  Long-range e.g. Coulomb-like interactions for (quantized) topological charges
in liquid crystals are observed experimentally, bringing open question this
article is exploring: how far can we take this similarity with particle
physics? Uniaxial nematic liquid crystal of ellipsoid-like molecules can be
represented using director field $\vec{n}(x)$ of unitary vectors. It has
topological charge quantization: integrating field curvature over a closed
surface $\mathcal{S}$, we get 3D winding number of $\mathcal{S}\to S^2$, which
has to be integer - getting Gauss law with built-in missing charge quantization
if interpreting field curvature as electric field. This article proposes a
general skyrmion-like mathematical framework to extend this similarity with
particle physics to biaxial nematic: additionally recognising intrinsic twist
of uniaxial nematic, having 3 distinguishable axes - e.g. allowing to construct
hedgehog configuration with one of 3 axes: with the same topological charge,
but different energy/mass - getting similarity with 3 leptons. The proposed
approach has vacuum dynamics as: electromagnetism from 3D rotation dynamics +
Klein-Gordon-like equation for twist corresponding to quantum phase. If
extending to 4D field with 4 distinguishable axes, e.g. with time axis
rotations replaced by boosts , we also get second set of Maxwell equations for
GEM (gravitoelectromagnetism) approximation of general relativity.
