the facets of ∆. The contribution to h 1,1 (M +2 ) is where Θ [1] is the dual face of Θ [2] . The possible values of (v Figure 1-have been studied in section 3.2.1. Interior points to facets of ∆ F which are not interior to facets of ∆ likewise give rise to reducible divisors for any K3-fibred Calabi-Yau threefold realized as a toric hypersurface. A brief explanation for this statement is given in Appendix A, but this is well-known already in the case of having an E 8 -elliptic K3 surface in the fibre [41]. When ν 6 F is chosen from (17), but not from (18), then the two lattice points interior to the two-dimensional face ν 2,3,4 F ∩(∂ ∆ F ) result in a singular fibre (or even several singular fibres) with three components, whereas ν 1,3,4 F ∩ (∂ ∆ F ) potentially gives rise to singular fibres with two components.
Coming back to the case of a degree-2 K3 surface as the fibre, let us take (v 1 −v 2 −v 3 ) = 1 as an example. 15 This is e.g. realized for ν 6 = (−ν 1 F , 1) = (−1, 0, 0, 1); we denote the resulting threefold by M −ν 1 F +2 . Its defining equation is of the form where the F (i,j) are homogeneous polynomials of degree i in [X 2 : X 3 : X 4 ] and degree j in the [X 5 : X 6 ] coordinates of the base P 1 A . At the point t 0 ∈ P 1 A defined by F (0,1) (X 5 , X 6 ) = 0 the fibre geometry S t 0 is singular and consists of the two irreducible components We can think of either one of them as the ; their sum is homologous to the class of the generic fibre.
The K3-fibration degenerates at the point t 0 ∈ P 1 A . This is an example of a Type II degeneration; background material on the theory of degeneration of K3 surface is summarized in Appendix B for the convenience of readers. In this particular example of Type II degeneration of a lattice-polarized K3 surface ( blown-up at eighteen points (F (3,2) The two surface components intersect along the elliptic curve {F (3,2) | t 0 = 0} ⊂ P 2 , see also [42].
Let us parametrise the Kähler cone of this threefold M where t 2,I , t vrt,I and t 5,I are real valued; the subscripts I are a reminder that they are meant to be the imaginary part of the complexified Kähler parameter B + iJ. This parametrization respects the filtration structure in the space of divisors and curves associated with fibration. The polytope ∆ for the ambient space of M −ν 1 F +2 has a unique triangulation and the Kähler cone of the toric ambient space is bounded by three walls: At the wall t vrt,I = 0, the eighteen (−1) curves in V 1 = Bl 18 (P 2 ) in the central fibre shrink to zero volume, while the volumes of P 1 ⊂ P 2 = V 0 and V 0 = P 2 itself go to zero at the wall t 2,I − 3t vrt,I = 0. The last inequality does not concern us, as we will stay within the large base P 1 A regime t 5,I |t 2,I |, |t vrt,I | of type IIA compactification (which is dual to the weak 4D dilaton regime in heterotic compactification) in this article. At the wall t vrt,I = 0, a flop transition 16 turns V 1 = Bl 18 (P 2 ) into P 2 , and V 0 = P 2 into Bl 18 (P 2 ). The phases found at the two sides of the wall can be regarded as the two small resolutions of a geometry given by From the perspective of the gauged linear sigma model (type IIA string theory), therefore, the Kähler parameter phase diagram is like Figure 3 (a) in the large base P 1 A regime. From the perspective of classical geometry, on the other hand, there is a holomorphic biregular map from M −ν 1 F +2 in the t vrt,I < 0 phase to that in the t vrt,I > 0 phase so that the singular fibre components V 0 = Bl 18 (P 2 ) and V 1 = P 2 in the t vrt,I < 0 phase are identified with V 1 = Bl 18 (P 2 ) and V 0 = P 2 , respectively. This isomorphism effectively cuts out the t vrt,I < 0 part of the Kähler moduli space. This is consistent with the fact that only one triangulation is found for the polytope ∆ under consideration.
Applying the adiabatic argument of duality to the fibre over generic points around the degeneration point t 0 , we find that the heterotic interpretation is to have a soliton (defect) localized at real codimension-two in the base P 1 Het . We cannot say much about what happens at the centre of the soliton (as the adiabatic argument breaks down there), but duality indicates that there is a U(1) vector multiplet associated with this soliton, at least for generic choice of moduli. An extended discussion on the heterotic string interpretation is provided in section 4.4. Before we get there, we study a few more examples of degeneration in latticepolarized K3-fibration in sections 4.2 and 4.3.

Corridor Branches and Reducible Fibres
Once we allow the K3 fibre to degenerate and become reducible at isolated points in the base A , we do not need to restrict to a construction where we choose to include just one vector 16 It is not a straightforward task to find a toric construction for the geometry on the other side of the flop transition of a Calabi-Yau hypersurface. Different choices of triangulation of the polytope ∆ sometimes do the job (just as they do for the transitions of the toric ambient space), but that is not always the case. See e.g. [43,44] for more examples.
The moduli spaces of type IIA compactifications for this broader class of threefolds M Λ S form bridges (or corridors) between the branches of the moduli space corresponding to the multiple fibration choices of a given algebraic K3 surface Λ S ∼ NS K3 discussed in the previous section.
This section will cover the geometry of threefolds where the reducible fibre is a Type II degeneration (as in section 4.1). Examples with a reducible fibre other than a Type II degeneration are postponed to section 4.3.

Warm-up
In the context of heterotic-type IIA duality, the best-known example of a K3-fibred Calabi-Yau threefold with a reducible fibre corresponding to a Type II degeneration is the case with Λ S = U (i.e., E 8 -elliptic K3 is in the fibre in type IIA compactification). When we include all the vectors ν 6,n Figure 4: The toric vectors for the base B (a), and the graph of intersection of curves . The latter can also be regarded as the dual graph of a Type II degeneration of an E 8 -elliptic K3 surface.
vector multiplets and fewer hypermultiplets than any one of the moduli spaces for connects the moduli spaces of M n U with −n v ≤ n ≤ n h by a trade-off between the Coulomb and Higgs branch degrees of freedom [3].
In the context of F-theory compactification, this threefold geometry M A Hirzebruch surface F n h is a P 1 fibration over P 1 A . Blowing up a Hirzebruch surface to B , the fibration is modified in such a way that the fibre curve P 1 degenerates into (n h + n v + 1) curves, C 0 ∪ · · · ∪ C n h +nv ; the graph of intersection of those curves in B is shown in Figure 4 (b). For the more general heterotic-type IIA duality, however, it is more suitable to describe the geometry of threefold M has been tuned so much (relatively to that of M n U for any −n v ≤ n ≤ n h ) that the fibre K3 surface-E 8 -elliptic K3 surface generically-is forced to degenerate to a collection of (n h + n v + 1) irreducible nonsingular surfaces V 0 ∪ V 1 ∪ · · · ∪ V n h +nv over one point in the base P 1 A . The surfaces over the curves C 0 and C n h +nv are rational elliptic surfaces, V 0 = V n h +nv = RES (also known as dP 9 in physics community), while the surfaces over C 1 , · · · , C n h +nv−1 are all T 2 × P 1 = V 1,··· ,n h +nv−1 . This is an example of a Type II degeneration of a (Λ S = U )-polarized K3 surface. This is the language suitable for heterotic-type IIA duality.

