  Twisted bilayer graphene (TBG) is known for exhibiting highly correlated
phases at magic angles due to the emergence of flat bands that enhance
electron-electron interactions. In the TBG chiral model, electronic wave
function properties depend on a single parameter ($\alpha$), inversely
proportional to the relative twist angle between the two graphene layers. In
previous studies, as the twist angles approached small values, strong
confinement, and convergence to coherent Landau states were observed. This work
explores flat-band electronic modes, revealing that flat band states exhibit
self-duality; they are coherent Landau states in reciprocal space and exhibit
minimal dispersion, with standard deviation $\sigma_k=\sqrt{3\alpha/2\pi}$ as
$\alpha$ approaches infinity. Subsequently, by symmetrizing the wave functions
and considering the squared TBG Hamiltonian, the strong confinement observed in
the $\alpha\rightarrow\infty$ limit is explained. This confinement arises from
the combination of the symmetrized squared norm of the moir\'e potential and
the quantized orbital motion of electrons, effectively creating a quantum well.
The ground state of this well, located at defined spots, corresponds to Landau
levels with energy determined by the magic angle. Furthermore, we demonstrate
that the problem is physically analogous to an electron attached to a
non-Abelian $SU(2)$ gauge field with an underlying $C_3$ symmetry. In regions
of strong confinement, the system can be considered as Abelian. This allows to
define a magnetic energy in which the important role of the wave function
parity and gap closing at non-magic angles is revealed. Finally, we investigate
the transition from the original non-Abelian nature to an Abelian state by
artificially changing the pseudo-magnetic vector components from an $SU(2)$ to
a $U(1)$ field, which alters the sequence of magic angles.
