The Cremmer-Scherk Mechanism in F-theory Compactifications on K3 Manifolds

It is well understood --- through string dualities --- that there are 20 massless vector fields in the spectrum of eight-dimensional F-theory compactifications on smooth elliptically fibered K3 surfaces at a generic point in the K3 moduli space. Such F-theory vacua, which do not have any enhanced gauge symmetries, can be thought of as supersymmetric type IIB compactifications on P1 with 24 (p,q) seven-branes. Naively, one might expect there to be 24 massless vector fields in the eight-dimensional effective theory coming from world-volume gauge fields of the 24 branes. In this paper, we show how the vector field spectrum of the eight-dimensional effective theory can be obtained from the point of view of type IIB supergravity coupled to the world-volume theory of the seven-branes. In particular, we first show that the two-forms of the type IIB theory absorb the seven-brane world-volume gauge fields via the Cremmer-Scherk mechanism. We then proceed to show that the massless vector fields of the eight-dimensional theory come from KK-reducing the SL(2,Z) doublet two-forms of type IIB theory along SL(2,Z) doublet one-forms on the P1. We also discuss the relation between these vector fields and the"eaten"world-volume vector fields of the seven-branes.


Introduction and Summary
Ever since its discovery, F-theory [1][2][3] has played a prominent role in understanding the landscape of string vacua. F-theory provides a very rich, if not the richest, range of string vacua in various dimensions. 1 This "versatility" comes from the fact that F-theory provides a framework to work with strongly coupled string -or to be exact, type IIB -backgrounds. An F-theory vacuum can be thought of as a compactification of a twelve-dimensional theory on an elliptically fibered manifoldM over some baseS. What this background is actually describing is a type IIB compactification on the manifoldS with a varying axio-dilaton profile -the value of the axio-dilaton is encoded in the complex structure of the elliptic fiber.
The strongly coupled nature of F-theory, however, makes it difficult to study global Ftheory backgrounds directly from the point of view of type IIB string theory. There are many different approaches to understand these vacua. One approach is to use string dualities with M-theory or heterotic string theory [1][2][3]. Another is to study weakly coupled "orientifold" limits [5][6][7] of F-theory vacua. Yet another is to study local backgrounds to gain insight into global backgrounds [8][9][10][11]. By now there are many aspects of F-theory that are well understood based on these approaches. Some features, however, remain unclarified from the point of view of type IIB string theory.
A subject that begs for better understanding is abelian gauge symmetry. For example, the abelian gauge symmetry of eight-dimensional or six-dimensional F-theory backgrounds can be deduced using F-theory/M-theory duality. Its interpretation in the original type IIB framework, however, has not been explored extensively. Let us elaborate the issue with K3 compactifications of F-theory, which is the subject of this paper.
The simplest F-theory backgrounds are eight-dimensional -they come from compactifying the theory on an elliptically fibered K3 manifold with a section. When the K3 manifold only has I 1 singularities, these backgrounds describe type IIB compactifications on P 1 with 24 (p, q) seven-branes. K3 compactifications of F-theory were thoroughly investigated from -and arguably even before [12] -the birth of F-theory [1,5,[13][14][15][16][17][18] and are very well understood based on the aforementioned methods. In particular, these compactifications are dual to T 2 compactifications of heterotic string theory, which are perturbative string vacua. The eight-dimensional F-theory compactification has 20 vector fields in its massless spectrum at a generic point in the F-theory moduli space, 18 of which belong to the vector multiplets. It is understood that these vector fields are related to the world-volume vector fields of the 24 seven-branes present in the background. The precise relation between the massless vector fields of the 8D theory and the vector fields living on the world-volume of the seven-branes, however, has not been explored further. For example, while qualitative explanations on the discrepancy between the number of the branes and the number of vector fields in the eightdimensional theory have been given [1,13], these arguments have not been made very sharp. In this paper, we expand on the idea of [13] on how the vector field spectrum of F-theory compactified on K3 can be obtained. In particular, we take the point of view that these backgrounds are type IIB supergravity compactifications on a P 1 with seven-branes in it. 2,3 We focus on the interaction of the bulk two-form fields of the type IIB theory and the worldvolume gauge fields, ultimately showing that the gauge degrees of freedom are eaten by the tensor fields through the Cremmer-Scherk (CS) mechanism 4 [23].
The Cremmer-Scherk mechanism is a generalized version of the Stückelberg mechanism to the tensor/vector field pair. Let us first review the Stückelberg mechanism before we describe its generalized version. An abelian vector field A µ can become massive by coupling to a scalar Stückelberg field φ by (1.1) The gauge symmetry of the theory is given by and hence the Stückelberg field can be gauged away. In the end, the degrees of freedom of the scalar field are eaten by the gauge field -one is left with one massive vector field in the theory.
One can readily generalize this mechanism for tensor-vector interactions. That is, given a two-form field B µν and vector field A µ , the two-form field becomes massive by the covariant coupling given the gauge symmetry Now the vector is "eaten" by the tensor field -this is the Cremmer-Scherk mechanism. The tensor-vector interaction (1.3) and the gauge symmetry (1.4) is a familiar one -it is precisely the way gauge fields living on branes interact with bulk tensor fields. The two-forms of the type IIB theory, which form a doublet under the global SL(2, Z) action of the theory, couple to world-volume gauge fields in this way. In this paper, we show that for F-theory K3 compactifications -type IIB compactifications on P 1 with 24 seven-branes -all the 24 world-volume gauge fields are "eaten" by the two-form fields. More precisely, we find 24 linearly independent SL(2, Z) doublet gauge transformations where a = 1, · · · , 24. Here, M, N /µ, ν are ten/eight-dimensional indices, respectively, while z andz denote the coordinates on the internal P 1 manifold. The I is an SL(2, Z) index. We have used i = 1, · · · , 24 to index the branes, while using (z i ,z i ) and (p i , q i ) to denote their positions and brane charges, respectively. A i is the gauge field living on the i-th brane. Hence due to the CS gauge symmetry of the system, we can work in a "unitary gauge" where the vector degrees of freedom are pulled from the branes into the "bulk." Although the world-volume gauge fields are eaten by the tensor fields through the CS mechanism, it turns out that there are still massless vector fields -in fact, 20 of themin the eight-dimensional effective theory. These vector fields come from KK-reducing the SL(2, Z) doublet two-form along SL(2, Z) doublet one-form zero modes on the compact P 1 : ξ k,I m a k µ .
(1. 6) Here, m is the two-dimensional index along the compact P 1 direction, while we have used k to enumerate the zero modes. These zero modes are harmonic along the compact directions, i.e., dξ k,I = 0 , d * M IJ ξ k,J = 0 , (1.7) while they must exhibit certain monodromies around seven-brane loci. Here, * denotes the Hodge dual with respect to the metric of the base manifold, while M IJ is a SL(2, Z) covariant Hermitian metric which depends on the axio-dilaton. We count the number of these zeromodes by relating them to elements of the cohomology group of a certain sheaf living on the baseS of the elliptic fibration.
To introduce this sheaf, let us review the F-theory backgrounds at hand in more detail. As before, let us denote the 24 seven-branes as B i with i = 1, · · · , 24. From the point of view of the K3 geometry, these branes sit at the loci of the base P 1 where the elliptic fiber degenerates. Picking an "A-cycle" α and a "B-cycle" β along the fiber, we can determine the type of brane sitting at B i . 5 B i is a (p i , q i ) seven-brane when the cycle p i α + q i β degenerates at the brane locus. Now the A and B-cycle exhibit monodromies around the brane locusthese are precisely the monodromies that SL(2, Z) covariant fields must exhibit around the branes in order for the field values to be well-defined.
We see that one way to view the K3 manifold is to see it as a family of elliptic curves parametrized by the base manifoldS. From this point of view, the harmonic one-forms (1.7) represent elements of the first cohomology group of "the sheaf of local invariant one-cycles" 6 living on S. Hence, the dimension of this cohomology group, h 1 (S, j * H Q ), can be identified with the number of linearly independent doublet harmonic one-forms. Cohomology groups of such sheaves have been examined systematically in the mathematics literature [29], and have been shown to have Hodge structures compatible with that of the elliptically fibered manifold itself. We use the results of [29] to show that h 1 (S, j * H Q ) = 20 (proposition B.1).
In this paper, we further relate the SL(2, Z) doublet harmonic one-forms with the cohomology of the K3 manifold in the following way. We show that the doublet one-forms can be constructed by integrating certain closed two-forms of the underlying K3 manifold along the A and B-cycles of the fiber. Let us be more precise. There exists a 20-dimensional subspace of the second cohomology of the elliptically fibered K3 manifold -which we denote H 2 (M ) ⊥ -that is transverse to the fiber and the base. For each element of H 2 (M ) ⊥ , we show that there exists a certain two-form Ξ k in the class whose projection to the zero section and to every fiber vanishes, i.e., Such a choice can always be made in a dense open patch of the base of the fibration. 6 Given the elliptic fibrationf :M →S, this sheaf is obtained by pushing forward a certain sheaf living in a dense open subset S ofS with respect to the inclusion map j : S ֒→S. S is obtained by excising the points onS where the fiber degenerates. Then one can consider the elliptic fibration f : M → S over S. The sheaf living on S is denoted by H Q = R 1 f * Q for the locally constant sheaf Q on M . The cohomology group of interest is H 1 (S, j * H Q ) [29]. Explanation of the notation we use can be found in standard texts on Hodge theory such as [30].
In fact, we can choose Ξ k to be harmonic with respect to the "semi-flat metric" [31] of elliptically fibered K3 manifolds constructed in [12]. In this case, a doublet of one-forms on the base manifold can be defined. Note that these doublets automatically exhibit the required monodromies around each brane locus due to the behavior of the cycles around these points. Also, these one-forms can be shown to be harmonic as defined in (1.7). A more mathematical formulation, as well as a proof of these facts are presented in appendix E. The massless vector field excitations are equivalent to a collective excitation of sevenbrane vector fields and bulk fields by CS gauge transformations. A particularly useful gauge is one in which the tensor field components are turned on along directions transverse to the compact space. In such a gauge, the tensor field excitations decouple from the string junctions [17,[24][25][26][27][28] -which are webs of (p, q) strings ending on the various seven-branes -stretching between the seven-branes, as the junctions lie along the compact P 1 . Therefore, in this gauge, one can identify the linear combinations of the seven-brane vector fields that reproduce the charges of the string junctions under a particular vector field a k .
In this sense, there is a correspondence between the massless vector fields constructed by KK-reduction and the world-volume vector fields living on the seven-branes. To be more precise, it can be shown that turning on an eight-dimensional vector field a k , i.e., turning on the ten-dimensional tensor field is gauge equivalent to turning on some linear combination of seven-brane vector fields along with a tensor field transverse to the compact directions: Here, ϕ k,I is a doublet scalar living on the P 1 that satisfies Hence the tensor field background (1.10) is equivalent to turning on the background gauge fields (1.11) from the point of view of the string junctions. With further "CS gauge fixing," we can in fact show that there is a invertible linear map between the vector fields a k and a moduli-independent 20-dimensional linear subspace L of the seven-brane vector fields. This paper is organized as follows. In section 2, we review basic facts about K3 compactifications of F-theory and show how they can be described from the type IIB point of view.
In section 3, we show that all the seven-brane world-volume vector fields are eaten by the type IIB tensor fields through the Cremmer-Scherk mechanism, and identify the responsible gauge transformations. In section 4, we find the 20 SL(2, Z) doublet harmonic one-forms of the type IIB geometry. We relate these one-forms to the elements of the cohomology group H 1 (S, j * H Q ), as well as H 2 (M ). We show that the type IIB doublet two-forms can be reduced along these one-forms to yield 20 massless vector fields in the 8D effective theory. We proceed to establish the aforementioned correspondence between these harmonic one-forms and world-volume vector fields. Further discussions and future directions are presented in section 5. In particular, we discuss the possibility of developing our approach further towards understanding more general F-theory backgrounds. We elaborate on some technical details that we have omitted in the main text in the appendix.