Corridor branches among models with a degree-2 K3 surface
Examples of this kind are also available in the case of Calabi-Yau threefolds with a degree-2 K3 surface (Λ S = +2 ) as the fibre over They are obtained as toric hypersurfaces for which the polytope ∆ contains all of the ν 6 's with n ≥ k 4 ≥ m (and k 2 = k 3 = 0). An example of a "short top" [40] is found as a part of this polytope ∆. It turns out that Hodge numbers of those threefolds are as follows: depend only on (n − m), just like they do on (n h + n v ) in (45,46).
Let us first focus on the geometry of M {n,n−1} +2 . A generic fibre S t.A (t ∈ P 1 A ) in those threefolds is a degree-2 K3 surface, but this K3 surface degenerates at one point (X 6 = 0) in the base P 1 A . The singular fibre at the degeneration, referred to as the central fibre and denoted by S 0 , consists of two irreducible pieces, V 0 ∪V 1 . Let V 0 be the divisorD 6,n in M {n,n−1} +2 for definiteness; V 1 =D 6,n−1 then. Both of these surfaces combined,D 6,n +D 6,n−1 = S 0 are linearly equivalent to the generic fibre class ∼D 5 . We found, by using computation techniques available for toric hypersurfaces [45] (plus additional formula in Appendix A.3), that those two irreducible surfaces satisfy Those two components meet along a curve of genus 1. This information is summarized in the first line of Table 2.
and that of M n +2 are connected. The transition locus between these two branches is reached from M {n,n−1} +2 by tuning a Kähler parameter such that the surface V 1 =D 6,n−1 collapses to a point, and it is reached from M n +2 by tuning 17 complex structure parameters. At the transition, the geometry has a point-like singularity  Table 2: Data of irreducible components of the central fibres in threefolds M +2 obtained by using two "neighbouring" points in Figure 1 for ν 6 F . A pair of points corresponding to the first two rows are connected by a solid line in Figure 1, while a pair corresponding to the next two rows by a dotted line in Figure 1. A pair corresponding to the last row are connected by a dashed line in Figure 1. In the last three rows, the value of h 1,1 (V 0 ) and h 1,1 (V 1 ) can be 9 and 11, or 11 and 9, respectively, depending on the choice of triangulation of the polytope ∆ (see the text for more information).
of type E 7 , which is captured by with a = 4 is used for this expression). As discussed in [46,3], this type of singularity is reached by collapsing a dP 7 . This can also be seen explicitly by observing that V 1 =D 6,n−1 is described as a hypersurface of degree 4 in P 3 2111 , which is a well-known realization of dP 7 .
The moduli space of M {n,n−1} +2 is also connected to that of M n−1 +2 . Here, we can reach the transition point from M {n,n−1} +2 by collapsing the surfaceD 6,n to zero volume to form a singular threefold, whereas we need to tune 17 complex structure moduli of M n−1 +2 to reach it. At the transition, M n−1 +2 develops an A 1 singularity along a curve X 1 = X 4 = X 6 = 0, and the surface V 0 =D 6,n comes out as the exceptional divisor when this singularity is resolved. 17 It turns out that V 0 can be regarded as Bl 10 (F 2 ), see appendix D for more information.
Therefore, type IIA compactification on M {n,n−1} +2 has transitions both to a compactification on M n +2 and a compactification on M n−1 +2 . The moduli spaces of all of the M n +2 with n = 2, 1, 0, −1 are connected in this way.
Similarly, the threefold M provides a branch of moduli space connecting M 1 +2 and M −1 +2 . In fact, it turns out that the geometry of the degenerate singular fibre in M is the same as in the M {n,n−1} +2 branch connecting those of M n−1 +2 and M n +2 . Although the S 3 symmetry of the graph of Figure 1 does not explain why the central fibre geometry is the same for those two transition channels, certainly the [−1 2 ]- [1 4 ] pair is a nearest neighbour link in the graph in Figure 1 (just like the pairs [n a ]-[n − 1 a ] are), and this agreement of the central fibre geometry may be just a trivial consequence of M 2 +2 = M −1 +2 as complex geometry.
By tuning more complex structure moduli and subsequent resolution, we can also reach the manifolds M {n,n−1,··· ,m} +2 . Here, the degree-2 K3 surface in the fibre degenerates over one point (X 6 = 0) of the base P 1 A to a central fibre S 0 = V 0 ∪ V 1 ∪ · · · ∪ V n−m . This is another example of Type II degeneration of degree-2 K3 surface. The dual graph of the irreducible components, V 0 , · · · , V n−m is a chain starting from a node for V 0 and ending with a node for V n−m . This graph comes directly from an edge (at the height= +1) of the polytope ∆. This is an example of a theorem in [40].
The rational surfaces at the end of the chain remain unchanged, V 0 = Bl 10 (F 2 ), V n−m = dP 7 , while the surface components in the middle, V 1 , · · · , V n−m−1 are all identical surfaces that are ruled over the elliptic curve C ∼ = (V i ∩ V i+1 ). They are isomorphic 18 to P[O C ⊕ L] for some degree (−2) line bundle L on C (cf Chap.V.2 of [47]). The value (−2) is tied to the self-intersection of the double curves C = V i ∩ V i+1 (and the degree of dP 7 ). This ruled surface does not admit an elliptic fibration morphism, see e.g. [48]. More information is provided in the appendices B and D. and have a closer look at the Kähler moduli space. Recall that the K3 fibration has a single reducible fibre with irreducible surface components corresponding to D 6,0 and D 6,−1 with χ(D 6,0 ) = 14 and χ(D 6,−1 ) = 10. There are 5 different fine, regular, star triangulations of ∆. Accordingly, there are 5 chambers in the Kähler moduli space of the toric ambient space, see Figure 5. Only two of those-phase B and C in Figure 5-result in a toric ambient space with an (obvious) toric fibration morphism to P 1 A from which the K3 fibration follows. At the wall between these two phases, there are surfaces and curves of the ambient space which collapse, but it seems the Calabi-Yau hypersurface we are interested 00 00 00 11 11 11 00 00 00 11 11 11 00 00 00 11 11 11 00 00 00 11 11 11 00 00 00 11 11 11 00 00 00 11 11 11  . The figure shows a projection preserving the relative location of the rays of the five three-dimensional cones corresponding to different triangulations. We have labelled the five phases by A to E and indicated where the various walls mentioned in the text are located.
in stays perfectly smooth. 19 In such a case, we can glue these two cones together and treat them as a single phase [49].
Let us parametrize the Kähler moduli space by 19 While it is not hard to work out that none of the curves obtained by intersecting a surface in the ambient space with M . An example of this phenomenon is given by the 18 (−1) curves contained in one of the fibre components for the model discussed in Section 4.1. While we do not have a candidate for a similar behaviour in this case, this is of course not enough to rigorously exclude such a thing.
The five geometric phases as a whole are delineated by the walls The wall separates the phases A and B+C from the phases D and E. The phase B is distinguished 20 from the phase A by the wall Similarly to W 5 , there seems to be no curve inside M which collapses at W 4 , so that we expect this wall to be fictitious and phase A should be combined with the phase B+C. At the level of the ambient space, however, the triangulation in the phase A is not compatible with the projection to P 1 A , and hence we cannot obtain K3-fibration morphism M is most likely still K3 fibred after we cross W 4 , we cannot confidently speak about the projection to P 1 A in phase A. The walls W 1 and W 2 are dual to curve classes C 1 and C 2 , where C 1 is represented by one of seven (−1) curves inD 6,−1 = dP 7 , and the class C 2 by one of ten (−1) curves in D 6,0 = Bl 10 (F 2 ). The volume ofD 6,−1 = dP 7 also vanishes at the wall W 1 , and that of D 6,0 = Bl 10 (F 2 ) at the wall W 2 . There is no flop available in the compact manifold M {0,−1} +2 acting on those (−1) curves in the central fibre S 0 =D 6,0 +D 6,−1 . There is not even a limit of Kähler parameters where the volume of those (−1) curves vanish while keeping the volume ofD 6,−1 andD 6,0 non-zero.
In the large base regime t 5,I |t 4,I |, |t vrt,I |, only the phases B and C can be realized. 21 The phase diagram in this context is given by Figure 3 (b). 20 The phase B is distinguished from the phase C by the wall W 5 : 2t 4,I + 3t vrt,I > 0. 21 Here, we exclude A as we cannot rigorously establish the existence of a K3 fibration in this phase. .
It is worth noting that the geometric phases D and E are available only outside of the large base regime (56). Given the fact that one can take a detour around the wall of Kähler cone by turning on B-fields, the heterotic-type IIA duality map should extended (at least via analytic continuation) to the geometric phases which are not compatible with a K3-fibration, at least not in an obvious way. Because the large base regime (56) corresponds to the weak coupling regime in heterotic string compactifications, the geometric phases D and E should be mapped to strongly coupled phase of heterotic string compactifications. It would be interesting to explore this territory, but this is beyond the scope of this article.

More transitions and degenerate fibres
Besides the singular transitions we have discussed, there are others which connect different threefolds along the "links of length has Hodge numbers which signals a single reducible fibre with two irreducible components. By construction, this Calabi-Yau threefold sits in between the threefolds M n=2 +2 and M n=−1 +2 . As before, these can be reached by blowing down one of the fibre components, followed by a subsequent deformation.
The polytope ∆ has a two-dimensional face which contains the lattice points ν 2 , ν 3 , ν 6 2 4 , ν 6 −1 2 and ν 5 . The three different triangulations of this face ( Figure 6) give rise to three different torically realized phases of M . The geometries corresponding to the triangulation on the left and in the middle only differ in how the two components of the singular fibre are distributed among D 6,2 4 and D 6,−1 2 . These two divisors are rational for any triangulation, and they obey χ(D 2 4 ) = 13 χ(D −1 2 ) = 11 (58) for the triangulation shown on the left of Figure 6 and for the triangulation shown in the middle of Figure 6. The two phases are connected by a flop which brings two (−1) curves from one fibre component to the other. In each of the two cases, the divisor with χ = 11 is a dP 8 realized as a hypersurface of degree 6 in P 3 3211 , whereas the divisor with χ = 13 is a blowup of dP 8 at two points.
The phase corresponding to the triangulation shown on the right hand side of Figure 6 does not respect the K3 fibration we intend to use for the duality between type IIA and heterotic string theory. Starting from the phase in the middle of Figure 6, we can reach the phase on the right hand side by passing through a wall of the Kähler cone. On the boundary of the Kähler cone in question, the curve D 3 · D 4 is collapsed, before another small resolution takes us to the phase corresponding to the triangulation shown on the right. This curve is projected surjectively on the base P 1 A of the K3 fibration, which means that we can get there only outside of the large base regime.
The same fibre geometries, including the flops discussed above, are realized by threefolds connected along other edges of length √ 3 such as M . The 3rd and 4th rows in Table 2 speak about that. However, the different phases cannot all be seen torically for all of these models. One sometimes has to go beyond toric hypersurfaces to realize the extended Kähler cone of the Calabi-Yau manifold M , as we have remarked already in footnote 16.
As before, the fibre components with Euler characteristic 11 are dP 8 surfaces, and those with 13 are dP 8 surfaces blown up in two points. The χ = 0 component in the middle has a ruling over an elliptic curve C = V i ∩ V i+1 ; it is in the form of P[O c ⊕ L] for a degree (−1) line bundle L on C, because (C) 2 = +1 in dP 8 . Starting from the first triangulation, a flop blows down two (−1) curves in V 2 , so both V 0 and V 2 turn into dP 8 in the second triangulation. At the same time, two points in V 1 along V 1 ∩ V 2 are blown up, so the "ruling" (P 1 -fibration) in V 1 splits into P 1 + P 1 over two points in C and now χ(V 1 ) = 2. The phase for the last triangulation is reached by a flop along the other P 1 in the P 1 + P 1 fibre (simultaneously at the two such fibres). Given the symmetry between the [2 4 ] and [2 3 ] vertices in Figure 1, it is reasonable that these flops exist, so that there is no asymmetry between V 0 =D 6,2 4 and V 2 =D 6,2 3 . There exists a fourth phase accessible via triangulation for which D 5 · D 6,−1 2 = 0. so that this phase cannot respect the K3 fibration. Again, this non-fibred phase can be reached outside of the large base regime.
The general feature of all the examples with degenerate fibres discussed so far is that they correspond to Type II degenerations in the sense of Kulikov. The degenerate fibre is always in the form of V 0 ∪ V 1 ∪ · · · ∪ V µ . There are sometimes multiple geometric phases that are compatible with a K3-fibration, however, and details of the fibre geometry change from one phase to another. The monodromy of a generic fibre around the degeneration locus in the base P 1 A , on the other hand, remains invariant under such birational transformations of the degenerate fibre. The Clemens-Schmid exact sequence extracts such an invariant part of information from the degenerate fibre (cf the appendix D). Three out of four inequivalent Type II degenerations of degree-2 K3 surface in (132), classified in terms of the lattice (W 2 ∩ Λ T )/W 1 , have been realized in the compact models in this section. The one in section 4.1 is for (W 2 ∩ Λ T )/W 1 = A 17 ; Z 3 , the one in sections 4.2.2-4.2.3 is for (E 7 ⊕ D 10 ); Z 2 (see the appendix D for derivation), while the (E ⊕2 8 ⊕ A 1 ) case is realized in the examples in section 4.2.4. It is also possible to construct polytopes for which these different fibre types coexist within a single threefold.

