On finiteness of Type IIB compactifications: Magnetized branes on elliptic Calabi-Yau threefolds

The string landscape satisfies interesting finiteness properties imposed by supersymmetry and string-theoretical consistency conditions. We study N=1 supersymmetric compactifications of Type IIB string theory on smooth elliptically fibered Calabi-Yau threefolds at large volume with magnetized D9-branes and D5-branes. We prove that supersymmetry and tadpole cancellation conditions imply that there is a finite number of such configurations. In particular, we derive an explicitly computable bound on the number of magnetic flux quanta, as well as the number of D5-branes, which is independent of the continuous moduli of the setup. The proof applies if a number of easy to check geometric conditions of the twofold base are met. We show that these geometric conditions are satisfied for the almost Fano twofold bases given by each toric variety associated to a reflexive two-dimensional polytope as well as by the generic del Pezzo surfaces dP_n with n=0,...,8. Physically, this finiteness proof shows that there exist a finite collection of four-dimensional gauge groups and chiral matter spectra in the 4D supergravity theories realized by these compactifications. As a by-product we explicitly construct all generators of the Kaehler cones of dP_n and work out their relation to representation theory.

1 Introduction M-theory or superstring compactification to four dimensions remains the most promising framework for the unification of the fundamental forces in Nature. The set of associated low energy effective theories which can arise in consistent compactifications is known as the string landscape. There have been many efforts to quantify this space, with the hope of uncovering observable properties shared by large classes of vacua which lead to novel insights in particle physics or cosmology. However, this has proven to be a very difficult problem deserving a multi-faceted approach. The traditional one is to study the effective scalar potential on moduli space and to examine its associated vacua; in general a variety of perturbative and non-perturbative effects are utilized to this end. For example, in the much studied moduli stabilization scenarios of Type IIB compactifications [1,2], these effects include superpotential contributions from background Ramond-Ramond flux and D-instanton effects. Increasingly more detailed realizations of these constructions have been studied in recent years; for progress on vacua with explicit complex structure moduli stabilization, see [3,4], and on constructing explicit de Sitter flux vacua with a chiral spectrum, see the recent [5]. While this progress is significant and provides excellent proofs of principle, a clear caveat to explicit construction of vacua is the enormity of the landscape.
Another approach is to study properties of the landscape more broadly. In Type IIB flux compactifications this has included, for example, the importance of four-form fluxes in obtaining the observed value of the cosmological constant [6]; issues of computational complexity, including finding vacua in agreement with cosmological data [7] and the systematic computation of non-perturbative effective potentials [8]; and the distribution and number of various types of supersymmetric and non-supersymmetric vacua [9,10]. Progress has also been made in understanding vacua in strongly coupled corners of the landscape. For example there has been much progress in F-theory, beginning with [11,12].
A final approach, which will be the one utilized in this paper, is to understand how consistency conditions and properties of the landscape differ from those of generic quantum field theories. The former case is motivated in part by the existence of a swampland [13] of consistent effective theories which do not admit a string embedding. There are a number of examples of limitations on gauge theories in the landscape not present in generic gauge theories. In weakly coupled theories with D-branes, Ramond-Ramond tadpole cancellation places stronger constraints [14][15][16][17] on low energy gauge theories than anomaly cancellation, which include additional anomaly nucleation constraints [18] on SU(2) gauge theories; see also [19] for a recent analysis of anomalies at strong coupling in F-theory; ranks of gauge groups are often bounded [20,21]; and the matter representations are limited by the fact the open strings have precisely two ends. While more matter representations are possible at strong coupling, the possibilities are still limited. For example, in F-theory the possible non-Abelian [22][23][24][25][26][27][28][29] and Abelian [30][31][32][33][34][35][36][37][38] matter representations are limited by the structure of holomorphic curves in the geometry.
In [39], Douglas and Taylor studied the landscape of intersecting brane models 1 in Type IIA compactifications on a particular Z 2 × Z 2 -orientifold 2 . They found that the conditions on D6-branes necessary for N = 1 supersymmetry in four dimensions, together with the D6-brane tadpole cancellation condition required for consistency of the theory, allow only a finite number of such D6-brane configurations 3 . In each configuration, the four-dimensional gauge group and matter spectrum can be determined explicitly. Thus, the finite number of D6-brane configurations gives a finite number of gauge sectors in a 4D supergravity theory that arise from these compactifications, and their statistics were studied explicitly. It is expected that the finiteness result which they obtained is a much more general consequence of supersymmetry and tadpole cancellation conditions, rather than a phenomenon specific to their construction. In fact, they proposed a potential generalization of their result to theories with magnetized D9-and D5-branes on smooth elliptically fibered Calabi-Yau threefolds, which can also be motivated by mirror symmetry, for example.
In this paper, we demonstrate that finiteness results are indeed much more general 1 See [40][41][42] for reviews of these compactifications and their implications for particle physics. 2 See [17,43] for a first construction of chiral N = 1 supersymmetric three-family models. 3 See [44] for a counting of three family vacua, that yields eleven such vacua.
phenomena, providing further evidence that the string landscape itself is finite. Specifically, in large volume Type IIB compactifications on many smooth elliptically fibered Calabi-Yau threefolds π : X → B, we prove that there are finitely many configurations of magnetized D9-and D5-brane satisfying Ramond-Ramond tadpole cancellation and the conditions necessary for N = 1 supersymmetry in four dimensions. We formulate a general, mathematical proof showing the existence of a computable, explicit bounds on the number of magnetic flux quanta on the D9-branes and on the number of D5-branes, which only depends on the topology of the manifold B and is in particular independent of its Kähler moduli, as long as they are in the large volume regime of X. These bounds involve simple geometric quantities of the twofold base B of X and the proof applies to any base B that satisfies certain geometric conditions, that are easy to check and summarized in this paper. Furthermore, we show that these conditions are met by the almost Fano twofold bases B given by the toric varieties associated to all 16 reflexive two-dimensional polytopes and the generic del Pezzo surfaces dP n for n = 0, . . . , 8. This list in particular includes also the Hirzebruch surfaces F 0 = P 1 × P 1 , F 1 = dP 1 , and F 2 . In this work, we focus on the finiteness question only, leaving the analysis of gauge group and matter spectra for this finite set of configurations to future work. This paper is organized is follows. In section 2 we provide the relevant background Type IIB compactifications with magnetized D9-and D5-branes and a elliptically fibered Calabi-Yau threefolds at large volume. We first discuss the tadpole and supersymmetry conditions of general such setups, then present a basic account on elliptically fibered Calabi-Yau threefolds and end with a detailed discussion of the geometries the twofold bases B = F k , dP n and the 16 toric twofolds. In section 3 we prove the finiteness of such Dbrane configurations. We begin by rewriting the tadpole and supersymmetry constraints in a useful form for the proof and make some definitions, then show the power of these definitions by proving finiteness on P 2 . Finally, we prove the existence of explicit bounds on the number of fluxes and D5-branes, that apply certain geometric conditions on B are satisfied. In section 4 we conclude and discuss possibilities for future work. In appendix A we discuss the detailed structure of the Kähler cone of generic del Pezzo surfaces dP n and give the proof of positive semi-definiteness of certain intersection matrices on these Kähler cones, which is essential for the proof. In appendix B we summarize the geometrical data of the considered almost Fano twofolds which is necessary to explicitly compute the bounds derived in this work.
While finishing this manuscript we learned about the related work [45] in which heterotic compactifications and their F-theory duals are constructed systematically.

Magnetized Branes on Elliptically Fibered Calabi-Yau Manifolds
We consider an N = 1 compactification of Type IIB string theory on a Calabi-Yau threefold X with spacetime-filling D5-branes, magnetized D9-branes, i.e. D9-branes with magnetic fluxes 4 , and an O9-plane. We group the D9-branes into stacks of N α branes and their orientifold image branes. The corresponding line bundle magnetic fluxes in H (1,1) (X, Z) are denoted by F α , respectively, −F α for the image brane. In addition, we add stacks of D5-branes wrapping a curve Σ D5 .
In the following discussion of these models 5 we mainly follow the notations and conventions of [39], to which we also refer for more details. For a concise review see [41].