Review of F-theory Compactifications on Smooth K3 Surfaces
In this section, we review eight-dimensional backgrounds coming from compactifying F-theory on a smooth generic elliptically fibered K3 manifold with a section. 7 We we take the point of view that these backgrounds are supersymmetric solutions of type IIB theory, that is, as a type IIB compactification on P 1 with 24 (p, q) seven-branes. We first review its massless matter content using F-theory/heterotic duality, focusing on the gauge fields, and proceed to describe the supergravity solution in more detail. The content of this section is a reorganization of facts presented in [1,4,12,15,17,18,[24][25][26][27][28], among other places. A great review of K3 geometry in the context of string theory is given in [32].
Eight-dimensional F-theory backgrounds with minimal supersymmetry come from compactifying F-theory on an elliptically fibered K3 manifoldf :M →S with a section. We denote the K3 manifold byM and the base manifold byS throughout this paper. The basē S of the fibration is a P 1 , and the manifold is parametrized by the Weierstrass equation (2.1) Here F 8 and G 12 are sections of 8H and 12H, where H is the hyperplane line bundle of the base P 1 manifold. In this paper, we assume that the complex structure ofM is at a generic point in the moduli space. We therefore assume generic values for the coefficients of F 8 and G 12 . When this is the case, the manifold is smooth and the elliptic fibration has 24 I 1 singularities at the loci ∆ is the discriminant of the elliptic curve; the locus ∆ = 0 is called the discriminant locus. Throughout this paper, we often choose work in a local patch of the ambient toric manifold, in which case the equation (2.1) can be written as where z is the local coordinate on the base manifold. f 8 and g 12 are polynomials in z with degree ≤ 8 and ≤ 12, respectively. There are thirty-seven moduli in the eight-dimensional theory -18 complex moduli parametrizing the complex structure of the elliptic fibration (2.1) and one real modulus that parametrizes the size of the base. These eight-dimensional theories are dual to E 8 × E 8 heterotic string compactifications on a two-torus. The complex structure moduli of the elliptically fibered K3 manifold map to the complex and Kähler moduli of the torus and the Wilson lines along the two T 2 directions. The modulus that parametrizes the size of the base of the K3 manifold maps to the value of the dilaton of the heterotic theory. At a generic point in the complex structure moduli space, the dual heterotic background has generic Wilson lines turned on. The massless spectrum of the heterotic background can be easily obtained by standard methods. In particular, the theory at such a point has 20 gauge fields in the massless spectrum. Sixteen of these gauge fields come from the Cartan subgroup of the E 8 × E 8 gauge group, while four -two of which are graviphotons -come from reducing the ten-dimensional graviton and tensor along the two "legs" of the torus.
The elliptic fibration (2.1) describes a supersymmetric background of type IIB string theory. Before we see how, let us first describe the low-energy effective theory of type IIB in more detail. The massless bosonic degrees of freedom are given by the graviton, a complex scalar, two two-forms, and one self-dual four form. Type IIB string theory is covariant under a global SL(2, Z) group. Following the conventions of [33], the bosonic part of the type IIB action can be written in an SL(2, Z) covariant way in Einstein frame: Here, τ = τ 1 + iτ 2 is the axio-dilaton, while F I 3 is the two-from field strength doublet: The SL(2, Z) group acts on these fields as where a, b, c and d are integers satisfying ad − bc = 1. We note that the dual six-form fields of B and C transform in the same way as the two-form fields under the SL(2, Z) action. The matrix M IJ is given by From the point of view of the type IIB theory, the elliptic fibration (2.1) parametrizes a supersymmetric solution to the equations of motion. To be more precise, it describes a compactification of type IIB theory on a P 1 with a varying axio-dilaton. Taking the base of the elliptic fibration (2.1) to be the compact P 1 , the axio-dilaton value at a point z in the base is related to the complex structure τ (z) of the fiber at the given point by [12] j is Klein's j-invariant. The metric on the P 1 can also be computed from the elliptic fibration [12] where z is the complex coordinate along the base P 1 . 8 The 24 loci where the fiber degenerates can be thought of as seven-brane loci. Let us denote these branes as B 1 , · · · B 24 . Now the j-invariant (2.9) is not a one-to-one function from the upper-half complex plane to the complex plane. In order to describe the F-theory vacuum from the point of view of type IIB, one must also choose two one-cycles -the A-cycle α and the B-cycle β -of the elliptic fiber that satisfy (2.11) and to compute where λ is the unique holomorphic one-form on the elliptic fiber. 9 The choice of different pairs of cycles that satisfy the conditions (2.11) result in different type IIB backgrounds related by SL(2, Z) transformations. The group of global SL(2, Z) transformations is nothing but the group of maps between different choices of cycles. The cycles of the elliptic fiber undergo monodromies as they go around the seven-brane loci -the value of the axio-dilaton transforms under the corresponding monodromies accordingly. Therefore the axio-dilaton profile of a non-trivial F-theory background cannot be defined globally on the base manifold -in fact, there are branch cuts emanating from the seven-brane loci. In the case the elliptically fibered manifold is a K3 manifold, the overall monodromy is trivial. Therefore we can "join" the 24 branch cuts emanating from each brane. We can then define the A-cycle α and B-cycle β of the elliptic fibration unambiguously in the dense open subsetS of the base P 1 manifold obtained by excluding these branch cuts. We note that the monodromy around each brane -and hence the type of each brane - depends on how one chooses these branch cuts [25]. 10 Unless an F-theory background has an orientifold limit, we must always pick such a patch to describe the backgrounds in the type IIB framework. In this sense, a useful way to view these F-theory backgrounds is to interpret them as type IIB compactifications on a dense open subset of P 1 rather than the full P 1 . We have depicted the situation in figure 1. We note that regardless of the way one chooses the cuts, the physics of the eightdimensional effective theory stays the same. Now the type IIB description of a given compactification can alter under drastic shifting of these cuts. For example, when one moves a cut through a brane locus so that their relative positions change, the (p, q) charge of the brane typically jumps. Under local variations of the cut, however, where no such "singular" shifts are made, the description of the background in terms of type IIB theory should remain invariant. This point turns out to be important in determining the monodromies of various fields of the type IIB theory.
It is worth noting that a choice of cuts defines an SL(2, Z) bundle on another dense open set S of the base, where The way to construct the bundle is the following. Let us choose to join branch cuts so that the tree of branch cuts only has trivalent vertices. Each edge of the tree has an assigned element of 10 In fact, one can only make sense of the monodromies as being an element of SL(2, Z) when A and B-cycles can be defined. Therefore a set of branes can have many different representations as (p, q)-branes depending on how one decides to "join" the cuts emanating from them. It is useful to note that two different (p, q) brane configurations obtained by choosing different ways of joining cuts are not in general related to each other by a global SL(2, Z) transformation. Such equivalences between different (p, q)-brane configurations have been extensively studied from the point of view of string junctions. Figure 2. An example of a two-cycle in an elliptically fibered manifold ending at seven-brane loci, for a system of three seven-branes B 1 , B 2 and B 3 . The three branes are are of type (1, 0), (0, 1) and (1, 1) respectively. At these branes, the cycles α, β and α + β of the elliptic fiber degenerate. As (1, 0) + (0, 1) − (1, 1) = (0, 0), there is a two-cycle C σ ending at the three branes with σ = (1, 1, −1). The cycle starts off at B 3 where the degenerate cycle (α + β) shrinks to a point. As we trace the cycle (α + β) through the manifold starting from B 3 , the cycle eventually splits into cycles α and β, which each shrink to a point where the branes B 1 and B 2 are located -the two-dimensional surface traced out in the process defines a closed two-cycle in the elliptically fibered manifold. The contours near each brane depict the cycles that shrink at the corresponding brane. SL(2, Z) that corresponds to the "monodromy" that would occur from crossing that cut in a designated direction. For every vertex of the tree of branch cuts, the clockwise "monodromies" m i (i = 1, 2, 3) of the three cuts joining at the vertex must satisfy the condition (2.14) This data corresponds to the transition functions of a principal SL(2, Z) bundle of the 24punctured sphere S. We study the sheaf associated to this bundle in detail later on. As noted before, choosing the cycles on the open setS corresponds to fixing a type IIB frame. Once the frame is fixed, the types of the seven-branes at each degeneration point can be determined. The seven-brane sitting at the point where an irreducible cycle pα + qβ is shrinking is defined to be a (p, q) brane -p and q must be mutually prime. The monodromy around a (p, q) brane is given in the following way. The cycle xα + yβ transforms as upon rotating the elliptic fiber a full cycle in the counter-clockwise direction around the (p, q) brane. Note that the vector (p, q) t -representing the shrinking cycle at the seven-brane locus -is left invariant by this monodromy. The (1, 0) brane is a D7-brane where a fundamental string can end at, while D1-strings can end at (0, 1) branes. In fact, (p, q) seven-branes are defined to be seven-branes at which (p, q) string can end. Let us denote the brane charge of each seven-brane B i as (p i , q i ).

