  Flat band electronic modes are responsible for superconductivity in twisted
bilayer graphene (TBG) rotated at magic angles. From there other magic angles
can be found for any multilayered twisted graphene systems. Eventually, this
lead to the discovery of the highest ever known electron-electron correlated
material. Moreover, the quantum phase diagram of TBG is akin to those observed
among high-$T_{c}$ superconductors and thus there is a huge research effort to
understand TBG in the hope of clarifying the physics behind such strong
correlations. A particularity of the TBG is the coexistence of
superconductivity and the fractional Quantum Hall effect, yet this relationship
is not understood. In this work, a simple $2\times 2$ matrix model for TBG is
introduced. It contains the magic angles and due to the intrinsic chiral
symmetry in TBG, a lowest energy level related to the quantum Hall effect. The
non-Abelian properties of this Hamiltonian play a central role in the
electronic localization to produce the flat bands and here it is proved that
the squared Hamiltonian of the chiral TBG model is equivalent to a single
electron Hamiltonian inside of a non-Abelian pseudo-magnetic field produced by
electrons in other layers. Therefore, the basic and fundamental elements in the
physics of magic angles are determined. In particular, a study is made on these
fundamental energy contributions at the $\Gamma$-point due to its relation to
the recurrence of magic angles and its relationship with the Quantum Hall
effect.