Tadpole Cancellation and SUSY Conditions
D-branes carry Ramond-Ramond charge and source flux lines that must be cancelled in the compact extra dimensions, in accord with Gauss' law. These give rise to the so-called tadpole cancellation conditions. The D5-brane tadpole cancellation conditions are (we note a sign difference between the D5-tadpoles 6 in [39] and [41]; here, we use the sign in [41]) where D I is a basis of divisors on X, K(·, ·, ·) is the classical triple intersection of three two-forms or their dual divisors, where we denote, by abuse of notation, a divisor D I and its Poincaré dual by the same symbol. Furthermore, we define the curvature terms where c 2 (X) is the second Chern-class on X and Σ D5 is the curve wrapped by all D5-branes. The integral wrapping numbers n D5 I are positive if Σ D5 is a holomorphic curve and the D I are effective divisors. Following [41], the D9-brane tadpole cancellation condition reads Compactification of Type IIB string theory on a Calabi-Yau manifold gives rise to a four-dimensional N = 2 supergravity theory. An O9-orientifold breaks half of these supersymmetries and give rise to an N = 1 supergravity theory at low energies. Only D9and D5-branes can be added in a supersymmetric way to this orientifold. However, this requires aligning the central charges Z(F α ) of the branes appropriately with the O9-plane. For consistency with the supergravity approximation, we have to assume that the Kähler parameters of the Calabi-Yau threefold X are at large volume. In this case, the conditions on the central charges 7 necessary for N = 1 supersymmetry, with J denoting the Kähler form on X, reduce to to which we will refer in the following as the SUSY equality respectively SUSY inequality.

Smooth Elliptic Calabi-Yau Threefolds
We study compactifications where X is a smooth elliptically fibered Calabi-Yau threefold over a complex two-dimensional base B, π : X → B, with a single section σ : B → X, the zero-section. The class of the section σ is the base B. By the adjunction formula and the Calabi-Yau condition, the section σ obeys the relation where c 1 denotes the first Chern class of the base B. For a smooth threefold the second cohomology is given by H (1,1) (X) = σH 0 (B)⊕π * H (1,1) (B). A basis of H (1,1) (X) generating the Kähler cone of X is given by with Poincaré duality implied when discussing divisors. The divisors D i , i = 1, . . . , p, are inherited from generators of the Kähler cone of the base, by abuse of notation denoted by the same symbol as their counterparts in B. The divisor D 0 is dual to the elliptic fiber E in the sense that it does not intersect any curve in B, i.e. D 0 · σ · D i = 0 by (2.5), and obeys D 0 · E = 1. We note that E is an effective curve. We emphasize that the requirement of a smooth elliptically fibered X, which means that the fibration can at most have I 1 -fibers, restricts the choice of two-dimensional bases B. The bases we consider here are smooth almost Fano twofolds, which are the nine del Pezzo surfaces dP n , n = 0, . . . , 8, that are the blow-ups of P 2 at up to eight generic points, the Hirzebruch surfaces F k , k = 0, 1, 2 and the toric surfaces described by the 16 reflexive two-dimensional polytopes. For these bases, the elliptic fibration X is smooth.
We abbreviate the triple intersections of three divisors on X as K IJK = K(D I , D J , D K ). In the particular basis (2.6), we obtain the following structure of the triple intersections, where the first equation is a property of the fibration and the second and third relations 7 In general, the central charge (and also the Kähler potential on the Kähler moduli space) receives perturbative and non-perturbative α ′ corrections. Recently it has been understood [48][49][50][51] that these corrections are captured by the so-called Gamma classΓX on X rather than √ T dX. Since we study compactifications at large volume, these corrections can be neglected.
can be derived using (2.5). We also introduce the p × p-matrix which defines a bilinear pairing on divisors on the base B. For the cases we consider here its signature is (1, p − 1) for F k and dP n , n = 1, . . . , 8, and (1, 0) for P 2 = dP 0 . Note that it will be convenient at some places in this work to view H (1,1) (B) as a p-dimensional vector space equipped with an inner product (2.8). We denote the inner product of two vectors v, w in H (1,1) (B) simply by C(v, w). In addition, we view the first Chern class c 1 of B, the fluxes F α and the Kähler form J as column vectors Here the components of these vectors are defined via the expansion w.r.t. the D I in (2.6), where b i ∈ Q + , m α I ∈ Q and j I ∈ R + . 8 We emphasize that the flux quantization condition F α ∈ H (1,1) (X, Z) can be equivalently written as where C is any curve in X. Noting that the elliptic fiber E and the Kähler generators D i of B are integral curves in X, this implies, using (2.10), We conclude by noting that for smooth elliptically fibered Calabi-Yau threefolds, the second Chern class c 2 (X) can be computed explicitly, see e.g. [52] for a derivation. By adjunction one obtains c 2 (X) = 12σ · c 1 + π * (c 2 + 11c 2 1 ) with c 2 the second Chern class on B, employing the relation (2.5). Using this and (2.7) we evaluate the curvature terms in (2.2) as which is straightforward to evaluate for concrete bases B. 8 We allow here for rational coefficients m α I in the expansion of the fluxes F α that are in the integral homology H (1,1) (X, Z) in order to account for the possibility of Kähler generators DI that only span a sublattice of H (1,1) (X, Z) of index greater than one. This can happen for non-simplicial Kähler cones.

Basic Geometry of Almost Fano Twofolds
In this section we briefly discuss those geometrical properties of the almost Fano twofolds B = F k , dP n and the toric surfaces. The discussion in this section is supplemented by the explicit computations of the Kähler cones of dP n in appendix A and the summary of the key geometric data of F k , dP n in Appendix B, which is critical for the proof in Section 3.

Hirzebruch Surfaces
The Hirzebruch surfaces F k are P 1 -bundles over P 1 of the form F k = P(O ⊕ O(k)). There is an infinite family of such bundles for every positive k ∈ Z + 0 . The isolated section of this bundle, S, and the fiber F are effective curves generating the Mori cone and spanning the entire second homology (2.14) Their intersections read From this we deduce that the generators D i , i = 1, 2, of the Kähler cone, which are defined to be dual to the generators in (2.14), read The Chern classes on F k read which implies that the vector b in (2.9) is b = (2 − k, 2) T . Using (2.15), we compute the triple intersections in (2.7), in particular (2.8), as from which the curvature terms in (2.13) immediately follow as We emphasize that F k by means of (2.17) is Fano for k < 2 and almost Fano for k = 2, since the coefficient b 1 = 2 − k ≥ 0. The general elliptic Calabi-Yau fibration X over F k with k = 0, 1, 2 is smooth and develops I 3 -singularities for k = 3 up to II * -singularities for k = 12, before terminal singularities occur for k > 12 [53]. Thus, we focus on the Hirzebruch surfaces with k = 0, 1, 2.

Del Pezzo Surfaces
The Fano del Pezzo surfaces dP n are the blow-up of P 2 at up to eight generic points. 9 Their second homology group is spanned by the pullback of the hyperplane on P 2 , denoted by H, and the classes of the exceptional divisors, denoted as E i , i = 1, . . . , n, (2.20) The intersections of these classes read The Chern classes on dP n read The Mori cone of dP n for n > 1 is spanned by the curves Σ obeying [56,57] where [K −1 dPn ] is the anti-canonical divisor in dP n , who is dual to c 1 (dP n ). By adjunction, we see that the curves obeying (2.23) obey the necessary condition for being P 1 's. By solving the conditions (2.23) with the ansatz a 0 H + n i=1 a i E i for a 0 , a i ∈ Z, we obtain a cone that is simplicial, i.e. generated by h (1,1,) (B) = 1 + n generators, for n = 0, 1, 2 and non-simplicial for n > 2. The number of generators, beginning with dP 2 , furnish irreducible representations of A 1 , A 1 × A 2 , A 4 , D 5 , E n , for n = 6, 7, 8, which concretely are 3, 2 ⊗ 3, 10, 16, 27, 56, 240. 10 For the simplicial cases the Mori cone reads and we refer to appendix A for more details on the non-simplicial cases. Consequently, also the Kähler cones of the dP n , which are the dual of the Mori cones defined by (2.23), are non-simplicial for n > 2. The Kähler cone is spanned by rational curves Σ obeying which again implies by adjunction that Σ = P 1 . The solutions over the integers of these conditions yield the generators of the Kähler cone of dP n which again follow the representation theory of the above mentioned Lie algebras. The number of generators, starting with dP 0 , is 1, 2, 3, 5, 10, 26, 99, 702 and 19440, see appendix A. In the simplicial cases, 9 See [54,55] for recent computations of refined BPS invariants on del Pezzo surfaces as well as their interpretation in M-/F-theory. 10 The genuine roots in H2(dPn) are the −2-curves orthogonal to [K −1 dPn ], i.e. αi = Ei − Ei+1, i = 1, . . . , n − 1, αn = H − E1 − E2 − E3 for n > 2. These act on H2(dPn) by means of the Weyl group, cf. [56].
the Kähler cone generators read (2.26) Generically, for n ≥ 2 the vector c 1 (dP n ) is the center both of the Kähler and Mori cone. This implies that for all del Pezzo surfaces, the coefficients b i are positive. For the simplicial Kähler cones, this can be computed explicitly. For the non-simplicial cases we will argue in appendix A, that a covering of the Kähler cone by simplicial subcones, i.e. subcones with h (1,1) generators, with all b i ≥ 0 always exists. We note that for all dP n , the defining property of the Kähler cone (2.25) implies with (2.7) the intersections For the three simplicial cases of P 2 , dP 1 and dP 2 , we compute the matrices (2.8) in the basis (2.26) as We emphasize that the del Pezzos dP n by means of (2.27) is Fano for n < 9 and almost Fano for n = 9, since c 2 1 = 0. The surface dP 9 is the rational elliptic surface. Its Mori cone is the Mordell-Weil group of rational sections by (2.23). Thus, it as well as its dual Kähler cone is infinite dimensional. We will only consider the Fano del Pezzo surfaces dP n , n < 9.