Let us denote a 24 dimensional vector
a "charge vector." 11 We note that the vector space of charge vectors is a 22 dimensional subspace of Z 24 due to the two constraints. For any charge vector σ, there is an oriented two-cycle that begins at the branes with σ i < 0 and ends at branes with σ i > 0. The end points of the cycle can be identified to be the singular point of the fiber at the seven-brane locus, where a cycle of the fiber is shrunk to a point. As we move along the open patch of the base manifold, we can trace the trajectory of such a one-cycle. As we do so, the cycles split and merge, thereby tracing the locus of the corresponding two-cycle inside the elliptically fibered manifold. σ uniquely determines a homology class of a two-cycle. Let us denote this cycle C σ . Note that C σ is defined such that |σ i | points of the cycle either end at (σ i > 0) or begin at (σ i < 0) the degenerate point of the fiber above brane B i when σ i = 0. Examples of C σ for two different σ are given in figures 2 and 3. Let us note that where the former equality means that the cycles are equal as homology classes. This can be easily confirmed, as C σ + C σ ′ can be smoothly deformed into C σ ′′ when σ + σ ′ = σ ′′ . The inverse statement, however, is not true. This is because two non-trivial charge vectors correspond to trivial homology classes. 12 The homology group of two-cycles C σi.e., cycles that interpolate between seven-brane loci -is in fact generated by 20 elements C 1 , · · · , C 20 . This is because the second homology group of a K3 manifold, when viewed as a vector space, is 22 dimensional -one of which corresponds to the class of the fiber and one of which corresponds to the class of the base 13 . The two-cycles we are interested in are generated by the elements that are orthogonal -with respect to the intersection product -to the base and fiber classes. Let us denote this 20dimensional space as H 2 (M ) ⊥ . The complex structure of an elliptically fibered K3 manifold is determined by the ray of the complex vector where Ω is the holomorphic two-form of the K3 manifold that is unique (up to a factor).
Using the local coordinates (2.3), Ω can be explicitly written as The complex structure of a generic K3 manifold -one without the restriction of being elliptically fibered or having a section -is parametrized by a 22 dimensional projective vector, obtained by integrating the complex two-form over all generators of the homology group. The K3 manifolds used for F-theory compactifications, however, can be parametrized by the 20 dimensional projective vector (2.18) due to the fact that Ω is orthogonal to the fiber and base directions, i.e.,

Ω =
Fiber Despite that we have parameterized the complex structure of an elliptically fibered K3 manifold by 20 projective coordinates, its moduli space is 18-dimensional. This follows from the fact that the vector (2.18) also satisfies the constraint [32] K3 Ω ∧ Ω = 0 . (2.21) The cycles C σ , as in figure 4, can be "thinned down" to a linear combination of string junctions -a collection of directed (p, q) string segments connected to each other at nodes -stretched between the seven-branes with net charge vector σ. Massive states of the eightdimensional theory can be obtained by quantizing modes of these string junctions and linear combinations thereof. The correspondence between homologically non-trivial two-cycles of the K3 manifold and string junctions therefore implies the correspondence between the twocycles and the massive particles of the 8D effective theory.
Let us end this section by commenting on the world-volume theory living on a sevenbrane. A seven-brane contains a dynamical gauge field coming from quantizing strings with Figure 4. Correspondence between string junctions and two-cycles in the elliptically fibered manifold. On the left side, we have depicted a string junction meeting at junction point P and ending at three seven-branes, each of type (1, 0), (0, 1) and (1, 1). One can "fatten" this junction to obtain an oriented two-cycle inside the elliptically fibered manifold in the F-theory picture.
both ends ending on that brane. The (p, q) seven-brane action can be written by first writing the D7-brane action in Einstein frame and performing an SL(2, Z) transformation. The linear terms relevant to investigating lifting of gauge fields come from the kinetic term. Being careful with the dilaton coupling, one can show that the gauge kinetic term for a (p, q) brane is given by where we have set 2πα ′ = 1 [13,33]. µ 8 does not depend on the (p, q) charge or the dynamical axio-dilaton in Einstein frame. From this, we see that the gauge coupling of the seven-branes are independent of (p, q) charge in Einstein frame -one could have already expected this, as type IIB theory in Einstein frame is SL(2, Z) covariant. Therefore the ten-dimensional type IIB action corresponding to the F-theory compactification on K3 has 24 gauge fields A i µ living on the world volume of seven-branes B i with the kinetic term In the next section, we proceed to show how these vector fields are absorbed by the tensor fields by the Cremmer-Scherk mechanism.

