  We study the finite-size spectrum of the O($N$) symmetric Wilson-Fisher
conformal field theory (CFT) on the $d=2$ spatial-dimension torus using the
expansion in $\epsilon=3-d$. This is done by deriving a set of universal
effective Hamiltonians describing fluctuations of the zero momentum modes. The
effective Hamiltonians take the form of $N$-dimensional quantum anharmonic
oscillators, which are shown to be strongly coupled at the critical point for
small $\epsilon$. The low-energy spectrum is solved numerically for $N =
1,2,3,4$. Using exact diagonalization (ED), we also numerically study explicit
lattice models known to be in the O($2$) and O($3$) universality class,
obtaining estimates of the low-lying critical spectrum. The analytic and
numerical results show excellent agreement and the critical low energy torus
spectra are qualitatively different among the studied CFTs, identifying them as
a useful fingerprint for detecting the universality class of a quantum critical
point.