Branches with Type III or Non-Kulikov Degenerations
It is also known, in toric language, how to construct a compact K3-fibred Calabi-Yau threefold that develops a Type III degeneration [40]. The simplest example is to consider M , where we collect three lattice points from Figure 1 to form a polytope ∆, a toric ambient space, and a Calabi-Yau hypersurface. A relevant toric graph is a twodimensional face Θ [2] with ν 6 2 4 , ν 6 1 4 and ν 6 −1 2 as the vertices, but this comes with a multiplicity * (Θ [1] ) + 1 = 2. The dual graph of this degenerate fibre is given by two copies of the triangle Θ [2] glued along the three edges, which topologically is a triangulation of a sphere S 2 . Since this threefold should be regarded as a common subset of two different "corridor" branches of the complex structure moduli for the [2 4 ]- [1 4 ] link and for the [2 4 ]-[−1 2 ] link, we should expect this degeneration to combine the Type II (E 7 ⊕ D 10 ); Z 2 and the Type II (E ⊕2 8 ⊕ A 1 ) degenerations. In light of the stratification structure of the boundary components of the Baily-Borel compactification of lattice-polarized K3 surfaces (see the appendix B.2.1), it is natural that a Type III degeneration develops in the common subset of the corridor branches.
It has been proved [40] that M {··· } Λ S → P 1 A has a Type III degeneration when the collection of lattice points {· · · } ⊂ 2 ∆ F ∩ N F forms a convex hull that is either 2-dimensional or 3dimensional. The collection of vertices {2 4 , 1 4 , −1 2 } is a minimal collection to have a Type III degeneration. The other extreme is to have the collection {· · · } all of 2 ∆ F ∩ N F . Despite this variety for construction of a threefold with a Type III degeneration (and the corresponding stratification of the moduli space) there is less richness in the classification of Type III degeneration of lattice-polarized K3 surface, primarily due to the indefinite signature of the lattice (W 2 ∩ Λ T )/W 0 (see the appendix B).
Degenerations of a lattice polarized K3 surface which correspond to Type I do not contribute to the story in this article although they are a very common phenomenon. To be more precise, all the K3-fibred Calabi-Yau threefolds we have discussed in this article have many degenerations that are not semi-stable, which would become Type I degeneration after base change of order-2 (see the appendix C). We call such degenerations "would-be Type I" in this article. In the threefolds M n +2 discussed in section 3.2.1, for example, there are N L 0,0 = 300 would-be Type I degenerations. These N L 0,0 = 300 degeneration points in the base P 1 A have also been known as the N L 2,1 = 300 Noether-Lefschetz loci that contribute to the Gromov-Witten invariant d β=2C 2 in (25) [20,50,51,52]. There is nothing new in particular.
In the context of string compactification over a compact Calabi-Yau threefold (that just happens to have a K3-fibration), we are not so happy to replace the threefold by its base change. Not all the degenerations in π M : M → P 1 A are semi-stable, or in Kulikov model, when we do not allow to "replace" them by their base changes. The would-be Type I degenerations above is the simplest example (cf. the appendix C). Such degenerations still come with the notion of monodromy on the generic fibre H 2 (S t ; Z). As one of the properties of Picard-Lefschetz monodromy of K3 fibration [53], the monodromy matrix T is quasi-unipotent, in that there exists an integer m so that In the case of semi-stable degenerations, m = 1 and the matrix N defined by is a nilpotent matrix. We call these degenerations would-be Type I, Type II and Type III, when N = 0, N 2 = 0 (but N = 0), and N 3 = 0 (but N 2 = 0), respectively, in this article. They may well be regarded as 1/m-Type I (Type II, Type III, resp.) degenerations, similarly to fractional D-branes. We are also tempted to call them fractional Type I, Type II and Type III degenerations for this reason. Such would-be Type II and would-be Type III degenerations will be constructed straightforwardly, given an observation in [40]. Recall that we started out in section 3.1 by allowing to use a vertex of the form ν 6 = (ν 6 F , +1) T in (16) to form a convex polytope ∆ ⊂ N ⊗ R. It is the definition of a short top in [40] to restrict the possibility of ν 6 to this form (placed at height +1); this restriction guarantees that all the irreducible components in the degenerate fibre appear with multiplicity +1, which is one of the conditions of semi-stable degeneration [40]. Allowing to involve vertices that are placed at height > 1, degenerations cease to be semi-stable as fibre components can now appear with multiplicity > 1. In case the degenerate fibre in question has at least two components, they must be either would-be Type II or would-be Type III. Placing a vertex at height > 1 this is guaranteed if there is at least a second lattice point 'above∆ F ' not contained in any face of dimension < 3. Indeed this happens in all examples known to us, but we do not have a general proof securing this in general.

Heterotic String Interpretation
We have seen many examples of Λ S -polarized K3-fibred Calabi-Yau threefolds (that are nonsingular) where the fibre K3 surface degenerates and forms multiple irreducible components over isolated points in the base P 1 A . The adiabatic argument can be used to translate type IIA compactifications over such threefolds to heterotic string compactifications at points in P 1 A away from such degeneration points. In this section, we discuss the heterotic dual description of degenerations of K3 fibrations.
There is an example of degenerations of lattice-polarized K3 surface whose heterotic dual is well-known. That is when we have an E 8 -elliptic K3 surface in the fibre (Λ S = U ), and the fibre undergoes Type II degeneration with (W 2 ∩ Λ T )/W 1 ∼ = E 8 ⊕ E 8 . The local geometry of M {n H ,··· ,−nv} U around a point of degeneration t = 0 ∈ P 1 A , discussed in page 26, corresponds to (n h + n v ) NS5-branes of heterotic string theory, wrapped on T 2 89 , extending along R 1,3 and localized at t = 0 ∈ P 1 Het , the base of T 2 67 -fibration of the K3 Het when we see it as an elliptic fibration.
The most direct way to see this duality dictionary is in terms of monodromy around the degeneration point t = 0. In type IIA language, period integrals of the generic fibre undergoes monodromy transformation T = exp[N ], as a point in the base t goes around t = 0 by t = t * × e 2πia , a ∈ [0, 1]. The nilpotent matrix N is given by (130), with δ 1 = δ 2 = 1 and µ = n h + n v : The lower two components near the degeneration point are the period integrals over 2-cycles that are obtained by fibering 1-cycles of the elliptic fibre along the long cylinder axis in the base P 1 of the elliptic K3 A .
The fibrewise heterotic-type IIA duality map simply replaces period integrals of a Λ Spolarized K3 surface in type IIA language by Λ T ⊗ C-valued Narain moduli in heterotic string language. Being away from the degeneration point, the Λ T ⊗C-valued period integrals / Narain moduli are allowed to vary over the base P 1 A / P 1 Het . Now, the standard parametrization of Narain moduli in the case of Λ where the first row corresponds to the rank-2 Λ S = U , the next four rows correspond to a basis {ê 1 ,ê 2 ,ê 1 ,ê 2 } and the last row to E 8 ⊕ E 8 ⊂ Λ T . τ andρ roughly correspond to the complex structure and complexified volume of the T 2 67 fibre of K3 Het , and a the E ⊕2 8 -valued Wilson lines along T 2 67 . The monodromy matrix N in (64) is equivalent to shiftρ →ρ + (n h + n v ) at the end of the monodromy. 22 The degeneration of E 8 -elliptic K3 surfaces is regarded in heterotic string theory as the presence of a magnetic source for the three-form field dB: where S 1 is a circle around the t = 0 point in P 1 Het  [40]. When the Λ S -polarized K3 surface in the fibre exhibits Type II degeneration at a point t = 0 ∈ P 1 A , Λ T ⊗ C-valued period integrals have monodromy around the degeneration point t = 0. Repeating the same argument as above, we find that such a degeneration in a threefold M Λ S for type IIA compactification corresponds to a soliton in heterotic string theory localized at the t = 0 point in P 1 Het . The monodromy matrix 22ρ behaves asρ(t) in the basis (126). The contribution [(τ, a)]-a quadratic in τ and a-needs to be determined by the unspecified part of the intersection form in (128), but does not depend onρ. The nilpotent matrix N in (130) implies that only theρ parameter of the Narain moduli in the basis (126) gets shifted byρ and all other Narain moduli parameters remain intact around the soliton localized at t = 0 ∈ P 1 Het . Type II degenerations with different δ 1 , δ 2 and (W 2 ∩ Λ T )/W 1 correspond to a shift (around a point in P 1 Het ) of different Narain moduli. The shift depends on µ and the δ i in the same way for each case. The value of µ in particular (the number of double curves in a Type II degeneration) is regarded as the number of coincident solitons of the same type.
What is the Narain modulusρ that shifts in terms of the weakly coupled heterotic E 8 ×E 8 string theory for each one of those solitons? Let us take the Λ S = +2 case as an example, and provide an explicit answer to this question.
In the Λ S = +2 case, we can always take δ 1 = δ 2 = 1 (see [55], or the appendix B.2), and the filtration structure {0} ⊂ W 1 ⊂ (W 2 ∩ Λ T ) ⊂ Λ T in (127) can be transformed into a direct sum, where At least as a question in mathematics, we can easily find how the structure (69) fits into Λ T given by (6) in the case of (W 2 ∩Λ T )/W 1 = A 1 ⊕E 8 ⊕E 8 . One just needs to identify (69) with (6). For three other choices of (W 2 ∩ Λ T )/W 1 in (132), it is useful to note that all the four  (6). See [55] and references therein for more information. Figure 8 contains all the information we need in this article. Now let us turn to the physics question. To get started, we fix the S 3 symmetry action. We have seen in section 3.2.3 that the heterotic ST -model is dual to type IIA compactification on M 2 4 +2 , where the weak coupling E 8 × E 8 in heterotic string theory corresponds to transcendental two cycles localized near X 4 = X 2 = 0 and X 4 = X 3 = 0. In Figure 8 (a), we regard the upper right and lower right corners as those two locations in P 2 , and therefore, the upper half triangle and the lower half triangle can be regarded as the weak coupling E 8 × E 8 in heterotic string. Now, at the transition from , theẼ 7 singularity appears at X 2 = X 3 = 0, as we saw in (49). This point corresponds to the left corner in Figure 8  to M 1 4 +2 , a curve of A 1 singularity forms along X 4 = 0, parametrized by [X 2 : X 3 ] ∈ P 1 ; this curve corresponds to the right edge in Figure 8 (a). Remembering that the E 7 algebra is contained in , which collapses at the transition to M 2 4 +2 , and that the D 10 algebra is in V 0 =D 6,2 4 , which collapses at the transition to M 1 4 +2 , we conclude that the E 7 ⊕ D 10 lattice in the Type II degeneration in M 8 ⊕ A 1 of the weak coupling heterotic string in a way that can be seen by superimposing Figure 8 (a) and (b). In particular, the E 7 current algebra associated with the E 7 -string at the M 2 +2 -M