The Cremmer-Scherk Mechanism
In this section, we show that the vector fields living on the world-volume of the seven-branes of the F-theory background are "eaten" by the type IIB tensor fields through the Cremmer-Scherk mechanism. We first review the Cremmer-Scherk gauge transformations of F-theory compactifications on K3. We proceed to identify the 24 gauge transformations responsible for absorbing the vector fields into the tensor degrees of freedom. As can be seen from the previous section, type IIB backgrounds with seven-branes are invariant under the local symmetry Figure 5. A zoom-in on a small neighborhood of a brane locus. The curvy line denotes the branch-cut around the brane. The concentric circles of radius d/2 and d are chosen to be small enough so that they do not intersect any other branch cuts.
where Γ I is a doublet of one-forms. As before, I is the SL(2, Z) index and i indexes the branes. Also, π i * is the push-forward of the projection map to the i'th brane. In order for (3.1) to make sense, a constraint on the doublet one-form Γ I must be imposed. The value must be unambiguously defined, i.e., it must be monodromy invariant at each brane locus. We stress that while the two-form field strength must exhibit certain monodromies, Γ I need not show such behavior. Upon investigation of the Lagrangian of the theory, one can verify that the field strengths of the two-form fields F I = dB I are required to undergo monodromies upon counter-clockwise rotation around -or, equivalently, crossing the branch cut in clockwise direction -a (p, q) brane. Note that this is the transpose of the monodromy (2.15). Such monodromies are imposed since we do not want the action to vary upon shifting the position of the branch cuts. The only other requirement the two-form field values themselves must satisfy is that be well-defined at brane loci B i . Any gauge transformation with well-defined values of (3.2) at B i preserve both requirements. Now let us find a gauge transformation that can be used to gauge away vector degrees of freedom living on a particular brane B i 0 . Let us assume a given seven-brane is a Dirichlet brane -i.e., a (1, 0) brane -and consider the gauge transformations of the form where φ I is a doublet scalar dependent on the internal coordinates while Λ is an eightdimensional one-form. Taking the local holomorphic coordinate around the brane locus w = 0 to be w, one can find some d such that the open neighborhood |w| < d of the brane does not include any other branch cut. Then one can find a real "bump function" f (w) that satisfies the following conditions: is C ∞ for |w| < d and hence so on the full manifold.
Then, for the gauge transformation (3.5) is well-defined on the full base manifold. In particular, the value (3.2) is well defined at each brane locus unambiguously. In fact, for each brane B i , The vector field A i 0 is thereby eaten by the two-form doublet by this CS gauge transformation by setting Λ µ = A i 0 µ . Likewise, for each brane B i of charge (p i , q i ), we can construct a gauge transformation with an appropriate bump function f i to absorb A i . Hence, in "unitary gauge" where all the gauge fields living on the world-volume of the seven-branes are eaten, the term (2.23) of the Lagrangian becomes Regardless of this term, which looks like an eight-dimensional mass term for the tensor fields, we still find modes of the tensor fields responsible for massless degrees of freedom of the 8D effective theory. These come from modes with components transverse to the seven-branes. We proceed to examine these modes in the next section.

SL(2, Z) Doublet Harmonic One-forms
In this section, we show how 20 massless vector fields arise in F-theory compactifications on K3 from the type IIB perspective. In section 4.1, we define what we mean by SL(2, Z) doublet harmonic one-forms and show that the doublet two-form of type IIB theory can be reduced along these one-forms to yield massless vectors in the eight-dimensional theory. In the following two sections, we study these zero modes from two different points of view. In section 4.2, we show that these harmonic forms represent elements of the first cohomology group of a certain sheaf living on the base manifoldS. We also derive that the dimension of this cohomology group is 20. In section 4.3, we establish the correspondence between doublet harmonic one-forms and certain closed two-forms -namely, the "semi-flat harmonic two-forms" -living inside the underlying K3 manifoldM . We end by showing how other particles and fields of the 8D theory couple to the vectors obtained by reducing along these one-forms in section 4.4. The various couplings are shown to encode the geometric data of M .

Definition
In this section, we show how massless vectors arise in F-theory compactifications on K3. These are obtained upon KK-reduction of the SL(2, Z) doublet two-forms of the type IIB theory along doublet one-form zero modes. In particular, we derive the "zero mode condition" or the "harmonicity condition" (1.7) a SL(2, Z) doublet one-form ξ I living in the compactification manifold S must satisfy. Let us consider the KK-reduction of the doublet two-form fields in the F-theory background along some doublet zero-mode. The KK-reduction ansatz is given as where φ is some doublet k-form aligned along the internal direction and b is an eightdimensional (2 − k)-form field. In order for b to be massless, φ must be closed: At the same time, φ I must respect the monodromies defined by the background, i.e., it must exhibit the same SL(2, Z) transformations that the field strengths experience as they cross the branch cuts. In particular, upon counter-clockwise rotation around a brane locus B i , φ I must transform as There are two ways of seeing why such behavior should be imposed on φ I . The first way is by examining the field strength of the doublet two-form -as explained in the previous section, in order for the action of the type IIB theory to be invariant under moving branch cuts, the field strength-doublet (dB, dC) must exhibit the monodromies (3.3). The constraint on φ I follows by applying this condition to the ansatz (4.1). Another way of seeing this constraint is to consider the normalization of the mode φ I , which is given by Unless φ I exhibits the correct monodromies, the normalization (4.4) is not well defined -in fact, it would vary as one moves the branch cuts around. On the other hand, when φ I exhibit the desired monodromies, we can define the norm It was pointed out in [1] that there is no way of turning on the two-form fields along the internal directions and also satisfying the required monodromies. Also, it is clear that there does not exist any non-zero closed zero-forms -i.e., constant scalars -that exhibit monodromic behavior. The interesting closed forms that yield massless particles in eightdimensions are the one-forms, which we denote by Hence we wish to find closed doublet one-forms living in the base manifoldS. These one-forms must be normalizable with respect to the Hermitian inner-product defined by the Lagrangian, i.e., ξ k , ξ k < ∞ . (4.8) Note that since ξ k have monodromies around the brane loci, their components exhibit logarithmic behavior at these points. Despite such singular behavior, the modes can nevertheless be normalizable. For example, when the singularities of ξ k are logarithmic, the integral near the brane loci is convergent. As with KK-reduction of any p-form, the closed one-forms ξ k we reduce along are defined up to an exact form dϕ, where ϕ is a doublet of zero-forms. This freedom comes as a remnant of the "CS gauge symmetry" of the type IIB background. Such an ambiguity is fixed by demanding that ξ k is harmonic with respect to the Hermitian inner-product defined by the Lagrangian, i.e., d * M IJ ξ k,J = 0 . (4.10) A simple computation shows that imposing this condition assures that ξ k has minimum norm given the "cohomology class" of ξ k is fixed. In other words, ξ k , ξ k < ξ k , ξ k + dϕ, dϕ = ξ k + dϕ, ξ k + dϕ (4.11) for any non-trivial CS gauge transformation dϕ = 0 when ξ k satisfies (4.10). Hence, the condition (4.10) singles out elements of a certain cohomology class, just as the usual harmonicity condition singles out a harmonic form in a given de Rham cohomology class. We describe this cohomology in more detail in the following section.
A source of worry for equation (4.11) is that integration by parts has been used in obtaining the equality -for the most general CS gauge transformation ϕ, the cross terms in the integral of interest may have boundary terms. Such boundary terms arise in the case that ϕ exhibit monodromic behavior that is affine rather than linear. To be more precise, let us first examine how ϕ is allowed to behave as we rotate around a brane locus. In order for the ξ k + dϕ to have a well defined norm, dϕ must exhibit the monodromy around brane B i . ϕ itself, however, does not have to display this monodromy -in fact, it is allowed to shift: Note that the shift must be in the direction (−q i , p i ) t , as the values p i ϕ 1 + q i ϕ 2 must be welldefined at the brane locus. In the event that C i = 0, the integration by parts we have used in (4.11) is no longer valid, as there would exist boundary terms living on ∂S -a contour encircling the "cuts" we have used to define a type IIB frame -that do not cancel out.
What makes the equality of (4.11) work is that we only allow gauge transformations such that at seven-brane loci (z i ,z i ). This is because we are computing the eight-dimensional massless spectrum in a "unitary gauge" where all the seven-brane world-volume vector fields are eaten by tensor degrees of freedom. The condition (4.14) ensures that the fluctuations we consider still satisfy the unitary gauge condition. In appendix A, we show that C i = 0 for gauge transformations ϕ that do not excite seven-brane vector fields and keep the mode ξ k + dϕ normalizable.
Let us conclude this section by summarizing the definition of a SL(2, Z) doublet harmonic one-form ξ: 1. ξ is doublet of one-forms exhibiting the monodromy (4.3); 2. It must satisfy the defining equations (1.7): 3. It must be normalizable with respect to the inner-product (4.7): In the subsequent sections, we go on to count and construct such harmonic one-forms.