Toric Surfaces from Reflexive Polytopes
Toric surfaces obtained from fine star triangulations of reflexive polytopes are smooth almost Fano twofolds. 11 There are 16 such polytopes in two dimensions, which are displayed in Figure 1.
A number of these twofolds are simply toric descriptions of previously described surfaces. Specifically, these are P 2 , dP 1 , dP 2 , dP 3 , F 0 and F 2 which are described by polytopes 1, 3, 5, 7, 2 and 4, respectively. From the form of some of the other polytopes it is clear that they can be obtained from P 2 , dP 1 , dP 2 , or dP 3 via toric blow-up. For example, reflecting polytope 7 through the vertical axis going through its center and performing a toric blow-up associated to the point (−1, 1), one obtains polytope 12. Thus, the smooth Fano surface associated to polytope 12 is a toric realization of dP 4 at a non-generic point in its complex structure moduli space.
The toric varieties associated to all these 16 reflexive polytopes can be constructed explicitly using the software package Sage [59]. The intersections (2.7), (2.8) are readily constructed in a given fine star triangulation and the Kähler cone can be obtained. We summarize the geometric data necessary for the computation of the bounds derived below in the proof in Appendix B.

Finiteness of Magnetized D9-& D5-brane Configurations
In this section we bound the number of possible gauge sectors arising in the considered compactifications of Type IIB string theory. As emphasized in section 2, the number N α of branes in a stack and their associated magnetic fluxes F α are subject to the consistency conditions imposed by tadpole cancellation conditions (2.1), (2.3) and the SUSY conditions (2.4). Since the numbers N α of D9-branes are bounded by (2.3), it is therefore the goal of this proof to bound the flux quanta F α and the number of D5-branes in Σ D5 .
Most of the proofs in this work have the same basic structure. The key point is to find a bound on the number of different flux configurations F α and D5-branes Σ D5 at an arbitrary point in the large volume regions of Kähler moduli space, i.e. a bound that is independent of the Kähler moduli. As we will see, proving this requires an intriguing interplay between both the tadpole conditions (2.1), (2.3) and the SUSY conditions (2.4), a general rubic which was also used in the proof of [39]. In addition, the following proof applies if a list of geometrical properties, listed at the beginning of Section 3.3, are satisfied. These are obeyed for the considered examples B = F k , dP n and the toric surfaces.
Before delving into the details of the proof, let us introduce a very important notation. Because of their fundamentally different contributions to (2.1), (2.3) and (2.4) it is useful to split D9-brane stacks into to qualitatively different types according to their flux quanta. We denote D9-brane stacks with m α 0 = 0 as β-branes, and those with m α 0 = 0 as γ-branes: In addition, in the rest of this section we label fluxes of a β-and γ-brane by m β I and m γ I , respectively.
We begin in Section 3.1 by preparing for the general finiteness proof by writing out the tadpoles and SUSY conditions of Section 2 for elliptically fibered Calabi-Yau threefolds X. We also make certain definitions and deduce a number of simple inequalities and bounds, that will be essential for the later discussion. Then, in Section 3.2 we prove finiteness for the special base B = P 2 , which will demonstrate the usefulness of the definitions of the previous section and serve as a warm-up for the general proof in Section 3.3.

Prerequisites: Definitions & Basic Inequalities
In this section we make some general definitions and observations necessary to formulate and organize the proof in Section 3.3.
As a starting point, we observe that the SUSY conditions (2.4) must be satisfied by each brane stack, but only involve the direction along the Kähler class, whereas the tadpole conditions (2.1) have to be obeyed for each divisor D I , but are summed across brane stacks. Thus, in order to bound each component m α I of every flux vector m α , labelled by the brane stack α, it is crucial is to identify quantities, that enter both type of constraints, when rewritten in a particular form.
To this end, we write out the tadpole conditions explicitly in the basis of divisors (2.6). The conditions (2.1) for I = 0, to which we will refer in the future as the 0 th -tadpole, reads where we have emphasized the respective contributions from β-branes and γ-branes. For I = i, to which we will refer as the i th -tadpole, the tadpole (2.1) reads 3) We note that the first term in t β k , can be written as j b j C ij = K 00i which is an integer by (2.18), (2.27) and Table 5. The quantities t β i can be defined for β-branes and play an important in the proof, because they naturally appear in the SUSY constraints. We emphasize that while both β-branes and γ-branes contribute to the 0 th -tadpole condition, only β-branes contribute to the i th -tadpole as is indicated by the braces in (3.2), (3.3).
We note that one can immediately deduce a lower bound on the left hand side of (3.2) and (3.3) by setting the positive numbers n D5 I = 0: (3.4) These lower bounds on the i th -tadpoles imply, as we will see, that if the t β i are bounded above, then they are automatically bounded below. This can be seen by bringing the bounded positive contribution to the left hand side of (3.4).
For β-branes, which have m β 0 = 0, it useful to divide the SUSY equality (2.4) by m β 0 . Using (2.8) and the vectors (2.9), we write the first condition in (2.4) as obtain (3.5) The inequality in (2.4) for β-branes can be combined with the SUSY equality (3.5) as follows. By dividing the SUSY inequality in (2.4) by j 0 and subtracting the SUSY equality (3.5), we obtain after a few lines of algebra the following inequality: This can equivalently be written in the form and we see that the expression t β i , which explicitly appears in the i th -tadpole conditions in (2.13), appears also in this manipulation of the SUSY constraints.
We note that (3.7) can be related to the tadpole conditions. By multiplying (3.7) by N β (m β 0 ) 2 and summing over β, we employ the right hand side of (3.3) to obtain (3.8) where we set n D5 i = 0 in the last inequality. This condition is used throughout the proof. Next, we demonstrate that it is possible to also rewrite the SUSY equality (3.5) and the 0 th -tadpole (3.2) in a form, that manifestly contains the quantities t β i . To this end, we first define for each distinct pair of indices {i, k}, i = k, the matrix M {i,k} whose (j, l)-th entry in the basis D i is: where x {i,k} ∈ Q + is a non-negative rational number. This number has to be chosen such that its corresponding M {i,k} is positive semi-definite. With this definition, the SUSY equality (3.5) and 0 th -tadpole (3.2) can be written as 10) respectively, where we indicated the contributions from β− and γ-branes by braces and used the short hand notationt As we will see, the proof of Section 3.3 applies whenever the M -matrices in (3.9) are all positive semi-definite. In fact, for all the bases B of the threefold X considered, this matrix is positive semi-definite. For P 2 , P 1 × P 1 , dP 1 , dP 2 , and F 2 the M -matrix can be readily computed in the Kähler cone basis, and indeed, it is positive semi-definite. However, for dP n with n ≥ 3 there exists a significant complication since in these examples, the Kähler cone is non-simplicial, as mentioned in Section 2. In these cases, we cover the Kähler cone by simplicial subcones consisting of h (1,1) generators and compute the M -matrix (3.9) for this choice. As demonstrated in Appendix A, for dP n , n < 9, the M -matrices are positive semi-definite for all such subcones. For the toric surfaces, we refer to Appendix B for positive semi-definiteness of the matrices (3.9). Thus, for the rest of the paper we can assume that all M {i,k} are positive semi-definite for these bases.