{2,1}
+2 transition (due to collapsed dP 7 ) is not from a subgroup of any one of the two weakly coupled E 8 's, but from somewhere in the middle of E 8 × E 8 (cf [56]). The (W 2 ∩ Λ T )/W 1 ∼ = E 7 D 10 ; Z 2 soliton squeezes instanton degrees of freedom in the ST -model from both of the two weakly coupled E 8 's, and the remaining E 8 factors do not have a free-choice in the instanton moduli anymore, as we saw in section 3.2.3. The soliton returns those degrees of freedom at the M +2 transition, but in a way that the free instanton interpretation is never restored in the M n +2 branches with n = 1, 0, −1 (at least not in an obvious way for n = −1).
Similarly, in the (W 2 ∩ Λ T )/W 1 = E ⊕2 8 ⊕ A 1 soliton that appears in the heterotic duals of M type IIA compactifications, the two E 8 algebras in the degenerate fibre correspond to the lower left half triangle and upper right half triangle in Figure 8 (a). Imagine Figure 8 (a) rotated by 2π/3 in a counter-clockwise direction. This soliton sucks away instanton degrees of freedom from a skewed combination of the two E 8 's (not diagonally as in the (E 7 ⊕ D 10 ); Z 2 soliton), and releases them somewhere else. Table 1 summarizes the consequence of this chain of transitions. The E 8 -string that emerges at the transition points is not associated simply with any one of the two weakly coupled E 8 's.
By now, the question "what is theρ modulus that shifts for these solitons" is not more than a technical question that is not particularly illuminating. So, we are not presenting technical details here. Roughly speaking, the U factor of U ⊕ (W 2 ∩ Λ T )/W 1 ∼ = U ⊕ (E ⊕2 8 ⊕ A 1 ) picks up the nodes in the Coxeter diagram that have not been used for (W 2 ∩ Λ T )/W 1 .
A similar reasoning can be applied to Type III degenerations of K3 surfaces. When a type IIA compactification on a K3-fibred Calabi-Yau threefold M has a Type III degenerate fibre at one point t = 0 ∈ P 1 A in the base, the adiabatic argument (fibre-wise duality) can be applied to any points away from the degeneration point. The holomorphic dependence of the period integrals of the K3 fibre in type IIA over P 1 A is translated into the holomorphic dependence of the Narain moduli of the T 4 fibre in heterotic string theory over P 1 Het . Any monodromy action on the generic fibre K3 surface T : H 2 (S t.A ; Z) → H 2 (S t.A ; Z) is directly translated into that on the Narain lattice. Parametrizing the Narain moduli / period integrals on the Λ T ⊂ II 4,20 part as in the basis (134), we find that the monodromy due to T = exp[µN III 0 (δ, u, v, x)] in (138) amounts to The heterotic dual of a Type III degeneration is to involve a soliton that is a magnetic source of the moduli field X and µ is interpreted as the number of soliton of this type.
If we are to ignore the distinction between Λ S and Λ T within II 4,20 in choosing the parametrization of the Narain moduli, then we can always use the monodromy matrix T = exp[µN III 0 (t 0 )] in (124) for (the heterotic dual of) a Type III degeneration. Using a parametrization (1, ρ 1 , ρ 2 , −ρ 1 ρ 2 ) T for the Narain moduli in the basis adopted in (124), the soliton in question is regarded as a magnetic source introducing a twist Degenerations of K3 fibration in type IIA compactification that are not in the Kulikov model are also regarded as solitons in heterotic string, and are magnetic sources of the Narain moduli fields precisely for the same reason as in the cases of Type II and Type III degenerations. A case-by-case study is necessary for the explicit form of the monodromy matrix in an integral basis for the would-be Type II and would-be Type III degenerations. Fractional powers of exp[N II 0 ] and exp[N III 0 ] need to be taken in an integral basis.

6D Perspectives
Degenerations in the K3 fibre of type IIA compactifications and their heterotic duals are both regarded as solitons, localized in codimension two in the base P 1 . In this section, we attempt at recapturing those solitons in terms of 6D (1,1) supergravity by taking the decompactification limit of the base P 1 . This approach provides a more bottom-up (more general, less constructive) perspective, and makes it possible to extract the intrinsic nature of those solitons unaffected by anything associated with the compactness of the base P 1 . This section is therefore meant to provide a complementary perspective to the study in the previous section.

6D (1,1) Supergravity and Half-BPS 3-branes
Both T 4 compactification of the heterotic string and K3 compactification of the type IIA string leads to a 6D effective theory at low energy with (1, 1) supersymmetry. The massless field contents of the effective theory in supergravity consists of one supergravity multiplet and n = 20 vector multiplets. The 32 bosonic degrees of freedom in the supergravity multiplet are represented by the 6D metric (9 DOF), B µν (3+3 DOF), 1 scalar 23 σ and four 6D vectors (16 DOF). The fermionic degrees of freedom consist of the gravitinos ψ where the last SU(2) factor corresponds to the holonomy group of a K3 surface. The SUSY transformation parameters There are 4 × n scalar fields, apart from σ, and they are known to parametrize a coset space (modulo quotient). Let φ x (x = 1, · · · , 4n) be a set of local coordinates in M .
is a vielbein on M , from which a metric g xy = f aγ x f aγ y is obtained. This metric is used in the 6D (1, 1) sugra action as the non-linear sigma model of φ x 's.
Here is a little more about the geometry of the coset space M . First, let C IJ be the intersection form of II 4,20 . Each point in M corresponds to some choice of {h γ I } γ=1,2,3,4 satisfying For such a choice of {h γ I } γ=1,2,3,4 , one can uniquely find one choice of {h a ∈ II 4,20 ⊗ R | a = 1, · · · , 20} modulo the SO (20) action that satisfies the orthonormality conditions above.
An SO(4) × SO(20) connection on M is defined by For the vielbein on M introduced earlier, we used the following: The dictionary between the SO(4) vector indices γ, δ and SU(2) × SU(2) doublet indices i, i is given by Therefore, the orthonormality condition becomes Using these geometric data, the supergravity transformation law is written down in eq.(3.2) of [57]. The SUSY variation of fermionic fields is where ellipsis stands for terms that involve multiple fermions, or H µνρ or F I µν . We consider codimension R = 2 defects of this 6D (1, 1) supergravity that preserves SO(1, 3) Lorentz symmetry and half of the SUSY charges. In particular, we consider field configurations where the 6D metric and scalars have a non-trivial configuration in (x, y) ∈ R 2 . Under the unbroken SO(1, 3) Lorentz symmetry, the supersymmetry transformation parameters where Spin L is a left-handed spinor of SO (1,3), and + · · · is the other term involving a righthanded spinor Spin R of SO (1,3). We are interested in defects where Spin L ⊗ ↑ ⊗(↑↑) i=1 in the first line and Spin L ⊗ ↓ ⊗(↓↓) i =2 in the second line remain as transformation parameters of the unbroken supersymmetry. Let us first take to be the metric configuration in the directions R 2 transverse to the defect. The BPS conditions from dilatino variations are For all of these to vanish, we need (∂ z σ) = (∂zσ) = 0. That is, the value of σ must remain constant.
The BPS conditions from gaugino variations are From the conditions in the first line, we find that while the conditions on the second line yield on M , pulled back by the scalar φ x (z,z) field configuration of a half-BPS configuration, satisfy These conditions are satisfied, when the values of (h ] cc takes its value in Λ T ⊗ C. Moreover, the orthonormality condition (85) implies that the space of h 2 1 's is an S 1 -fibration over the period domain of Λ T , The S 1 fibre corresponds to a complex phase multiplication for h 2 1 (SO(2) rotation on the 2-plane), which does not change a point in M . Thus, we can use a natural set of complex coordinates of D(Λ T ) for the subspace of M where the soliton has a non-trivial field configuration. The BPS condition means that the map φ : We are interested in the 6D (1, 1) supergravity where the target space M is replaced by Isom(II 4,20 )\M n=20 , because Isom(II 4,20 ) is the modular group of both heterotic/T 4 and type IIA/K3 compactifications [11,12]. Since the field configuration φ does not have to be well-defined at the centre of the soliton, the holomorphic map φ : [C\{z = 0}] → D(Λ T ) may have a branch cut emanating from the origin {z = 0}, and the field configuration may be identified along the branch cut by some element T ∈ Isom * (Λ T ).