Coordinate-free Description of Doublet One-forms
In this section, we give a coordinate-free description of SL(2, Z) doublet one-forms on S in terms of certain sheaves. We also show that the dimension of the space of doublet harmonic one-forms is equal to 20, by applying results by Zucker [29] about polarized variations of Hodge structure on curves.
Recall thatf :M →S is an elliptically fibered K3-surface with a section; hereS is the complex projective line. We assume that there are exactly 24 singular fibers, each with a single ordinary double point -for degree reasons, the section cannot pass through any of the 24 special points. If we denote by S ⊆S the complement of the 24 singular values, and by M =f Now the first cohomology group H 1 (E, Z) carries a polarized Hodge structure of weight 1. The Hodge structure is given by the decomposition according to type; the polarization is given by the intersection form It is a polarization because of the Riemann bilinear relations: the Hermitian form α → i p−q Q(α, α) is positive-definite on the subspace H p,q (E), and the above decomposition is orthogonal with respect to the resulting Hermitian inner product on H 1 (E, C).
Because The connection ∇ can be extended to an operator from doublet k-forms to doublet (k + 1)forms, and the resulting complex computes the cohomology groups H k (S, H C ) of the locally constant sheaf H C , by a version of the Poincaré lemma. One might expect naively that the number of zero-modes defined in the previous section can be obtained by computing the dimension of the cohomology group H 1 (S, H Q ). This is not true, since elements of this cohomology group can exhibit singular behavior near the discriminant locus, and may therefore not be normalizable with respect to the inner-product defined in the previous section. In fact, the dimension can be shown to be 20 + 24 = 44. Instead, the correct cohomology group to consider is where j : S ֒→S denotes the inclusion map of S intoS. In appendix B, we use some theorems by Zucker [29] to prove that H k S , j * H Q = 0 for k = 1, and that dim H 1 S , j * H Q = 20, as expected. Moreover, it is shown in [29] that the first cohomology group is isomorphic to the subspace of A 1 (S, H) consisting of forms that are square-integrable and harmonic (with respect to the Hodge metric on H and the Poincaré metric on S). These two conditions are exactly the same as in the previous section, and so we deduce that the space of SL(2, Z) doublet harmonic one-forms is indeed 20-dimensional.
The work of Zucker also endows H 1 S , j * H Q with a polarized Hodge structure of weight 2. This Hodge structure is compatible with that on H 2 (M , Q); more precisely, there is a natural morphism and this morphism is an isomorphism of polarized Hodge structures. We will see below how this statement about cohomology groups can be sharpened to a result about spaces of harmonic forms.

Construction from Harmonic Two-forms on M
In this section, we explain how to the construct the doublet harmonic one-forms from the point of view of the K3 manifoldM . To be more precise, we show that there is a correspondence between the doublet harmonic one-forms and two-forms of the K3 manifold that are harmonic with respect to the semi-flat metric [12]. We conjecture that these "semi-flat harmonic twoforms" can be obtained as a limit of harmonic two-forms of the K3 manifold with respect to the Calabi-Yau metric. Let us begin by noting that a natural way to obtain a doublet of closed one-forms on the base manifold S of the elliptic fibration, that displays the monodromies of the cycles of the fiber, is by using two-forms of the underlying K3 geometry. Consider a smooth closed two-form Ξ living inside the M obtained by excising the 24 singular fibers of the K3 manifold M . Now for each point z on the base manifold, let us define a doublet of forms where the integration cycles are taken to lie within the holomorphic fiber above the point z. α and β are the A and B-cycle of the fibration used to define the type IIB frame. If ξ is "well defined," it is a doublet of one-forms that exhibit the correct monodromies. It is also closed due to the closedness of Ξ.
In order for ξ to be well defined, the projection of Ξ to each fiber must vanish. Meanwhile, components of Ξ with both legs parallel to the base would not affect ξ and should be "gauged away" if one wishes to establish a one-to-one correspondence between one-forms on the base and two-forms in the full manifold. Let us hence assume the components of Ξ with both legs along either the fiber or the base direction vanish. A better presentation of this condition is given shortly.
For ξ to be harmonic as defined in (1.7), an additional condition on Ξ must be imposed. It is in fact that Ξ should be harmonic with respect to the semi-flat metric constructed in [12]. More precisely, we consider the family of semi-flat metrics whose Kähler form J t is given by for some function W [31]. The one forms θ z and θz are defined to as where z is the holomorphic coordinate on the base. θ w and θw, which we define shortly after, are one-forms aligned along the fiber direction. The semi-flat metric is a Ricci-flat Kähler metric on the K3 manifold, which is locally defined by the hypersurface equation λ = dx/y here is the unique holomorphic one-form of the fiber while the integral is taken to be along the elliptic fiber at z. This relation is derived in appendix C -the normalization constant is added for aesthetic reasons. The one-form θ w can be expressed using the canonical holomorphic coordinate -where the starting point of the integral is the locus of the zero section on the fiber -by The one-form ∂λ living on the fiber can be explicitly written as While ∂λ shows singular behavior at certain points on the elliptic curve, it is a "differential of the third kind," i.e., its integral over closed cycles of the elliptic curve are nevertheless well-defined. Hence the values (∂λ,λ) and (∂λ, λ) are well-defined for the inner-product The one form θ w may at first sight look rather peculiar -it, however, can be re-written in a simple way, using the coordinates x 1 and x 2 which parametrize the flat elliptic fiber such that Upon acknowledging that it can be shown that We hence see that J t is aligned in the base and fiber directions -it is orthogonal to cycles of the manifold that are orthogonal to the class of the fiber and the base. Despite the nice properties of the semi-flat metric, it fails to be a smooth Calabi-Yau metric, as it degenerates at the discriminant locus of the elliptic fibration. It has, however, been shown that it is a good approximation to the Calabi-Yau metric on the K3 manifold with fiber size t as t approaches zero [31]. It also is a smooth, non-degenerate Calabi-Yau metric of the open manifold M . In order for a doublet one-form ξ constructed from a two-form Ξ living inside this manifold to be well defined, Ξ zz = Ξ ww = 0, as discussed at the beginning of this section. This condition can be expressed using the following harmonic two-forms with respect to J t : Although B and F behave singularly at discriminant loci, they nevertheless represent cohomology classes of the manifoldM . While de Rham cohomology is defined by using smooth differential forms, a form Ξ that is not necessarily smooth still represents a cohomology class as long as its integrals along homology classes are well-defined. 15 In this case, the cohomology class of Ξ can is given by the dual cohomology class of

29)
15 A representative example of non-smooth forms with a well-defined cohomology class is a differential form of the third kind on a algebraic curve. Such one-forms are singular, but have only higher order poles and no residues. Hence the integrals of such a form along closed cycles of an algebraic curve are well-defined.
with respect to the canonical pairing between forms and cycles. 16 Here C i is the basis of the homology group of the manifold. The forms B and F are in fact dual to the base and the fiber class ofM . In particular, it can be shown that while the integrals of B and F over cycles orthogonal to the base and fiber vanish. It is worth noting that the forms B and F are normalizable with respect to the semi-flat metric at finite t; Here, * sf denotes the Hodge dual with respect to the semi-flat metric while Ω is the holomorphic two-form.
In appendix D, we show that when Ξ is harmonic with respect to the semi-flat metric, and its components Ξ zz and Ξ ww vanish, * M IJ ξ J = β * sf Ξ − α * sf Ξ (4.33) for ξ constructed by (4.15). In this case, the condition is satisfied since * sf Ξ is also closed. Given this result, it is straightforward to obtain the innerproduct (4.7) of harmonic one-form doublets ξ and η constructed from harmonic two-forms Ξ and H as an integral on M . It is a simple exercise to show in fact that using properties of harmonic forms of the semi-flat metric derived in appendix D. Hence in order for ξ to be normalizable as defined in section 4.3, Ξ must also be normalizable with respect to the canonical inner-product defined by the semi-flat metric.
Before we carry further on, let us sum up the properties of the two-forms Ξ that produce doublet harmonic one-forms upon integrating along cycles of the fiber: 1. Ξ is harmonic with respect to the semi-flat metric, i.e., both Ξ and * sf Ξ are closed.

