A Quadrillion Standard Models from F-theory

We present an explicit construction of ${\cal O}(10^{15})$ globally consistent string compactifications that realize the exact chiral spectrum of the Standard Model of particle physics with gauge coupling unification in the context of F-theory. Utilizing the power of algebraic geometry, all global consistency conditions can be reduced to a single criterion on the base of the underlying elliptically fibered Calabi--Yau fourfolds. For toric bases, this criterion only depends on an associated polytope and is satisfied for at least ${\cal O}(10^{15})$ bases, each of which defines a distinct compactification.


I. INTRODUCTION AND SUMMARY
As a theory of quantum gravity that naturally gives rise to rich gauge sectors at low energies, string theory is a leading candidate for a unified theory. Achieving unification is an ambitious goal that requires accounting for all aspects of our physical world, which includes not only a rich cosmological history, but also the detailed structure of the Standard Model of particle physics.
In this paper we present an explicit construction that guarantees the existence of O(10 15 ) fully consistent string compactifications which realize the exact chiral particle spectrum of the minimally supersymmetric Standard Model (MSSM). This construction is performed in the framework of F-theory [1], a strongly coupled generalization of type IIB superstring theory. It captures the nonperturbative back-reactions of 7-branes onto the compactification space B 3 in terms of an elliptically fibered Calabi-Yau fourfold π : Y 4 → B 3 over it. Gauge symmetries, charged matter, and Yukawa couplings are then encoded beautifully by Y 4 's singularity structures in codimensions one, two, and three, respectively. 1 In the present work, we consider a class of elliptically fibered Calabi-Yau fourfolds giving rise to precisely the three-generation MSSM spectrum provided certain geometric conditions on the base of the fibration are satisfied. We perform a concrete analysis, finding O(10 15 ) such bases. All these models come equipped with modulidependent quark and lepton Yukawa couplings, as well as gauge coupling unification at the compactification scale.
The existence of a very large number of Standard Model realizations in string theory could perhaps be anticipated within the set of an even larger number of string compactifications (see, e.g., [3]) that form the so-called string landscape. Indeed, though Standard Model realizations within the landscape could potentially be scarce [4], recent works hint towards an astronomical number of them [5]. Our construction explicitly demonstrates this possibility, increasing the number of concretely known, global Standard Model compactifications in string theory by about ten orders of magnitude.
There are also explicit constructions of the Standard Model in other corners of string theory. Some of the early examples of globally consistent intersecting brane models [6] in type II compactifications (see also [7] and references therein) were strongly constrained by global consistency conditions such as tadpole cancellation. In the heterotic string, the typical difficulties of constructions like [8][9][10] arise from having a stable hidden bundle and the existence of Yukawa couplings. These issues are solved elegantly in F-theory through the geometrization of non-perturbative stringy effects: (almost all 2 ) global conditions analogous to tadpole cancellation or bundle stability are automatically taken care of by having a compact, elliptic Calabi-Yau fourfold Y 4 , and the presence or absence of Yukawa couplings can be easily read off from codimension three singularities of Y 4 .
Despite these advantages, only a handful [11,12] of F-theory compactifications that realize the exact chiral spectrum of the MSSM are currently known, due to focusing on a very simple base, B 3 = P 3 . This limitation will be avoided in the current work by instead studying smooth toric varieties, which provide a much larger class [13,14] of geometries. To take advantage of this large class, we first restrict to one class of elliptic fibrations (known as P F11 [15]) in such a way that it can be consistently fibered over all such toric threefolds.
Every such fibration realizes the precise Standard Model gauge group [SU (3) × SU (2) × U (1)]/Z 6 as well as its matter representations and Yukawa couplings [11,15,16]. Interestingly, the restriction automatically leads to gauge coupling unification at the compactification scale, compatible with the existence of a complex structure deformation to a geometry realizing the Pati-Salam model with unified gauge coupling [11,15].
