Generic Construction of the Standard Model Gauge Group and Matter Representations in F‐theory

Abstract

We describe general classes of 6D and 4D F-theory models with gauge group (SU(3) × SU(2) × U(1))∕ℤ 6 . We prove that this set of constructions gives all possible consistent 6D supergravity theories with no tensor multiplets having this gauge group and the corresponding generic matter representations, which include those of the MSSM. We expect, though do not prove, that these models are similarly generic for 6D theories with tensor multiplets and for 4D  = 1 supergravity theories. The largest class of these constructions come from deforming an underlying geometry with gauge symmetry SU(4) × SU(3) × SU(2).

General F-theory models with tuned (SU(3) × SU(2) × U(1))/ℤ6 symmetry

“…As discussed in [15], these two classes of models appear to have good constructions in F-theory that generalize naturally to 4D supergravity theories. Specifically, the β ≥ 0 (Class (A)) models have spectra consistent with a Higgsing deformation of an SU(4) × SU(3) × SU(2) model with respective gauge anomaly coefficients B 4 = b 3 , B 3 = b 2 , B 2 = β, and we would thus expect that many of these models could be constructed in F-theory by starting with a Tate-type tuning [22,44] of SU(4) × SU(3) × SU(2) (see section 8.2) and carrying out the deformation in the Weierstrass model.…”

“…Such constructions of theories with the standard model gauge group have been considered in [7][8][9][10]; recently, Cvetič, Halverson, Lin, Liu, and Tian (CHLLT) [3], building on a toric construction identified in [11] and aspects of global group structure studied in [12][13][14], considered one class of such models that can be realized over any weak Fano base, giving a large number of possible standard model-like constructions in F-theory. In the paper [15], two of the authors of this paper described what seems to be the most generic class of tuned F-theory models that give the gauge group (SU(3) × SU(2) × U(1))/Z 6 ; these models include the CHLLT models as a particular subclass. As reviewed further in section 2.2, we focus on constructions with the group (SU(3) × SU(2) × U(1))/Z 6 , since there is a well-defined sense in which the generic matter content of models with the gauge group SU(3) × SU(2) × U(1) does not match the observed matter in the standard model [16].…”

“…As reviewed further in section 2.2, we focus on constructions with the group (SU(3) × SU(2) × U(1))/Z 6 , since there is a well-defined sense in which the generic matter content of models with the gauge group SU(3) × SU(2) × U(1) does not match the observed matter in the standard model [16]. A limitation of [15], however, is that the description given there of the generic (SU(3) × SU(2) × U(1))/Z 6 model was somewhat indirect. In this paper, we make this class of models more concrete by giving an explicit general form of the Weierstrass model that covers much of the range of models that were identified through more indirect means in [15].…”

“…This is the approach we consider in this paper, in which the full gauge group is tuned in the Weierstrass model. As discussed in the introduction, models of this type were previously considered in [3,[7][8][9][10]15]. While these models can largely arise from deformations (which always can be understood in terms of Higgsing processes in the 6D context) from larger groups, the breaking in these cases of any possible GUT depends upon deformations of the Weierstrass model.…”

“…The split condition is also satisfied for the {b 1 = 0} and {d 0 = 0} singular loci. Therefore, the enhanced gauge group is SU(4) × SU(3) × SU(2), with the gauge group factors tuned on the exact divisor classes predicted in [15]. The divisors {b 1 = 0}, {d 0 = 0}, and {s 1 = 0} do not have any singular structure, so one would expect this model to have a generic matter spectrum.…”