Ξ satisfies
for the two-forms B and F defined in equation (4.27).
3. Ξ is normalizable with respect to the inner-product Let us denote such harmonic two-forms, "semi-flat harmonic two-forms." It is worth commenting that the definition of semi-flat harmonic forms is independent of the parameter t. This is because the Hodge dual * sf acting on a two-form Ξ is independent of t when Ξ satisfies (4.36). This, in particular, implies that these harmonic forms can be defined at the singular point t = 0. So far, we have shown that there is a map from semi-flat harmonic two-forms of a dense open subset M of the K3 manifoldM to doublet harmonic one-forms on the baseS. We show that this map is actually bijective in appendix D. This implies that the 20-dimensional space of doublet one-forms H 1 (S, j * H R ) can be lifted to a 20-dimensional space of semi-flat harmonic two-forms. These two-forms are a priori defined only on M .
A natural question to ask is whether these 20 semi-flat harmonic two-forms are related to cohomology classes of the K3 manifold H 2 (M ). In appendix E we show that this is in fact the case. To be more precise, Let us provide an intuitive sketch of why the dual homology elements of {Ξ k } must lie within the image of H 2 (M ) ⊥ in H 2 (M ). This turns out to be a consequence of imposing normalizability on Ξ k . We note that since the semi-flat harmonic two-forms are orthogonal to the base and fiber directions, it is enough to show that {Ξ k } are orthogonal to the cycles T i described in the preceding paragraph. To show this, let us assume that a semi-flat harmonic two-form Ξ has a non-trivial integral over some cycle T i , i.e., Recall that T i is constructed by rotating the invariant cycle (p i α+q i β) around the degeneration locus B i . Denoting the doublet harmonic one-form constructed from Ξ as ξ, this implies that for a contour c surrounding B i . Following the latter part of appendix A, it can then be shown that such ξ cannot be normalizable with respect to the norm defined for doublet one-forms, due to the divergent behavior of ξ near B i . It follows that Ξ is not normalizable with respect to the semi-flat metric, hence concluding the proof. The fact that the semi-flat harmonic two-forms form a basis for H 2 (M ) ⊥ suggests that they can be related to harmonic forms of the K3 manifold with respect to a class of smooth Calabi-Yau metrics in the following way. Let us consider a class of smooth, non-degenerate Ricci flat metrics with Kähler form K t that satisfies the following conditions: 1. The dual homology class of K t is aligned in the direction of the class of the base and the fiber.
2. K t ∧ K t = 2Ω ∧Ω for the holomorphic two-form Ω of the K3 manifold. In terms of the one-form λ we have been using, this condition can be re-expressed as 3. f −1 (z) K t = t for any point z in the base, i.e., the fiber size with respect to this metric is given by t. Recall thatf is the projection map of the fibration.
The semi-flat metric is also a Ricci flat metric whose Kähler form J t satisfies these conditions. While J t is degenerate at the discriminant locus, J t and K t are closely related -in fact, tJ t and tK t have been shown to coincide in the limit t → 0 [31]. Since J t approximates K t well in the small-t limit, we can expect that there is a one-to-one correspondence between the semi-flat harmonic two forms and harmonic two-forms of the Calabi-Yau metric in the subspace H 2 (M ) ⊥ . More precisely, we can put forth the following Proposal : For the stated class of Calabi-Yau metrics K t , let {ω 1 , · · · , ω 20 } denote the linearly independent harmonic forms spanning the 20-dimensional subspace H 2 (M ) ⊥ of the cohomology group spanned by elements orthogonal to the classes [B] and [F ]. In the limit t → 0, these 20 harmonic forms stay linearly independent and form a basis {Ξ 1 , · · · , Ξ 20 } of the semi-flat harmonic two-forms.
Given that the forms {ω 1 , · · · , ω 20 } do not develop singularities that obstruct the normalizability condition, it can be shown that ω k stay linearly independent. This is done by examining the duals of the cohomology classes of ω k . Since the duals of {ω k } are linearly independent in the homology group, {ω k } also remain linearly independent as two-forms.
A crucial test for the validity of the proposal would be to verify that the limits of ω k satisfy the orthogonality condition (4.36). This is because orthogonality at the level of cohomology does not guarantee orthogonality at the level of forms. Let us consider harmonic two-forms ω k with respect to the Calabi-Yau metric whose cohomology class are orthogonal to [B] and [F ], i.e., where B t and F t are harmonic forms in the cohomology classes [B] and [F ]. We have added the subscripts to emphasize that while the cohomology classes are defined irrespective of the metric, the harmonic forms have a metric dependence. While the conditions (4.41) do not imply that the two integrands vanish at every point, they imply "half" of these two conditions. Since the Kähler form K t of the metric -which is harmonic -is given by a linear combination of B t and F t by assumption, and since the Lefschetz action commutes with the Laplacian operator, it follows that It would be interesting to verify that the other half of the constraint is satisfied as the fiber size is taken to zero. A natural way to map the semi-flat harmonic forms {Ξ k } to the corresponding harmonic forms {ω k } at finite t is provided by M-theory/F-theory duality [1]. Let us consider the eightdimensional theory obtained by compactifying F-theory on K3 manifoldM , and let a k be the massless vector fields obtained by KK-reducing the doublet two-form fields of type IIB along one-forms ξ k constructed from Ξ k . The seven-dimensional effective theory obtained upon further compactification on a S 1 of radius ∼ α ′ /t −1/2 -where α ′ is the Regge slope of the type II string -is dual to M-theory compactified onM with Calabi-Yau metric K t . The sevendimensional vector fieldsã ′ k -which are modes of a k constant along the S 1 -are obtained by KK-reducing the M-theory three-form along harmonic forms ω k . The inverse coupling of the vector fields of the 8D theory are given by the inner-products of the semi-flat harmonic forms Ξ k (4.37). Meanwhile, upon reduction on a circle, the corresponding 7D couplings receive quantum corrections from charged particles winding around the compactification circle. The quantum corrected inverse couplings can be computed by the inner-product of the harmonic forms ω k with respect to the Calabi-Yau metric. Hence, in this sense, ω k can be thought of as a "quantum corrected" version of Ξ k .
In the next subsection, we investigate various properties of the vector fields of the 8D theory whose construction we have been studying up to this point. Let us conclude this section by summarizing what we have learned so far about the massless vector field spectrum of the effective 8D theory of the F-theory compactification onM , and setting the conventions for the next section: 1. The massless vector spectrum comes from reducing the type IIB doublet two-forms along doublet harmonic one-forms. We denote the massless vectors a k and the one-forms ξ k .
3. The components of ξ k can be obtained by integrating the "semi-flat harmonic twoforms" Ξ k ofM along the A and B-cycles of the elliptic fiber.