Furthermore, for each compatible B 3 there exists a G 4flux that induces three families of chiral fermions. These models have a particularly pleasant feature: all global consistency conditions on the flux (including quantization and D3-tadpole cancellation) can be reduced to a single criterion on the intersection number K 3 of the anticanonical class K of the base B 3 . For toric threefolds which have a description in terms of a reflexive polytope ∆, K 3 depends only on the point configuration of ∆ and not its triangulation. On the other hand, for a single polytope there can be multiple different toric threefolds associated with the different fine regular star triangulations (FRSTs) of ∆, the number of which grows exponentially with the number of lattice points in the polytope [13]. Putting together these different components, we find that the number N toric SM of globally consistent threefamily Standard Models in our construction is 7.667 × 10 13 N toric SM 1.622 × 10 16 . (1) We emphasize that this number is construction dependent; F-theory could realize more Standard Models. The detailed derivation of this count first requires a short review in section II of the flux consistency conditions as well as a brief summary of the class P F11 of elliptic fibrations. In section III we then present a restriction of P F11 that allows us to consistently fiber it over all toric threefolds B 3 subject to a single criterion on B 3 . To count how many B 3 satisfy this criterion, we discuss the methods to construct FRSTs of 3D polytopes in section IV, which ultimately lead us to O(10 15 ) possibilities. We close in section V with some geometric and physical comments, as well as future directions.

II. F-THEORY FIBRATION WITH STANDARD MODEL SPECTRUM
We begin by constructing F-theory geometries that realize the Standard Model matter representations. These arise on elliptically fibered fourfolds π : Y 4 → B 3 in the class P F11 , described by P :=s 1 e 2 1 e 2 2 e 3 e 4 4 u 3 + s 2 e 1 e 2 2 e 2 3 e 2 4 u 2 v + s 3 e 2 2 e 3 3 u v 2 +s 5 e 2 1 e 2 e 3 4 u 2 w + s 6 e 1 e 2 e 3 e 4 u v w + s 9 e 1 v w 2 = 0.
The generic fiber is embedded inside a compact complex surface with homogeneous coordinates u, v, w, and e k . For Y 4 to be Calabi-Yau, the coefficients s i have to be sections of line bundles on B 3 with first Chern classes [s i ] ∈ H 1,1 (B 3 , Z) given by [11,15]: where K ≡ c 1 (B 3 ) is the anti-canonical class of B 3 . The classes S 7,9 ∈ H 1,1 (B 3 , Z) parametrize different fibrations over the same base, on which {s i = 0} define effective divisors.
For irreducible sections s i , F-theory on Y 4 has [SU (3)× SU (2) × U (1)]/Z 6 gauge symmetry [15,16]. The global gauge group structure is reflected in the precise agreement between the geometrically realized matter representations and those of the Standard Model:

II.1. Consistency conditions for G4-flux
A chiral spectrum in F-theory requires a non-zero flux G 4 ∈ H 2,2 (Y 4 ), which must also be specified. To enable a feasible Standard Model search over a large ensemble of base manifolds, it is advantageous to perform a base-independent flux-analysis first. For that, we make the key requirement that the sections s i are generic and do not factorize, thus there is no enhancement of gauge symmetries and no additional factorization of the matter curves.
Using the techniques of [17,18] (see also [11,19,20]) it may be shown that the subspace of so-called vertical G 4 -fluxes is spanned by U (1)-fluxes of the form π * ω ∧ σ, where ω ∈ H 1,1 (B 3 ) and σ being the (1, 1)form Poincaré-dual to the Shioda-divisor associated with the U (1) [21], as well as the (2, 2)-form where [x] denotes the (1, 1)-form Poincaré-dual to the divisor {x = 0} ∩ {P = 0} ⊂ Y 4 . In other words, the G 4 -flux is parametrized by a rational number a ∈ Q and a vertical (1, 1)-form ω ∈ H 1,1 (B 3 ): For a consistent compactification the flux must be quantized [22,23]: where c 2 (Y 4 ) is the second Chern class of Y 4 . Since explicitly verifying this condition for concrete geometries is difficult, we will content ourselves with the usual consistency checks [11,12,18,19,24], which are the integrality of the following integrals: This condition is necessary for G 4 +c 2 /2 to be an integral four-form. Over a generic base, the set of integral fourforms must contain the Poincaré-duals (PD) of all matter surfaces γ R as well as intersection products D 1 · D 2 of all integral divisors D i , which by the Shioda-Tate-Wazir theorem [25] are known to be generated by the fibral divisors ({u = 0}, {v = 0}, {w = 0}, {e i = 0}) and integral vertical divisors. These integrals may be evaluated using the second Chern class c 2 (Y 4 ) and matter surfaces given in [11,15]. Another consistency condition is a D3-tadpole satisfying [26], where χ(Y 4 ) is the Euler number of the fourfold Y 4 computed in [11]. The condition aids in ensuring the stability of the compactification. We must also require that the electroweak hypercharge U (1) Y is massless, which is guaranteed if the D-terms vanish [27,28]: Lastly, to obtain the chiral index χ(R) of a representation R, one simply computes the integral [29] A realistic chiral Standard Model requires finding flux configurations which lead to χ(R) = 3 for all representations R in (4). Each of the topological calculations discussed in this section may be readily computed using well-known techniques from the F-theory literature. The rather lengthy expressions are omitted, for brevity.