Warm Up: Finiteness for Elliptic Fibrations over P 2
Before proceeding on to more difficult examples, let us prove finiteness in the simplest example of B = P 2 . In particular, in this example we will demonstrate the usefulness of the derived inequality (3.6) respectively (3.8).
For an elliptically fibered Calabi-Yau threefold X over B = P 2 , the relevant geometrical data following from (2.22), (2.27), (2.28) and (2.29) is: (3.12) Using this the inequality (3.6) reduces to for each β-brane and the tadpole for D 1 reads (3.14) By (3.13), the right hand side of (3.14) must be negative. Thus we have a bound for n D5 1 , given by n D5 1 < T 1 . In addition, for each β-brane we deduce from (3.13) that Notice that |K 001 m β 0 + 2K 011 m β 1 | is a non-zero integer by virtue of the strict inequality (3.13). This implies the bound Thus we have shown that the magnetic flux quanta m β associated to β-branes are bounded. A bound on the flux quanta of γ-branes is straightforward to obtain. The SUSY equality in (2.4) for each γ-brane is K 011 is strictly positive, we must have m γ 1 = 0. Since a γ-brane by definition has m γ 0 = 0, the flux quanta of γ-branes are trivially bounded. This completes the proof for B = P 2 .

Proving Finiteness for Two-Dimensional Almost Fano Bases
In this section we present the general proof of the finiteness of the number of consistent Type IIB compactification with magnetized D9-branes on elliptically fibered Calabi-Yau threefolds. As discussed before the bases B for which the presented proof has been developed are the two-dimensional almost Fano varieties. These are the del Pezzo surfaces dP n , n = 0, . . . , 8, with the case of dP 0 = P 2 discussed in the previous section 3.2, the Hirzebruch surfaces F k , k = 0, 1, including the almost Fano F 2 , as well as the toric surfaces.
The geometrical properties that are essential for the following proof are the smoothness of the generic elliptic Calabi-Yau fibration over them, as well as the following list of properties: • positive semi-definiteness of matrices M {i,k} , for i = k, defined in (3.9).
• positivity of the coefficients b i in (2.10), i.e. b i ≥ 0 for all i.
• positivity and integrality of K 00i as defined in (2.7) i.e. has one positive and n negative eigenvalues.
• positivity of the Kähler parameters j i and validity of the large volume approximation, i.e. j i ≫ 1 for all i.
We claim that the proof presented below applies to all bases B that obey these conditions. The validity of these properties for the considered bases is shown in the Appendices A and B. As discussed there, the only subtlety arises for the higher del Pezzos dP n , n > 2, which have non-simplicial Kähler cones. In this case, the indices i refer to the generators of a suitably chosen simplicial subcone, such that the above list of properties hold. As argued in appendix A it always exists a covering of the Kähler cones of the dP n by simplicial subcones, such that for each subcone in the covering all the above properties hold.
The following proof is organized as follows. We already introduced the two types of branes, denoted β-and γ-branes, to distinguish between branes with respectively without fluxes along the fiber E, i.e. E F β = 0 respectively E F γ = 0. First we prove in Section 3.3.1 that there is only a finite number of flux configurations on β-branes. Then in Section 3.3.2 we show finiteness of the numbers of D5-branes n D5 I . Finally, we conclude the proof in Section 3.3.3 by showing finiteness of the number of flux configurations on γ-branes.

Bounds on β-branes
Bounds on m β 0 In the following we obtain the bound on the flux component m β 0 for all β-branes where the maximum is taken over all generators of the specific subcone of the Kähler cone. We note that here and in the rest of the paper, all minima and maxima on T i and x {i,k} are taken across generators of the specific subcone we are in. However, except the minimum on T i in theorem 4, the reader is free to take all other maxima and minima across all generators of the Kähler cone, for easy computation purposes. For del Pezzo surfaces this yields max(T i ) = 36, for the Hirzebruch surfaces F k it is max(T i ) = 24 + 12k and for the toric surfaces we can read off this bound from Table 5. We begin considering inequality (3.7). In fact, since K 000 ≥ 0, (3.7) implies Next we multiply this by N β (m β 0 ) 2 and sum over β to obtain, using (3.3), where we set the positive n D5 i = 0 for all i in the last inequality. This lower bound on the sum over β also implies because by (3.19) all summands are negative. This motivates the following definition: For special branes, we immediately conclude from (3.21) that Here we have used (3.21) in the second inequality, and that t β 3) in the last inequality: it is an integer because both its first term, K 00i m β 0 , and the second term, the flux F β integrated over the integral class D i , are integers by (2.12). It is non-zero because t β i is non-zero by the definition of special branes, and m β 0 is non-zero by the definition of β-branes. Thus for special branes, the flux quantum m β 0 is bounded as We will show that mixed branes have a even smaller bound for their |m β 0 |. Let us first make an observation that will facilitate the identification of special branes.
Suppose it is not a special brane. Then by definition we cannot have t β i < 0 ∀i. Remark 1 also forbids t β i ≥ 0 ∀i. Thus there exists a pair of i, k such that t β i and t β k are of opposite signs 12 . Writing the RHS of (3.10) in terms of this particular pair of t β i , t β k , we observe that RHS of (3.10) where in the first inequality we dropped all negative terms on the RHS of (3.10) and in the last inequality we employed the lower bound on (3.8). Now (3.24) shows that the LHS of (3.10) is at least quadratic in the j i 's and grows as the Kähler volume of B. However, inequality (3.25) implies that the RHS of (3.10) is at most on the order of j i /j 0 . In the limit of all j I large, which in particular implies large volume of B, the LHS of (3.10) has to be greater than the RHS of (3.10). Thus, the SUSY equality (3.10) is violated. Our initial assumption that this β-brane is not a special brane must be wrong; it must be a special brane.
Remark 2. The argument in Lemma 1 about the growth of the two sides of the SUSY equality (3.10) can be further substantiated for concrete bases B. For all F k , we can check that we have LHS of (3.10)> RHS of (3.10) when j I ≥ 3 ∀I. This is clearly the case if the supergravity approximation is supposed to be valid. For dP n , the matrix C(·, ·) has signature (1, n), i.e. we can have C(j, j) = 0 for j = 0 and the above argument might be invalidated. However, we can only have C(j, j) = 0 if the Kähler form j B = i j i D i on B is on the boundary of the Kähler cone. This means that the Kähler volume of B is zero or cycles in B have shrunk to zero which clearly invalidates the supergravity approximation.
Thus, it remains to bound m β 0 for β-branes satisfying 0 . For such β-branes, we observe The following argument still applies if one of them is zero.
-17 -by β ′ , multiply the above inequality by N β ′ (m β ′ 0 ) 2 and sum over β ′ : Here in the second line we extended the sum over β ′ to the sum over all β-branes; by (3.19) each summand is negative, thus, extending the sum only decreases it. In the third line we have used (3.3). With (2.13) and the last line of the above inequality we obtain This is an even smaller bound than (3.23) derived previously for special branes satis-  Table 1.

Branes
Special branes with Special branes with Mixed branes γ-branes

Bounds on the number of Solutions to the Vector m β
We begin by noting that (3.3) can be viewed as the following matrix multiplication equation The invertible matrix 2C gives a 1-1 correspondence between the vector m β and the vector t β . Thus, in order to show that there are finitely many solutions for the vector m β , we can equivalently show that there are finitely many solutions for the vector t β . We can accomplish this by showing each component t β i is bounded. We recall that it suffices to prove each t β i is bounded above: since (m β 0 ) 2 is bounded as we have just shown, an upper bound also implies a lower bound by the second inequality in (3.4). Since the t β i of special branes are by definition bounded above by 0, see Definition 1, we only have to bound the t β i of mixed branes. It is helpful for finding this upper bound on the t β i , to first to summarize the positive and negative contributions to a tadpole. We obtain the table 2, where we have indicated in parenthesis where the corresponding result will be proven.

