Modular Forms as Classification Invariants of 4D N=2 Heterotic--IIA Dual Vacua

We focus on 4D $\mathcal{N}=2$ string vaccua described both by perturbative Heterotic theory and by Type IIA theory; a Calabi--Yau three-fold $X_{\rm IIA}$ in the Type IIA language is assumed to have a regular K3-fibration. It is well-kwown that one can assign a modular form $\Phi$ to such a vacum by counting perturbative BPS states in Heterotic theory or collecting Noether--Lefschetz numbers associated with the K3-fibration of $X_{\mathrm{IIA}}$. In this article, we expand the observations and ideas (using gauge threshold correction) in the literature and formulate a modular form $\Psi$ with full generality for the class of vacua above, which can be used along with $\Phi$ for the purpose of classification of those vacua. Topological invariants of $X_{\mathrm{IIA}}$ can be extracted from $\Phi$ and $\Psi$, and even a pair of diffeomorphic Calabi--Yau's with different K\"{a}hler cones may be distinguished by introducing the notion of ``the set of realizable $\Psi$'s''. We illustrated these ideas by simple examples.


Introduction
The duality between the Heterotic string theory and the Type IIA string theory has been known for a long time. The dualty with SO(5, 1) symmetry and 16 supersymmetry chargesthe duality at 6D-comes with just one piece of moduli space [Isom (4,16)\ SO (4,20)/ SO(4)× SO (20)]×R >0 , and its various aspects are understood very well [1]. The duality with SO(3, 1) symmetry and 8 supersymmetry charges [2]-the duality at 4D-is less understood. The moduli space of Heterotic-IIA dual 4D vacua forms a complicated network of branches. It is desirable that those individual branches are characterized both in the language of Heterotic string and Type IIA string, and the dictionary between the branch-characterizing data on both sides are understood. At the moment, we do not have one for the 4D Heterotic-IIA duality 1 as clear as Batyrev's dual polyhedra for mirror symmetry. For a systematic approach, we need to find invariants characterizing the branches of the moduli space. A lattice pair Λ S ⊕ Λ T that fits into II 4,20 is assigned for those branches [3,4,5,6], and a modular form Φ of certain type that depends on Λ S is also assigned [7,8]. It is known, however, that there are physically distinct branches of vacua that cannot be distinguished by the triple of invariants (Λ S , Λ T , Φ). The primary purpose of this article is to introduce more invariants by using modular forms to improve the state of affairs.
From the perspective of pure mathematics, this task is equivalent to classification of Calabi-Yau three-folds with a K3-fibration. The modular forms introduced in this article can be used therefore for study of such a geometry classification. It should be mentioned, however, that we consider only regular K3-fibrations in this article.
The organization of this article is as follows. We begin in section 2.1 with a short review on the Heterotic-Type IIA duality and a summary of technical limitations on the class of vacua to be considered in this article. The modular form Φ is parametrized by low-energy BPS indices, which are bounded from below as we argue in section 2.2; in the Heterotic language, the bounds come from the quantization of the level of current algebra and the spin under the SU(2) action. In sections 2.3 and 2.4, those bounds and the modular nature are combined to constrain the possibilities of Φ and the Euler number of a Calabi-Yau three-fold X IIA that compactifies IIA theory.
We formulate in section 3 a modular form Ψ for X IIA and a little more data, and use it to define new invariants for X IIA in addition to (Λ S , Λ T , Φ). Section 3.1.1 includes the basic definition of Ψ. This modular form appears in the integrand of 1-loop gauge threshold correction. In section 3.1.2 we comment on the restrictions on the degree of freedom of Ψ that comes from some physical constraints. We introduce a map (diff coarse , diff fine ) in section 3.1.3, which extracts from (Φ, Ψ) the full information in specifying the diffeomorphism class of a Calabi-Yau three-fold X IIA . Combining this map with the degree-of-freedom study of Φ and Ψ in sections 2 and 3.1.2, we can use modular nature of Φ and Ψ to obtain non-trivial results in the diffeomorphim classes of real six-dimensionoal manifolds realized by Calabi-Yau threefolds. We also propose to use the notion of "the set of Ψ's realized by Higgs cascades" or of "the set of Ψ's associated with curve classes" as an invariant that resolves a diffeomorphic pair of Calabi-Yau three-folds with different cone of curves. In section 3.2, all those ideas are illustrated by using simple examples.
In section 4 we comment on some open questions. The appendix A contains basics about (vector-valued) modular forms and explicit Fourier expansions of those in the main text. In the appendix B we review the lattice unfolding method and the embedding trick of Borcherd's [9], presented in a form we need for threshold ccalculations in Heterotic theory. The embedding trick is used in explicitly evaluating the integrals, for example in the case of Λ S = +2 in the appendix B.3.1.

A Brief Review
Let us first review what is known in the literature about the classification using the new supersymmetry index / the generating function of the Noether-Lefschetz number.

Heterotic Description: the New Supersymmetry Index
A Heterotic string compactification to 3+1-dimensions has an unbroken N = 2 supersymmetry (8 supersymmetry charges), if and only if the right-mover of the internal worldsheet CFT contains N = 4 superconformal algebra (SCA) with central chargec = 6 and N = 2 free SCA withc = 3 corresponding to a flat space of one complex dimension [10]. We restrict our attention in this article only to compactifications without an NS5-brane or its generalizations discussed in §5 of [11].
Let ρ be the number of free chiral bosons in the left-mover in such a compactification. There are vertex operators of the form e ip L ·X L +ip R ·X R in the CFT, where X L and X R are the ρ + 2 chiral bosons in the left mover and right mover; the set of U(1) charges {(p L , p R )} = Λ S forms a lattice with the quadratic form given 2 by p 2 R /2 − p 2 L /2, so its signature is (+, −) = (2, ρ). This lattice Λ S should be even, p 2 R /2 − p 2 L /2 ∈ Z for any element of Λ S , since the contribution of the state e ip L ·X L +ip R ·X R to the partition function should be invariant under T : τ → τ + 1. The U(1) charge of any worldsheet operator should lie in 3 Λ ∨ S , the dual lattice of Λ S . Note that we deal with the case Λ S is not necessarily unimodular, so that the discriminant group G S := Λ ∨ S / Λ S may be non-trivial. We assume, however, that Λ S is a primitive sublattice of 4 II 4,20 = U ⊕4 ⊕ E 8 [−1] ⊕2 ; the orthogonal complement [ Λ ⊥ S ⊂ II 4,20 ] is denoted by Λ T .
For example, when we compactify Heterotic theory on K3 × T 2 with instantons in g ⊂ The Hilbert space of the internal CFT can be decomposed using the action of the free boson algebra and the N = 4 right-mover SCA [12,7]: where superscripts show the central charge (c,c). The rank-(ρ + 2) U(1) charges in Λ S are denoetd by w, and H  2 We use the convention α ′ = 2. 3 We assume all the charge w ∈ Λ ∨ S is realized by some state. The Type IIA counter part of this assumption is that the pairing (H 2 (X; Z)/ZD s ) × [H 2 (X; Z)] vert → Z is represented by the unit matrix; see page 9 for notations. 4 The even unimodular lattice of signature (1,1) is denoted by U . We use the same notation R for one of ADE types, its Lie algebra, and its root lattice with positive definite signature in this article. specified by the central charge p C R : Λ ∨ S → C, which appears in the 4D N = 2 supersymmetry algebra; At the Heterotic string perturbative level, p C R is governed by the Coulomb branch moduli space which constitutes the special geometry along with the dilaton complex scalar s := 4πiS; the weak coupling limit 5 is s 2 ≫ 1. The rest of spectrum information (i.e. the spectrum of H (22−ρ,0) w,(h,Ĩ) for each possible (w;h,Ĩ) depends also on the hypermultiplet moduli space.
The new supersymmetry index [13,7] of the internal CFT is defined by 6 Z new (τ,τ ) := −i η(τ ) 2 Tr (22,9) R sector e πiF R F R q L 0 − c 24qL 0 −c 24 , where F R := 2(Jc =6 3 ) 0 +(Jc =3 ) 0 is the total U(1) current in the right-mover;Jc =6 3 is the SU(2) Cartan operator in N = 4 SCA andJc =3 the U(1) current in N = 2 SCA. The important point is that this index does not depend on any continuous deformations of the hypermuliplet moduli, but on that of vector multiplet moduli. Furthermore, it is used in computing the 1-loop correction ∆ grav to the gravitational coupling √ −gR 2 in the 4D effective theory; where the integration is over the fundamental region of SL(2; Z) in the upper complex half plane (of the torus world sheet complex structure τ ), andÊ 2 := E 2 − 3 πτ 2 is a non-holomorphic modular form of weight (2, 0). The constant b grav is set to the q 0 coefficient of B grav , to cut off the IR divergent 1-loop contributions and brings the massless degrees of freedom back into the path integration in low energy effective theory. From the modular invariance of the integrand, Z new has weight (−1, 1).
The action of free boson algebra on H int leads to the following decomposition 5 The real and imaginary components of s are denoted by s 1,2 . Similar notations (t 2 , τ 1,2 , ρ 2 etc.) are used throughout this article. 6 To be more precise, only the trace part is called as the new supersymmetric index of the internal CFT. The 1/η 2 factor is included within Z new here because the trace part appears in ∆ grav in the combination (3); the 1/η 2 factor is from the 4D Minkowski part in the light-cone gauge where θ Λ S [−1]+γ (τ,τ ) is the Siegel modular form, which describe the dependence on the vector multiplet moduli. θ Λ S [−1] = γ∈G S e γ θ Λ S [−1]+γ is a vector valued modular form in Mod((ρ/2, 1), ρ Λ S [−1] = ρ ∨ Λ S ), while Φ ∈ Mod(11 − ρ/2, ρ Λ S ). 7 (See appendix A.1 for our notations.) In particular, Φ is holomorphic at cusps, i.e. has no negative power of q in its expansion. Transformation law under T : τ → τ + 1 fixes the fractional part of power of q, so Φ γ /η 24 can be expanded as 8 c γ (ν)q ν , c γ (ν) = 0 for ν < −1.
The second equality in each line comes from the spectral flow of N = 4 SCA that brings the representations in R-sector to those in NS-sector. The coefficient −2, +1 in (8) corresponds to the Witten index of the representation (1/4, 1/2), (1/4, 0) [14]: It follows from (8), in particular, all the Fourier coefficients c γ (ν) are integers, so we have only discrete choice of Φ. In fact, since Φ/η 24 has negative weight as a modular form, it can be uniquely specified 9 by the coefficients of negative power of q Here the fractional part We can deduce n γ=0 = −2 from the supersymmetry constraints. First, the uniqueness of the ground state forces n V 0 = 1: n V 0 is the number of states in H V γ=0 with conformal weight 0. Such states, tensored with the right-moving highest weight state of (h,Ĩ) = (0, 0), gives the ground state in the Hilbert space of internal CFT. Second, ∂X µ ∈ H V γ=0 tensored with the right-moving Ramond ground states (inc = 3 sector) and the highest weight state of (h,Ĩ) = (1/4, 1/2) (inc = 6 sector) gives the exactly required number of gravitino states for 4d N = 2 supersymmetry. If n H 0 > 0, the 4d effective theory would have N = 2 + n H 0 supersymmetry by similar way. Since we focus on the case with only 8 supercharges, n H 0 should be zero, so n 0 = −2. For the expalantion of n 0 = −2 from the Type IIA perspective, see section 2.1.2. n γ for γ = 0 also has simple relation with spacetime effective theory: for each w ∈ γ ⊂ Λ ∨ S such that −2 ≤ w 2 < 0, 10 there exist n V γ BPS vector multiplets and n H γ BPS halfhypermultiplets 11 , both of which have U(1)-charge w and BPS mass p 2 R (w). w 2 < 0 implies that these states 12 will be massless at some points in the (weak coupling) Coulomb branch moduli space D( Λ S ). For this reason, we call n γ 's the low-energy BPS indices in this article.
In this paper, we impose some technical constraints on Λ S and Φ for simplicity. First, we only consider the case for some even lattice Λ S of signature (1, ρ − 1). Note that the direct summand U[−1] does not contribute to the discriminant group: of the generic fibre of K3-fibred Calabi-Yau manifold that compactifies Type IIA theory. See also section 2.1.2. 13 9 In fact, we assume some n γ to be zero. See the comments around (14). 10 This condition comes from the left-right matching of conformal weights. 11 If 2γ = 0 ∈ G S , these n H γ + n H −γ = 2n H γ half-hypermultiplets are combined to become n H γ full hypermultiplets. 12 In section 2.2, we show that there is non-trivial constraints about γ for BPS massless vector multiplet states to exist, coming from charge and level quantization conditions. 13 Λ S that does not have U [−1] as a direct summand may correspond to some non-geometric background of Type IIA, such as mirror-folds, etc.
Second, we assume that if γ 2 /2 ≡ 0 mod Z and γ = 0 then n γ = c γ (−1) = 0. In other words, Φ has expansion −2q 0 e 0 + (strictly higher in q). We denote this condition 14,15 as We define h min (γ) so that the Fourier expansion of Φ γ begins with O(q h min (γ) ) (or higher): We denote After all, the modular form Φ is specified by at least d < − 1 integers Sometimes, the modular properties predict linear relations among the low-energy BPS indices {n |γ| }; see section 2.3.2.