Recap and Speculations
In the 6D (1, 1) supergravity with M = SO(4, 20)/ SO(4) × SO (20), strings are classified by their electric and magnetic charges under B µν , while particles / 2-branes are classified by their electric / magnetic charges under the 4 + 20 vector fields. Counting of BPS states has been carried out for those objects as a check of heterotic-type IIA duality. 3-branes (real-codimension-2 defects) are magnetic source of scalar fields.
A half-BPS 3-brane comes with a choice of a pair of primitive sublattices Λ S and Λ T of II 4,20 that are mutually orthogonal in II 4,20 ] (modulo complex phase) takes its value in D( Λ S ) and remains constant over the real 2-dimensional space transverse to the 3-brane. 24 On the other hand, [(h) i =2 i=1 ] (modulo complex phase) takes its value in 25 Isom * (Λ T )\D(Λ T ), and is allowed to vary over the transverse space C holomorphically. 26 The 6D metric configuration in the real 2-dimensional transverse space R 2 is assumed to be Kähler, and a holomorphic coordinate is introduced in R 2 to turn it into C. The holomorphic configuration of [h i =2 i=1 ] may be twisted around the defect (3-brane) by T ∈ Isom * (Λ T ). Such 3-branes are therefore classified by a choice of lattices, Isom(II 4,20 )\( Λ S , Λ T ), and conjugacy classes of Isom * (Λ T ) are to be used for the twist T around the defect. Note that we have not yet assumed that the transverse space is compact.
The type IIA string compactified on an family of Λ S -polarized K3 surfaces provides an example of such half-BPS 3-branes whenever the fibre K3 surface has a Type II, Type III, would-be Type II or would-be Type III degeneration. The classification of such degenerations, reviewed in the Appendix B.2, is regarded as a study of a subset of possible twists in Inn[Isom(Λ T )]\Isom * (Λ T ). At least for the choices of T that correspond to those degenerations we know that there is holomorphic solutions to h i =2 i=1 . It may turn out that the BPS 3-branes of the 6D (1, 1) theory from degenerations of lattice polarized K3 surface is only a small subset of all possible BPS 3-branes characterized above. We leave it as an open problem for which T ∈ Isom * (Λ T ) a holomorphic solution to h i =2 i=1 exists. 27 It is not obvious to us purely from the perspective of solitons in 6D (1, 1) supergravity whether the monodromy matrix T should have the quasi-unipotent property or not. We also note, from the 6D soliton perspective, that Λ S = U ⊕ Λ S is not necessarily required. For more general choices of Λ S ⊂ II 4,20 , we cannot expect to obtain such a 3brane in a family of K3 surface in the geometric phase. Whether such a 3-brane solution to supergravity has a UV completion is yet another open question.

Analogy and Difference to 7-branes in F-theory
The 3-branes in the 6D (1, 1) supergravity share many aspects with 7-branes in Type IIB string/F-theory. Both are magnetic sources of scalar fields, a branch cut emanates from the centre of the soliton, and scalar fields are identified by an element of the modular group, Isom(II 4,20 ) or SL(2; Z), along the branch cut.
As an isolated object, (p, q) 7-branes are all alike, in that a (p, q) 7-brane can be taken into a (1, 0) 7-brane by SL(2; Z) transformation in Type IIB string theory (and if the asymptotic value of Type IIB axi-dilaton is not referred to). It is associated with a shift of a scalar field by an integer unit around the defect. 3-branes in 6D associated with a Type II degeneration are also all alike (if the asymptotic value of the scalar fields φ ∈ M is not referred to), in that one and the same matrix (120) is used in describing the monodromy.
The crucial difference is that the 3-branes in 6D have a lot more variety. 3-branes associated with a Type III degeneration (monodromy T = exp[N III 0 (t 0 )]) are labelled by an invariant t 0 , and are not equivalent to the 3-branes associated with a Type II degeneration. Solitons associated with a would-be Type II or would-be Type III degeneration also constitute a collection of solitons that are inequivalent from one another.
Furthermore, when the information of the asymptotic value of the scalar fields φ ∈ M is brought back into the discussion, there is a notion of ( Λ S , Λ T ) even for a 3-brane isolated in C = R 2 . If a pair of 3-branes share the same ( Λ S , Λ T ), they can form a BPS configuration together. If they do not, they cannot be BPS together. Such a notion is absent in the case of (p, q) 7-branes in F-theory.
Ramond-Ramond 7-brane charge cancellation condition in a Type IIB orientifold is re-placed by a condition in F-theory that where T i is the SL(2; Z)-valued Picard-Lefschetz monodromy matrix associated with a discriminant point z i ∈ P 1 F of the F-theory base. This condition is not additive anymore. Similarly, the heterotic string Bianchi identity for the B-field is the condition to be imposed in the supergravity regime (the gauge field F is Hermitian, here). This condition yields non-trivial constraints for any choice of compact four-cycle. Relevant to the present discussion is the four-cycle K3 Het , which is a T 2 67 -fibration over P 1 Het (here, we use Λ S = U , so that we are in the supergravity regime in heterotic language). This condition is replaced by (104), but now with T i ∈ Isom(Λ T ) ⊂ Isom(II 4,20 ). The Type IIB additive condition is reproduced from (104) in F-theory when we consider the orientifold limit. Two 7-branes come so close to one another, that we can collectively treat them as an O7-plane. The monodromy i T i around the two 7-branes (O7-plane as a whole) commutes with the monodromy matrix for a D7-brane. The multiplicative condition (104) for 3-branes in 6D (1, 1) supergravity also becomes additive in the same way. To see this, note first that monodromy from NS5-branes is given by T = exp[N II 0 ] acting non-trivially on (U ⊕ U ) ⊂ U ⊕2 ⊕ E ⊕2 8 = Λ T . Secondly, the instanton number is the same as the zero of g ±1 in (31), which is a section of O(η ± ). The monodromy locus was worked out in [36]. Each one of the zeros of g ±1 splits into multiple monodromy points, and the splitting remains small when we take η and K in section 3.2.2 small, just like in the orientifold limit of F-theory. The monodromy i T i from a set of those monodromy points associated with a given zero of g ±1 is block diagonal in E 8 ⊕ (U ⊕ U ) ⊕ E 8 = Λ T ; it is a Weyl reflection on one of the two E 8 's, while it is trivial on the other E 8 , and it acts as exp[−N II 0 ] on (U ⊕ U ), according to footnote 11 of [36]. 28 The monodromy i T i from the NS5-branes and instantons combined-the left-hand side of (105)-therefore splits into the two E 8 's and (U ⊕ U ). The monodromy in the E 8 ⊕ E 8 takes values in the Weyl group, and will probably cancel after all of the would-be Type I monodromies (cf the appendix C) are taken into account. The monodromy on the (U ⊕ U ) 28 We adopt a heterotic-F dictionary −ρ ↔ ± C 2 component has become additive, i T i = exp[(#(NS5)−#(inst.))N II 0 ]. We believe that there is a mistake somewhere and the correct result is i T i = exp[(#(NS5) + #(inst.))N II 0 ] = exp[24N II 0 ], since there can be trade-off between instantons and the NS5-branes [58], but we have not managed to identify an error. This U ⊕ U part of the monodromy will presumably be cancelled against contributions from other would-be Type I monodromy points that are attributed to the contribution on the right-hand side of (105). unique cone of the fan of P 1 . For the examples discussed in this work, this is easy to verify, Section 4.2 contains several interesting examples.
With a triangulation admitting a fibration morphism, we may then describe the coordinates of the base P 1 by One can think of all but one of the coordinates ν i for which f, ν i > 0 (and similarly for < 0) as corresponding to the exceptional divisors of blow ups of singular fibres.

A.1 Geometry of Generic Fibres
Fixing the coordinates at a generic point of the base P 1 , we find a generic fibre S t described as an algebraic hypersurface. The defining polynomial is found from the defining polynomial of M upon fixing all X i for which ν i , f = 0 and this hypersurface is embedded in an ambient toric variety with rays ν i F on ∆ F . Equivalent to ∆ F = ∆ ∩ (N F ⊗ R) being a lattice polytope is the existence of a projection P : ∆ → ∆ F induced by translations along f such that ∆ F is the polar dual to ∆ F [62]. This means that a generic fibre S t is described in the usual way by a pair of reflexive polytopes ∆ F , ∆ F . In particular, the Picard lattice of a generic fibre is the same as the Picard lattice of a generic toric K3 hypersurface.

