What is the minimal supersymmetric Standard Model from F-theory?

We construct gauge theory of SU(3)xSU(2)xU(1) by spectral cover from F-theory and ask how the Standard Model is extended under minimal assumptions on Higgs sector. For the requirement on different numbers between Higgs pairs and matter generations (respectively one and three) distinguished by R-parity, we choose a universal G-flux obeying SO(10) but slightly breaking E_6 unification relation. This condition forces distinction between up and down Higgs fields, suppression of proton decay operators up to dimension five, and existence and dynamics of a singlet related to mu-parameter.

Microscopic phenomena up to sub-TeV scales are successfully described by the Standard Model (SM). Its overlooked but important feature might be being a gauge theory based on a particular group. Quantum consistency of anomaly cancellation requires very special matter contents, and it is best understood by spontaneous symmetry breaking of a unifying theory extending the gauge group [1]. The minimal anomaly free chiral representation 16 of SO(10) contains all the observed fermions of SM including right-handed neutrino. With the aid of supersymmetry (SUSY), Higgs bosons belong to a hypermultiplet and can be treated on equal footing as matter. Thus matter and Higgs pair (as well as colored Higgs pair) are unified to a single representation 27 of E 6 , predicting another kind of singlet related to µ-parameter of SUSY [2]. Continuing, larger groups in higher dimension can even unify vector multiplets. This is well-known chain of E n unifications in which the position of the SM group is unique E 3 × U (1).
However, it is difficult to realize the unification in field theory, yielding the observed vacuum configuration by the scalar potential using only scalar fields. Magnetic flux or gauge bundle in extra dimensions supplements the idea, but anomaly cancellation is only guaranteed if it is embedded in consistent string theory, free of one-loop divergence. So we construct the model from heterotic string or its dual F-theory, which explains the origin of E 8 gauge theory where the E n series ends [3], and provides all the above ingredients. Further constraints, such as repetition of matter generations and gauge coupling unification, show this unification pattern itself is compelling in any UV completion [4].
Our strategy here is to construct the SM gauge group from the beginning, nevertheless the matter generations and Higgses can follow unification relation of a larger group. This feature is clearly realized by spectral cover construction [5]. It enables us to explicitly construct (poly)stable vector bundle on elliptic fibered manifold, using two data. The first is spectral cover that specifies the structure group 'broken part of the E 8 ' hence determines unbroken gauge group, here to be SU (3) × SU (2) × U (1). To obtain chiral spectrum in four dimension, we also have to turn on so called G-flux. If we partly turn on the flux on S[U (3) ⊥ × U (1)] spectral cover, the resulting identities of chiral matter fall into 27 of E 6 , the commutant of SU (3) in E 8 , and the numbers of matter generations and Higgs pairs obey a unification relation of G = SO(10), the commutant to SU (4). Same things happen in heterotic dual theory. Those two data are translated into a vector bundle V of H in the heterotic side by Fourier-Mukai transformation [6,7]. So, first the vector bundle V of the same structure group yields the unbroken group G. Then there is a Wilson line on the elliptic fiber, breaking G down to the SM group. However, the language of spectral cover is easy to distinguish matter from Higgs and to control their multiplicities separately.
Finally, the four dimensional interactions follow from gauge invariant terms of the higher dimensional effective Lagrangian by dimensional reduction [8,9]. The invariance under the mother group E 8 plays the role of selection rule.

