An F-theory realization of the chiral MSSM with ℤ2-parity

Abstract

Using F-theory we construct 4D N = 1 SUGRA theories with the Standard Model gauge group, three chiral generations, and matter parity in order to forbid all dimension four baryon and lepton number violating operators. The underlying geometries are derived by constructing smooth genus-one fibered Calabi-Yau fourfolds using toric tops that have a Jacobian fibration with rank one Mordell-Weil group and SU (3)×SU (2) singularities. The necessary gauge backgrounds on the smooth fourfolds are shown to be fully compatible with the quantization condition, including positive integer D3-tadpoles. This construction realizes for the first time a consistent UV completion of an MSSM-like model with matter parity in F-theory. Moreover our construction is general enough to also exhibit other relevant Z 2 charge extensions of the MSSM such as lepton and baryon parity. Such models however are rendered inconsistent by non-integer fluxes, which are necessary for producing the exact MSSM chiral spectrum. These inconsistencies turn out to be intimately related to field theory considerations regarding a UV-embedding of the Z 2 into a U (1) and the resulting discrete anomalies.

F-theory models with U(1) × ℤ2, ℤ4 and transitions in discrete gauge groups

“…There are situations in which a genus-one fibration has a global section and in which it does not have a global section; when a genus-one fibration does not have a global section, a discrete gauge group forms in F-theory on this fibration, as mentioned. Recent discussions of F-theory on genus-one fibrations without a global section can be found, for example, in [20,19,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45] 2 . When a genus-one fibration has a global section (in which case, the fibration is often called an elliptic fibration 3 in the F-theory literature), the U (1) gauge group forms in F-theory if the fibration has two or more independent global sections.…”

“…We consider in this study Calabi-Yau four-section geometry built as complete intersections 5 See also, e.g., [22,77,39] for discussions of F-theory models in which both U (1) and a discrete gauge symmetry form. 6 The relation of the moduli of genus-one fibrations and those of the Jacobian fibrations and Weierstrass models was discussed in [19] in the context of F-theory.…”

Yukawa hierarchies in global F-theory models

“…The action of Φ (3.25) on this field only corresponds to a multiplication by D 1 . Therefore, the rank of Φ drops along the matter curve5 6 :…”

“…One can equally obtain a parametrization for the5 −4 curve, which however is slightly more cumbersome because of the more complicated expression. For the purpose of exhibiting the higher rank Yukawa structure, we will fix one chiral "generation" of this matter, and compute the coupling matrix W ij for different generations h i 10 −2 and h j 5 6 . For the residue formula, we then simply insert for h ī 5 −4 a constant, which is always a valid basis element for holomorphic sections (unless there are none).…”

“…A promising regime where we can construct and study globally consistent, four-dimensional (4d) string compactifications is F-theory [1], an extension of weakly coupled type IIB string theory that incorporates non-perturbative back-reactions of 7-branes. There has been a lot of compact model building efforts in this framework, ranging from supersymmetric GUTs 1 to Pati-Salam models or Standard-Model-like examples [3][4][5][6][7]. Along with advancements in our understanding of abelian symmetries in F-theory (see [8] and references therein for a review), these works also developed conceptual and practical tools that give us relatively good control over the chiral spectrum.…”