III. UNIVERSALLY CONSISTENT FIBRATIONS WITH THREE FAMILIES
Since the fiber geometry of Y 4 is fully specified by the hypersurface (2), one can reduce all relevant integrals (8)-(11) to integrals of (1, 1)-forms on the generic base, which by Poincaré-duality are equivalent to intersection numbers of divisors on B 3 (see appendix A of [18] for details). Their actual values depend critically on B 3 .
In addition, we must also specify the cohomology classes S 7,9 that parametrize the hypersurface fibration (2). For a fixed B 3 one can scan over all allowed S 7,9 as well as the flux parameters (a, ω) to find consistent fluxes satisfying the above conditions. This approach has been utilized in [11,12,18] to produce a dozen (almost or exactly) MSSM-like F-theory models over simple bases.
For more complex base manifolds, this procedure is not feasible, as the number of possible choices for S 7,9 grows exponentially in h 1,1 (B 3 ). Furthermore, for many S 7,9 -choices there is "accidental" gauge enhancement over rigid divisors {r k = 0} ⊂ B 3 . This occurs when the S 7,9 choice forces the sections s i to factorize into s i = k r m k k s i . For phenomenological simplicity, we consider cases where the additional gauge sectors are absent.
We make two choices to ensure the non-factorization of the sections s i . First, by the choice S 7,9 = K, each s i is a section of K by (3). Second, given this choice, no s i factorizes if B 3 is a so-called weak Fano toric threefold associated to a reflexive polytope, since the anti-canonical divisor in such a B 3 defines smooth, non-rigid, and irreducible K3 surfaces [30]. By these two choices the resulting F-theory model has the Standard Model gauge group with SU (3) and SU (2) on separate anti-canonical divisors {s 9 = 0} and {s 3 = 0}, respectively, and no additional gauge sectors.
This simple yet universal solution to eliminate undesired enhancements has another pleasant feature: It vastly simplifies the expressions for the chiral indices, Dterm for the U (1), as well as the integrality checks necessary for flux quantization. The resulting streamlined consistency conditions deserve a more detailed analysis.
First, let us examine the masslessness condition (10) for the U (1) gauge boson. With S 7,9 = K, the corresponding D-terms expressed as integrals on B 3 are This can always be satisfied if ω = −5K/a, regardless of the base. In this case, all gauge anomalies are also canceled, which for the Standard Model spectrum (4) is known to enforce a family structure. That is, all representations R have the same chiral index: For exactly three families, we thus need a = −15/K 3 .
This fixes the flux parameters (a, ω) in (6) completely. We turn to flux quantization conditions, which require the flux integrals (8) to be integers. Again, these expressions simplify drastically for fibrations with S 7,9 = K, taking also flux parameters ω = −5K/a and a = −15/K 3 .
After reducing the integrals over Y 4 to intersection numbers on B 3 , the only non-manifestly integer quantities of (8) are and where c 2 (B 3 ) is the second Chern class of B 3 . For smooth threefolds B 3 that appear as a base manifold of a Y 4 , it is known [23] that B3 c 2 (B 3 )∧K = 24 and that c 2 (B 3 )+K 2 is an even class. Thus, the only remaining condition is that K 3 is even.