Special branes
Mixed branes γ-branes  Next, we proceed with proving the results of this table. We begin with the following Proposition 1. γ-branes only contribute negatively to the 0 th -tadpole (3.2). Furthermore, any γ-brane contributing zero is the trivial brane, i.e. m γ I = 0 for all I.

Proof.
A γ-brane's contribution to the 0 th -tadpole is proportional to C(m γ , m γ ) by (3.2). In addition, for γ-branes, the SUSY equality in (2.4) reads as can be seen by setting m γ 0 = 0 and using the intersection relations (2.7). We recall that C has Minkowski signature (1, 1) for F k and (1, n) for dP n and the toric surfaces. The vector b 2 + j j 0 is time-like, since Here, the first term on the RHS of (3.32) is positive because K 000 = 8 for F n , 9 − n for dP n and Table 5 applies for toric surfaces. The second term is positive because j I > 0 and for F k , K 001 = 2, K 002 = 2 + k; for dP n , K 00i = 2, 3; for toric surfaces, all relevant entries in Table  5 are positive. Finally, the third term is positive because it is proportional to the volume of B. By (3.31) the vector m γ is orthogonal to a time-like vector, thus, it is space-like, i.e. 0 > C(m γ , m γ ), unless it is the zero vector, which trivially has C(m γ , m γ ) = 0.
Proposition 2. Only special branes contribute positively to the 0 th -tadpole. This is equivalent to the fact, that mixed branes only contribute negatively to the 0 th -tadpole.
Proof. We recall that the 0 th -tadpole can be written in the form (3.10) for arbitrary choices of {i, k}. Focusing on its RHS, we note that the second term is always negative by the positive semi-definiteness of the matrices M {i,k} . Furthermore, the third term is always negative by Proposition 1. Thus, the RHS of (3.10) can only be positive, if the first term on the RHS is positive. This implies that allt β i = K 00i + t β i , cf. (3.11), have to be of the same sign: if not, there exists a pairt β i ,t β k of opposite sign. Writing the RHS of (3.10) in terms of this pair, the first term is negative and the entire RHS of (3.10) would be negative.
If allt β i are negative, all t β i have to be strictly negative since each K 00i are strictly positive. By Definition 1, a β-brane with this property is a special brane. If thet β i are all positive, then we have 1 2 p i=1t ≥ 0, and by Lemma 1 it is also a special brane.

Corollary 1.
A special brane that contributes positively to the 0 th -tadpole must havet β i < 0 for all i.
Proof. Recall from the proof of Proposition 1 that a special brane which contributes positively to the 0 th -tadpole must have allt β i of the same sign. If they are all negative, we are done. Thus, assume allt β i ≥ 0. We prove this is not possible using a similar argument as in the proof of Lemma 1.
Since K 00i > 0 ∀i and we are considering a special brane, i.e. all t β i < 0, havingt β i = (K 00i + t β i ) ≥ 0 ∀i means |t β i | ≤ K 00i ∀i. Now consider the SUSY equality (3.10). Since the M -matrix is positive semi-definite, the RHS of (3.10) is at most Also, by the last inequality in (3.8), we have i.e. (m β 0 ) 2 is smaller than a linear combination of j i /j 0 , so is 1 4 K 000 + 3 2 x {i,k} K 00i K 00k (m β 0 ) 2 , since the prefactor 1 4 K 000 + 3 2 x {i,k} K 00i K 00k ∼ K 000 . However,t β i > 0 for all i means ≥ 0, which implies that the LHS of (3.10) is at least 3C(j, j) which is quadratic in the j i . Thus, in the limit that all j I are large, the LHS of (3.10) will always be greater than its RHS, thus violating the SUSY equality. 13 This concludes the proof of the results in Table 2. We prove three more important Lemmas before we finally derive the bounds on t β i . For the rest of the proof, we will label special branes that contribute positively to the 0 th -tadpole by β s , and mixed branes by β m . We also use the simplified notation (3.34) Lemma 2. For any index i, we have the following inequality: Proof. By (3.4), we have a lower bound for the i th -tadpole. Thus, we have the following inequality for the i th -tadpole: where in the first inequality, we split terms in the sum of (3.4) into positive and negative contributions, as indicated in the summation by t βm i ≥ 0 and t βm i < 0. In the second inequality, in the first term, we only kept those special branes in the sum that contribute positively to the 0 th -tadpole, which are labelled by β s . In the second line, we added three more negative terms and in the next equality, we combined them into three sums using (3.11), that yield the two sums in the last line.
Lemma 3. For any pair of a special brane that contributes positively to the 0 th -tadpole and a mixed brane, there exists an index k such thatt βm k is strictly negative and |t βm k | > |t βs k |. In particular |t βm k | − |t βs k | ≥ 1 6 . (3.37) Proof. Suppose the converse is true, i.e. for some pair of a special brane that contributes positively to the 0 th -tadpole and a mixed brane, there does not exist an index k such that t βm k is strictly negative and |t βm k | > |t βs k |. Then, consider the difference of the SUSY equality (3.10) for the mixed brane and for the special brane: LHS of (3.10) for the mixed brane -LHS of (3.10) for the special brane = RHS of (3.10) for the mixed brane -RHS of (3.10) for the special brane (3.38) We will show that (3.38) will be violated. To simplify our notation, we will in the following denote the difference of the LHS and RHS in (3.38) by ∆ LHS and ∆ RHS , respectively. First consider the difference ∆ LHS . The first term, 3C j j 0 , j j 0 j 2 0 , is the same for both branes. Thus, by expanding everything out and using (3.3) and (3.11), we obtain By Corollary 1, since the special brane contributes positively to the 0 th -tadpole,t βs i < 0 for all i. Also notice that the mixed brane must have at least one i for whicht βm i > 0, because by definition, a mixed brane must have at least one i for which t βm i ≥ 0, and for this i, by (3.11) and the positivity of K 00i ,t βm i > 0. Labelling those i for whicht βm i > 0 as i+, and those i for whicht βm i ≤ 0 as i−, (3.39) becomes Here in the last step we dropped the second sum, which is positive, because by assumption there does not exist an index k such thatt βm k is strictly negative and |t βm k | > |t βs k |. By Table 1, we have |m βm 0 | ≤ 3. For the special brane, sincet βs i < 0 for all i, we have Thus, the bound |m βs 0 | ≤ 3 in the third column of Table 1 applies. Thus, both |t βs i | > 0, t βm i+ > 0 are at least a third of a non-zero integer: Hence, (3.40) becomes where in the last step we dropped the term containing the positive b i . We have discussed that at least one index i+ exists. With j I ≫ 1 for all I, (3.43) shows that the difference between the LHS of (3.10) for the two branes is large. Next, we show that the difference ∆ RHS between the RHS of (3.10) for the two branes is much smaller. Starting from the RHS of (3.10) for the special brane we note the identity Since the special brane contributes positively to the 0 th -tadpole, the LHS of (3.44) is positive. We also recall thatt β i < 0 for all i by Corollary 1, which implies that the second last term on the RHS of (3.44) is strictly negative, as b i ≥ 0. In addition, the last term on the RHS is always negative for the bases B we consider. Thus, the term in square brackets on the RHS of (3.44), which is the RHS of (3.10), must be strictly positive. In particular, it must have a bigger magnitude than that of (the next to last term and) the last term: RHS of (3.10) for the special brane > K 000 (m βs 0 ) 2 .
Next, consider the RHS of (3.10) for the mixed brane. Since it is a mixed brane, we can pick a pair of t βm i , t βm k of opposite signs to make the second term of the RHS of (3.10) negative. By the positive semi-definiteness of the M -matrix, the third term of the RHS of (3.10) is always negative. Thus RHS of (3.10) for the mixed brane ≤ 1 4 K 000 (m βm 0 ) 2 .
-23 -Hence, we obtain, using again the bounds on m β 0 from Table 1, By comparison of (3.43) and (3.45), using the property j I ≫ 1 for all I, we see that we will always have ∆ LHS > ∆ RHS , (3.46) which clearly violates (3.38). Finally we prove (3.37). Recall botht βs k ,t βm k are either integers, half integers or a third of an integer. Since |t βm k | > |t βs k |, their difference is at least a non-zero integer divided by their common denominator, which is 6, i.e. (3.37) applies.
We make two useful definitions for the next lemma before stating it. Recall that the contribution of a mixed brane to the 0 th -tadpole is negative, cf. Table 2, and is given by the first term in (3.10): Similarly, for a special brane that contributes positively to the 0 th -tadpole, its contribution is also given by the first term in (3.10): holds, then h 1 βm, + |R βm | > h 2 βs S βs . In particular, where the minimum is taken across all pairs {i, k} of indices of Kähler cone generators.
Proof. We introduce a partition of unity {f βs } βs , and, for every index β s , a partition of unity {g βs,βm } βm, + with 15 βs f βs = 1 , βm,+ g βs,βm = 1 , f βs , g βs,βm ∈ Q , 0 < f βs , g βs,βm ≤ 1 , (3.52) and the property Inserting unity as 1 = βs f βs = βm,+ , g βs,βm , we obtain the obvious identity  Here, the pair {i, k} is chosen so that the index i is the one for which (3.50) holds, and the index k is chosen such that the inequality |t βm k | − |t βs k | ≥ 1 6 of Lemma 3 holds for the pair (β s , β m ) of special and mixed brane in (3.55). We emphasize that the choice of this index k depends on the brane pair (β s , β m ) and thus might be different for each summand in (3.54). Next, we drop the positive semi-definite M -matrix terms in (3.55) to get where we have used that the coefficients of |t βm k |, |t βs k | in the first line are equal by (3.53). In addition, we have removed the aforementioned implicit dependence of the index k on (β s , β m ) by taking the minimum over all {i, k}. where we performed the sum over β s and used βs f βs = 1, cf. (3.52).
Now we are finally ready to show that every t β i has an upper bound.
Theorem 2. The t β i for all i and β are bounded from above as where the minimum and maximum have to be taken over all pairs {i, k} of indices of Kähler cone generators.
Proof. We derive the above bound for t β i for an arbitrary index i. By Lemma 2, we either have βs N βs t i βs (m βs We consider each case separately: In other words, we have a relation as in (3.50) Starting with the first inequality in (3.4) and employing (3.47), (3.48), we obtain where in the first inequality we only kept negative contributions to the 0 th -tadpole from mixed branes witht βm i ≥ 0 (see Table 2). In the second line we used Lemma 4. In the third line we plugged in the definition (3.47) of R βm and dropped the M-matrix term. Moreover, in (3.47) we picked our choice of the pair {i, k} so that i is the same index i that we want to derive a bound for t β i , and k such that |t βm k | ≥ 1 3 . Indeed, since a non-zerot βm k is at least a third of an integer, we only have to argue that a k with a non-zerot βm k exists. But this is true since otherwise which by Lemma 1 implies that this brane would be a special, not a mixed brane. The remaining two lines of (3.60) are just algebra. Thus, we have the following bound on t i : . (3.61) In this case we are in a special case of (3.50) with h 1 = 1, h 2 ≤ 1 and (3.62) Analogous to (3.60) of Case 1, we obtain We digress to consider the following inequality: where in the first equality we used (3.11), in the second inequality, we extended the sum across β s to the sum across all β because each summand is negative by (3.21), and in the last inequality we used (3.20). Comparing coefficients of b i 2 + j i j 0 between the first and last term in (3.64), we note that there has to exist an index k such that If the index i for which we want to bound t β i coincides with such an index k, we have an obvious bound on t β where in the first inequality, we used (3.65) with k = i, and in the second inequality we used the assumption that βs N βs t βs i (m βs 0 ) 2 > βm,+ N βmt βm i (m βm 0 ) 2 . Thus we only need to consider i = k with k satisfying (3.65). Then, the last term on the second line of (3.63) becomes where in the first inequality we plugged in the definition (3.48) of S βs and picked the pair {i, k} such that i is the index for which we want to show boundedness for t β i , k is the index such that (3.65) is satisfied and dropped the negative M -matrix term. In the second inequality we used (3.65) fort βs k , as well as the second inequality in (3.62). Combining (3.63) and (3.67), we obtain (3.68) and arrive at the final bound