A.2 Singular Fibres
Over specific points in the base P 1 A , the K3 fibre may become reducible. Individual components of such reducible fibres contribute to h 1,1 , which can also be computed combinatorially. We hence expect to be able to describe in terms of combinatorial data when reducible fibres occur. From the point of view of ∆, these come in two types, which we discuss now. The first type can already be seen from (106): whenever ∆ contains 29 more than one lattice point with f, ν > 0 (more than one lattice point with < 0), there is a reducible fibre over z 0 = 0 (z 1 = 0). In particular, we may write the total fibre class Note that some of these fibre components will contribute with a multiplicity greater than one. This means in particular that fibres for which this happens are not reduced. A closely related discussion is given in [40]. The second type of singular fibres stems from interior points of facets of ∆ F which do not lie in facets of ∆. Denoting such a facet of ∆ F by Θ [2] F , this means that Θ [2] F is also a face of ∆. As ∆ is reflexive, there is hence a dual one-dimensional face Θ [1] on ∆ and each interior point of Θ [2] gives rise to a divisor that has ( * (Θ [1] ) + 1) irreducible components. Those ( * + 1) irreducible pieces, however, do not form a single reducible fibre of the K3 fibration but are distributed among several reducible fibres separated in the base P 1 , as we now explain. First of all, calling the dual vertex (under polar duality of ∆ F , ∆ F ) of Θ [2] F by m F , the dual one-dimensional face Θ [1] of Θ [2] F is contained in the line m F + l · f (l ∈ R). For any point ν interior to a two-dimensional face Θ [2] F , the defining equation of the associated divisor is hence of the form It follows from the theory of [45] that P (z 0 , z 1 ) has * (Θ [1] ) + 1 roots p i , so that D ν has * (Θ [1] ) + 1 components. As is apparent from the above equation, these components are sitting over * (Θ [1] ) + 1 different locations in the base P 1 . Note that the same monomials (the ones related to Θ [1] ) will appear in the defining equation of each of the divisors corresponding to interior points of the face Θ [2] F , so that the same P (z 0 , z 1 ) will appear for each of them. We can turn the argument around and investigate the geometry of the K3 fibre over the points p i in the base. As we have learned above, the defining equation of M must have the form where only ν interior to Θ [2] F are considered. Hence the singular fibre over any of the p i has the components We cannot exclude that R is reducible and gives an non-trivial multiplicity to some of the D ν . Examples of this second type of singular fibre are found in [41].

A.3 Hodge Numbers of Divisors of a Calabi-Yau Threefold
Let M be a Calabi-Yau n-fold obtained as a hypersurface of a toric (n + 1)-dimensional ambient space, ν a lattice point ∆ ∩ N , and D ν the corresponding divisor of M . The Hodge numbers of the (n−1)-fold D ν can be derived using the methods of [45]. Formulas for h 1,1 (D ν ) are found in [9], but they are applicable only to cases with n ≥ 4. While the same reasoning can be applied to the n = 3 case, the formula looks different. This appendix provides a summary of the result for n = 3. Let Θ [k] , k = 0, 1, 2 be the face containing ν in its relative interior (a vertex corresponds to a zero-dimensional face and our convention is to consider it as its own relative interior) and let us denote the dual face of Θ [k] by Θ [3−k] . We can then summarize the Hodge numbers h 0,i (D ν ) by [45] k h 0,0 where * (Θ [l] ) counts points in the relative interior of the face Θ [l] . Note that these are already determined without having to specify the details of the fan of the ambient space (triangulation). Since divisors of a threefold are surfaces, we only need to determine h 1,1 (D ν ) now. It depends on the triangulation data of ∆ and can be described as 1 ν ( Θ [1] ) + ( Θ [2] ,Θ [1] ) 1 ν ( Θ [2] ) * (Θ [1] Here, 1 ν ( Θ [l] ) counts the number of ν-containing one-simplices in the relative interior of a face Θ [l] . The last line only contributes if ν is a vertex.

B Mathematics of Degenerations of K3 Surfaces
A lot is known about the degeneration of K3 surface in the mathematics literature. Here is a quick summary of what we use in this article, for convenience of readers. The largest fraction of material in this appendix B originates from [65,66,55]. Whenever we do not refer to a reference for a non-trivial statement, at least some clue is provided in one of these papers.

B.1 Degenerations of K3 Surfaces
B.1.1 Kulikov models and the geometry of the central fibre [Def.] A one parameter family of K3 surface consists of (X , π, Disc), where X is a complex threefold, Disc := {t ∈ C | |t| < 1}, and π : X → Disc is a morphism such that S t := π −1 (t) for ∀ t = 0 is a non-singular K3 surface. A central fibre of a degeneration is π −1 (t = 0) which is often denoted by S 0 .
One can think of a degeneration over a multi-dimensional parameter space, where Disc ⊂ C is replaced by Disc ⊂ C n . We do not deal with multi-parameter degenerations, and we will drop "one parameter", though it is always assumed implicitly in this article.
[Def.] A degeneration of a K3 surface (X , π, Disc) is semi-stable, if the following conditions i)-iii) are satisfied.
i) X is non-singular ii) the central fibre S 0 consists of irreducible components S 0 = V 0 ∪ V 1 ∪ · · · ∪ V µ , and all the singularity in the variety S 0 corresponds to normal crossing loci of the divisors V i 's in X iii) each one of V i 's appear in S 0 with multiplicity 1 (S 0 is reduced) [Theorem (Kulikov [67], Persson-Pinkham [68])] For a semi-stable degeneration of a K3 surface (X , π, Disc), one can always find a chain of birational transformations and base changes of the degeneration so that the resulting degeneration (X , π , Disc) is a Kulikov model..
[Def.] A degeneration of (X , π , Disc) is a birational transformation of another degeneration of a K3 surface (X , π, Disc) and vice versa, if there is a birational morphism between X and X that commutes with the projections π and π , and the birational morphism induces isomorphism between X \(π ) −1 (0) and X \π −1 (0). [Def.] When (X , π, Disc) is a degeneration of a K3 surface, one can construct another degeneration of a K3 surface, (X , π , Disc), by using a base change of order n. Let f : Disc t → (t ) n = t ∈ Disc; then X is the fibre product X × Disc Disc of π : X → Disc, f is the base change morphism, and π is the projection to the second factor. A degeneration (X , π , Disc) constructed in this way is a base change of a degeneration of a K3 surface. The theorem above makes Kulikov models into a well-motivated class of degenerations to study. Kulikov model degenerations of K3 surface are classified into three types. Let (X , π, Disc) be a Kulikov model degeneration of a K3 surface, and S 0 the central fibre. The classification is stated in terms of the geometry of the central fibre and also in terms of the monodromy of a generic fibre. Let T be the monodromy matrix acting on H 2 (S t ; Z) of a generic non-singular K3 fibre at t = 0 around the point of degeneration t = 0, from which a matrix N is defined as its log, T = exp [N ]. Now, • Type I: the central fibre S 0 consists of a single irreducible component. N = 0.
• Type III: the central fibre S 0 consists of multiple irreducible components, and their dual graph is a triangulation of S 2 (two-dimensional sphere). N 2 = 0, and N 3 = 0.
Even in Type III degeneration, it is known [69] that the matrix N is integer valued, when represented in the integral basis of H 2 (S t ; Z).
A choice of Kulikov model is not necessarily unique, in that there may be two Kulikov model degenerations of a K3 surface (X , π, Disc) and (X , π , Disc) that are birational transforms of one another. Those Kulikov models are always classified into the same type, because birational morphism between X and X do not modify the properties of the monodromy matrix of a generic fibre.
In this article, examples of Type II degeneration are discussed in sections 4.1, 4.2, those of Type III degeneration in section 4.3, while a brief discussion is given on a "cousin" of Type I degeneration in the appendix C.
The geometry of the central fibre of a Type II degeneration has the following properties.
{a} V 0 and V µ at the ends of the dual graph are both rational surfaces, while the surfaces in the middle, V 1 , · · · , V µ−1 , have a minimal model that is ruled over an elliptic curve.
{b} The curve V i ∩ V i+1 =: C i,i+1 of a pair of adjacent irreducible pieces V i and V i+1 is often referred to as the double curve. Within X , there is a normal crossing singularity at each C i,i+1 . The double curve C i,i+1 is always an elliptic curve.
{d} All the double curves C i,i+1 with i = 0, · · · , µ − 1 in a Type II degeneration share the same complex structure, which is ensured by the ruling of the surfaces V i+1 .
When there is a pair of Type II degenerations of K3 surfaces that are birational transformations of one another, the number of irreducible components µ + 1 is common to both. The rational surfaces V 0 and V µ of one degeneration may not be isomorphic to those of the other degeneration. The value of (C i,i+1 ) 2 | V i is not necessarily preserved in the birational transform either. The geometry of the central fibre of a Type III degeneration has the following properties.
{a} Each one of the irreducible components, V i , is a rational surface.
The number of triple points (V i · V j · V k )-the number of triangles in the dual graph-remains invariant under flops. This invariant is denoted by t.