Lastly, we turn to the D3-tadpole (9), which with the above simplifications becomes This number also only depends on the value of K 3 . Basic arithmetic further shows that if n D3 ∈ Z, then K 3 must be even, thus all of (14) are integer. The smallest values of K 3 such that n D3 ∈ Z ≥0 are K 3 ∈ {6, 10, 18, 30, 90 ...}.
To summarize, we have shown that the elliptic fibrations (2) with parameters S 7,9 = K lead to the Standard Model gauge group and matter representations in F-theory when placed on a smooth base threefold B 3 with non-rigid irreducible anti-canonical divisors. In such compactifications, we have further constructed a vertical G 4 -flux which induces the exact chiral MSSM spectrum. Flux quantization and D3-tadpole cancellation requires that only a single condition is satisfied:

IV. COUNTING STANDARD MODEL GEOMETRIES
From above, any smooth threefold B 3 with non-rigid anti-canonical divisors satisfying (16) realizes a globally consistent three-family MSSM in F-theory. A subset of such spaces, which can be enumerated combinatorially, is the set of so-called weak Fano toric threefolds encoded by 3D reflexive polytopes ∆. While there are "only" 4319 such polytopes [31], each ∆ can specify inequivalent manifolds B 3 through different fine-regular-star triangulations (FRSTs) of the polytope, whose numbers can be very large [13].
What makes this ensemble particularly attractive for our purpose is the fact that the intersection number K 3 is determined solely by the polytope ∆, and is completely triangulation-independent. Therefore any B 3 associated to an FRST of ∆ gives rise to a consistent chiral threegeneration MSSM by our construction, provided that the triangulation-independent constraint on K 3 is satisfied.
A quick computation reveals that there is a set S with 708 polytopes that satisfy (16). By our construction we immediately have where N FRST (∆) is the number of FRSTs of ∆. Hence, the problem of counting the number of consistent F-theory models that admit the chiral MSSM spectrum by our construction reduces to counting FRSTs of reflexive polytopes.
In S, there are 237 polytopes with less than 15 lattice points. For these, we can explicitly compute N FRST using SageMath [32], resulting in a total of 41 430 consistent MSSMs. For the 471 polytopes with at least 15 lattice points, the exponentially growing computation time makes it unfeasible to explicit determine all triangulations. Instead, we shall follow the strategy put forward in [13] to provide bounds on the number of FRSTs in these cases. The basic idea is to reduce the complexity by first counting the number of fine-regular triangulations (FRTs) of each facet of a polytope ∆. Since the facets are two dimensional polytopes, it is possible to brute-force the combinatorics of FRTs for (almost 3 ) all polytopes' facets. By virtue of the reflexivity of ∆, any combination of FRTs of all its facets yields a fine triangulation starred around the only point internal to ∆.
The drawback of this approach is that the triangulation of ∆ obtained this way is not guaranteed to be regular. To tackle this issue, we randomly pick 1.3 × 10 4 samples out of O(10 9 ) fine-star triangulations constructed by gluing together FRTs of the facets of a specific polytope, ∆ 8 [31]. Out of these samples, we find roughly 2 3 to be also regular triangulations. The reason for picking specifically ∆ 8 is because this polytope, with K 3 = 6 (and thus n D3 = 12), has the largest number (39) of lattice points. As a consequence, the landscape of toric threefolds satisfying (16) is dominated by the triangulations of ∆ 8 (see also [13]). Combining the factor 2 3 with the bounds of finestar triangulations (FSTs) for ∆ 8 [13], we then obtain 2.570 × 10 13 ≤ N FRST (∆ 8 ) ≤ 1.617 × 10 16 . We believe that the reduction factor 2 3 is also applicable to the other large polytopes in S where we can only explicitly enumerate FSTs. With this estimation, these other polytopes would sum up to "only" ∼ 5 × 10 13 FRSTs. Hence, even in the worst case scenario where these polytopes have much less regular triangulations, we would still have an astronomically large number of consistent toric threefolds coming solely from ∆ 8 . We therefore expect the number of consistent three-family F-theory Standard Models in our construction over toric threefold bases to be 7.667 × 10 13 N toric SM 1.622 × 10 16 .