Type IIA Description: the Generating Function of the Noether-Lefschetz Numbers
In this article, we consider Type IIA string compactified on a non-singular Calabi-Yau threefold X = X IIA that has K3-fibration over P 1 = P 1 IIA , π : X → P 1 ; complexified Kähler parameters may be analytically continued out of a geometric phase, but otherwise we remain in a geometric phase. 16 This restriction means, in particular, that we do not treat T-folds or mirror folds [15]. This restriction corresponds to (13) in the Heterotic side. We also assume that The effective theory on 3+1-dimensions has stricly N = 2 supersymemtry, not more, not less.
K3-fibres in π : X → P 1 may degenerate at isolated points on the base P 1 . Degeneration of K3-fibration is classified (by allowing base change locally) into Type I, Type II, and Type III [16]. When a K3-fibration π : X → P 1 only has degeneration classified as Type I, 17 such a K3-fibration is said to be regular. In this article, we only consider regular K3-fibrations, because that is when one can find Heterotic dual descriptions without (generalization of) NS5-branes [11].
Let π : X → P 1 be a regular K3-fibration. Then the cohomology groups of X have the following filtration: where D s is the total K3-fibre divisor class, and [H 2 (X; Z)] vert is the subgroup generated by curves that are projected to points on P 1 . The free abelian group Λ S is a subgroup of the Neron-Severi lattice L S of X p , the fibre K3 surface over a generic point p ∈ P 1 . So, an intersection form is introduced on Λ S by restricting the intersection form of L S ; Λ S is now regarded as a lattice. The natural pairing between Λ S and [H 2 (X; Z)] vert is non-degenerate, and we have an isomorphism [H 2 (X; Z)] vert ∼ = ab Λ ∨ S as abelian groups. We reserve ρ for the rank of Λ S , not for L S . Λ S [resp. L S ] is a primitive sublattice of II 3,19 ∼ = H 2 (K3; Z); the orthogonal complement lattice is denoted by Λ T [resp. L T ].
For a regular K3-fibration π : X → P 1 , one can think of a generating function Φ of the number of Noether-Lefschetz points on the base P 1 IIA . When it is defined appropriately (see below), it is known to be a modular form [8]. First, there is a holomorphic map where Γ T := Ker (Isom(Λ T ) → Isom(G T )), because D(Λ T )/Γ T is the coarse moduli space of Λ S -polarized K3 surfaces. At points on the base P 1 IIA where the ι π -image hit the Noether- Think of the Heegner divisor [8] Φ pre := where {e γ } γ∈G T is the set of formal basis elements of the vector space C[G T ], q = e 2πiτ a formal variable, and N the level of the quadratic discrirminant form of the lattice Λ T .
T . An extra term e 0 q 0 D N L(0) /Γ T term is included in the definition of Φ pre ; we do not provide a description of the divisor "D N L(0) /Γ T " (see [8] for details), but all the necessary properties are provided later on. Note that in this definition Φ pre does not have a term 18 proportional to e γ q 0 for a non-zero isotropic γ ∈ G T . Φ pre is independent of ι π . Given ι π , we obtain a C[G T ]-valued function of the formal variable q = e 2πiτ by pairing Φ pre with the image of the base P 1 IIA mapped into 19 D(Λ T )/Γ T : Define Ref. [8] arrives at a statement (by using earlier math results in [17], but not relying on the duality with Heterotic string) See footnote 18 for why Φ is in Mod 0 , not in Mod. The coefficient NL ν,γ with γ = 0 and ν = 0 in Φ does not describe the number of Noether-Lefschetz points on P 1 IIA . Following the definition of the divisor "D N L(0) /Γ T " in [8], one arrives at where we used h 0,3 (X) = 1 and h 0,2 (X) = 0 at the last equality. So, as a consequence 20 of (18), we have Φ ∼ −2q 0 e 0 + (strictly higher in q). 18 This leads to the subtle fact that Φ lies in Mod 0 rather than Mod in (24). 19 Precisely, this procedure is only well-defined for smooth fibrations. In the case of Calabi-Yau X with finitely many nodal singular K3-fibres (these degenerations are classified as Type I), one has to take double cover of X and resolve the conifold singularities, so that non-singular fibration is obtained. One can apply the procedure to this fibration and divide the modular form by 2. We denote as Φ the modular form defined in this way. See [18,19] for details. 20 If . This is consistent with the Heterotic string description (n 0 = −2 + n H 0 = 0 in the N = 4 supersymmetry situation).

Heterotic-Type IIA Duality and Effective Theory
When branches of moduli space of Heterotic theories and Type IIA theories associated with a pair of primitive sublattices Λ S and Λ T of II 4,20 are identified, both descriptions should give the same modular form: The isomorphism G S ∼ = G T as Abelian groups is the one specifying the embedding relation 21 . That is because all the Fourier coefficients of Φ/η 24 determine physical quantities in the N = 2 supersymmetric effective theory on 3+1-dimensions; the helicity supertrace is defined by on the Hilbert space H(0, 0; (1, w), q 0 ) of particles on R 3,1 with a given pure electric charge under the (ρ + 2) U(1) gauge fields; q 0 ∈ Z and w ∈ Λ ∨ S . J 3 is the 3-component of the angular momentum of this space-time. In Heterotic language [20] In Type IIA language, we apply the electromagnetic duality transformation in the effective theory for the one of the (ρ + 1) U(1) gauge fields originating from the Ramond-Ramond 3-form field, the one associated with the base 2-form; then a D4-brane wrapped on the fibre class along with a 2-form F ∈ H 2 (K3; Z) and N ≥0 units of anti-D0-brane gives rise to a particle on R 3,1 with a pure electric charge (Mukai vector on the K3 surface) Here, F is the projection of F ∈ H 2 (K3; Z) to ı * (H 2 (X; Z)) ∼ = Λ ∨ S , and ı : (a fibre K3) ֒→ X. The helicity supertrace is [21,22,23,24] (and [25]) The value of v 2 /2 in the set of states above exhaust 22 all ν ∈ h min (γ) + Z ≥0 , so all the Fourier coefficients of Φ Het /η 24 and Φ IIA /η 24 should be the same. The Coulomb branch moduli space D( Λ S ) in (2) is parametrized by t ∈ Λ S ⊗ C with (t 2 , t 2 ) > 0, which is interpreted as Narain moduli in Heterotic description and as complexified Kähler parameter in Type IIA description. where , v) . We will focus only on the s 2 ≫ 1 and s 2 ≫ t 2 region of the special geometry throughout in this article; that is the weak coupling region in the Heterotic description, and the large base P 1 region in the Type IIA description. 23 To wrap up, moduli space of Heterotic-Type IIA dual vacua with 4D N = 2 supersymmetry form branches, and each branch is labeled by a pair of lattices Λ S and Λ T , and a modular form Φ ∈ Mod 0 (11 − ρ/2, ρ Λ S ). It is known [27] that one can find a C-basis {φ i } of Mod 0 (11 − ρ/2, ρ Λ S ) so that all the Fourier coefficients of φ i,γ (τ ) of φ i = γ∈G S e γ φ i,γ (τ ) are integers. So, Φ must be in the free abelian group within Mod 0 (11 − ρ/2, ρ Λ S ) whose rank is the same as the dimension of Mod 0 (11 − ρ/2, ρ Λ S ). This free abel group is denoted by Mod Z 0 (11 − ρ/2, ρ Λ S ). Without relying on explicit constructions (such as toric complete intersection), we can therefore hope to use properties of the free abelian group in Mod 0 (11 − ρ/2, ρ Λ S ) to derive some properties of Λ S -polarized K3-fibred Calabi-Yau three-folds.

