Building SO 10 -models withD4symmetry

Abstract

Using characters of finite group representations and monodromy of matter curves in F-GUT, we complete partial results in literature by building SO 10 models with dihedral D 4 discrete symmetry. We first revisit the S 4 -and S 3 -models from the discrete group character view; then we extend the construction to D 4 . We find that there are three types of SO 10 × D 4 models depending on the ways the S 4 -triplets break down in terms of irreducible D 4 -representations: (α) as 1 +,− ⊕ 1 +,− ⊕ 1 −,+ ; or (β) 1 +,+ ⊕1 +,− ⊕1 −,− ; or also (γ) 1 +,− ⊕2 0,0 . Superpotentials and other features are also given.

MSSM-like from SU 5 × D 4 models

“…Below, we give another manner to approach the spectrum of S 4 -model. By help of the standard relation 24 = 1 2 + 1 2 + 2 2 + 3 2 + 3 2 showing that S 4 has 5 irreducible representations R i and 5 conjugacy classes C i [39,40,41,42]; and by using properties of the irreducible R i representations of S 4 given in appendix; eq(2.26) may be expressed in terms of the R i 's and their χ (a,b,c) R characters as follows…”

“…Notice that S 4 has three generators denoted here by (a, b, c) and chosen as given by 2-, 3-and 4-cycles; they obey amongst others the cyclic properties a 2 = b 3 = c 4 = I id ; these three generators are non commuting permutation operators making extraction of full information from them a difficult task; but part of these information is given their χ (a,b,c) R 's; these characters are real numbers as collected in following table [39,40,41,42],…”

“…For other features see [41]. With these tools at hand, we turn to engineer the SU 5 × D 4 × U ⊥ 1 models with dihedral monodromy symmetry.…”

“…This diagrammatic description is very helpful in dealing with S 4 representation theory [40,41,42]; it teaches us a set of useful information; in particular helpful data on the three following: i) Expressions of (3.5) In the representation 3 of the permutation group S 4 , the three x i -weights in (3.5) read in terms of the t i 's as…”

On building superpotentials in F-GUTs

“…By extending this spectral covers construction to the other representations of SU (5) ⊥ involved in the breaking of E 8 down to SU (5) × SU (5) ⊥ , in particular to the antisymmetric and adjoint ones, one can also describe in quite similar manner the localisation of the other matter multiplets namely the (5, 10 ⊥ ) and (1, 24 ⊥ ); for explicit details see [38] and refs therein.…”