Bounds on n D5
I In this section, we employ the results from the previous Section 3.3.1 to derive bounds on the numbers n D5 I of D5-branes. These bounds are formulated in two theorems.
Theorem 3. For all i we have the following bound on n D5 i : 71) where in the last equality we used (3.11). If β N βt β i (m β 0 ) 2 ≤ 0, then we have the obvious where we dropped negative terms in the last inequality. Thus, we are in case 1 in the proof of Theorem 2, i.e. βs N βs t βs i (m βs 0 ) 2 ≤ βm,+ N βmt βm i (m βm 0 ) 2 , and can use results derived previously for that case. Using the fraction h 1 defined in (3.59), (3.72) becomes (3.73) By the third line of (3.60), we obtain by dropping the second term on the RHS of the first line. By rearranging and combining with (3.73), we arrive at , (3.75) which in combination with (3.71) gives the desired bound (3.70).

Remark 3.
We note also, that the first inequality of (3.8) forbids (n D5 i − T i ) ≥ 0 for all i, i.e. although each n D5 i is bounded above by (3.70), together they are further constrained by this condition.
Theorem 4. We have the following bound on n D5 0 : where the minimum of T i has to be taken over the Kähler cone generators in the chosen subcone of the Kähler cone.
Proof. Using (3.10), we obtain where we dropped some negative contributions of the first term on the RHS of the first line as well the negative γ-brane contribution and used S βs , R βm as defined in (3.48), (3.47), respectively. We see that the coarsest bound on n D5 0 occurs when βs S βs − βm,+ |R βm | is maximized. By (3.60), since its last line is positive, this expression is always negative in case 1 of Theorem 2. To maximize it, we look at case 2 of Theorem 2. Starting from (3.63) in case 2 of Theorem 2, we obtain where in the second inequality, we dropped the positive first term. In the third inequality, we plugged in the definition (3.48) of S βs and picked the pair {i, k} such that k is an index so that (3.65) is satisfied, and i is the particular index such that T i = min(T l ) if this i = k. If i = k, pick any other index as i, and drop the M -matrix term. In the fourth inequality we used the second inequality in (3.62). In the last inequality, we note that if we have used the first way of choosing the pair {i, k}, then T i = min(T l ) and T k ≤ max(T l ); if we have used the second way of choosing the pair {i, k}, then T i ≤ max(T l ) and T k = min(T l ). Combining this result with (3.77), we get the desired bound (3.76) on n D5 0 .

Bounds on γ-branes
Finally, we derive a bound on the number of γ-brane configurations, i.e. we bound the flux quanta m γ . The contribution of γ-branes to the 0 th -tadpole is fixed by (3.10) as (3.79) As by Proposition 1, the LHS of this equation is positive, a solution to it only exists if also the right hand side is positive. Thus, this is the equation of an ellipsoid and the vector m γ of discrete flux quanta is given by the finite number of integral points on this ellipsoid. We denote the positive RHS of (3.79) by r 2 with r ∈ R.
Consequently, the question of boundedness of m γ translates into showing boundedness of r 2 . By (3.79) we have where we set n D5 0 = 0 and dropped some negative terms in the sum over β to obtain the second line and used (3.78) for the last inequality.
This argument and also Proposition 1 require that the matrix C is of negative signature (0, n) when restricted to the subspace of vectors obeying (3.31). As we have argued before, for the bases B = F k , dP n , n > 1 and the toric surfaces the matrix C is of Minkowski signature and the vector b 2 + j j 0 is time-like. Thus, the above argument applies.

