  In twisted bilayer graphene, long-wavelength lattice fluctuations on the
scale of the moir\'e period are dominated by phason modes, i.e., acoustic
branches of the incommensurate lattice resulting from coherent superpositions
of optical phonons. In the limit of small twist angles, these modes describe
the sliding motion of stacking domain walls separating regions of partial
commensuration. The resulting soliton network is a soft elastic manifold, whose
reduced rigidity arises from the competition between intralayer (elastic) and
interlayer (adhesion) forces governing lattice relaxation. Shear deformations
of the beating pattern dominate the electron-phason coupling to the leading
order in $t_{\perp}/t$, the ratio between interlayer and intralayer hopping
parameters. This coupling lifts the layer degeneracy of the Dirac cones at the
corners of the Moir\'e Brillouin zone, which could explain the observed 4-fold
(instead of 8-fold) Landau level degeneracy. Electron-phason scattering gives
rise to a linear-in-temperature contribution to the resistivity that increases
with decreasing twist angle due to the reduction of the stiffness of the
soliton network. This contribution alone, however, seems to be insufficient to
explain the huge enhancement of the resistivity of the normal state close to
the magic angle.
