BFmodels, duality, and bosonization on higher genus surfaces

Abstract

The generating functional of two dimensional $\mathrm{BF}$ 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 $L=dB\ensuremath{\wedge}A$ is written in terms of a globally defined 1-form A and a multivalued scalar field B. Consistency conditions on the periods of $\mathrm{dB}$ 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 $4\ensuremath{\pi}n,$ 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.