  The physics of nematic liquid crystals has been subject of intensive research
since the late 19th century. However, because of the limitations of chemistry
the focus has been centered around uni- and biaxial nematics associated with
constituents bearing a $D_{\infty h}$ or $D_{2h}$ symmetry respectively. In
view of general symmetries, however, these are singularly special since nematic
order can in principle involve any point group symmetry. Given the progress in
tailoring nano particles with particular shapes and interactions, this vast
family of "generalized nematics" might become accessible in the laboratory.
Little is known since the order parameter theories associated with the highly
symmetric point groups are remarkably complicated, involving tensor order
parameters of high rank. Here we show that the generic features of the
statistical physics of such systems can be studied in a highly flexible and
efficient fashion using a mathematical tool borrowed from high energy physics:
discrete non-Abelian gauge theory. Explicitly, we construct a family of lattice
gauge models encapsulating nematic ordering of general three dimensional point
group symmetries. We find that the most symmetrical "generalized nematics" are
subjected to thermal fluctuations of unprecedented severity. As a result, novel
forms of fluctuation phenomena become possible. In particular, we demonstrate
that a vestigial phase carrying no more than chiral order becomes ubiquitous
departing from high point group symmetry chiral building blocks, such as $I$,
$O$ and $T$ symmetric matter.