Conclusions
We have studied Type IIB compactifications on smooth Calabi-Yau elliptic fibrations over almost Fano twofold bases B with magnetized D9-branes and D5-branes. We have proven that the tadpole cancellation and SUSY conditions imply that there are only finitely many such configurations. We have derived explicit and calculable bounds on all flux quanta (Table 1, Theorem 2, Section 3.3.3) as well as the number of D5-branes (Theorem 3, Theorem 4), which are independent on the continuous moduli of the compactification, in particular the Kähler moduli, as long as the supergravity approximation is valid. The presented proof applies for any geometry that meets the geometric conditions listed at the beginning of Section 3.3. We have shown explicitly in Section 2.3 and Appendix A that these geometric conditions are obeyed for the twofold bases B given by the Hirzebruch surfaces F k , k = 0, 1, 2, the generic del Pezzos dP n , n = 0, . . . , 8 as well all toric varieties associated to the 16 reflexive two-dimensional polytopes. This in particular required showing the positive semi-definiteness of the matrices M {i,k} defined in (3.9). To this end we studied the Kähler cones of the generic dP n and explicitly constructed their Kähler cone generators, which are listed in Table 3 and reveal useful geometric properties of these Kähler cones.
Physically, we have proven that there exists a finite number of four-dimensional N = 1 supergravity theories realized by these compactifications. Most notably, there arise only finitely many gauge sectors in these theories with finitely many different chiral spectra. The details of these gauge sectors are determined by the bounded number of branes in a stack and the bounded magnetic flux quanta. Concretely, this means that the ranks of the gauge groups are bounded, that only certain matter representations with certain chiral indices exist (which is always true in weakly coupled Type IIB) and that for fixed gauge group there exist only a finite set of possible multiplicities for the matter fields. These finiteness properties, and more broadly similar results elsewhere in the landscape, are particularly interesting when contrasted to generic quantum field theories.
While we have shown finiteness of these compactifications and provided explicit bounds, we have not explicitly constructed all of these compactifications. It would be interesting to systematically construct this finite set of configurations and extract generic features of the four-dimensional effective theories in this corner of the landscape. In addition, we have not systematically explored the bases B for which the proof applies, i.e. there may exist additional algebraic surfaces satisfying the geometric conditions of Section 3.3. Other points of interest would be to determine whether a simple modification of our proof exists for blow-ups of singular elliptic fibrations or elliptically fibered Calabi-Yau manifolds which do not satisfy the supergravity approximation; in the latter case the supersymmetry conditions receive corrections of various types. Of most interest would be to find a general proof for a general Calabi-Yau threefold X. It seems plausible that there are even more general proof techniques which utilize SUSY and tadpole cancellation conditions to prove finiteness for a general X. For example, some of the arguments in the proof presented here, e.g. the ones used to eliminate the dependence of the SUSY conditions (2.4) on the Kähler moduli, should still apply for general Calabi-Yau manifolds X. In addition, string dualities of the considered Type IIB configurations extend our finiteness proof to the dual theories, for example to the heterotic string on certain elliptic fibrations with specific vector bundles and to F-theory on certain elliptic K3-fibered fourfolds. It is very important to work out the details of the duality maps and the analogs of the bounds we found in the dual theories.
The presented proof is based on tadpole and supersymmetry conditions at weak coupling and large volume of X. It is crucial for a better understanding of the string landscape to understand string consistency conditions away from large volume and weak coupling. This requires the understanding of perturbative and non-perturbative corrections 16 both in α ′ and in g S ; for example, the supersymmetry conditions receive α ′ -corrections from worldsheet instantons. Avenues towards a better understanding might be provided by applications of N = 1 mirror symmetry, i.e. mirror symmetry, and S-duality.
It is particularly interesting that the finiteness results we have proven and similar results elsewhere in the landscape do not have known analogs in generic quantum field theories. Such differences are one of the hallmarks of string compactifications, and it seems reasonable to expect that similar finiteness results can be proven for even the most general string compactifications, in particular those at small volume and strong coupling. This would have profound implications for our picture of the landscape: while it is larger than originally thought, our results provide further evidence that it may, in fact, be finite.

