  It was suggested recently that the study of 1-dimensional QCD with fermions
in the adjoint representation could lead to an interesting toy model for
strange metals and their holographic formulation. In the high density regime,
the infrared physics of this theory is described by a constrained free fermion
theory with an emergent {\cal N}=(2,2) superconformal symmetry. In order to
narrow the choice of potential holographic duals, we initiate a systematic
search for chiral primaries in this model. We argue that the bosonic part of
the superconformal algebra can be extended to a coset chiral algebra of the
form {\cal W}_N =\SO(2N^2-2)_1/\SU(N)_{2N}. In terms of this algebra the
spectrum of the low energy theory decomposes into a finite number of sectors
which are parametrized by special necklaces. We compute the corresponding
characters and partition functions and determine the set of chiral primaries
for N \leq 5.
