  In this paper, we investigate the theory of dynamical axion string emerging
from chiral symmetry breaking in three-dimensional Weyl semimetals. The chiral
symmetry is spontaneously broken by a charge density wave (CDW) order which
opens an energy gap and converts the Weyl semimetal into an axion insulator.
Indeed, the phase fluctuations of the CDW order parameter act as a dynamical
axion field $\theta({\vec{x}},t)$ and couples to electromagnetic field via
$\mathcal{L}_{\theta}=\frac{\theta(\vec{x},t)}{32\pi^2}
\epsilon^{\sigma\tau\nu\mu} F_{\sigma\tau} F_{\nu\mu}.$ Additionally, when the
axion insulator is coupled to the background geometry/strain fields via
torsional defects, i.e., screw dislocations, there is a novel interplay between
the crystal dislocations and dynamical axion strings (i.e., vortices of the CDW
order parameter). For example, the screw dislocation traps axial charge, and
there is a Berry phase accumulation when an axion string is braided with a
screw dislocation. In addition, a cubic coupling between the axial current and
the geometry fields is non-vanishing and indicates a Berry phase accumulation
during a particular three-loop braiding procedure where a dislocation loop is
braided with another dislocation and they are both threaded by an axion string.
We also observe a chiral magnetic effect induced by a screw dislocation density
in the absence of chemical potential imbalance between Weyl points and describe
an additional chiral geometric effect and a geometric Witten effect.
