  The surprising insulating and superconducting states of narrow-band graphene
twisted bilayers have been mostly discussed so far in terms of strong electron
correlation, with little or no attention to phonons and electron-phonon
effects. We found that, among the 33492 phonons of a fully relaxed
$\theta=1.08^\circ$ twisted bilayer, there are few special, hard and nearly
dispersionless modes that resemble global vibrations of the moir\'e supercell,
as if it were a single, ultralarge molecule. One of them, doubly degenerate at
$\Gamma$ with symmetry $A_1+B_1$, couples very strongly with the valley degrees
of freedom, also doubly degenerate, realizing a so-called
$\text{E}\otimes\text{e}$ Jahn-Teller (JT) coupling. The JT coupling lifts very
efficiently all degeneracies which arise from the valley symmetry, and may
lead, for an average atomic displacement as small as $0.5~$mA, to an insulating
state at charge neutrality. This insulator possesses a non-trivial topology
testified by the odd winding of the Wilson loop. In addition, freezing the same
phonon at a zone boundary point brings about insulating states at most integer
occupancies of the four ultra-flat electronic bands. Following that line, we
further study the properties of the superconducting state that might be
stabilized by these modes. Since the JT coupling modulates the hopping between
AB and BA stacked regions, pairing occurs in the spin-singlet Cooper channel at
the inter-(AB-BA) scale, which may condense a superconducting order parameter
in the extended $s$-wave and/or $d\pm id$-wave symmetry.