Conditions for n
As seen in the section 2.1.1, n V γ is the number of BPS vector multiplets of fixed charge w ∈ γ subject to −2 ≤ w 2 < 0. Since charged massless gauge boson 24 should be a part of 22 The Gopakumar-Vafa invariants of vertical curve classes β ∈ [H 2 (X; Z)] vert ∼ = Λ ∨ S are equal to c γ (β 2 /2), and are also physical in the effective field theory because of its appearance in the prepotential (81, 84); see [7,8,26]. Not all the coefficients c γ (ν) of Φ correspond to those Gopakumar-Vafa invariants for some Λ S (e.g., Λ S = +2 ). 23 That is when we expect little contributions to low-energy physics from the NS5-branes in Heterotic string and D-branes wrapped on cycles that are mapped surjectively to P 1 IIA . BPS states of such origins are not used in defining the modular form Φ. 24 Heterotic string non-perturbative effects modify the infrared dynamics and the moduli space, as in Seiberg-Witten theory, but they are all known story [28]. non-abelian gauge bosons for some compact Lie group, its multiplicity must be one for each possible charge. This implies that n V γ = 0, 1 for any nonzero γ ∈ G S , so that n γ ≥ −2. Let us consider what happens when n V γ = 1. Suppose n V γ 0 = 1 for given nonzero γ 0 ∈ G S . Fix w 0 ∈ γ 0 ⊂ Λ S such that −2 ≤ (w 0 , w 0 ) < 0. Let us consider the points in the vector moduli space D( Λ S ) where the states of charge w 0 become massless: p R (w 0 ) = 0. At these points, the massless vector bosons of charge w 0 should be a part of non-abelian gauge bosons. In the language of worldsheet theory, this gauge symmetry are described by some current algebra carried by the left-mover. Therefore we have SU(2) current algebra (that may be a sub-algebra of larger current algebra) that consists of where X L is the internal free bosons in c = ρ sector, O ± are the vertex operators of conformal weight [(γ 0 , γ 0 )/2] frac in H γ 0 ,V . Note that J ± have conformal weight 1 as well as J 3 , by the assumption −2 ≤ (w 0 , w 0 ) < 0. Normalization of J 3 is set so that we have OPE This means that 25 the level of current algebra is k = −2/(w 0 , w 0 ). Let us consider some constraints from unitarity. First, the level k needs to be a positive integer. Second, any state (with U(1)-charge w ∈ Λ ∨ S ) should have half-integral spin in terms of this SU(2): this results in because that 26 is the charge of e i(p L (w)·X L +p R (w)·X R ) under J 3 . These mean the following constraints 27 for γ 0 ∈ G S = Λ ∨ S / Λ S : 25 In other words, O ± are from the coset SU (2) k /U (1) k . This coset model is known as parafermion theory with Z k symmetry. 26 This condition is required for all w ∈ Λ ∨ S , because we assumed in section 2.1 (footnote 3) that there exists at least one state of charge w for any w ∈ γ ⊂ Λ ∨ S . 27 It follows from these constraints that, for any nonzero isotropic γ in G S , we get n V γ0 = 0. The assumption n γ0 = 0 in (14)-automatic in Type IIA (footnote 18)-further implies n H γ0 = 0 then. n 1 2 3 4 5 6 · · · (j, k) ∅ (2, 2) (3, 4) ∅ (±4, 5) (±4, 3), (±6, 2) · · · Table 1: list of (γ 0 , k) = ( j 2n e, k) satisfying the condition (35) when Λ S = +2n The possibility k = 1 does not have to be included here, because k = 1 would simply mean γ 0 = 0 in (35), where we know that n V γ 0 =0 = 1 from the beginning. Not all γ ∈ G S satisfy the conditions (35), but those that satisfy (35) are not extremely rare. Table 1 shows the list of such γ's for some of the ρ = 1 lattices, Λ S = +2n ∼ = ab Ze. In the case of lattices of the form Λ S = U ⊕ −2m =: ab U ⊕ Ze, at least (γ 0 = 1 2m e, k = 4m) and (γ 0 = 2 2m e, k = m) satisfy (35). There are a couple of different behaviours in the Type IIA geometry X IIA that correspond to appearance of a massless non-abelian gauge boson in the low-energy effective field theory.
Let v = re 0 + q 0 e 4 + F ∈ Λ ∨ S be the U(1) charge of such a gauge boson (as in (29)). Suppose rq 0 = 0. Then the Calabi-Yau X IIA should have a curve class F ∈ Λ ∨ S ∼ = [H 2 (X IIA ; Z)] vert realized algebraically over a generic point of the base P 1 IIA (B ∞ in §3.2 of [8]); this is because is possible only when ι π (P 1 ) stays within a Noether-Lefschetz divsior ( F ⊥ D N L(F ⊥ ) )/Γ T with the sum ranging over those with F 2 +F 2 ⊥ = −2. The algebraic curve is F = (F +F ⊥ ) ∈ II 3,19 , which must be a (−2) curve. The vector boson on the spacetime R 3,1 is massless when this (−2) curve collapses to zero volume. F ⊥ must be nonzero since we think of a case γ 0 = 0 ∈ G S . Then there must be non-trivial monodromy on F ⊥ ∈ Λ T so that F is in L S , but not in Λ S . Λ S is a proper subset of L S then.
In the case of Λ S = U ⊕ −2m and γ 0 = ±e/m (the level is k = m), the following interpretation seems to work: in terms of lattice, L S ∼ = U ⊕ A ⊕m 1 and the lattice −2m ⊂ Λ S is embedded diagonally in A ⊕n 1 ; in terms of geometry, each K3 fibre of X → P 1 has m points of A 1 singularity, and those singular points forms a curve in X that is a m-fold cover over the base P 1 . The gauge kinetic term of the vector field is ms, which agrees nicely with the the gauge kinetic function ks for gauge fields associated with the level-k current algebra in Heterotic constructions [29].
There should also be geometry / lattice interpretation along the line of Λ S L S also for the case Λ S = U ⊕ −2m and γ 0 = ±e/(2m), but we have not been able to find a functioning interpretation yet.
Suppose instead that rq 0 = 0. The massless vector boson is then a D4-D2-D0 bound state, not just of D2-and D0-branes (r = 0). The Kähler parameter t must be of order unity for the vector boson to become massless (for moderate choices of r, q 0 , and F ), so the base P 1 IIA may be large (Im(s) ≫ 1), but the fibre K3 is not safely in the large radius geometric regime. In any case of Λ S = +2n with γ 0 ∈ G S satisfying (35), the U(1) charge w 0 for such a massless vector boson should be the one of this category, because F 2 /2 is positive definite in Λ S = +2n .

Examples of Φ
One can list up modular forms Φ ∈ Mod Z 0 (11 − ρ/2, ρ Λ S ) that satisfies n 0 = −2 and the lower bounds n γ ≥ −2 or n γ ≥ 0 depending on γ ∈ G S , as seen in section 2.2. The easiest and well-known case is when Λ S is unimodular: The Picard number ρ is 2, 10, 18 for each case and Φ should be a scalar-valued modular form of weight (22 − ρ)/2 with n 0 = −2. So Φ = −2E 4 E 6 and −2E 6 for the first and second case, respectively. There is no candidate of Φ for the third; Λ S = U ⊕2 ⊕ E 8 [−1] ⊕2 (i.e., zero instantons in E 8 × E 8 in Heterotic string) cannot be realized at least in our setup reviewed in section 2.1.
Similar procedure can be worked out for Λ S = +4 , +6 . See appendix A.2.1 for detail. Here we just cite the results for Euler number in the table 2. Λ S = +4 , +6 also allow only finite possibilities for χ, {n γ } and Φ.
We could not rule out n 1/4 , n 1/6 , n 2/6 that are non-zero, 34 or n 3/6 = 2, 4, 6. We may have missed some additional physical/mathematical constraints, 35 or it is possible that some of such Calabi-Yau three-folds may exist, maybe outside of the scanned range of the combinatorial data in [24], or as those that do not allow their realization by complete intersections in toric varieties.

Linear Relations on the Spectrum of Local Effective Field Theory
As stated already in section 2.1.1, the classification invariant Φ is in the free abelian group whose rank is the dim C (Mod 0 (11 − ρ/2, ρ Λ S )), and is also completely determined by the lowenergy BPS indices {n |γ| } ±γ∈G < S (because the weight (−1 − ρ/2) of Φ/η 24 is strictly negative for any ρ = 1, 2, · · · , 20). This implies immediately that there is a linear relation among n γ 's when dim C (Mod 0 (11 − ρ/2, ρ Λ S )) is strictly less than d < = |G < S /{±1}|. As is clear from the Heterotic description, 36 the low-energy BPS indices n γ 's with γ = 0 are the multiplicies of fields with purely electric charge under the (ρ + 2) gauge bosons. So, such a linear relation is that of the spectrum of Lagrangian-based effective field theory on R 3,1 with vector-like matter representations; it cannot be related to the 4D triangle anomaly (possibly to 6D box anomaly if Λ S ⊃ U), but it originate from the modular invariance of Φ.
For a general divisor P in X, however, such a linear relation is among the multiplicities of states whose U(1) charges are not necessarily mutually local. So, it cannot be regarded as a prediction on a spectrum of a Lagrangian-based local effective field theory. In the set-up discussed in the main text, P is the total fibre class D s , where P · P = 0, and r = h 1,1 (X) − 1, not h 1,1 (X). This property makes all the states from a D4-brane on P free from magnetic charge (obvious in Heterotic description from the start). see appendix A.2.2 for necessary details. In a series of ρ = 3 lattices Λ S = U ⊕ −2m , the m = 2 case already has a prediction, because dim C (Mod 0 (19/2, ρ −4 )) = 2 and d < = d = 3.
Details are found in the appendix A.2.1. In both of the series Λ S = +2n and Λ S = U ⊕ −2m , we confirmed that the dimension of Mod 0 (11 − ρ/2, ρ Λ S ) lies strictly below d < and also above zero for large n's and m's, by evaluation of the dimension formula (143, 144). Swampland surely exists within the space of local effective field theories, 37 if we restrict our attention to the class of Heterotic-Type IIA dual vacua reviewed in section 2.1. It is also found that the vector spaces Mod 0 (21/2, ρ +2n ) and Mod 0 (19/2, ρ −2m ) continue to have strictly positive dimensions 38 for large n and m (by numerically evaluating (143, 144)). So, the modular invariance of Φ and the integrality of its coefficients alone do not rule out existence of Calabi-Yau three-folds X with Λ S = +2n -polarized K3-fibration for an arbitrary large n, or also with Λ S = U ⊕ −2m for an arbitrary large m.

Cases with ρ = 20 and ρ = 19
Here, we have a look at a few families of choices ( Λ S , Λ T ) with ρ = 20 and 19. In some of them, we will see that the vector space Mod(11 − ρ/2, ρ Λ S ) is empty, and that there cannot be such a lattice-polarized regular K3-fibration in a Calabi-Yau three-fold, so studies from both sides agree nicely.
When ρ = 20, the lattice Λ T is rank-2, positive definite, and even. One can see that the vector space Mod(1, ρ Λ S ) is empty in the following way. For any Φ = 0 in this vector space, θ Λ T · Φ must be a scalar-valued weight-2 modular form starting with −2 + O(q). Because there is no such weight-2 modular form, the vector-valued modular form Φ should have been zero.
One can also arrive at alomst the same conclusion independently by using geometry vaialable in Type IIA language. If a Calabi-Yau three-fold X IIA has a K3-fibration with a generic fibre having ρ = 20 Neron-Severi lattice, the fibre K3 surface has a fixed complex structure over the entire base P 1 , so X IIA must be of the form (ρ = 20 K3) × P 1 IIA . This is not a Calabi-Yau three-fold, so there should not be such a K3-fibred Calabi-Yau three-fold. It 37 The lattice Λ S is characterized within the language of local effective field theory on R 3,1 ; it appears in the prepotential (81). 38 There are more modular forms of a fixed weight and for Γ(4n) for large n. So, that is not surprising.
should be noted, however, that this second argument does not rule out non-geometric phase 39 Type IIA constructions in a ρ = 20 case, and hence the first argument is stronger. Similar arguments also rule out a family of ρ = 19 cases where n ∈ Z >0 . The first argument for the ρ = 20 cases can be repeated by replacing θ Λ T with θ +2n , to see that the vector space The absence of such a Φ (with n 0 = −2) is also understandable in geometry language. The Fourier coefficients of Φ are the intersection numbers of the Noether-Lefscetz divisors in D(Λ T )/Γ T (they are points in the ρ = 19 cases), and the image of the holomorphic map ı π : P 1 IIA → D(Λ T )/Γ T determined by the K3 fibration map π : X IIA → P 1 IIA . The modulai space 40 D(Λ T )/Γ T = H/Γ 0 (n) contains the large complex structure limit point; if ı π is surjective, then there are points in the base P 1 IIA where the K3-fibration is not regular. All the ρ = 19 K3-fibration studied in [33]. If ı π were to be a constant map, then X IIA would not be a Calabi-Yau three-fold (see the second argument for the ρ = 20 case).
The arguement for the absence of Λ S -polarized regular K3-fibred Calabi-Yau three-folds holds true for all the ρ = 19 cases, not necessarily for the Λ T = U ⊕ +2n cases discussed above. The authors do not have a proof yet that Mod(3/2, ρ Λ S ) is empty for more general Λ T 's in the ρ = 19 case, however.
The consequence that the dimension of Φ is smaller for larger ρ is understandable intuitively in itself. K3 fibration is specified, after all, by specifying a map from the base P 1 to the period domain D(Λ T ); less complicated geometry D(Λ T ) allows less variety in the map from P 1 to D(Λ T ).

Lower Bound on Euler Number
We have seen in Table 2 that the Euler number χ(X IIA ) of the Calabi-Yau three-fold X IIA that admits Λ S -polarized K3-fibration is given by a linear sum of the low-energy BPS indices {n |γ| }, for a few choices of Λ S . In Table 2, all the coefficients of {n |γ| } are positive, from which it follows that χ(X IIA ) is bounded from below. Actually, this is true for any choice of Λ S , as we see below.
for φ. The lower bounds on {n γ }'s imply a lower bound on χ(X): If there appear no higher-level current algebras (i.e. there are no γ ∈ G S that satisfy the condition (35)) then n γ ≥ 0 for γ = 0 ∈ G S , and It is not necessary to work out a basis of Mod 0 (11 − ρ/2, ρ Λ S ) (where Φ is in) in deriving such relations as those in Table 2.
The relation (48) reproduces those in the Table 2 by using φ = θ L[−1] for L determined as in footnote 43. We applied the same procedure for some Λ S = +2n not covered in section 2.3, and obtained the result summarized in Table 3; calculations that led to Table 3 are found in the appendix A.2.2. In the cases of Λ S = U ⊕ W for some even lattice W , the relation (48) In particular, All the lower bounds of χ(X IIA ) for individual Λ S are safely above the absolute lower bound for all Calabi-Yau three-folds X IIA [35] The bound (52) was derived [35] by exploiting the modular invariance of the fundamental string partition functions of Type II compactification (with X IIA as the target space), while the bound (49) is due to the fact that the generating function Φ of the helicity supertraces of BPS D4-D2-D0 brane bound states on X IIA is a modular form, and the fact that the new supersymmetry index Z new of the fundamental string in the Heterotic description must be a modular form of weight (-1,1).
One also finds from the relation (48) that the low-energy BPS indices also have an upper bound. This is because This is a generalization of the same observation made already in section 2.3.1. Note, however, that all of the coefficients n γ do not necessarily appear in the equation (48); in the case of Λ S = U ⊕ −2n =: ab U ⊕ Ze, for example, the linear relation (48) among O(n) low-energy BPS indices. So, we cannot use this argument to claim that only a finite set of the low-energy BPS indices {n γ } corresponds to a geometric phase in Type IIA description.

