BF models, duality, and bosonization on higher genus surfaces

The generating functional of two dimensional BF field theories coupled to fermionic fields and conserved currents is computed in the general case when the base manifold is a genus g compact Riemann surface. The Lagrangian density L=dB∧A is written in terms of a globally defined 1-form A and a multivalued scalar field B. Consistency conditions on the periods of dB have to be imposed. It is shown that there is a nontrivial dependence of the generating functional on the topological restrictions imposed to B. In particular if the periods of the B field are constrained to take values 4πn, with n any integer, then the partition function is independent of the chosen spin structure and may be written as a sum over all the spin structures associated with the fermions even when one started with a fixed spin structure. These results are then applied to the functional bosonization of fermionic fields on higher genus surfaces. A bosonized form of the partition function which takes care of the chosen spin structure is obtained.