Properties of KK-reduced Vector Fields
In this section, we explore the properties of the massless vector fields of the 8D theory obtained by KK-reduction. We first compute the charges of string junctions under these vector fields. We go on to relate the vector fields to seven-brane world-volume vector fields through a particular CS gauge transformation. We conclude the section by computing the Chern-Simons couplings of the 8D effective theory involving the massless vector fields. The massive charged states of the 8D theory come from string junctions stretching inside of the base manifoldS and ending on the seven-branes. Any junction with charge vector σ can be represented by a tree of directed segments {s l } of n l (p l , q l ) strings -where p l and q l are mutually prime -that either begin/end at (p l , q l ) branes or junction points. The segments {s l i } meeting at the junction points P i must satisfy the charge conservation condition where the notation l → P (l ← P ) is used to denote that the segment l is ending at (emanating from) the point P , respectively. The charge of any such a junction under the 8D vector field a k is given by as a (p, q) string couples to the doublet two-form fields along its world-sheet Σ. π is the embedding of the world-sheet in space-time. The charge q σ,k can be expressed in terms of Ξ k as q σ,k = l s l n l p l α+n l q l β where C σ is the two-cycle of the K3 manifold that is obtained by "fattening" the junction. Hence the electric charge of a junction with charge vector σ under the 8D gauge field a k is given by the topological pairing between the cycle C σ ∈ H 2 (M ) ⊥ and the cohomology class There is a correspondence between these vector fields and world-volume vector fields A i living on the seven-branes B i . This correspondence can be established by "pushing" the eight-dimensional massless vector modes coming from exciting the two-form tensor ξ k ∧ a k back into the branes via a CS gauge transformation. The gauge transformation we use is given by Λ = −ϕ k a k where ϕ k is a "doublet" one-form such that (4.47) The two-form fluctuation is then gauge equivalent to This particular gauge transformation is implemented so that B does not have components tangent to the internal directions, so that none of the string junctions are charged under the B components. Now the map from a k to A i defined by (4.49) can be expressed in terms of the topological charges (4.46) either by using Stokes' theorem or by the following observation. Given that the fields (4.49) are turned on, a massive state of the 8D theory coming from quantizing a string junction with charge vector σ is coupled to the fields via σ i A i . Meanwhile, this is gauge equivalent to turning only B = ξ k ∧ a k on. Under this field configuration, the 8D state is coupled to a k via q σ,k a k . Since the two field configurations are gauge equivalent, the following identity holds: This identity clearly cannot define a one-to-one mapping between the gauge fields, as there are 20 of the vector fields a k , while there are 24 seven-brane vector fields A i . There is, however, a linear subspace of all the vector fields A i that one can identify with the space of physical massless vector fields of the eight-dimensional theory.
To identify this subspace, we first observe that there is an ambiguity in defining the gauge transformations ϕ k in (4.47) -it is defined up to a constant. As noted in section 3, a CS gauge transformation Λ is allowed as long as the values (p i Λ 1 + q i Λ 2 ) at the branes are well defined -this allows gauge transformations constant in the internal directions. Using this, we can gauge away two linear combinations of A i , namely p i Λ and q i Λ ′ without affecting charges of string junctions. We can therefore project away the linear combinations p i A i and q i A i , i.e., impose Next, we recall that among the remaining 22 linearly independent charge vectors, there exist two charge vectors Z 1 and Z 2 whose corresponding cycles are trivial homologically. These two vectors are the charge vectors of "null junctions." Hence, for these vectors, (4.52) Hence for A i satisfying (4.50), it must be that Since the cohomology classes of Ξ k are linearly independent and span the full space H 2 (M ) ⊥ , equation (4.50) defines a bijective linear map between the vector space spanned by a k and a subspace of the 24-dimensional space spanned by A i . We note that the usual SO(24) invariant Euclidean inner-product is used in defining this subspace, as it is inherited from the kinetic term of the gauge fields (2.23). This is the advertised correspondence between bulk and brane vector fields. Let us end the section with computing the Chern-Simons couplings of the eight-dimensional theory k lm da l ∧ da m ∧C 4 , (4.55) whereC is the 8D four-form, which is the mode of the type IIB self-dual four-form C 4 that is constant along the internal direction. This term comes from reducing the type IIB Chern-Simons term written in equation (2.4). k lm is then given by where ω l are harmonic forms with respect to the Calabi-Yau metric onM with fiber size ∼ α ′ 2 /r 2 . While the semi-flat harmonic forms Ξ l become "quantum corrected" into harmonic forms ω l , the Chern-Simons couplings k lm are still given by the topological intersection numbers of the cohomology classes [Ξ l ] = [ω l ], and thus remain the same.

Future Directions
In this paper, we have examined a thoroughly studied F-theory background through a rather uncommon approach. Namely, we have examined K3 compactifications directly from the point of view of type IIB string theory, only focusing on the degrees of freedom present there. While this approach did not reveal anything we did not know about K3 compactifications, we have demonstrated that much that we know about them can be recovered without referring to any dualities. Hopefully, the methods employed in this paper can be expanded to more complicated backgrounds to address problems that are hard to resolve by using other techniques. Let us conclude this paper by discussing directions in which to improve and expand our results.
Further Study of Semi-flat Harmonic Forms. In section 4.3, we have conjectured that harmonic forms of an elliptically fibered K3 manifoldM sitting inside the subspace H 2 (M ) ⊥ of the second cohomology group behave "nicely" in the semi-flat limit, i.e., the limit the fiber size of the manifold shrinks to zero. We have further proposed that, in this limit, they become harmonic forms with respect to the semi-flat metric [12]. Although this proposal is quite natural from the point of view of string theory, it seems quite non-trivial from the perspective of geometry. It would be interesting to see if this proposal can be proved with mathematical rigor. Another interesting direction of research would be to approach the semi-flat harmonic forms numerically. The physical quantities associated to massless modes of F-theory backgrounds are, at least classically, computed by using semi-flat harmonic forms. As we have demonstrated in this paper, these harmonic forms are much simpler beasts than the forms that are harmonic with respect to the Calabi-Yau metric. For example, our analysis shows that the Hodge duals and the norms of the semi-flat harmonic two-forms are "well-behaved" in the case of the K3 manifold. It would be interesting to compute these quantities explicitly using numerical methods. Hopefully, these methods can be developed further to apply to more complicated backgrounds, which we now discuss.
Backgrounds Constructed from Higher Dimensional Calabi-Yau Manifolds. We have dealt with the simplest non-trivial F-theory compactification in this paper. Generalizing our approach to more complicated backgrounds would be interesting. An immediate generalization would be to understand F-theory compactified on elliptically fibered Calabi-Yau threefolds [2,3]. In this case, the base of the elliptic fibration f :M →S is two-complexdimensional. Although these backgrounds are very well understood, the abelian gauge symmetry of the low-energy effective theories of these compactifications are rather mysterious from the point of view of type IIB string theory.
For example, let us consider a Calabi-Yau threefold elliptically fibered over P 2 . The lowenergy effective theory is a six-dimensional (1, 0) supergravity theory. At a generic point in moduli space, the fiber has an I 1 singularity along a degree-36 curve in the base manifold. In the type IIB picture, there is a single seven-brane wrapping this curve. There are no vector fields in the massless spectrum of the theory. At various points in the complex structure moduli space, however, the number of massless vector fields jump. The jump can be understood when there is enhanced non-abelian gauge symmetry -in this case multiple branes become coincident and the new particles added to the massless spectrum can be understood from the point of view of the world-volume theory of the coincident branes [8][9][10][11]. The picture is less clear when only abelian vector fields are added to the massless spectrum, mainly due to global issues. For example, certain loci of the Calabi-Yau moduli space with abelian gauge symmetry can be described by a single brane wrapping a single curve with self-crossings in the base. 18 The approach described in this paper seems promising when restricted to understanding theories with only abelian gauge symmetry -a naive generalization of the results of [29] on computing the cohomology H 1 (S, j * H R ) seems to produce the correct number of massless vector fields. Some subtleties, however, should be worked out. For example, the discriminant locus of the fibration have singular points which are not treated in [29]. From the physics point of view, one must formulate the world-volume theory of the seven-brane wrapping the singular curve, and also keep track of the behavior of the bulk fields and their interactions with the world-volume fields. 19 It would be interesting to achieve an understanding of such backgrounds and compute their massless spectrum from type IIB string theory. Hopefully, insight gained from this process can shed light on the more sophisticated Calabi-Yau fourfold backgrounds relevant to F-theory phenomenology. 20 Singular Backgrounds and Exploration of Non-Geometric Moduli. Another direction to expand our results is to consider its extension to F-theory compactifications on singular manifolds that give rise to non-abelian gauge symmetry. While the massless fields of type IIB supergravity would not fully account for the degrees of freedom of these backgrounds, one can nevertheless ask questions about the family of deformations of such backgrounds to gain insight into the singular backgrounds themselves. Observing how various modes of the type IIB fields behave in the singular limit might shed light on novel string backgrounds that are difficult to probe using dualities.
In particular, the approach of viewing F-theory backgrounds from the type IIB perspective may shed light on non-geometric vacua with "F-theory fluxes." An interesting class of such vacua can be described locally in terms of "T-branes" coined in [45], and also studied in [46][47][48][49][50][51]. From the point of view of the local seven-brane, T-branes are particular solutions to the Hitchin-type equations of the non-abelian fields living on coincident branes. To be more precise, they are solutions in which the Higgs field vacuum expectation values are upper-diagonal. An interpretation of these configurations in terms of the global geometry of the elliptically fibered manifold has been given [48,49,51]. It would be interesting to see how this geometric data translates into the type IIB data living on the base of the elliptic fibration.
Here, f is a multivalued function that exhibits the shift upon rotation around the brane locus. This implies that in fact for any contour c encircling the brane locus (z i ,z i ) close enough. Hence denoting where z is the holomorphic coordinate on the base, F has a singularity at the brane locus that behaves like 21 Near a given (p i , q i ) brane B i , the axio-dilaton τ behaves as where r i and s i are integers such that It follows that the behavior of the integrand of the norm (4.7) is at best given by near the brane locus (z i ,z i ). (r, θ) are polar coordinates centered at the brane. It follows that the mode ξ k + dϕ becomes non-normalizable upon such a gauge transformation, unless C i = 0. Therefore, in order to keep ξ k + dϕ normalizable, all C i must vanish. ✷ B Computation of h 1 (S, j * H C ) Here we provide proofs for the results in section 4.2, based on the work of Zucker [29].
Now the idea is to use the Leray spectral sequence to compare the vector space in question to the cohomology of the K3-surface. We observe that R 0f * Q ≃ R 2f * Q ≃ Q, because the cohomology in degree 0 and 2 is one-dimensional for all fibers off , including the 24 singular ones. The E 2 -page of the spectral sequence therefore has the following shape: It is obvious that the spectral sequence degenerates at E 2 ; in fact, by Corollary 15.15 of [29], this is always the case. Because we know the cohomology of a K3-surface, we get the desired result.
The degeneration of the Leray spectral sequence in (B.2) has the following additional consequence.