B.1.2 Monodromy action
The monodromy group action T = exp[N ] : H 2 (S t ; Z) → H 2 (S t ; Z) and the geometry of the central fibre S 0 are related by the Clemens-Schmid exact sequence: Reference [66] provides background material for the Clemens-Schmid exact sequence including the definition of other homomorphisms (such as α, β and i * ). The monodromy matrix N introduces a filtration (called the monodromy weight filtration) into the cohomology groups of a generic fibre. A filtration is also introduced into the cohomology and homology groups of the central fibre by using the Mayer-Vietoris spectral sequence computation. In the Clemens-Schmid exact sequence for the degeneration of a K3 surface, the morphisms α, i * , N and β respect the filtration structure while shifting the weight by +6, +0, −2 and −4, respectively.
The monodromy weight filtration on the middle dimensional cohomology H 2 (S t ; Z) is given by where The following properties are useful: The monodromy is trivial on H 4 (S t ; Z) and H 0 (S t ; Z) and the filtration is formally defined here by In a Type II degeneration of a K3 surface, where The W 1 subspace is always of rank-2, and W 2 always of rank-20, within the rank-22 space H 2 (S t ; Z). Restriction of the intersection form of H 2 (S t ; Z) to its primitive sublattice W 1 is trivial, because an element of W 1 ⊂ W 2 is orthogonal to any element in W 1 . That is, W 1 is a rank-2 isotropic primitive sublattice of H 2 (S t ; Z) ∼ = II 3,19 .
Because of the self-dual nature of II 3,19 , one can always find a sublattice so that and It follows that Because the matrix T = exp[N ] = 1 + N needs to be an isometry of the lattice II 3,19 , the nilpotent matrix N for a Type II degeneration of K3 surface is always in the form of for some integer µ, in the basis of {ê 1 ,ê 2 ,ê 1 ,ê 2 }; N acts trivially on W 2 , because N (W 2 ) ⊂ W 0 = {0}. The integer µ-taken always positive-is called index of a Type II degeneration of K3 surface.
It is known that the integer µ is the same as the number of double curves C i,i+1 (i = 0, · · · , µ − 1) in the geometry of the central fibre [42]. It is reasonable that a birational invariant of the geometry of the central fibre is also captured in the language of monodromy acting on a generic fibre.
The central fibre is regarded as a limit of complex structure of the fibre K3 surface in such a way that the period integral is dominated by W 1 ⊗ C ⊂ II 3,19 ⊗ C. The limiting value Ω ∈ W 1 ⊗ C of the period integrals obviously satisfies Ω 2 = 0, because the intersection form on W 1 is trivial.
In a Type III degeneration of K3 surface, where N 3 = 0, The W 0 subspace is always of rank-1 and W 2 always of rank-21 within the rank-22 space H 2 (S t ; Z). The restriction of the intersection form of H 2 (S t ; Z) to its primitive sublattice W 0 is trivial, because an element of W 0 ⊂ W 3 is orthogonal to any element in W 0 . The self-dual nature of the lattice II 3,19 can be exploited to find a sublattice so that It follows that W 2 /W 0 ∼ = (U ⊕2 ⊕ E ⊕2 8 ) and (W 4 /W 2 ) ∼ = W 0 . In the case of a Type III degeneration, both N : W 2 /W 0 → W 0 and N : W 4 /W 2 → W 2 are non-trivial. Because of the self-dual nature of W 2 /W 0 ∼ = U ⊕2 ⊕ E ⊕2 8 , one can always find a sublattice U = Span Z {f ,f } isometric to U such that N (W 4 ) ⊂ U ⊂ W 2 /W 0 . In the basis of {ê,f ,f ,ê }, the monodromy matrix N is always in the form of for some integer µ. Here, µ 2 t 0 = t = (N (ê), N (ê)). It is known that this t is the same as the birational invariant t of the central fibre geometry explained earlier. The period integral in this degeneration limit is dominated by the components in W 0 ⊗ C.

B.2 Degeneration of Lattice-Polarized K3 Surfaces
[Def.] A degeneration of a K3 surface (X , π, Disc) is called lattice-polarized, if the restriction of divisors D i=1,··· ,ρ of X to a generic fibre S t generates a subset of Pic(S t ) that is isometric to a lattice Λ S . Such a degeneration is also called a degeneration of Λ S -polarized K3 surface.
The isotropic sublattice W 1 in a Type II degeneration and W 0 in a Type III degeneration is a primitive sublattice of Λ T := Λ ⊥ S ⊂ II 3,19 , because this is where the limiting values of period integrals reside. The lattice Λ S sits within the W 2 component for a Type II (resp. Type III) degeneration, because the algebraic component in Λ S must be orthogonal to W 1 (resp. W 0 ). The monodromy matrix T (and hence N ) acts non-trivially on the lattice Λ T , and trivially on Λ S . Scattone [55] formulated a classification problem of Type II and Type III degenerations of Λ S -polarized K3 surface as follows: • Type II: Classify rank-2 primitive isotropic sublattice W 1 in Λ T , modulo Γ, • Type III: Classify rank-1 primitive isotropic sublattice W 0 in Λ T , modulo Γ.
Two well-motivated choice of the quotient group Γ are Isom(Λ T ) and its normal subgroup Classification under Γ = Isom * (Λ T ) achieves a finer classification than that under the choice Γ = Isom(Λ T ). It is often easier to think of classification by Γ = Isom(Λ T ) first, and then to refine the classification later. In the rest of this appendix, we only refer to classification under Γ = Isom(Λ T ). The Isom(Λ T ) classification for Type II degenerations can be worked out in this way [55]. Note first, that one can always find a basis ê 1 ,ê 2 ,f 1 , · · · ,f 18−ρ ,ê 1 ,ê 2 (126) of Λ T in such a way that and the intersection form of Λ T is given by 30 for some positive integers δ 1 and δ 2 satisfying δ 1 |δ 2 . The two integers δ 1 and δ 2 (with a constraint δ 1 |δ 2 ) and a lattice (W 2 ∩ Λ T )/W 1 (modulo isometry) are uniquely determined for a given Isom(Λ T )-equivalence class. Once a pair (Λ S , Λ T ) is given, possible choices of δ 1 , δ 2 and an isometry class of (W 2 ∩ Λ T )/W 1 can be worked out systematically as follows. The discriminant group G Λ T is supposed to allow this substructure, first of all: The (Z δ 1 × Z δ 2 ) subgroup is an isotropic subgroup of (G Λ T , q Λ T ), and furthermore the (Z δ 1 × there are only finitely many options for such a substructure in (G Λ T , q Λ T ). In particular, there are only finitely many choices of δ 1 , δ 2 and isometry classes of the signature (0, 18 − ρ) lattice (W 2 ∩ Λ T )/W 1 . There can be multiple isometry classes for a given discriminant form (G (W 2 ∩Λ T )/W 1 , q) because of negative definite signature. The nilpotent matrix N for the monodromy matrix T = exp[N ] is determined uniquely. When it is presented in the basis (126), 30 The presentation in [55] corresponds to δ 1 = 1 and δ 2 = e. Since concrete examples of Type II degeneration treated in [55] were all for ρ = 1 lattice polarization, the discriminant group G Λ S is always a cyclic group. It was thus safe to set δ 1 = 1 for that reason. The presentation here is a straightforward generalized of that.
Here, µ ∈ Z is the index. This µ is the same as that in (120), when we allow to choose a basis without respecting the distinction between Λ S and Λ T .
Here are some examples. The first one is for Λ S = U , the E 8 -elliptic K3 surface. In this case, there are only two Type II degenerations of Λ S = U -polarized K3 surface in the Isom(Λ T ) classification. δ 1 = δ 2 = 1 (obviously because Λ T is self-dual), and For Λ S = +2 and +4 (i.e., degree-2 and quartic K3 surface), there is no choice but δ 1 = δ 2 = 1, and there are four choices for Λ S = +2 , whereas there are nine choices for Λ S = +4 . For the ρ = 1 cases Λ S = +2k , δ 1 = δ 2 = 1 are the only possibility, if k is not divisible by a square of an integer. See [55] for more information.
Similarly, the Γ = Isom(Λ T )-classification of Type III degenerations can be worked out as follows. Note first that one can choose a basis ê ,f 1 , · · · ,f 20−ρ ,ê of Λ T so that and the intersection form of Λ T is given in this basis as for some positive integer δ. Other parts of the intersection form, a, B, B T and C are also integer valued. Once a pair (Λ S , Λ T ) is given, one can systematically work out possible values of δ and isometry classes of the lattice (W 2 ∩ Λ T )/W 0 as follows. First, the discriminant group needs to allow the substructure The Z δ subgroup is isotropic under the discriminant form, and the (Z δ .G (W 2 ∩Λ T )/W 0 ) subgroup is orthogonal to the Z δ subgroup under the discriminant bilinear form b. There are only a finite number of such options for a given (G Λ T , q Λ T ) = (G Λ S , −q Λ S ). In the classification of Type III degenerations, the lattice (W 2 ∩ Λ T )/W 0 has signature (1, 19 − ρ), which is not negative definite. Due to a theorem of Nikulin [70] (Thm 1.14.2), any two even lattices (W 2 ∩ Λ T )/W 0 that reproduce the same discriminant form (G (W 2 ∩Λ T )/W 0 , q) are mutually isometric provided that ρ ≤ 11 [55]. The fact that the monodromy matrix T = exp[N ] is an isometry translates to the skew-symmetry condition on N with respect to the intersection form above. Therefore, where u, v, x are assumed to be integral. Allowing to choose a basis that does not respect the distinction between Λ S and Λ T , this µ here becomes the index µ in (124), and t 0 = v T ·C ·v = (v, v)| (W 2 ∩Λ T )/W 0 .

