  We investigate theoretically the collective dynamics of a suspension of low
Reynolds number swimmers that are confined to two dimensions by a thin fluid
film. Our model swimmer is characterized by internal degrees of freedom which
locally exert active stresses (force dipoles or quadrupoles) on the fluid. We
find that hydrodynamic interactions mediated by the film can give rise to
spontaneous continuous symmetry breaking (swarming), to states with either
polar or nematic homogeneous order. For dipolar swimmers, the stroke averaged
dynamics are enough to determine the leading contributions to the collective
behaviour. In contrast, for quadrupolar swimmers, our analysis shows that
detailed features of the internal dynamics play an important role in
determining the bulk behaviour. In the broken symmetry phases, we investigate
fluctuations of hydrodynamic variables of the system and find that these
destabilize order. Interestingly, this instability is not generic and depends
on length-scale.
