String models of glueballs and the spectrum of $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ gauge theories in $2+1$ dimensions

The spectrum of glueballs in $2+1$ dimensions is calculated within an extended class of Isgur-Paton flux tube models and compared to lattice calculations of the low-lying $\mathrm{SU}(N>~2)$ glueball mass spectrum. Our modifications of the model include a string curvature term and a new way of dealing with the short-distance cutoff. We find that the generic model is remarkably successful at reproducing the positive charge conjugation, $C=+,$ sector of the spectrum. The only large (and robust) discrepancy involves the $0^{-+}$ state, raising the interesting possibility that the lattice spin identification is mistaken and that this state is in fact $4^{-+}.$ Additionally, the Isgur-Paton model does not incorporate any mechanism for splitting $C=-$ from $C=+$ (in contrast with the case in $3+1$ dimensions), while the “observed” spectrum does show a substantial splitting. We explore several modifications of the model in an attempt to incorporate this physics in a natural way. At the qualitative level we find that this constrains our choice to the picture in which the $C=\pm$ splitting is driven by mixing with new states built on closed loops of adjoint flux. However, a detailed numerical comparison suggests that a model incorporating an additional direct mixing between loops of opposite orientation is likely to work better, and that, in any case, a nonzero curvature term will be required. We also point out that a characteristic of any string model of glueballs is that the $\mathrm{SU}\left(\overrightarrow{N}\infty \right)$ mass spectrum will consist of multiple towers of states that are scaled up copies of each other. To test this will require a lattice mass spectrum that extends to somewhat larger masses than currently available.