Corollary B.3. The Leray spectral sequence induces an isomorphism
which is in fact an isomorphism of polarized Hodge structures of weight 2.
Proof. We denote by the subspace of cohomology classes that restrict trivially to all the fibers off . Because (B.2) degenerates at E 2 , this subspace has dimension 21, and the natural mapping As mentioned above, H 1 S , R 1f * Q ≃ H 1 S , j * H Q ; this clearly implies the desired isomorphism because The isomorphism respects the polarized Hodge structures on both sides according to the results in Section 15 of [29].
Let us also compute the dimension of H 1 (S, H Q ). By the Picard-Lefschetz formula, the local monodromy around each of the 24 points of the discriminant locus is given by the matrix 1 1 0 1 in a suitable basis for the first cohomology of a nearby elliptic curve. It follows that the sheaf R 1 j * H Q is supported on the discriminant locus, and that its stalk at each of the 24 points is equal to Q. For dimension reasons, the Leray spectral sequence degenerates at E 2 ; this leads to a short exact sequence It follows that dim H 1 (S, H Q ) = 20 + 24 = 44.

C The Thomae Formula
We write components of the semi-flat metric in terms of the unique holomorphic one-form In particular, we show that Here, g zz is the metric of the elliptically fibered K3 manifold in the limit the fiber shrinks to zero-size. As is always, for a choice of A and B-cycles, α and β. The integral in (C.2) is taken along the elliptic fiber.
Recall that the local equation defining the K3 manifold is given by where z is the base coordinate. The discriminant ∆ is defined to be Now τ satisfies the relation Hence one finds that |η(τ )| 4 |∆| 1/6 = 6912 −1/6 |θ 2 (τ ) 8 Meanwhile, by the Thomae formula, Note that both sides of this equation vary with respect to modular transformations that come from choosing of different A and B-cycles. We therefore arrive at -we obtain Since Ξ zz = Ξ ww = 0, the Hodge dual * sf Ξ also have vanishing components along these directions, as the semi-flat metric factors in the base and fiber directions. Hence * sf Ξ can be written in the form * sf Ξ = ω ′ ∧ dz +ω ′ ∧ dz + f ′ dzdz . (D. 13) As before, since * sf Ξ is also closed, ω ′ takes the form ω ′ = A ′ λ + B ′λ + (∂ ζ C ′ dζ + ∂ζC ′ dζ) (D.14) where A ′ and B ′ are functions independent of the ζ andζ coordinates. Meanwhile, applying the Hodge dual to the expression (D.4) shows that upon a particular normalization of the antisymmetric four-tensor ǫ wwzz . 22 Taking partial derivatives with respect toζ and ζ on the first and second equation, respectively, we arrive at ∂ ζ ∂ζC = ∂ ζ ∂ζC ′ = 0 . (D. 16) Since the only harmonic function on a compact elliptic curve is the constant function, it must be that It also follows that ω ′ = −Aλ + Bλ . upon integration along cycles. This coincides with (D.11).
A corollary of this proof is that a semi-flat harmonic two-form takes the form Ξ = A(z,z)θ w ∧ θ z + B(z,z)θw ∧ θ z +B(z,z)θ w ∧ θz +Ā(z,z)θw ∧ θz , The construction is based on the following observation. Let X = V /Γ be a compact complex torus of dimension g; here V ≃ T 0 X is a g-dimensional complex vector space, and Γ ≃ π 1 (X, 0) is a lattice of rank 2g in V . In particular, Γ ⊗ Z R = V . By the Hurewicz theorem, H 1 (X, Z) ≃ Γ; now the universal coefficients theorem shows that H 1 (X, C) ≃ Hom Z (Γ, C) ≃ Hom R (V, C).
The isomorphism works like this: given an R-linear functional ϕ : V → C, we get a closed one-form dϕ ∈ A 1 (V, C); it is translation-invariant, and therefore descends to a translationinvariant closed one-form on X. Note that a closed form on a compact complex torus is translation-invariant if and only if it is harmonic for the flat metric. Now we return to our family of elliptic curves f : M → S. As a complex manifold, M is isomorphic to a quotient B/B Z , where p : B → S is a holomorphic line bundle on S, and B Z ⊆ B is a one-dimensional complex submanifold that intersects every fiber of B in a lattice of rank 2. It is easy to see that H is isomorphic to O S (B * ), the sheaf of holomorphic sections of the dual bundle. According to the discussion above, a section σ ∈ H C (U ) over an open subset U ⊆ S gives rise to a smooth function ϕ σ : p −1 (U ) → C 24 These doublets are non-vanishing at every point of S except for z = ∞. Given that the behavior of A and B are specified at this point, it is enough to check the equations that they satisfy at S \ {z = ∞} for our argument to hold. whose restriction to every fiber of B is R-linear. Its derivative dϕ σ descends to a smooth oneform on f −1 (U ) that is closed, restricts to a translation-invariant closed one-form on every fiber of f , and vanishes identically on the preferred section of f : M → S.
More generally, let A k (S, H) denote the space of smooth k-forms on S with coefficients in the holomorphic vector bundle H. By applying the construction from above on a suitable open covering -consisting of simply connected open sets on which H C is trivial -we obtain linear mappings with the property that the following diagram commutes: Let us consider a smooth one-form ξ ∈ A 1 (S, H) with ∇ξ = 0; it goes to a closed two-form Ξ ∈ A 2 (M, C). By construction, the restriction of Ξ to the fibers of f (and the section) is identically zero, and so Ξ ∈ L 1 A 2 (M, C) lies in the first step of the Leray filtration. We therefore get a well-defined linear mapping is an isomorphism, since L 2 H 2 (M, C) ≃ H 2 (S, f * C) vanishes. One can deduce from the construction above that the composition of the edge mapping with ℓ is the identity. This implies that ℓ is also an isomorphism. is spanned by the class of a fiber. In particular, H 2 (M , C) ⊥ injects into L 1 H 2 (M, C). Now the functoriality of the Leray spectral sequence gives us the following commutative diagram: It was shown in appendix B thatε is an isomorphism; we also know that ε is an isomorphism. Because ℓ = ε −1 , the assertion follows.
To illustrate the meaning of this result, let us consider an SL(2, Z) doublet harmonic one-form ξ ∈ A 1 (S, H) and the corresponding closed two-form Ξ ∈ A 2 (M, C). The cohomology class of Ξ lies in the image of the subspace H 2 (M , C) ⊥ , but unless one knows more about its behavior near the 24 singular fibers, one cannot say whether Ξ itself extends to a smooth closed form onM . It would be interesting to understand this point better.