Finer Classifications
The modular form Φ (new supersymmetry index (Het)/ NL number generating function (IIA)) is not enough discrete data for classification of families (branches) of moduli space of Het-IIA dual vacua. Let us take the Λ S = U case as an example. In the Heterotic language, all the K3 × T 2 compactifications with the 24 instantons on K3 distributed by (12 − n, 12 + n) to the two weakly coupled E 8 gauge groups share the same Φ = (−2E 4 E 6 ) for all 0 ≤ n ≤ 2, but they form three distinct branches of moduli space. In the Type IIA language, the modular form Φ determines the Gopakumar-Vafa invariants of all the vertical curve classes of elliptic-K3 fibred Calabi-Yau three-folds X IIA , but the classical trilinear intersection numbers of the divisors are not [7] (for precise statements, see section 3.1.3). X IIA can be any one of the elliptic fibrations over F n with n = 0, 1, 2. Presence of such multiple branches of moduli space sharing Φ has been reported also for the case of Λ S = +2 and +4 [24] (see also [11]). We introduce invariants of branches of moduli space of Het-IIA dual vacua that can distinguish those sharing a common Φ. That is done by developing observations and ideas that are found in the literatures. Those invariants do not rely on supergravity approximation or explicit construction of geometries, but use modular forms.

The Idea
Consider a branch of moduli space of Het-IIA dual vacua, where we have special geometry and hypermultiplet moduli space of fixed dimensions (h 1,1 (X IIA )-and (h 2,1 (X IIA ) + 1)-dimensions, respectively, if the branch contains a geometric phase in the Type IIA language). We call it the original branch. It often comes with special loci in the hypermultiplet moduli space where non-abelian gauge symmetry R is enhanced in the effetive theory on R 3,1 ; one ventures into other branches of moduli space by turning on non-zero Coulomb vevs in R. Modular forms denoted by Ψ and Φ are assigned to such a symmetry-enhanced branch (see below for more); the idea is to use the set of such modular forms as an invariant of the original branch. We will see in this section 3 that the set of Ψ's or the set of Φ's distinguish multiple branches sharing the same Λ S , Λ T , and Φ; moreover, the modular form Ψ or Φ of even just one symmetry-enhanced branch attached to the original branch already improves classification by the modular form Φ alone.

Higgs Cascades and Modular Forms
Let us first assign two modular forms Φ and Ψ for a symmetry-enhanced branch. We will discuss in section 3.1.3 the information of target-space geometry that we can extract from such a modular form Ψ, or from the set of Ψ's associated with all the symmetry-enhanced branches.
We restrict our attention to the case of R as one of ADE types, and its non-abelian gauge bosons are given by left-mover level k = 1 current algebra in the Heterotic language. 45 The lattice R[−1] is chosen within Λ T ; now we introduce lattices for the symmetry-enhanced branch. The lattice Λ S is Λ S ⊕ R[−1] or its extension. It is assumed here that the symmetry-enhanced branch is also realized without NS5-branes and the likes in the Heterotic description, or without a degeneration of K3 fibre classified as Type II or III in the Type IIA description. A related discussion is found at the end of section 3.2.2.
Under this assumption, there must be a modular form it describes the BPS indices of the Heterotic description and the Noether-Lefschetz numbers of the K3-fibre in the Type IIA description in the symmetry-enhanced branch, just like Φ does for the original branch. Here ρ := ρ + rank(R), and ρ Λ S the representation of Mp(2, Z) associated with the lattice Λ S . The modular form Φ of the symmetry-enhanced branch should be related to Φ of the original branch in the following way. At the entrance of the symmetry-enhanced branch (so the Coulomb branch moduli still stays within the subset D( Λ S ) of D( Λ S )), the nonabelian symmetry R remains unbroken. Since this vacuum belongs to both of the original and symmetry-enhanced branch, the new supersymmetry index Z new of the both branches should be equal at this point: Here Therefore, the modular form Φ of the symmetry-enhanced branch should reproduce Φ of the original branch through In the case of Λ S = Λ S ⊕ R, this simplifies to [38] The modular form Ψ is associated with 1-loop 46 threshold correction ∆ R to the coupling constant of the enhanced non-abelian gauge group R. It is given by [39] The corrections to special geometry that is regarded as 1-loop contributions in Heterotic string language are those that neither diverge (tree in Het) nor vanish (non-perturbative in Het) in the large base limit.
where the Heterotic string is used as a language. Here, I, J ∈ {1, · · · , rank(R)} label rank(R) left-moving free bosons 47 X I , and Q I is the zero-mode momentum in the expansion X I (z) = x I +Q I ln z +oscillators. Q I works as the Cartan charge operator of R. b R = ( hyper 2T rep )− 2T R is the 1-loop beta function of the probe gauge group R. Contracting the indices I, J, we arrive at Here ∂ S is the Ramanujan-Serre derivative (see the appendix A.1). Immediately from (55, 63), Obviously the modular form Ψ/η 24 (for not necessarily unimodular Λ S ) is the generalization . The modular form Ψ of the symmetry-enhanced branch captures only a part of information in Φ, because Ψ can be determined from Φ as in (63). In fact, when there is a chain of symmetry enhancements R 1 R 2 · · · acompanied by chain of tunings in the hypermultiplet moduli space, the chain of the invariants ( Λ S , Λ T , Φ) all reproduce one common modular form Ψ through (63). This is because the corrections to the gauge coupling constants remain unchanged by continuous change in the hypermultiplet vevs in N = 2 supersymmetric gauge theories on R 3,1 , an observation implicit already in [7]. For this reason, the modular form a Φ is assigned to each symmetry-enhanced branch, but Ψ to a chain of symmetry-enhanced branches attached to the original branch (such a chain is called a Higgs cascade).
We will see in section 3.1.3 that the modular form Ψ for one Higgs cascade attached to the original branch-an arbitrarily chosen cascade is fine-specifies the diffeomorphism class of X IIA of the original branch (of the individual geometric phase chambers of the original branch, to be more precise); the modular form Φ alone does not have enough information for this purpose in general. Furthermore, the set of Ψ's for the set of Higgs cascades attached to the original branch can also be used as a classification invariant of the original branch. This viewpoint is sometimes useful for distinguishing two different branches of moduli space with the same diffeomorphism class of Λ S -polarized K3-fibred Calabi-Yau three-folds. The modular forms Φ are more useful in capturing the network of symmetry-enhanced branches.

The Space of the Modular Form Ψ's
For a given branch of Het-IIA dual moduli space characterized by ( Λ S , Λ T , Φ), the modular form Ψ of a Higgs cascade of the original branch is not completely arbitrary element of the vector space (64). We will derive a few constraints on Ψ in the following.
General Constraints First, recall the definition: . Choose a basis of the left-moving free bosons X I so that roots of a fixed subalgebra su(2) ⊂ R have charges only in I = 1; Q I = ± √ 2δ I 1 . Now set I = J = 1 in the above equation. Since all the states in the Hilbert space with a definite charge under R have Q I=1 Q J=1 ∈ 1 2 Z, contributions from states with a charge (w, Q 1 ) under Λ S ⊕ su(2) and those with a charge (w, −Q 1 ) add up to be an integer. This implies that As an immediate consequence, all the Fourier coefficients are all integers.
In other words, this comes just from the properties of lattice theta functions of simple Lie algebra: Using the relations (63, 58), we see that Defining a (R) δ (ν) by we have The integrality of a δ (ν) can be seen in essentially the same way as the discussion above (see also the appendix A.2.3). So we have d γ (ν) ∈ 12Z, because the BPS indices c [γ,δ] (ν) are also integers.
As we see later, consistency in the low-energy effective field theory on R 3,1 implies that d 0 (0) ∈ 24Z ≥0 . We have not tried much to think whether this condition can be derived directly from consistency of string theory.
Not all the d γ (ν)'s are arbitrary integers divisible by 12 (or 24). It follows immediately 48 from the fact a (R) We can say a little more. Define m γ := d γ ([γ 2 /2] frac − 1) for (ΦE 2 − Ψ), like we defined n γ for Φ: Here only δ ∈ G R such that [(γ, γ)/2] frac > ν δ can contribute to the sum. In particular, Let us introduce a space denoted by Mod Φ 0 (13 − ρ/2, ρ Λ S ) for a given Φ, which is the set of modular forms Ψ satisfying d γ (ν) ∈ 12Z, d 0 (0) ∈ 24Z ≥0 , and (73). When the modular form Ψ is for a Higgs cascade attached to the original branch with Φ, then Ψ must be in this set. For example, in the case of Field-theory Argument for d 0 (0) ∈ 24Z ≥0 : We realize that d 0 (0) ∈ 24Z, not just in 12Z, by remembering that it is related to the 1-loop beta function b R through b R = d 0 (0)/24 (eg. [7]); this relation itself is obvious also from the expression (70): where we made a replacement a as the (effective) number of half-hyper multiplets in the representation. We see b R ∈ Z/2 comes from d 0 (0) ∈ 12Z. But b R can be in 1/2 + Z only when R = A 1 and there are odd number of hypers in the fundamental representation, which is not allowed because it would cause the SU(2) global anomaly. So b R ∈ Z, and d 0 (0) ∈ 24Z.
In addition, b R should be non-negative because there must be plenty of matter fields to Higgs the gauge symmetry R completely; the Higgsing brings the symmetry-enhanced branch back to the original branch. In the case of Λ S = U with the probe symmetry set in one of the two weakly coupled E 8 of the Heterotic string, for example, it is known that there are I := 10 + 12 −2 d(0) instantons in the rest of the E 8 . The condition that d(0) ≥ 0 corresponds to the fact that at least 10 instantons on K3 are necessary to break E 8 completely.
As a side remiark, one notices (see the appendix A.2.3) that the coefficients a (R) δ (ν) for smaller values of ν are divisible by 2 in D r≥4 , by 6 = 2T 27 in R = E 6 , 12 = 2T 56 in R = E 7 , and by 60 = 2T 248 in R = E 8 , although the authors have not know a proof that this property may persist for arbitrary large values of ν. So, a modular form Ψ is such that all the d γ (ν)'s are divisible not just by 12, but by 24 [resp. 12 × 6, 12 × 12, or 12 × 60] if the Higgs cascade to which Ψ is assigned has an enhanced symmetry R as large as 50 Extra Degrees of Freedom The set Mod Φ 0 (13−ρ/2, ρ Λ S ) is parametrized by finite number of d γ (ν)'s in 12Z. Those with ν < 1-the m γ 's-are enough in the case of ρ > 2, and those of q ν with ν < 2 are enough if ρ = 1, 2, because Ψ/η 24 [resp. Ψ/η 48 ] has negative weight when ρ > 2 [resp. ρ = 1, 2]. Those d γ (ν)'s (or equivalently the Fourier coefficients c Ψ γ (ν)'s) may be subject to some linear constraints, just like we discussed for Φ in section 2.3.2.
In the case ρ = 2, the remaining freedom in the space Mod Φ 0 (0, ρ Λ S ) not specified by the m γ 's is in the free abelian group Mod Z (k = 0, ρ Λ S ). This abelian group is equivalent to that of a (τ -independent 51 It is enough to make sure that {φ γ } is invariant under ρ Λ S (T ) and ρ Λ S (S). The invariance under ρ Λ S (T ) implies that φ γ = 0 only if (γ, γ)/2 ∈ Z. Imposing the invariance under the ρ Λ S (S), it follows that non-zero φ is possible only when G S contains a non-zero isotropic element γ =0 ∈ G S or G S = {0} (so Λ S = U). For most of rank-2 Λ S , therefore, there is no extra degree of freedom for Ψ/η 24 .
Suppose that G S contains an isotropic 52  In the case of ρ = 1, namely, Λ S = +2n for some n ∈ N >0 , the 2(n + 1) integers {∆m γ } and {∆d γ ( 0≤ ν <1 )} must be enough to parametrize Ψ for a given Φ. They are often redundant, however. 53 We do not have a general theory about how many linear constraints exist within {∆(m γ )} without relying on a case-by-case analysis. About {∆d γ ( 0≤ ν <1 )}, at least we know that there is one degree freedom not captured by {m γ }'s; we stay within the 51 Consider lifting φ ∈ Mod(k = 0, ρ ΛS ) to a modular curve H/Γ(N ) where Γ(N ) is in the kernel of the representation ρ ΛS . The lift φ should be C-valued functions on the compact curve, so it has to be τ -independent. 52 A subgroup H of a discriminant group G is isotropic, if the restriction of the discriminant quadratic form on H is trivial. 53 because the dimension of Mod 0 (25/2, ρ +2n ) does not grow as fast as ∼ (2n).

