  We compute and analyse the low-lying spectrum of 2+1 dimensional $SU(N)$
Yang-Mills theory on a spatial torus of size $l\times l$ with twisted boundary
conditions. This paper extends our previous work \cite{Perez:2013dra}. In that
paper we studied the sector with non-vanishing electric flux and concluded that
the energies only depend on the parameters through two combinations: $x=\lambda
N l /(4\pi)$ (with $\lambda$ the 't Hooft coupling) and the twist angle $\tilde
\theta$ defined in terms of the magnetic flux piercing the two-dimensional box.
Here we made a more complete study and we are able to condense our results,
obtained by non-perturbative lattice methods, into a simple expression which
has important implications for the absence of tachyonic instabilities, volume
independence and non-commutative field theory. Then we extend our study to the
sector of vanishing electric flux. We conclude that the onset of the would-be
large-volume glueball states occurs at an approximately fixed value of $x$,
much before the stringy torelon states have become very massive.
