Topological sectors of spin 1 theories in 2+1 dimensions

It is shown that the topological massive and “self-dual” theories, which are known to provide locally equivalent descriptions of spin 1 theories in 2+1 dimensions, have different global properties when formulated over topologically non-trivial regions of space-time. The partition function of these theories, when constructed on an arbitrary Riemannian manifold, differ by a topological factor, which is equal to the partition function of the pure Chern-Simons theory. This factor is related to the space of solutions of the field equations of the topological massive theory for which the connection is asymptotically flat but not gauge equivalent to zero. A new covariant, first order, gauge action, which generalizes the “self-dual” action, is then proposed. It is obtained by sewing local self-dual theories. Its global equivalence to the topological massive gauge theory is shown.