V. DISCUSSION AND OUTLOOK
We have presented a construction that ensures the existence of O(10 15 ) explicit, globally consistent string compactifications having the exact chiral spectrum of the Standard Model within the framework of F-theory. To our knowledge, this is the largest such ensemble in the literature, outnumbering existing results by about 10 orders of magnitude. The models arise by varying the base of one "universal" class of elliptic fibrations introduced in [11,15]. We have only focused on the set of toric bases, which already produces around a quadrillion examples. However, we expect that the ensemble of Standard Models arising from our construction is of orders of magnitude larger than this, as might be shown, for instance, by including non-toric bases.
All these models have in common that the Higgs and lepton doublets are localized on the same matter curve. As such, this curve must have non-zero genus to allow for the existence of vector-like pairs [33]. Given the homology class of the doublet curve [11] and our restriction S 7,9 = K, the genus in question is indeed g = 1 + 9/2K 3 > 0, since K 3 ≥ 6 by (16). It would be very interesting, albeit extremely difficult with current methods, to study the precise complex structure dependence of the number of Higgs doublets and other charged vector-like pairs in this ensemble. Furthermore, since our models have no additional (possibly massive) abelian gauge symmetries, all Yukawa couplings relevant for the Standard Model are automatically realized perturbatively, as can be shown by an explicit study of codimension three singularities [15]. However, this in turn also implies that certain proton decay operators compatible with the Standard Model gauge group will in general be present [11]. We expect that in some corners of the moduli space, which incidentally could also support high-scale SUSY breaking, these operators can be suppressed. Another avenue could be to instead focus on "F-theory Standard Models" that have additional (U (1) [18,34] or R-parity [12]) selection rules, and estimate their numbers in the toric base landscape. We leave this for future work.
One interesting aspect of our ensemble is gauge coupling unification without a manifest GUT-origin at the compactification scale. It can be easily read off geometrically from the divisors on B 3 , which the 7-branes supporting the gauge symmetries in the type IIB picture wrap. Due to our restriction S 7,9 = K, both SU (3) and SU (2) gauge symmetries are realized on anti-canonical divisors {s 9 = 0} and {s 3 = 0} with class K. Therefore, the gauge couplings are g 2 3,2 = 2/vol(K) [27,35]. 4 The U (1) Y coupling is determined by the volume of the socalled height-pairing divisor b ⊂ B 3 [36], which has been computed in [6,15] and reduces to b = 5K/6 in our ensemble. Therefore, we have the standard MSSM gauge coupling unification, 4 The factor of 2 arises because in F-theory, the normalization dictated by the geometry is one where the Cartan generators satisfy tr fund (T i T j ) = C ij with C the Cartan matrix [27,35]. On the other hand, the particle physics convention necessary to determine the coupling is tr fund (T i T j ) = which for our models is achieved at the compactification scale. While this scale as well as the actual values of the couplings will depend on the details of moduli stabilization, the relationship (18) is independent of Kähler moduli. It would be interesting to see if this relationship originates from an honest geometric realization of a GUT-structure. Given the known connection of our ensemble to a Pati-Salam [SU (4) × SU (2) 2 ]/Z 2 model [11,15], we expect an underlying SO (10). Lastly, because the gauge divisors in our ensemble are all proportional to the anti-canonical class, the gauge sector has no limit where gravity decouples, implying the existence of a non-trivial interplay between gravity and the visible sector. This interplay can perhaps lead to interesting interactions between particle physics and cosmology in our ensemble. There is a non-trivial relationship due to the details of our construction. At the level of toric geometry, the models differ from one another by how the facet interior points are triangulated. The associated divisors do not intersect the anti-canonical divisors that realize SU (3) and SU (2), and thus the particle physics is relatively insensitive to details of the triangulation; it is, after all, what gives rise to the large number of Standard Models in our construction. The details of the triangulation are critical, however, for the closed string sector and moduli stabilization. For instance, the classical Kähler potential on Kähler moduli space is determined by triangulation-dependent topological intersections. This affects numerous aspects of the cosmology of these models, including possibilities for inflation.