GAUGE GROUP
The model is obtained from F-theory compactification on elliptic Calabi-Yau fourfold with a section, admitting heterotic dual. We construct the spectral cover for the structure group S[U (5)×U (1) Y ] as follows [10,11]. First, embed a twofold B 2 into a projectivized threefoldŽ = P(K B2 ⊕ O) π → B 2 using the canonical bundle K B2 [6,13] (Its global embedding using heterotic dual is easily done [7]). The spectral cover in it is described by where a 1 = −b 1 /b 0 and s is an affine coordinate parameterizing the normal direction to B 2 = {s = 0}. Each coefficient b m , parameterizing the positions of the covers, is related to the elementary symmetric polynomial of degree m, out of weights of the fundamental repre- The surviving group on B 2 is the commutant, the SM group SU (3) × SU (2) × U (1) Y . This a sufficient specification, but it also provides the information on the unbroken gauge group [12].
The singularity equation corresponding to the SM  group is given by where x, y are affine coordinates of P 2 and z is the normal coordinate to B 2 = {z = 0} inside the base of elliptic fibration B in F-theory side. At the discriminant locus ∆, we have the the SM gauge group [11]. Referring to Tate's table [12], already (2) is a special form of the SU (3) singularity whose parameters are tuned up to O(z 5 ). A change of coordinate a 1 b 5 + z → z shows the other SU (2) part also special up to O(z 5 ). The U (1) Y part is the relative position between two linearly equivalent components. The Calabi-Yau conditions require that the b m are sections of η − mc 1 , where η = 6c 1 (B 2 ) + c 1 (N B2/B ) and c 1 = c 1 (B 2 ) are combinations of tangent and normal bundle to B 2 . The leading order locus of ∆ in z coincides with B 2 .
To distinguish Higgs from lepton doublets, we need an extra U (1) X symmetry inside U (5) cover, which is the famous continuous version of matter parity, a cousin of U (1) B−L . Shortly we will see, for the observed number of Higgses in four dimension, we need one more parameter from an extra U (1) Z , so that the structure group should be factorized as The resulting spectral cover, respectively with the constraint f 0 (a 1 + d 1 + e 1 ) + f 1 = 0. InŽ, their classes are respectively C 3 = 3σ + π * η and C Z ∼ C X ∼ C Y = σ.

MATTER CONTENTS
Accordingly the adjoint 248 of E 8 branches, as summarized in Table I. We identify the fields by charge assignments in the basis {t 1 , t 2 , t 3 , t 4 , t 5 , t 6 }, the weight vectors where τ is involution flipping the orientation of the cover.
The chiral spectrum in four dimension is completed by specifying G-flux. The field strengths along the Cartan direction come from the dimensional reduction of fourform field strength G of the dual M-theory and this induces vector bundle on the spectral cover [5]. Although the minimal SU (4) G-flux preserves unification relation of its commutant SO(10) in E 8 , the number of Higgs pairs turns out to be completely fixed to be twice the matter multiplicity [10]. Here we have one more parameter ζ, the trace part of U (3) ⊥ ⊂ SU (4) vector bundle [14], to relax the condition. So we turn on a universal flux where σ is the class for B 2 insideŽ and π 3 , π Z are projections from U (3) ⊥ and U (1) Z covers to B 2 , respectively. We do not turn on flux on X and Y cover thus the components t 5 and t 6 can be neglected in calculating the spectrum. The number n R of chiral R zero modes minus antichiral R ones of the Dirac operator inŽ is a topological number and counted by index theorem. It is simply given by the intersection between matter curve class and Poincaré dual of the G-flux, projected on B 2 [7,15] n R = P R ∩ Γ| B2 .
Because of identical geometry of spectral cover as in Ref. [16,17], and we refer to it for the calculation of matter curves n q = n u c = n e c = n d c = n l = n ν c Here we omitted pullback and the product is for the divisors of B 2 . We defined σ ∞ = σ + π * c 1 . The identities of matter are those in 27 of E 6 , and their multiplicities manifest the SO(10) unification relation: it is a nontrivial check that h u and h d gives the same number in (8) and (9). The numbers of matter generations and Higgs pairs can be individually controlled, depending on the topological data on B 2 . We require three generations of matter and one pair of Higgs doublets λη · (η − 3c 1 ) = − 7 3 , η · ζ = 2, c 1 · ζ = 0. (13) However this relation restricts the number of SM neutral field S be five. They are subject to quantization condition 3( 1 2 + λ) ∈ Z, ( 1 2 − λ)η + (3λ − 1 2 )c 1 + 1 3 ζ ∈ H 2 (S, Z). We find a solution λ = 1 6 and the base as del Pezzo two surface with η = 2H, ζ = H − 3E 1 do the job, where H is hyperplane divisor and E 1 is one of the exceptional divisor. In addition, because a 1 in (1) transforms as a section of −c 1 , we have a scalar field O charged under SU (3) and another under SU (2), belonging to H 2,0 (B 2 ) + H 0,1 (B 2 ) [6]. The other E 8 serves as hidden sector and is completely decoupled in smooth compactification and it can serve as supersymmetry breaking sector.

