Flipped SO(10)

Flipped SO(10) is a grand unified theory which is to standard SO(10) as flipped SU(5) is to SU(5).

Recall that in conventional SO(10) models, the fermions lie in three spinorial 16 representations, one for each generation, which decomposes under [SU(5)× U(1)_χ] /Z₅ as

:

This can either be the Georgi-Glashow SU(5) or flipped SU(5).

In flipped SO(10) models, however, the gauge group is not just SO(10) but SO(10)_F×U(1)_B or [SO(10)_F×U(1)_B] /Z₄. The fermion fields are now three copies of

:$16\_1oplus\; 10\_\{-2\}\; oplus\; 1\_4$

These contain the standard model fermions as well as additional vector fermions with GUT scale masses. If we suppose [SU(5)×U(1)_A] /Z₅ is a subgroup of SO(10)_F, then we have the intermediate scale symmetry breaking $[SO(10)\_F\; imes\; U(1)\_B]\; /mathbb\{Z\}\_4\; ightarrow\; [SU(5)\; imes\; U(1)\_chi]\; /mathbb\{Z\}\_5$ where

:$chi=-\{Aover\; 4\}+\{5Bover\; 4\}$

In that case,

:::$1\_4\; ightarrow\; 1\_5$

note that the standard model fermion fields (including the right handed neutrinos) come from all three [SO(10)_F× U(1)_B] /Z₄ representations. In particular, they happen to be the 10₁ of 16₁, the of 10_-2 and the 1₅ of 1₄ (my apologies for mixing up SO(10)×U(1) notation with SU(5)×U(1) notation, but it would be really cumbersome if we have to spell out which group any given notation happens to refer to. It is left up to the reader to determine the group from the context. This is a standard practice in the GUT model building literature anyway).

The other remaining fermions are vectorlike. To see this, note that with a 16_1H and a $overline\{16\}\_\{-1H\}$ Higgs field, we can have VEVs which breaks the GUT group down to [SU(5)×U(1)_χ] /Z₄. The Yukawa coupling

:

will pair up the 5_-2 and fermions. And we can always introduce a sterile neutrino φ which is invariant under [SO(10)×U(1)_B] /Z₄ and add the Yukawa coupling

:

OR we can add the nonrenormalizable term

:

Either way, the 1₀ component of the fermion 16₁ gets taken care of so that it is no longer chiral.

It has been left unspecified so far whether [SU(5)×U(1)_χ] /Z₅ is the Georgi-Glashow SU(5) or the flipped SU(5). The reason why is because both alternatives lead to reasonable GUT models.

One reason for studying flipped SO(10) is because it can be derived from an E₆ GUT model.