Modular Forms and Topological Invariants
Let X [resp. X] be a Calabi-Yau three-fold with a regular Λ S -polarized [resp. Λ S -polarized] K3 fibration, and suppose that X with some cycles collapsed is regarded as a limit of complex structure of X in a way a complex codimension-2 singularity of type-R emerges along a curve C R ; lim cpx str X = lim Kahler X. The modular form Φ and Φ assigned for X and X determine such information as Noether-Lefschetz numbers of X and X, but there are also some topological invariants of X and C R that can be determined from the Fourier coefficients of Φ and Ψ [7]. We start off with quickly reviewing traditional calculation of the matching between the data Φ and Ψ and the low-energy effective theory, and proceed to discuss how we can use such modular forms for classification of such Λ S -polarized K3-fibrations.
A Quick Summary of the Matching The low-energy effective theory on R 3,1 has a prepotential F , gravitational coupling F 1 , and the gauge kinetic function f R of the enhanced symmetry R, when the Type IIA string is compactified on lim cpxstr X = lim Kahler X. Those functions of the effective theory in the Im(s) ≫ 1 limit 54 are determined 55 from the microscopic data ∆ R and ∆ grav through the relation 56 Those low-energy functions are of the following form, 57 because we already assume a geometric phase Type IIA compactification: Here, a component description {t a=1,··· ,ρ } is given to t ∈ Λ S ⊗ C by choosing an integral basis {D s , D a=1,··· ,ρ } of H 2 (X, Z) consistent with the filtration structure 58 in (19); the divisors {D a=1,··· ,ρ } modulo +ZD s may be regarded as a basis of Λ S . The complexified Kähler class 54 We consider only the cases where the curve C R covers the base P 1 just once. 55 See footnote 46. 56K = − ln(t 2 , t 2 ) + const. is the (Heterotic string) tree-level Kähler potential of the non-dilaton vectormultiplet scalars. 57 In N = 2 field theory on R 3,1 , the prepotential itself is not physical (e.g., [40]); different choices of (2n V + 2)-tuple of symplectic sections (X I , F I ) related by a Sp(2n V + 2; Z) duality transformation may have different prepotentials (a prepotential does not exist for some frames). The prepotential here is for a frame where a D2-brane wrapped on any real 2-dimensional cycle (in Type IIA language) is treated as an electrically charged particle in R 3,1 . cf section 2.1.3.
58 For a Calabi-Yau three-fold, the structure of K3-fibration π : X IIA → P 1 IIA is in one-to-one with a divisor class D s of X IIA satisfying D 2 s = 0 and XIIA D s · c 2 (T X IIA ) = 24. The divisor class charcterized in this way is the topological class of the K3 fibre over a generic point in the base P 1 . Choice of a divisor D a has ambiguity D a → D a + δn ′ a D s with δn ′ a ∈ Z. In terms of the coefficients, this corresponds to s → s − δn ′ a t a and t a unchanged. of X is t CY = sD s + t a D t = sD s + t when e 2πis corrections are ignored (as we will everywhere in this article).
The sums of exponential terms run over effective vertical curve classes β eff , because we retain only the terms that remain non-zero in the large base (Im(s) ≫ 1) region of the moduli space. n r β eff is Gopakumar-Vafa invariant. A β eff is related r = 0 Gopakumar-Vafa invariants of X IIA . It is well-known that the matching relations (78, 79, 80) determine those parameters in terms of the coefficients of Φ and Ψ as We have nothing to add or discuss about them in this article, however.
The non-exponential part of them captures topological invariants of X IIA and the curve C R of enhanced singularity of type R in lim cpx str X = lim Kahler X; χ = χ(X) is the Euler number, and 24s The coefficients d ′ a can also be regarded as trilinear intersection numbers in X among D a and a pair of exceptional divisors that emerge after resolving the type R singularity.
Those invariants are determined by 59 by the matching conditions (78, 79, 80) as where P 3 (t) and P 1 (t) are polynomials of t given by the integrals over the fundamental region 59 Here we have useds = s + d ′ a t a for convenience. Since d ′ a is an integer, using (s, t a ) instead of (s, t a ) corresponds to the integral basis change D a → D a − d ′ a D s .

SL(2, Z)\H:
See appendix B for details of the integrals; evaluation method for the case Λ S has a nontrivial null element is reviewed in appendix B.1. Appendix B.3 explains how to reduce a case of Λ S without such an element to cases with such an element.
Discussion 1 For a given X, choose any Higgs cascade attached to the branch of moduli space 60 of Type IIA compactification of X. We see in the following that the pair of modular forms Φ and Ψ contains complete information in specifying the diffeomorphism class of X.
Let us recall Wall's theorem [41], which states that the set of diffeomorphism classes of real six-dimensional, simply-connected, spin, oriented manifolds with a torsion free cohomology and a given set of Betti numbers b 2 and b 3 are in one-to-one with the set where H 2 ∼ = Z ⊕b 2 and (a) µ(x, x, y) + µ(x, y, y) ≡ 0 mod 2 for ∀ x, y ∈ H 2 , the relation is (µ, p 1 ) ∼ (µ ′ , p ′ 1 ) if and only if there is an isomorphism φ : H 2 → H 2 such that (µ ′ , p ′ 1 ) = (µ, p 1 ) · φ. For a manifold X, the trilinear symmetric form µ is the wedge product of H 2 (X; Z), and p 1 the linear form X p 1 (T X) ∧ x for x ∈ H 2 (X; Z).
Its subset of interest in this article is those where H 2 contains an element D s of the property described in footnote 58. It is given by where Λ S = Z ⊕ρ and (a') d aab + d abb ≡ 0 mod 2 for any a, b ∈ {1, · · · , ρ}, (b') 4d aaa + 2(c 2 ) a ≡ 0 mod 24 for any a ∈ {1, · · · , ρ}; d abc and (c 2 ) a for a, b, c = 1, · · · , ρ are the component description of d and c Λ S 2 for some basis of Λ S ; the relation is 61 given by setting (d, ) if and only if they become identical for some combination of isometries of Λ S and the basis changes in footnote 58.
The modular form Φ of X determines the combinations 62 and hence the combinations They remain invariant under the shifts D a → D a + (δn ′ a )D s with δn ′ a ∈ Q for a basis {D S , D a=1,··· ,ρ } ∈ H 2 ⊗ Q. So the modular form Φ of a Calabi-Yau three-fold X determines an element of There may be a pair of three-folds X and X ′ sharing the same modular form Φ that are not diffeomorphic to each other. They must have the same combination d abc − [(c 2 ) a C Λ S bc + · · · ]/24 but (d abc , (c 2 ) a ) of X may be converted to that of X ′ only by allowing the shifts with (δn ′ a + Z) = 0 ∈ Q/Z.
With just the modular form Ψ of one arbitrary chosen Higgs cascade of X (along with Φ), however, the dictionary (88, 89, 90) determines (d abc , (c 2 ) a ) precisely with the relation ∼ Λ S , because the integrality of d ′ a allows only the shifts s → s − δn ′ a t a with δn ′ a ∈ Z. To summarize, the modular forms Φ and Ψ may be seen as information of the spectrum of BPS states of string theory, or that of Noether-Lefschetz numbers and curve counting invariants, but they also carry full information the diffeomorphism class of the original manifold 61 Note that an element D s ∈ H 2 with the property in footnote 58 is mapped by φ : H 2 ∼ = H 2 for the relation (µ, p 1 ) ∼ (µ ′ , p ′ 1 ) to an element φ(D s ) ∈ H 2 that also has the same property. 62 It also determines χ(X). Under the assumption that X is a Calabi-Yau three-fold, now all the Betti numbers are specified by ρ and χ(X).
X. The way to extract has already been described.
The following disccussions explain why some of the mapps are drawn in double lines. For a given three-fold X, there may be multiple Higgs cascades attached to the original branch of the moduli space, and hence multiple modular form Ψ's. The arrow from {X's} to Mod Φ 0 (12 − ρ/2; ρ Λ S ) in (99) is shown in a double line because of that. Those Ψ's should yield the same element in Diff d ′ Λ S . It follows that the difference ∆Ψ must be such that the resulting ∆P 1 (t) is of the form 24(∆d ′ a )t a with (∆d ′ a ) ∈ Z. One may also ask which subset of the diffeomorphism classes Diff Λ S of real six-dimensional manifolds are realized under the restriction that X is a Calabi-Yau three-fold. Because we do not know well the set of such Calabi-Yau three-folds, {X's}, one may think of applying the procedure of assigning (d ′ abc , χ) to an abstract general element Φ ∈ Mod Z 0 (11 − ρ/2; ρ Λ S ). First, it is not true that the resulting d ′ abc can be iterpreted as some d abc ∈ Z and (c 2 ) a ∈ Z, if we just require that Φ ∈ Mod Z 0 (11 − ρ/2; ρ Λ S ) is subject to the inequalities we derived in section 2; see an example in the appendix B.1.3. So, the subset of the Φ's whose d ′ abc backed by integer (d abc , (c 2 ) a ) subject to (a,b) is denoted by [Mod Z 0 (11 − ρ/2; ρ Λ S )] r.mfd . Similarly, not a general element of Ψ ∈ Mod Φ 0 (13 − ρ/2; ρ Λ S ) yields an element of Diff d ′ Λ S (see section 3.2), so those that fall into Diff d ′ Λ S forms a subset denoted by [Mod Φ 0 (13 − ρ/2; ρ Λ S )] r.mfd . We have the map see (99). The set of diffeomorphism classes represented by Calabi-Yau three-folds must be 63 within the image of the map (diff coarse , diff fine ) in Diff Λ S .
The map (diff coarse , diff fine ) can be worked out by dealing with purely mathematical objects. In setting up the relation between the modular forms and diffeomorphism classes, however, we have combined two physics observations under the Heterotic-Type IIA string duality; the parameters d abc and (c 2 ) a in the low-energy effective theory is deterimned i) by the topology of the target space X in Type IIA string compactification, and ii) also by 1-loop integrals in the Heterotic string where the integrands are modular forms. The Rindependence of Ψ in a given Higgs cascade may well be proved purely in math, although physics reasonings are enough; the claim that the difference among the Ψ's from different Higgs cascades of a given original branch disappear in the image of diff d ′ Λ S also relies on physics reasonings.
Discussion 2 For a given X with its modular form Φ, one may specify a curve class C ∈ H 2 (X; Z) and ask whether complex structure of X can be tuned to have singularity of some type R along C (and its resolution is still a regular K3-fibration). In general, there is no guarantee that a modular form Ψ exists in Mod Φ 0 (13 − ρ/2; ρ Λ S ) so that the RHS of (88, 89, 90) reproduce all the input data on the left-hand sides. In such cases, we learn that complex structure cannot be of X cannot be tuned in that way.

