Abelian gauge theory on Riemann surfaces and new topological invariants

An Abelian gauge field theory framed on a complex line bundle L over a compact Riemann surface M is developed which allows the coexistence, simultaneously in the same model, of magnetic vortices and antivortices represented by the N zeros and P poles of a section of L. The quantized minimum energy E is given in terms of the first Chern class ci(L) and by a certain intersection number obtained from the multivortices. We show that E = 2ir(N + P). To realize such topological invariants as minimum energies, an existence and uniqueness theorem is established under the necessary and sufficient condition that |ci(L)| = \N -P\ < (total volume of M)/27r.