A Kähler Cones of del Pezzo Surfaces & their M {i,k} -Matrices
In this Appendix we discuss in detail the structure of the Kähler cone of the del Pezzo surfaces dP n for n ≤ 8. We are interested in the extremal rays, i.e. the generators, of these in general non-simplicial cones, and the existence of coverings of these cones by simplicial subcones so that all conditions listed at the beginning of Section 3.3 are obeyed. In particular, we show that all b i are positive and that the matrices (3.9) for any choice of two Kähler generators are positive semi-definite.
First, we expand the Kähler cone generators D i of dP n in the basis (2.20) of H 2 (dP n , Z) which maps every D i to a vector v i in Z n+1 . With this definition, we obtain the matrices (3.9) in this basis as where v T denotes the transpose of a vector, '·' denotes the matrix product and η = diag(1, −1, . . . , −1) is the standard Minkowski matrix in n + 1 dimensions. We note that in order to check positive semi-definiteness of the matrices in (A.2), it suffices to prove it for the matrices η · M {i,k} · η, which is the matrix in the square brackets in (A.2). Next, we need the explicit form for the Kähler generators of dP n . We present these by listing the corresponding vectors v i defined via (A.1). We explicitly solve (2.25) over the integers to obtain the Kähler cone generators. For the simplicial cases dP 0 , dP 1 , dP 2 we obtain (2.26) as discussed earlier. In the non-simplicial cases dP n , n > 2, we summarize the generators in Table 3.
Here, the second column contains the schematic form of the vectors v i , with each row containing all vectors of the same particular form. In each row, the explicit expressions for the v i are obtained by inserting the values listed in the third column for the place holder variables in the entries of v i that row and by permuting the underlined entries of the vector v i . The number of different vectors in each row is given in the fourth column, where the two factors are given by the number of elements in the list in the third column and the number of permutations of the entries, respectively. The fifth column contains a list of the Minkowski length of all vectors in a given row. We note that this column precisely contains the self-intersection of the curves associated to the D i . All are either 0 or 1 and it can be checked that the intersections of the v i with c 1 (dP n ) = 3H − i E i ≡ (3, −1, . . . , −1) are precisely 2 or 3, respectively, as required by (2.25).
For example, in the second row of Table 3, all vectors v i are of the form v i = (a, b, b, b) by the second column. By the third column, there are two different vectors of this type, namely v 1 = (2, −1, −1, −1) and v 2 = (1, 0, 0, 0). Thus, there are precisely 2 vectors as indicated in the fourth column and the Minkowski length of the two vectors is 1, 1, respectively, as in the last column of the second row. We note that the Kähler cone generators and their grouping as in Table 3 can be understood representation theoretically, recalling that the Weyl group naturally acts on H 2 (dP n , Z). For instance the Kähler cone generators of dP n , n = 2, . . . , 6 form the representations 3, (3 ⊗ 1) ⊕ (1 ⊗ 2), 5 ⊕5, 16 ⊕ 10 and 78 ⊕ 27 under the corresponding groups A 1 , A 2 × A 1 , A 4 , D 5 and E 6 , respectively. Here the first representation in all direct sums is formed by all generators with Minkowski length 1 and the the second one is formed by generators with Minkowski length 0. These results can be worked out explicitly by computing the Dynkin labels of the generators in Table 3 for the canonical roots α i , which are the −2-curves in H 2 (dP n , Z) orthogonal to c 1 (dP n ). Thus, the zero weight vector is identified with c 1 (dP n ). For dP 7 only the union of the generators of the Kähler and Mori cone have a representation theoretical decomposition as 912 ⊕ 133, where the first representation contains the length 1 and the second one the length 0 Kähler cone generators. 17 Next, we make one important observation. As one can check explicitly from Table 3 and (2.26), for every del Pezzo dP n with n > 1, the first Chern class c 1 (dP n ) ≡ (3, −1, . . . , −1) is proportional to the sum of all Kähler cone generators v i where N denotes the total number of Kähler cone generators of dP n , cf. Table 3. The positive proportionality factor A n depends on n and reads for dP 3 , dP 4 , dP 5 , dP 6 , dP 7 and dP 8 , respectively. This means that c 1 (dP n ) is in the center of the Kähler cone of all del Pezzo surfaces with n > 1. This implies that we can find a cover of the Kähler cone by simplicial subcones so that all properties listed at the beginning of section 3.3 are satisfied. We present two such covers: Cover 1: This cover is constructed as follows. Intersect the Kähler cone with a hyperplane that is normal to c 1 (dP n ) and passes through c 1 (dP n ). This yields a n-dimensional polytope with vertices corresponding to the generators of the Kähler cone. Triangulate this polytope with star being c 1 (dP n ). This triangulation induces a decomposition of the Kähler cone into simplicial subcones. As the generators of a particular simplicial subcone, we take c 1 (dP n ) and those generators v i of the Kähler cone that go through the vertices of this n-dimensional cone of the triangulated polytope.
In this covering of the Kähler cone, we obviously have b i all positive, because c 1 (dP n ) is one of the generators in every simplicial subcone. From (2.10) we actually get b i = 0 for all D i = c 1 (dP n ) and b K = 1, where K denotes the index such that D K = c 1 (dP n ). In addition, we have C KK = K 000 = 9 − n and C iK = K 00i = 2, 3 for i = K by (2.7) and (2.27) and T K = 12 B c 2 1 = 12K 000 = 12(9 − n) by (2.13) and (2.27). 17 A number of weights in the 912 have higher multiplicities yielding only 576 different weights.
In order to show that the matrices (3.9) are positive semi-definite, we only have to prove that the matrices (A.2) for all possible choices of two vectors v i , v j of Table 3 are positive semi-definite (which we prove at the end of this appendix). In fact, in every simplicial subcone of this covering, all matrices M {i,j} with i, j = K involve only the generators v i , v j . Thus, we only have to consider the matrices M {i,K} with i = K 18 . For these we use Proof. Using the first Chern class c 1 (dP n ) ≡ (3, −1, . . . , −1) and λ j = 1 An·N , we obtain where we have used (A.3). Next, choose x {i,K} for every i so that the following equality is satisfied where 1 x {i,j} j denotes the average of 1 x {i,j} with i kept fixed and j varied over all Kähler cone generators. Then, (A.5) can be written as where we set λ ′ j = λ j Cover 2: Although the above cover 1 obeys all the required properties listed at the beginning of Section 3.3, it slightly increases the bounds because it increases max(T i ) for n ≤ 6 in which case max(T i ) = T K = 12(9 − n) is larger than the T i found in (2.28).
Thus, we provide the following alternative cover which exists if the Kähler cone is sufficiently symmetric, in addition to c 1 (dP n ) being its center. Take a vertex of the polytope constructed in cover 1. Construct the line through that vertex and the star, i.e. c 1 (dP n ). This line has to intersect the boundary of the polytope at another point. This point lies on a certain facet of this polytope. Take the vertices of this facet together with the original vertex we have started with to define a simplicial subcone of the Kähler cone. Notice that this subcone contains c 1 (dP n ) and the cone formed by the vertices of this facet and c 1 (dP n ), i.e. a subcone in cover 1. Repeat this procedure for all vertices of the polytope. If the Kähler cone is sufficiently symmetric, each facet will be hit exactly once. Thus, each subcone in cover 1 is contained in a corresponding subcone defined in this way. Consequently, since cover 1 covers the Kähler cone completely, so does cover 2.
The advantage of this cover is that all generators of all simplicial subcones are generators of the Kähler cone. Thus, this cover satisfies all the conditions at the beginning of Section 3.3: the matrices (3.9) for all choices of pairs of vectors v i , v j are positive semidefinite, which we show below; also c 1 (dP n ) is contained in each subcone, thus b i ≥ 0 for all i, and by (2.27) all K 00i are positive integers. In addition, in all bounds derived in this work, we have that max(T i ) is given precisely by (2.28). Given the fact that the generators of the Kähler cone sit in representations of Lie algebras, which implies that the Kähler cone is symmetric, and that c 1 (dP n ) lies in its center, we expected this cover 2 to exist.
Finally, we show the positive semi-definiteness of the matrices M {i,k} defined in (A.2) for any choice of two Kähler cone generators of dP n from Table 3. We note that the number of necessary computations is greatly reduced by noting that the Kähler cone generators of dP n are obtained by permutations of the vectors in Table 3. Most of these permutations simply interchange the rows and columns of the matrices (A.2), which does not affect their eigenvalues. Thus, we only have to consider matrices (A.2) that do not only differ by such a permutation. Computing all such inequivalent matrices can be done in many cases very quickly by hand or on a computer. We provide an efficient algorithm making use of this permutation symmetry of (A.2) at the end of this appendix.
We obtain that the matrices (A.2) are positive semi-definite for any choice of two Kähler cone generators in Table 3 We have the following Proof. Let P jl be the permutation matrix that permutes the j th and l th rows/columns. Then we have the identitỹ k are obtained from a sequence of swaps of corresponding entries at the same position in v i , v k , respectively. The above argument shows that we just need to check positive semi-definiteness of a singleM {i,k} defined by any representative (v i , v k ) in its equivalence class [(v i , v k )].
In order to find the set of all such equivalence classes, start by picking an arbitrary pair of vectors v i , v k from Table 3. Then, we can carry out the following algorithm: (1) Fix v i in the pair (v i , v k ) and only permute the entries of v k . Indeed, for any pair where v ′ i , v ′ k are obtained from arbitrary permutations of the entries in v i and v k , respectively, we have ( Here v ′′ k is obtained by performing the same permutations on v ′ k that transform v ′ i to v i . (2) Given a pair (v i , v k ) with fixed v i , only permute those entries in v k for which the corresponding entries in v i are different from each other. Permuting two entries in v k when the corresponding two entries in the fixed vector v i are the same is equivalent to the action of permuting these two entries for both vectors. Thus, the resulting pair of vectors (v ′ i , v ′ k ) is in the same equivalence class as the old pair of vectors (v i , v k ). For example, consider dP 8 . Suppose we begin by picking v i = (a, b, b, b, b, b, b, b, b) and v k = (s, t, u, u, v, v, v, v, v) from the second column of Table 3, and we want to find {(v ′ i , v ′ k )}/ ∼. By (1) above, we can fix v i and only consider permutations in the last eight entries of v k . By (2), however, we do not need to consider any permutation in the last eight entries in v k , because the last eight entries in the fixed vector v i are the same; they are all equal to b. Thus, no permutation yields anM ′ {i,k} with a new set of eigenvalues. From the third column of Table 3, there are two sets of different values for v i = (a, b, b, b, b, b, b, b, b), and six sets of different values for v k = (s, t, u, u, v, v, v, v, v). Thus there will be 2 · 6 = 12 differentM ′ {i,k} matrices to check for positive semi-definiteness.

B Geometric Data of almost Fano Twofolds for computing Explicit Bounds
In this appendix, we summarize the geometric data of Hirzebruch surfaces F k , k = 0, 1, 2, the del Pezzo surfaces dP n , n = 2, . . . , 8, and the toric varieties associated to the 16 reflexive polytopes that is necessary to explicitly compute the various bounds derived in this paper. We begin with the bases F k and dP n . The following results in Table 4 are derived employing (2.19), (2.28), the two covers of the Kähler cones of dP n constructed in Appendix A, (A.4) and the values of x {i,k} listed below (A.8).
First, we list the maximal and minimal values of x {i,k} and T i for the bases F k and dP 2 that have a simplicial Kähler cone. For the non-simplicial Kähler cones, we obtain different results for the two different covers of their Kähler cones. We note that for both cover 1 and 2 the values below (A.8) apply. Indeed, this is precisely what we get in the second and third column under cover 2. However, for cover 1, these numbers have to be multiplied by appropriate A n in (A.4). Indeed, by (A.6) we have x {i,K} = A n ( x {i,k} j ) −1 . By (A.4), we have A n ≤ 1 for n ≤ 6, i.e. the minimum value of x {i,K} is bounded by A n · min(x {i,K} ), but the maximum is unaffected, as indicated in the first four rows of the second and third column in Table 4 under cover 1. For dP 7 and dP 8 , we have A n > 1, thus x {i,K} ≤ A n max(x {i,k} ) = A n and the minimum is unaffected, as displayed in the last two rows of the second and third column in Table 4 Table 5. Displayed are some of the relevant data for the smooth almost Fano toric surfaces obtained from fine star triangulations of the two-dimensional reflexive polytopes in Figure 1.
can not determine the explicit value min(T i ) for both covers. Therefore, depending on the chosen subcone, employing (2.28), we either obtain 24 or 36 as indicated in the last column of Table 4. However, in the case of cover 1 we have T K = 24, 12 for dP 7 and dP 8 , respectively. Since by construction, the first Chern class c 1 (dP n ) is in every subcone, we know that min(T i ) = T K = 24, 12 for dP 7 and dP 8 , respectively. Finally, in Table 5 we display the relevant topological data of the toric varieties constructed from the 16 reflexive two-dimensional polytopes which is relevant to our finiteness proof in section 3.3. In addition to this data, we also checked that the matrices (A.2) are all positive semi-definite for x {i,k} of the form x {i,k} = 1 a with a ∈ {1, . . . , 6}. In addition, we confirmed that the first Chern class c 1 (B) is inside the Kähler cone in all these cases, i.e. a cover of the non-simplicial Kähler cones constructed like the cover 1 of Appendix A exists. In this cover all b i are positive and all required conditions listed at the beginning of Section 3.3 are obeyed.