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We study the $\theta$ dependence of the continuum limit of 2d $U(N)$ gaugetheories defined on compact manifolds, with special emphasis on spherical($g=0$) and toroidal ($g=1$) topologies. We find that the coupling between$U(1)$ and $SU(N)$ degrees of freedom survives the continuum limit, leading toobservable deviations of the continuum topological susceptibility from the$U(1)$ behavior, especially for $g=0$, in which case deviations remain even inthe large $N$ limit.

Topological effects in continuum 2d $U(N)$ gauge theories