Discussion 3
Suppose that a pair of Calabi-Yau three-folds X and X ′ have a diffeomorphism between them, but not a holomorphic one-to-one map. Type IIA string compactifications over X and X ′ form two different branches of moduli space then. Such a pair of branches of moduli space cannot be distinguished by the invariants (diff coarse , diff fine ).
Instead of finding the modular form Ψ for one Higgs cascade of the branch of X and specify diff fine (Ψ), we can specify the subset of [Mod Φ 0 (13 − ρ/2; ρ Λ S )] r.mfd of all the Ψ's of the Higgs cascades attached to the original branch of X. An example of such a pair is discussed in section 3.2.1.
That idea of extracting an invariant of a branch of moduli space is faithful to the way we analyze the moduli space by using the low-energy effective field theory. In practice, however, it is not easy to work out all the possible ways to tune complex structure of a manifold to obtain singularity. A close alternative to the idea of using the set of Ψ's of all the Higgs cascades is i) to think of all the holomorphic curves in H 2 (X; Z), ii) apply the reasoning in Discussion 2 to eliminate some of those curves, and finally, iii) to extract the set of Ψ's for those remaining curves. The latter set of Ψ's contain the former set of Ψ's.
The latter idea detects difference in the Kähler cone, or in the cone of curves. Let f : X → X ′ be a diffeomorphism; if C ∈ H 2 (X; Z) is in the cone of curves of X, but f * (C) is not in that of X ′ , then the modular form Ψ for C may be in the set of Ψ's for X, but not in the the set for X ′ . In the example of section 3.2.2, we discuss this latter set of Ψ's.

Discussion 4
The positive cone (t 2 , t 2 ) > 0 of Λ S ⊗ R may contain multiple chambers separated by walls orthogonal to some elements in Λ ∨ S ; some of those chambers correspond to three-folds with different topology [42]. The modular form Φ remains the same on both sides of the wall, but the integral in (95)  There are not many things we can say with confidence about a symmetry-enhanced branch available on one side of the wall continues to exist on the other side of the wall. We think it is likely, however, if lim cpx.str X has A 1 singularity along a curve C ⊂ X, then a flop transition on X along a curve disjoint from C yields a three-foldd X ′ that continues to have a symmetry-enhanced phase with singularity along C. Even in such cases, the map diff fine depends on the choice of a chamber, because the integrals P 1 and P 3 have singularity along the walls.
Such invariants as diff coarse (Φ) and diff fine (Ψ) are not assigned to branches of Coulomband-hyper moduli space, but for branches of individual chambers-and-hyper modul space.
Discussion 5 For Heterotic string compactifications reviewed in section 2.1.1, the invariants such as the set of Ψ's and the set of Φ's are assigned for (individual chambers of the) branches of moduli space. Their Type IIA dual do not necessarily have a geometric phase, so we may think of Φ in Mod Z (11 − ρ/2; ρ Λ S ) than in [Mod Z 0 (11 − ρ/2, ρ Λ S )] r.mfd . Those ivariants are given in terms of the CFT of the fundamental string, and are well-defined, without relying on Heterotic supergravity approximation, or a geometric phase.
Those invariants beyond (Λ S , Λ T , Φ) generalize the integer n of the (12 + n, 12 − n) instanton number distribution in the case of Λ S = U, and detect difference among branches of Heterotic string moduli space already present at perturbative level (with corrections of order (e 2πis ) ignored); see also [24,11].
Remark: This is a small side remark before closing this section 3.1. The modular forms Φ (R i ) for a chain of symmetry enhancement {0} ⊂ R 1 ⊂ R 2 · · · also contain how many hypermultiplet moduli need to be tuned to have the enhanced symmetries in the chain: This ∆h 2,1 R was used in [11] to distinguish four different branches of moduli space that share the same Λ S = +2 and the modular form Φ (those that are discussed in section 3.2.2). But it is enough to have Φ (R) without ∆h 2,1 R as an invariant of branches of the moduli space.

Examples
Let us see in simple examples how Φ, Ψ and "the set of possible Ψ" as an invariant work in distinguishing different branches of the moduli space.
where n ′ := 2 − 12 −2 d(0) = 2 − b R /6. Here, the component description of t ∈ Λ S = U is that of (194) associated with an obvious null element z ∈ U. The expressions above is for the chamber 0 ≤ ρ 2 ≤ u 2 ; those for the other chamber 0 ≤ u 2 ≤ ρ 2 are obtained by exchanging ρ and u. As we will take a quotient by Isom(U), which includes the ρ ↔ u exchange, it is enough to focus on the 0 ≤ ρ 2 ≤ u 2 chamber in the following.
One can see that after working out details by using the expressions of P 3 and P 1 above. The restriction on the value of b R is from the integrality of d abc 's; once n ′ ∈ Z is imposed, then the condition (a') and (b') are automatically satisfied in this ρ Λ S = U case. Corresponding to the shift s →s + (∆n ′ a )t a with ∆n ′ a ∈ Z is ∆n ′ = 2, ∆b R = −12, which mods out [Mod Φ 0 (12, ρ U )] r.mfd in passing to Diff d ′ Λ S by the map diff fine . The image of the map (diff coarse , diff fine ) must be in Both of ν = 0, 1 of Diff d ′ Λ S are realized by the images of even n ′ and odd n ′ . Only just one element of Diff Λ S (Q) = Z ⊕4 /(ρ ↔ u) × {χ ∈ Z} is in the image of diff coarse , however. It is the element represented by d ρρρ = 2, d uuu = 0, N ρ = 4 + ν, N u = 4 + ν, and χ = −480. The modular forms Φ behind the scene indicates that the diffeomorphism classes realized in the form of Calabi-Yau three-folds are significantly less. 65 As is well-known, there are Calabi-Yau three-folds for both of even n ′ and odd n ′ . Think of a Weierstrass-model elliptic fibration over the Hirzebruch surface F n that is Calabi-Yau, and denote it by X (n) ; we denote by D 7 the zero-section divisor of X (n) . The base surface F n is a P 1 -fibration over P 1 IIA , where the D f is the P 1 -fibre class, and the two sections denoted by D + and D − have self-intersection +n and −n, respectively. The pull-back of the divisors D f , D + , and D − to X (n) are denoted by D s , D 3 , and D 4 , respectively. D 3 ∼ D 4 + nD s . 65 In the case of Λ S = U , this is not surprising, because a simple argument [43,44] shows that X (n) with n = 0, 1, 2 (explained shortly in the main text) are all the possibilities. First, a U -polarized K3-fibration X over P 1 IIA must be an elliptic fibration over a surface, which itself must be a P 1 -fibration over P 1 IIA . So, the base surface is F n ; once the base is fixed, then the Weierstrass model with f in O(−4K) and g in O(−6K) determines X (n) without an extra topological freedom. Those with n > 2 are ruled out, because the lattice Λ S is strictly larger than U for n > 2.