HIGGS SECTOR AND NUCLEON DECAY
The requirement of one pair of Higgs doublets fixed the factorization of spectral cover (3). It has the following phenomenological implications.
Firstly, it also distinguishes between up and down type Higgses. This is due to the structure of the U (3) ⊥ monodromy S 3 [18], the permutation of the elements {t 1 , t 2 , t 3 }. It is the natural Weyl group, without a special monodromy further selected by hand. In terms of the S 3 representations, the fields having the same quantum number of lepton doublet under the SM group are l : {t 1 + t 5 + t 6 , t 2 + t 5 + t 6 , t 3 + t 5 + t 6 }, h c u : {t 1 + t 2 + t 6 , t 2 + t 3 + t 6 , t 3 + t 1 + t 6 }, h d : {t 1 + t 4 + t 6 , t 2 + t 4 + t 6 , t 3 + t 4 + t 6 }, Effectively, the Higgs doublet is distinguished from the lepton doublet by an opposite matter parity or the U (1) X . It also forbids bare (super)renormalizable lepton and/or baryon number violating operators lh u , lle c , lqd c , u c d c d c . Further factorization ruins this one Higgs pair structure but we obtain three pairs of Higgses, so our factorization seems the unique for the U (n) type spectral cover with universal flux. Well-known is that the matter parity and U (1) X alone cannot forbid dimension five proton decay operators such as qqql and u c u c d c e c . However, the above structure group forbids these operators. For instance, the former is not allowed because of nonvanishing sum of the weights (t i ) + (t j ) + (t k ) + (t i + t 5 + t 6 ) and (t i + t 6 ) + (t j + t 6 ) + (t k + t 5 ) + (t i − t 6 ), required by U (3) ⊥ invariance, since one of S 3 index should appear twice [19]. Once forbidden at the tree-level, it is also known that the induced operators are highly suppressed, probably explained by worldsheet instanton contribution [20].
Another prediction is the presence of an SM singlet field S. Surveying the quantum number, it belongs to 27 representation of E 6 , therefore, its interaction is restricted and we can calculate the corresponding terms. Because Higgs doublet and triplets are not simply vectorlike and up and down Higgses live on different matter curves, bare masses are forbidden by U (3) ⊥ invariance. Instead we have Next-to-Minimal Supersymmetric SM (NMSSM) [21,22]. We can check that the only renormalizable superpotential for the surviving fields are (see also below) omitting the flavor dependent coefficients. We expect the terms involving D 1 and D 2 c are all decoupled, yielding the µ-like term Sh u h d . Bare quadratic or cubic terms in S are not allowed. Induced higher order s SSS(QU c E c L + QD c N c L) but they are to be suppressed by a string scale M s . A Majorana mass for the ν c does not appear up to dimension five. There is an interesting room for this from Euclidean D3-brane or gauge instanton in F-theory [23], which might as well generate similar potential for S. Remarkably, if one combination X − Z of U (1) X and U (1) Z is broken down, the orthog-onal combination X + Z inducing a Majorana neutrino mass also generates S 3 term e −Sinst (ν c ν c + SSS).
Since the Higgs fields also obey SO(10) unification relation, we have as many colored Higgs pairs D 1 , D c 2 as doublets. This doublet-triplet splitting problem should be solved by an effect evading the unification structure, close but below the compactification scale. It is an interesting possibility to consider a vectorlike extra generation of matter fields, without changing the Dirac indices. Using aforementioned SU (3)  On the other hand, we expect a coupling ( S + µ D )D 1 D 2 is generated, with a possible SUSY breaking effect µ D . The most strongly constrained nucleon decay operator is qqql, whose coefficient has upper bound 10 −5 M −1 P [25]. At low energy scale, integrating out heavy fields, qqD 1 and qlD c 2 may induce an operator ( S + µ D )/M −2 D qqql up to geometric suppression factor. If the numerator is naturally small or tuned, we can suppress this operator. A possible mixing from bare mass term d c d does not change this eigenvalue. The same argument goes to other induced operators for nucleon decay.

GAUGE COUPLING UNIFICATION
In type IIB language, Kaluza-Klein expansion of fourform field along the harmonic two-forms w i on B 2 , C 4 = C 2 ∧ w 2 , we have a term tr Q 2 M 4 F Q ∧ C 2 B2 w 2 ∧ ιF Q from Chern-Simons interactions and here F Q is the field strength for U (1) Q flux and ι is immersion to B. We turn on is for U (1) Z thus the corresponding gauge boson acquire mass by Stückelberg mechanism and the symmetry is broken. We do not turn on flux along hypercharge or X direction, so they are not broken and their gauge couplings receive no threshold correction from the flux [26,27]. The existence of U (1)'s is clearly seen in the heterotic dual side. Four gauge couplings unify with the correct normalization in SO(10)