  Two different families of abelian chiral gauge theories on the torus are
investigated: the aim is to test the consistency of two-dimensional anomalous
gauge theories in the presence of global degrees of freedom for the gauge
field. An explicit computation of the partition functions shows that unitarity
is recovered in particular regions of parameter space and that the effective
dynamics is described in terms of fermionic interacting models. For the first
family, this connection with fermionic models uncovers an exact duality which
is conjectured to hold in the nonabelian case as well.