Some of the triple intersection numbers are
which means that we can use {D s , (D 7 + D 3 ), D 3 } as a basis of H 2 (X (n) ; Z) in a way that Now, think of a symmetry-enhanced limit lim cpx str X (n) of this X (n) so a singularity of type R emerges in the fibre of D − ⊂ F n . Then (f R ) nonexp = s =s. The modular form Ψ for this Higgs cascade must be the one for n ′ = 2 − 12 −2 d(0) = n, becausesρu + P 3 (t; n ′ = n)/3! and 24s + P 1 (t; n ′ = n) reproduces the toplogical invariants (112, 113) of X (n) . In particular, the set Diff d ′ Λ S ≃ Z/2Z is realized by the diffeomorphism classes of X (n=0) ∼ X (n=2) and X (n=1) .
It is possible to find a broader class of symmetry-enhanced limits of X (n) by using Ftheory. Let C ′ be H 2 (F n ; Z) represented by an irreducible curve (we call it an irreducible curve class). Now, choose f and g of the Weierstrass model y 2 = x 3 + f x + g so where c ∈ Γ(F n ; O(−2K Fn )), a ∈ Γ(F n ; O Fn (−4K Fn − C ′ )), and σ ∈ Γ(F n ; O Fn (C ′ )) [45]; in this limit, the three-fold X (n) has a singularity of type R = A 1 along a curve C in the fibre of the σ = 0 curve in F n . Think of C ′ 's of the form 66 C ′ ∼ D − + mD f labeled by m ∈ Z, so that C ′ ·D f = 1 in F n , and C ·D s = +1 in X (n) . For this class of symmetry-enhanced limits (Higgs cascades), we have 67 (f R ) nonexp = s + m(ρ + u) =s. Therefore, we find that the modular 66 We have discussed the invariant Ψ for the Higgs cascades with C ′ = D − (m = 0); the choice C ′ = D + (m = n) corresponds to placing the probe gauge group R in the other weakly coupled E 8 in the Heterotic language, but there are more varieties (m) in the symmetry-enhancement limits (e.g., [43]). 67 In this construction of lim cpx str X (n) , the curve C of A 1 singularity is along (x, y) = (h, 0) in the elliptic fibre, so it does not touch the zero section divisor D 7 . So D 7 · C = 0. form Ψ of this Higgs cascade must be that of n ′ = n − 2m (b R /6 = 2m + 2 − n), which we find by requiring that the topological invariants (112, 113) of X (n) must be reproduced bỹ sρu + P 3 and 24s + P 1 (t), respectively.
The class C ′ is represented by a curve when m ≥ 0; the curve is not irreducible for a choice m = 1 in the case of n = 2, however, because the divisor D − + D f has D − as a base locus then. There is also an upper bound, m ≤ 8 + 4n; when the divisor class −4K Fn − C ′ is not effective, X (n) is singular in the fibre of any points on F n because f = −3c 2 and g = 2c 3 . The effectiveness is translated into the upper bound. So, we have found a class of Higgs cascades attached to the branch of the moduli space of X (n) whose invariants are The difference among Ψ's for one given X (n) is precisely of the form we expected in Discussion 1. The set of Ψ's of X (n=2) and X (n=0) are not the same, however, reflecting the fact that this pair of three-folds have a diffeomorphism but not a holomorphic one-to-one map between them, and the Kähler cones are not identical when H 2 (X (2) ; R) and H 2 (X (0) Z) are identified by using the diffeomorphism between them. One will wonder if there are other symemtry-enhancement limits of X (n) . At least we can rule out cases where singularity of type R emerges along a curve C satisfying C · D s = 1 and C · D 7 = 0 (Discussion 2). To see this, suppose that there is such a limit. Then (f R ) nonexp = s + m ρ ρ + m u u =s with m ρ = m u ; no choice of Ψ from [Mod Φ 0 (12, ρ U )] r.mfd for the polynomials P 3 (t) and P 1 (t) can reproduce F cub and F 1 appropriate for X (n) , and hence the assumption must be wrong. 68 We do not have an argument to rule out that there are other limits of X (n) for symmetry-enhancements with C · D s = C · D 7 = 0 that cannot be obtained in the form (114).
There is at least one pair of three-folds in the same diffeomorphism class but a holomorphic one-to-one map between them may or may not exist, also in the Λ S = +2 case. The threefolds denoted by M n +2 with n = 2, 1, 0, −1 in [11] are in the diffeomorphism classes with n 1/2 = 0 for all of them, and b R + 6Z = 0 +6Z , 2 +6Z , 4 +6Z , and 0 +6Z , respectively; the pair (with m ∈ Z ≥0 for n ≥ 0), we have f R = s + m(t a=1 ). The dictionary (120) with n 1/2 = 0 and −δn ′ a=1 = m reads 2n = 4 − b R + 6m. So, the modular form Ψ is that of b R = 6m + 4 − 2n, if it is possible to tune complex structure, and the corresponding Higgs cascade exists for this class F . The easy-going version of the set of Ψ's (using the cone of curves) is A Look at R-dependence The analysis up to this point relied on Ψ of a Higgs cascade as a whole, so it is independent of the choice of the symmetry R. Now let us use Φ and look into the information which type of singularity may develop in a given manifold.
Suppose that a three-fold M +2 is in the diffeomorphism class characterized by n 1/2 and b R + 6Z. Unless (b R + 6Z) = 0 ∈ Z/6Z, any Higgs cascade attached to the branch of Type IIA compactificataion on M +2 do not lead to an enhancement of singularity of type R = E 6 (or higher) whose resolution M +2 has a regular K3-fibration.
As a test for whether an enhancement of singularity of a given type R, one may ask whether an appropriate modular form Φ (R) can be found. For example, take Λ S = +2 ⊕ A 3 [−1]. Then the modular form Φ of the hypothetical branch of Type IIA on M is in the form of 73 Φ = −68 + 6n 1 + n 2 + n 3 72 parametrized 74 by n 1,2,3 ∈ Z and n 0 = −2. This parametrization is for an obvious reason: Φ = e 0 ⊗ e 0 (−2 + (324 − 56n 1 − 8n 2 − 6n 3 )q + · · ·) + e 0 ⊗ e 1 n 1 q 1/4 + · · · + (e 1 + e 3 ) ⊗ e 0 n 2 q 5/8 + · · · + (e 1 + e 3 ) ⊗ e 1 (16n 1 − 2n 2 + 8n 3 + 96)q 7/8 + · · · + e 2 ⊗ e 0 n 3 q 1/2 + · · · + e 2 ⊗ e 1 (8 + 10n By the discussion in section 2.2, the coefficients n 1,2,3 ∈ Z need to satisfy n 1 , n 2 ≥ 0, and n 3 , Φ and Ψ is given by Φ = θ A 3 · Φ and Ψ = ∂ S θ A 3 · Φ. Then we have For n 1/2 = 0 and b R = 0, 2, 4, for example, there are solutions (n 1 , n 2 , n 3 ) to (128). We do not have to rule out a possibility that there exists tuning of complex structure of M n=1,0 , and a 0 , b 0 , c 0 and d 0 are regarded sections of appropriate line bundles of the base P 1 ; those line bundles should have the degree specified in Table 5 for construction of M n +2 . More details are found in [11]. Singularity of type R = E 7 develops in M n +2 for any one of n = 2, 1, 0, −1, when all of the sections a 2,··· ,6 , b 2,··· ,4 , and c 1,2 are set to zero. The singularity is along the curve 76 (x 1 , x 4 , x 3 ) = (0, 0, 0), which is in the class It is only in the case of n = 2, however, that the K3-fibration in the three-fold M after the tuning remains regular; for n = 2, the coefficient d 0 is a section of a line bundle of the base P 1 of positive degree. The section d 0 vanishes at some points in the base, and the three-fold M has psuedo-Type II degeneration in the fibre of those points. So, this failiure in finding a tuning of complex structure for M n=1,0 +2 is consistent with b R + 6Z = 2, 4 + 6Z that does not allow interpretation as a 1-loop beta function of R = E 7 gauge group. On the other hand, singularity of type R = A 3 enhances along C R = Σ (n) B by setting a 3,··· ,6 and b 3,4 to zero, and we can see by using toric data (just like in [11]) that their M has a regular K3-fibration, as announced earlier.
Given the diffeomorphism f : M . For singularity of type R = E 7 for the latter, we can tune a ′ 4,··· ,0 , b ′ 1 , b 0 , a 1 , b 1 , and a 2 to zero, for example, because the hypersurface equation is ξ 2 1 + b ′ 2 ξ 3 3 ξ 2 + a 3 ξ 3 2 ≃ 0 near the curve 77 ξ 1 = ξ 3 = ξ 2 = 0; this is not difficult to see by using toric language. So, this Higgs cascade for M B . Those two symmetry-enhancement branches are still different, because a 3 is a section of O(3) on P 1 IIA in the former, whereas d 0 that of O(0) on P 1 IIA in the latter, and the three-fold M for the former does not have a regular K3-fibration. There is still a pair of branches of Heterotic-IIA dual 76 We used inhomogeneous coordinates x 1 = X 1 /X 3 2 , x 4 = X 4 /X 2 , and x 3 = X 3 /X 2 . 77 Now, the inhomogeneous coordinates are ξ 1 = X 1 /X 3 4 , ξ 2,3 = X 2,3 /X 4 .
vacua that cannot be distinguished by Λ S , Λ T , Φ, and the set of Ψ's.
The image of the map (diff coarse , diff fine ) restricts possible diffeomorphism classes of real six-dimensional manifolds that can be realized by Calabi-Yau three-folds with Λ S -polarized regular K3-fibrations. This method has been applied only for Λ S = U and +2 , and found that the image of diff coarse is much smaller than the set Diff Λ S (Q) for both Λ S , and the image of diff fine is all of Diff d ′ Λ S for (d ′ , χ) in the image of diff coarse . One can find out whether that remains to be true for various different Λ S 's, by working out the images of [Mod Z 0 (11 − ρ/2; ρ Λ S )] r.mfd and [Mod Φ 0 (13 − ρ/2 : ρ Λ S )] r.mfd . A few theoretical questions can also be put down. There are two possibilities for a pair of modular forms Φ and Ψ that are not in the subsets [Mod Z 0 (11 − ρ/2; ρ Λ S )] r.mfd and [Mod Φ 0 (13 − ρ/2; ρ Λ S )] r.mfd . One is that there are more theoretical constraints of string theory that we failed to capture in sections 2 and 3 and such a (Φ, Ψ) is in conflict with those constraints. The other is that such a modular form (Φ, Ψ) is for a branch of moduli space whose Type IIA description does not involve a geometric phase. It remains to be an open question how to determine the boundary between those two possibilities in the space of (Φ, Ψ).
We have already seen that the image of the map (Diff coarse , Diff fine ) is small compared with the set Diff Λ S . But not all the diffeomorphism classes of real six-dimensional manifold in the image are guaranteed to be realized as a Calabi-Yau three-fold. By taking advantage of large data base of topology of Calabi-Yau three-folds, one may try to get the feeling how much fraction of the diffeomorphism classes in the image of (Diff coarse , Diff fine ) are indeed guaranteed to be realized by Calabi-Yau three-folds. Such a study may provide hints in considering the "determining the boundary" issue above.
In pure mathematics literatures, some inequalities on topological invariants of Calabi-Yau three-folds have been derived (e.g., [47, §2]). It is beyond the scope of this article to study how those inequalities are related to the the bounds that we discussed in this article. Also in physics approach, various integer parameters are likely not just bounded from below, but also from above (for maintaining strictly a given lattice Λ S ). But we have not given enough thoughts on how this intuition is related or unrelated to the bounds and classifications discussed in this article.
A few ideas are also available in improving the effort of introducing invariants for classification of the branches of moduli space of Het-IIA dual vacua. We just introduced the idea of using the set of Ψ of all the Higgs cascade as an invariant of a branch (than just using its image by Diff fine ) in this article; more knowledge in the cone of curves (and tuning of complex structure to have certain singularity along a curve class) would make it possible to compute the set of Ψ's for general Λ S , not just for Λ S = U. Also, a part of the idea (using ∆h 2,1 R ) for invariants in the case Λ S = +2 in [11] has been incorporated as a part of Φ (R) for general Λ S in this article, but a bit more idea beyond ∆h 2,1 R in [11] has not been generalized to other Λ S 's, or brought into the language of world-sheet CFT in this article.
Finally, as a reminder, K3-fibration of a Calabi-Yau three-fold was assumed to be regular in this article. Classifications of Calabi-Yau three-folds with a non-regular K3-fibration (and their Heterotic duals) should be considered separately from this article. Furthermore, we also set some other technical liminations in section 2.1 on the class of Heterotic-IIA dual vacua to study in this article. Structure of branches of all those vacua is yet to be figured out.

Metaplectic Group Metaplectic group Mp(2, Z) is defined by
f is a holomorphic function in the upper half plane H ⊂ C. f specifies the choice of sign ± √ cτ + d, so Mp(2, Z) is a double covering of SL(2, Z). The multiplication of two elements in the group Mp(2, Z) is defined by The gruop SL(2, Z) is generated by two elements which satisfy S 2 = (ST ) 3 = −1. Similarly, Mp(2, Z) is generated by Vector valued modular form Let k ± ∈ 1 2 Z, and ρ : Mp(2, Z) → GL(V ) be a representation on a vector space V . A real analytic (but not necessarily holomorphic) function F : H → V is called a (vector valued) modular form of weight (k + , k − ) and type ρ if F satisfies the modular transformation laws A modular form F (τ,τ ) is said to be almost holomorphic, if it is a polynomial of (1/τ 2 ) with τdependent coefficients and has finite values at all the cusp points. The vector space of almost holomorphic modular forms of weight (k + , k − ) and type ρ is denoted by Mod((k + , k − ), ρ).
Here q = e 2πiτ . γ 2 /2 denotes the quadratic form over G M . We define Mod 0 (k, ρ) by imposing cusp condition at isotropic (but nonzero) γ: In this article, the Fourier coefficients Dimensional formula The dimension of the vector space Mod(k, ρ M ) is determined by the formula in the case of k > 2 and k/2 ≡ sgn(M)/4 mod Z [17]: where d := dim C C[G M ] + , and α(ρ M (g)) := d i=1 β j when {β j }'s are the complex phases of the eigenvalues of the representation matrix, ρ M (g)| C[G M ] + ∼ E(diag(β j )), set within the range 0 ≤ β j < 1. The restriction k > 2 of this formula is due to the fact that dim C Mod(k, The dimension of the subspace Here Eisenstein series E 4 (τ ) and E 6 (τ ) are modular forms of weight 4 and 6, respectively, for SL(2; Z), but E 2 (τ ) is not modular (it is Mock modular). The space of scalar valued modular form can be identified with the polynomial ring C[E 4 , E 6 ]. The q 0 -term vanishes in the combination where Ramanujan-Serre derivative and Rankin-Cohen bracket For a modular form F ∈ Mod(k, ρ), the Ramanujan-Serre derivative ∂ S : Mod(k, ρ) → Mod(k + 2, ρ) is defined by Vector-valued modular forms and their Ramanujan-Serre derivatives can be multiplied to produce yet another vector-valued modular form. The Rankin-Cohen bracket [F, G] n of a pair of modular forms F ∈ Mod(w F , ρ F ) and G ∈ Mod(w G , ρ G ) and n ∈ N is Lattice-polarized Jacobi forms and vector valued modular forms For a positive definite even lattice M, and k ∈ 1 2 Z, a holomorphic function φ : where the bilinear form (−, −) of M has been extended linearly to M ⊗ Z C. The classical definition of a weight-k index-m Jacobi form is regarded as that of a weight-k index-MJacobi form with the lattice M = +2m . To any Jacobi form φ(τ, z) of weight-k and index-M, one can assign a vector valued modular form of weight-(k − 1/2) and type ρ ∨ M . That is through and this is one-to-one. The modular form x e x f x (τ ) is holomorphic at the cusps if and only if the expansion φ(τ, z) = n∈Z λ∈M ∨ c(n, λ)q n e 2πi(λ,z) has support in n ≥ (λ, λ)/2. See [48,49] for more information.

A.2 Explicit Examples
Some of the modular forms used in the main text are written down explicitly here.
All the details so far in the appendix A.2.1 are used in section 2.3.1.
In the n = 7 case, there must be one linear relation among the 8 coeffcieints n |γ| with |γ| ∈ G S /{±1}, because the dimension formula indicates that the vector space Mod(21/2, ρ +14 ) (and Mod 0 (21/2, ρ +14 ) is of 7-dimensions. So, an expression of the form χ(X) = −c 0 (0) = |γ| κ |γ| n |γ| is not expected to be unique. Indeed, we can think of two choices of here, the lattice E 8 is expressed as the abelian group Z ⊕8 ⊕ (1/2, · · · , 1/2) + Z ⊕8 , and the intersection form on this is given by setting (e ′ i , e ′ j ) = +δ ij for the generators e ′ i of the i-th factor of Z ⊕8 . The lattices L[−1] ⊂ E 8 are worked out for both chices, and it turns out that the corresponding lattice theta functions are not the same. Two expressions for χ is obtained from the relation (48), and are shown in Table 2.
Multiple φ ∈ Mod 0 (3 + ρ/2, ρ × Λ S )'s result in multiple expressions for χ(X) in terms of {n γ | ± γ ∈ G S /{±1}}'s, as we have seen above in the Λ S = +14 case. When we form a linear combinations of such φ's so that the leading Fourier coefficient vanishes (i.e., a cusp form), then we obtain linear relations among {n γ }'s discussed in section 2.3.2. Multiple expressions for χ(X) are consistent because of the linear relations among those low-energy BPS indices.

