One dimensional nexus objects, network of Kibble-Lazarides-Shafi string
  walls, and their spin dynamic response in polar distorted B-phase of $^3$He
  The Kibble-Lazarides-Shafi (KLS) domain wall problem in the axion solution of
CP violation in QCD has condensed-matter based analogy in the nafen-distorted
superfluid $^3$He. Recent experiment in rotating superfluid $^3$He produced the
network of KLS string walls in human controllable system. In this system, KLS
string wall appears in two-step symmetry break transition from normal phase to
polar-distorted B phase and turn out to be the descendants of HQVs of polar
phase. Here we show the KLS string wall smoothly connects to spin solitons when
the spin orbital coupling is taken into account. This means HQVs are 1D nexus
which connects the spin solitons and the KLS domain walls. This is because the
subgroup $G=\pi_{1}(S_{S}^{1},\tilde{R}_{2})$ of relative homotopy group
describing the spin solitons is isomorphic to the group describing the half
spin vortices -- the spin textures KLS string wall. In the nafen-distorted
$^3$He system, 1D nexus objects and the spin solitons with topological
invariant $2/4$ form two different types of network, which are named as
pseudo-random lattices of inseparable and separable spin solitons. These two
types lattices correspond to two different representations of $G$. We discuss
the condition under which pseudo-random lattices model works. The equilibrium
configuration and surface densities of free energies are calculated by numeric
minimization. Based on the equilibrium spin textures of different pseudo-random
lattices, we calculated their transverse NMR spectrum, the resulted frequency
shifts and $\sqrt{\Omega}$-scaling of ratio intensity exactly coincide with the
experimental measurements. We also discussed the mirror symmetry in the
presence of KLS domain wall and the influence of the explicitly break of this
discrete symmetry. Our discussions and considerations can be applied to the
composite defects in other condensed matter and cosmological system.
