# Physical applications of the coset-space $E_6/Spin(10) \times S0(2)$

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The coset-space $E_{6}/Spin(10) \times SO(2)$ is Hodge manifold which can be imbedded into a complex projective  space.  Please let me know what are the possible physical applications of such coset-space.

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In Grand unification theory, $E_6$ appears as a possible gauge group which, after a suitable symmetry breaking, gives rise to  the gauge group of the standard model $$SU(3) × SU(2) × U(1)$$

This is achieving via the breaking to $Spin(10) × SO(2)$ and then to $SO(10) × U(1)$ .  According with this, the physical states will appear  in representations of the coset-space  $E_6 / Spin(10)×SO(2)$.  Explicitly we have that

$$78 \rightarrow 45_0 \oplus 16_{-3} \oplus \overline{16}_3 + 1_0$$

$$27 \rightarrow 1_4 \oplus 10_{-2} \oplus 16_1$$
$$\bar{27} \rightarrow 1_{-4} \oplus 10_2 \oplus \overline{16}_{-1}$$

Where the subscript denotes the $SO(2)$ charge. In such way, we can get the Standard Model's elementary fermions and Higgs boson.

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