A.2.3 The First Derivative of Some Lattice Theta Functions
For any one of ADE types, R, the coefficients a are all integers; R also stands for the positive definite lattice of type R. To see that they are integers, we use Lemma 6.1 and 6.2 of [50] for the lattice R. We have a formula a (R) for any a ∈ R ⊗ R. By using any root in R as the vector a, we see that any pair b and −b in the sum 79 contributes by The values of a (R) δ (ν) for ν < 1 are recorded here.

B Evaluation of Integrals by Lattice Unfolding
We need evaluation of integrals ∆ grav in (4, 3) and ∆ R in (61), just like in Ref. [52,7]. The evaluation method in [52] (with extension by [7]), however, is applicable immediately only for lattices Λ S of the form Λ S = U ⊕ W with a signature (0, ρ − 2) lattice W . Instead, we 79 b = 0 does not form a pair, but a 0 (0) = 0 obviously. rely on the evaluation method presented in [9]; here, we describe the outline of the evaluation method in [9], and quote results relevant to this article (a review is also found in [51, §3] Those integrals with M = Λ S are of immediate relevance, because where we used the right mover and left mover momenta p = (p R , p L ) in the Heterotic description to refer to a choice of v ∈ Gr( Λ S ). We can just take ν min = −1 81 and k max = 1, and both ∆ grav and ∆ gauge are within the class of integral (189, 190) introduced above.
In evaluating the integral I M (v, F ), Ref. [9] relates it to an integral in the same class, but with a lattice M ′ of b + (M ′ ) = b + (M) − 1, v ′ ∈ Gr(M ′ ), and F ′ characterized 82 as in (190) 80 When the integral shows some divergence, we understand the integral as reguralized by subtracting the integrand by const × τ b + /2−2 2 (equivalent of integrating in IR degrees of freedom) [52] or replacing dτ 2 /τ 2 2 → dτ 2 /τ 2+s 2 as in [9]. 81 We have to treat cases ν min < −1 when using the embedding trick in the appendix B.3. 82 The value of k max does not change under this reduction, but the value of ν min may not be the same as before if one applies the embedding trick (see section B.3).
with M replaced by M ′ , as we review in the appendices B.1 and B.3). This procedure is called the lattice unfolding method. At the end, we are left with evaluating integrals of the form (189, 190) for a negative definite lattice M ′′ . Since θ M ′′ (τ ) is holomorphic, the integral I M ′′ (F ′′ ) can be regarded as that for 0-dimensional lattice: . This type of integral can be evaluated by simple partial integrals. When F (τ ) = φ(τ )Ê m 2 with φ some scalar valued modular form of weight −2m, the formula is [53,52]

B.1 Lattice Unfolding Formula
When the lattice M with a signature (b + , b − ) has a nonzero element z with norm z 2 = 0, a lattice M ′ : is determined by the (b + − 1)-dimensional vector subspace of the b + -dimensional positive definite subspace corresponding to v orthogonal to z. Discussions in [9] rewrite I M (v; F ) as a sum of I M ′ (v ′ , F ′ ) for an appropriately chosen F ′ and additional terms that are completely determined in terms of v and F . Since U[−1] has a nonzero null vector, evaluation of I Λ S with Λ S = U[−1] ⊕ Λ S can be reduced to that of I Λ S . When the signature (1, ρ − 1) lattice Λ S also has a nonzero null vector, we can again reduce I Λ S to an integral for smaller lattice [z ⊥ ⊂ Λ S ]/Zz of signature (0, ρ − 2). Since this latice is negative definite, we can apply the formula (192).
When the lattice Λ S does not have a nonzero null vector, we can use the embedding trick (explained in appendix B.3) to think of I Λ S (v ′ , F ′ ) as I M (ṽ, F ) for some lattice M and some modular form F . Here M contains Λ S as a sublattice and b + ( M ) = b + (Λ S ), and it can be chosen so that it has a nonzero null element. In this way, the integrals for all Λ S can be reduced to integrals with some negative definite lattice.
Here (w · t) = (w, t 1 ) + i |(w, t 2 )| and Li 3 (e iz ) = Li 3 (e iz ) + Im(z)Li 2 (e iz ). The modular form F is used as F ′ in the first term on the right hand side. The details of the second and the third line are necessary when working out the matching calculation (84), but are not relevant to the matching (88, 89, 90) that we need to discuss in this article. The integral I Λ S (v, F ) as a function of v ∈ Gr( Λ S ) has singularity only along real codimension (b + ( Λ S ) = 2) subspace, whereas I Λ S as a function of v ′ ∈ Gr(Λ S ) has singularity ("wall-crossing") along real codimension (b + (Λ S ) = 1) walls; I Λ S (v, F ) has logarithmic singularities at the locus ℧(w) = 0 for some w ∈ Λ ∨ S , while I Λ S (v ′ , F ) has conical singularities at (w, t 2 ) = 0 for some w ∈ Λ ∨ S . This implies that there is (partial) cancellation of singularity between I Λ S (v ′ , F ′ ) and the third line of (193) so the sum of them remains non-singular at the codimension-1 walls (except the codimension-2 points). 84 See section B.2 for the wall-crossing formula of I Λ S . For more information, see [9, §6].

B.1.2 I Λ S for Λ S with a Null Element
In the rest of the appendix B.1, we discuss evaluation of I Λ S (v ′ , F ) for Λ S that has a nonzero null element z. Any Neron-Severi lattice Λ S of a K3 surface with ρ ≥ 5 are known to have such a nonzero null element, and the same is also true for lattices Λ S = U ⊕ W ′ for some even lattice W ′ with signature (0, ρ − 2).
Let z be a nonzero primitive null element of Λ S , and N be the GCD of the values of (z, λ) ∈ Z of λ ∈ Λ S . We then choosez ∈ Λ S so that (z,z) = N, and a free abelian subgroup W ⊂ Λ S such that z ⊥ W and Λ S ∼ = ab Zz ⊕ Zz ⊕ W ; here ∼ = ab stands for an When one expand the rhs, there appear some terms with ρ 2 in the denominator, but they should cancel out because of consistency with wall-crossing behavior of I Λ S (see [9,Thm. 10.3]). 88 After this cancellation is taken care of, the integral I Λ S (v ′ , F ) × |t 2 | 1+2kmax should become a chamber-wise homogeneous function of the components of t 2 of degree (1 + 2k max ).
The difference among Ψ's that is irrelevant in Diff d ′ Λ S after the map diff fine is ∆n ′ ∈ 2Z and (b,a 2 )>0 b∈W ∨ ∆d b (b 2 /2)b ∈ 24W ∨ . So, the set Diff d ′ Λ S is Z/2Z for any W .

B.2 Wall-crossing Behavior
We give some comments on wall-crossing behavior. See section 6 of [9] for details. I Λ S (v, F ) as a function of v = Rt 2 ∈ Gr(Λ S ) shows conical singularity at (w, t 2 ) = 0 for some w ∈ Λ ∨ S . 91 These points in Gr(Λ S ) forms real-codimension-1 walls that separate Gr(Λ S ) into many chambers; I Λ S is analytic in each chamber but shows jump from its analytic continuation when t 2 crosses a wall. Let I Λ S (v, F ; C) be the analytic continuation of the restriction of I Λ S (v, F ) to a chamber C. Then the difference of them for two different chamber C 1 and C 2 is given by Here (w, C) > 0 means (w, t 2 ) > 0 for any t 2 ∈ C. Note that the difference as shown above is a polynomial of v. Using this fact, [9,Thm. 10.3] shows that I Λ S (v, F ) gives chamber-wise polynomial of v with degree at most 3 in our case (in fact, without even degree terms). As a corollary, we obtain wall-crossing formulas for P 1 (t) and P 3 (t), that are directly relevant to the topological invariants of X IIA : 1 3! P 3 (t; C 1 ) − 1 3! P 3 (t; C 2 ) = w c w (w 2 /2) 3! (w, t) 3 + (t, t) 2 d w (w 2 /2) 12 (w, t) , (208) P 1 (t; C 1 ) − P 1 (t; C 2 ) = − w 2c Ψ w (w 2 /2)(w, t).
The summation over w is same as in (207).

B.3 Embedding Trick
Even when Λ S has no nonzero null elements, one can evaluate the integral I Λ S (v, F ) by embedding Λ S into a larger lattice(s) M with a nonzero null element so that I Λ S (v, F ) is equal to (linear combination of) IM (v, G) for suitable modular form G. One can then apply the lattice unfolding method to compute it. This method is called "embedding trick" in [9]. In this section, we explain the original embedding trick and its slight modifications. We also treat concrete example (the case Λ S = +2 ).
The original embedding trick is as follows [9,Thm. 8.1]. To begin with, choose a pair of (negative definite) Niemeier lattices M 1 , M 2 with different number of roots: r 1 − r 2 = 0, r i := the number of roots in M i .
Then we obtain this is because the rhs is a scalar-valued modualr form of weight 0 with the term q −1 vanishing and the coefficient of q 0 normalized. By inserting this equation to the integrand of I Λ S (v, F ), we get where G = F [(r 1 − r 2 )η 24 ] −1 and v ∈ Gr(Λ S ) is regarded as the same positive definite (b + = 1)-dimensional subspace v ∈ Gr(Λ S ) ⊂ Gr(Λ S ⊕ M i ). Since Λ S ⊕ M i has U as direct summand ([9, §8]), we can apply the lattice unfolding formula to evaluate the right hand side.
There are a few points to keep in mind. First, G has a pole of higher order at cusps than F (i.e. ν min (G) = ν min (F ) − 1 = −2). Second, in many cases (e.g., the appendix B.3.1), we need to care about wall-crossings of I Λ S ⊕M i for walls between v and the region in Gr(Λ S ⊕M i ) where the unfolding formula (197) is valid.
This embedding trick adds Niemeir lattice to the original lattice and increases the rank of lattice by as many as rank M i = 24; sometimes it is troublesome to handle such big lattice and consider all the relevant wall-crossings. But actually, it suffices to add smaller lattice: choose sublattices N i ⊂ M i such that N 1 ∼ = N 2 ( ∼ =: N). Then by decomposing the theta function of M i asθ we can rewrite (211) to {h δ } is a modular form of type ρ N . Inserting this equation to the integrand of I Λ S (v, F ), we get Here G is a modular form of type ρ Λ S ⊕N . Note that also in this case ν min (G) is less than ν min (F ) = −1 (but still ν min (G) ≥ −2). After all, an important point for embedding trick is to find a decomposition for some lattice N and modular form h, not necessarily associated with Niemeier lattices. There are many choices. For example, let 92 N = −2m , and ϕ(τ, z) be a weak Jacobi form of weight 0 and index m, satisfying ϕ(τ, z = 0) = 0. We can theta-expand ϕ using a modular form h: Since a modular form of weight 0 and holomorphic at cusps is necessarily just a constant, setting z = 0 in the above equation leads to the required decomposition (216) up to some normalization. If m ≥ 2, there are multiple choices for such ϕ. So, even for a given lattice Λ S , there are multiple different ways to use the embedding trick, I Λ S (v, F ) = i IM i (v, G i ). There is no unique choice forM = Λ S ⊕ N; even for a given N, the choice of G is not necessarily unique. One can use just any version of the embedding trick, so practical calculation of one's interest is easier.
As a practical implementation of the embeding trick, we choose N = −2 , soM = +2 ⊕ −2 , and GM = Z 2 × Z 2 . We will use the notation Z 2 ∼ = {0, 1} (instead of {0, 1/2}) 92 We can choose N to be rank-1 for any Λ S . This is because Λ S has necessarily an element of norm 2m for some positive integer m; the lattice Λ S ⊕ N with N = −2m has a nonzero null element then. 93 When F is holomorphic (e.g. Ψη −24 ), I +2n (F ) can be also calculated by using [9, Cor 9.6], which uses Zagier's modular form and Stokes' theorem.