B.2.1 Baily-Borel compactification
The period domain of a Λ S -polarized K3 surface is given by (102), and the moduli space of Λ S -polarized K3 surface is the quotient of this space by Γ = Isom * (Λ T ). This group mods out unphysical marking without touching the lattice polarization divisors in Λ S . This moduli space D(Λ T )/Γ is not compact. There are multiple different ways to make it compact by adding boundary components. The Baily-Borel compactification D(Λ T )/Γ is a minimal one. The boundary components D(Λ T )/Γ \ D(Λ T )/Γ form different strata, each of which corresponds to one of the Type II or Type III degenerations of a Λ S -polarized K3 surface. A stratum corresponding to a Type II degeneration comes with a variety of one complex dimension, while one corresponding to a Type III degeneration is a point. This is because P[W 1 ⊗ C] for Type II is of one dimension, while P[W 0 ⊗ C] for Type III is of zero dimension.
Multiple strata for Type II degenerations labelled by various choices of δ 1 , δ 2 and (W 2 ∩ Λ T )/W 1 can meet at a point (stratum) for a Type III degeneration. The structure of such a stratification of the boundary components is studied for ρ = 1 polarized K3 surfaces in [55]. In the case of Λ S = +2 , for example, there is just one Type III stratum and four Type II strata (appearing in (132)), and all the four Type II curve strata meet at the Type III stratum point.
Other compactifications of the moduli space make it possible to retain more information of a K3 surface at a degeneration limit [71,72]. Possibly interesting in the context of heterotic-type IIA duality is the one discussed in [72], which retains information in the [(W 2 ∩ Λ T )/W 1 ] ⊗ C component of the complex structure in the degeneration limit. The heterotic string "instanton" moduli can be translated into these moduli at the degeneration limit. A version for [(W 2 ∩ Λ T )/W 1 ] = E ⊕2 8 in Λ S = U is well-known in string theory community through [37], but this story may be generalized for other Λ S and (W 2 ∩ Λ T )/W 1 -Type II degenerations.
C Picard-Lefschetz Monodromy and Collapsing dP 7 One of the simplest forms of degenerations of a K3 surface is for an A 1 singularity to be formed. In a local geometry, X → Disc may be given by This degeneration at t = 0 is not semi-stable, since the fibre at t = 0 has an A 1 singularity, which is not a normal crossing singularity. We can turn this into a Kulikov model by a base change followed by a resolution. In the present case, this means to replace the coordinate of Disc from t to t = s 2 , and further replace X by a small resolution of the conifold singularity at (x, y, z, s) = (0, 0, 0, 0). We then arrive at a Kulikov model of Type I. In the context of string compactifications, however, we are interested in a compact threefold M fibred over a compact space P 1 A , instead of X → Disc. We are usually not happy to replace M → P 1 A by its base change, either. We would rather think of the degeneration above as a "would-be" Type I.
A 1 -singularities in the fibre, i.e. would-be Type I degenerations, are quite a common phenomenon. In fact, for any Calabi-Yau threefold M with a Λ S = +4 -polarized K3-fibration (quartic K3 in the fibre), there are 216 such would-be Type I fibres; the topological Euler characteristic of M is understood in a simple way then: where the singular fibre of each one of those would-be Type I degenerations has χ = 23. The number of would-be Type I fibres-216-remains the same for any one of ν 6 F chosen from 2 ∆ F ∩ N F . Similarly, for any Calabi-Yau threefold M with Λ S = +2 -polarized K3-fibration (degree-2 K3 is in the fibre) discussed in section 3.2.1, there are 300 would-be Type I singular fibres. Here is how the counting goes, then: which holds for all the choices of ν 6 F in Figure 1. The number of would-be Type I singular fibres can be determined by using the discriminant of the K3 fibre (similarly to elliptic fibration). See [73] for how to compute the discriminant, from which we can derive such values as 216 and 300 above.
Let C p be the two-cycle in the K3-fibre that shrinks at a discriminant point z p ∈ P 1 A of a would-be Type I singular fibre. The Picard-Lefschetz monodromy T p on H 2 (S t.A ; Z) around z = z p is given by T p : H 2 (S t.A ; Z) x −→ x + (C p , x)C p ∈ H 2 (S t.A ; Z).
This is a reflection, and the monodromy becomes trivial after base change.
Consider tuning the complex structure moduli of M 2 +2 so we approach the transition point to the branch of M {2,1} +2 . The hypersurface equation (49) for the local geometry of M 2 +2 at the transition can be deformed to by introducing four parameters z i . When all of the z i 's are set to zero, this hypersurface equation approaches (49) at the transition point. At each one of z i 's, this equation is in the form of a deformation of a parabolic singularity X 9 [74]. In the local geometry of the fibre K3 captured in this equation (deformed X 9 ), nine compact two-cycles and two non-compact two-cycles are identified [75]. Seven of them-α 1,2,··· ,7 -form E 7 , the two remaining compact two-cycles are denoted by e 1,2 , and the two non-compact ones by e 1,2 . The intersection form is (e i , e j ) = δ ij , (e i , e j ) = 0. The singular fibre at a given z i is regarded as nine would-be Type I fibres coming on top of another, and the Picard-Lefschetz monodromy T i := 9 p=1 T p can be computed by using the information in [75]. The monodromy from all of the four z i 's combined, T = 4 i=1 T i acts trivially on the E 7 part of the two-cycles in the fibre, and on the remaining cycles as +2 has also been captured. The Type II degeneration of a degree-2 K3 surface, with (W 2 ∩ Λ T )/W 1 ∼ = (E 7 ⊕ D 10 ); Z 2 , can be regarded as a certain limit of the complex structure where 4 × 9 = 36 would-be Type Here, the contribution of the degenerate fibre is χ(V 0 ∪ V 1 ) = 24, the same as a generic K3 fibre. Let S 0 be the central fibre in a Type II semi-stable degeneration of a degree-2 K3 surface found in M n,n−1 +2 . The central fibre S 0 consists of two irreducible components. Let us use the notation V 0 =D 6,n and V 1 =D 6,n−1 .
The surface V 1 has the property h i,0 (V 1 ) = 0 (i = 1, 2), which we found by using the techniques of [45] (and an additional formula (112)) available for divisors of a toric hypersurface. The restriction of divisors of M {n,n−1} +2 toD 6,n−1 , the first line of (112), accounts for only 1 generator within H 1,1 (V 1 ) and we can takeD 6,n | V 1 as this generator. 31 The double curve C =D 6,n | V 1 has (D 6,n | V 1 ) 2 = +2 in this surface. This curve class is also the restriction of the polarization divisorD 2 of the generic K3 surface (see footnote).
The surface V 1 =D 6,n−1 is a dP 7 . To see this, note first that it is a hypersurface of W P 3 The cohomology group H 2 (V 1 ; Z) is a unimodular lattice. It is an index-2 overlattice of where the rank-1 lattice +2 is generated by C =D 6,n | V 1 . There are elements of H 2 (dP 7 ; Z) that correspond to a Z 2 subgroup of the discriminant group Z 2 × Z 2 of the lattice above (H 2 (dP 7 ; Z) is not an even lattice, however). 31 For M {n,n−1} +2 , rational equivalence relations areD 1 | V1 ∼ 2D 2 | V1 andD 2 | V1 ∼D 3 | V1 ∼D 6,n | V1 .
Let us now turn our attention to V 0 . Blowing down this irreducible component V 0 in M {n,n−1} +2 , we obtain a threefold M n−1 +2 with a singularity which may be deformed so that we find a smooth Calabi-Yau threefold M n−1 +2 branch. The singularity in M n−1 +2 right after this transition is given by X 2 1 + X 2 4 F (4) G (2) + X 4 X 6 F (5) G (4−n) + X 2 6 F (6) G (6−2n) 0 ; F (d) 's and G (d) 's are homogeneous functions of [X 2 : X 3 ] and [X 5 : X 6 ], respectively, with the degree specified in the superscript. The singular locus is along the curve X 6 = X 1 = X 4 = 0, which is a P 1 [X 2 :X 3 ] . The A 1 singularity in the directions transverse to this curve gets worse at 10 points in this P 1 ; that is where the Hessian of the quadratic form in (X 4 , X 6 ) degenerates. The divisor V 0 =D 6,n in M {n,n−1} +2 before the blow-down is obtained as the exceptional locus of this singularity.
A computation using toric techniques (as in V 1 ) indicates that (the first line of (112) and we can useD 6,n−1 | V 0 andD 3 | V 0 for now. 32 The complete linear system of the divisorD 3 | V 0 can be used to construct a projection Φ |D 3 | V 0 : V 0 −→ P 1 . This P 1 can be identified with the curve of A 1 singularities in M n−1 +2 ; [X 2 : X 3 ] is the homogeneous coordinate of this P 1 andD 3 | V 0 ∼D 2 | V 0 is the fibre class in this projection. This fibre is generically a conic in P 2 [X 1 :X 4 :X 6,n−1 ] . Both of the divisor classesD 4 | V 0 andD 6,n−1 | V 0 are 2-sections in the fibration corresponding to the projection, they differ only by the fibre class. The fibre conic degenerates into P 1 + P 1 whenever the Hessian degenerates. The 2-sectionD 6,n−1 | V 0 intersects once with one P 1 and also once with the other P 1 in such singular fibres. Let one of those two P 1 's be E ± i (i = 1, · · · , 10 labels singular conic fibres and ± distinguishes the two components). We have thatD 3 | V 0 ∼ E + i + E − i for any i. From (D 3 | V 0 ) 2 = 0 and E + i · E − i = δ ij it follows that (E ± i ) 2 = −1. Then the intersection form of V 0 32 There are three rational equivalence relations among restriction of the five toric divisors in M {n,n−1} +2 : in the basis (D 6,n−1 | V 0 ,D 3 | V 0 , E + 1 , · · · , E + 10 ) is in the form  The discriminant of this intersection form is −4. Hence the unimodular lattice H 2 (V 0 ; Z) must be an index-2 overlattice of the lattice generated by the basis above. The intersection form above indicates that there must be an element that is topologically regarded as 1 2D 3 | V 0 The basis above, withD 3 | V 0 replaced by 1 2D 3 | V 0 can be regarded as a generator set of H 2 (V 0 ; Z). This surface V 0 is rational, because V 0 ends up with a Hirzebruch surface F 2 after blowing down all the E + i 's. The double curve C =D 6,n−1 | V 0 has self-intersection (−2) in V 0 . The polarization divisorD 2 of a generic fibre K3 surface is restricted on this surface to beD 2 The intersection form can be presented in any choice of basis one likes; we do so as preparation for study in Clemens-Schmid exact sequence later. When we chooseD 6,n−1 | V 0 , (D 6,n−1 +D 3 )| V 0 ,D 3 | V 0 − E + 1 − E + 2 and (E + i − E + i+1 ) (i = 1, · · · , 9) as a set of generators, it becomes This intersection form is that of the lattice −2 ⊕ +2 ⊕ D 10 . The unimodular lattice H 2 (V 0 ; Z) should be an index-4 overlattice of this one.

D.2 Cohomology and Homology Groups of the Central Fibre
Homology and cohomology groups of the central fibre, S 0 = V 0 + V 1 , can be determined by using the Mayer-Vietoris spectral sequence. Let us begin with the homology groups. First of all, H 4 (S 0 ; Z) ∼ = H 4 (V 0 ; Z)⊕H 4 (V 1 ; Z) ∼ = Z⊕Z. The filtration H 4 (S 0 ; Z) = W −4 ⊃ W −5 = {0} corresponds to the convention in [66].
On H 2 (S 0 ; Z), the spectral sequence introduces a filtration where W −2 is obtained by identifying the double curve E 2,1 in H 2 (V 0 ; Z) and H 2 (V 1 ; Z); The W −1 /W −2 part, which is isomorphic to H 1 (C; Z) ∼ = Z ⊕2 , are the two-cycles that are obtained by gluing discs in V 0 and V 1 along α or β cycle in the double curve C.