BF models, Duality and Bosonization on higher genus surfaces

TL;DR

This paper computes the generating functional of 2D BF theories with fermions on higher genus surfaces, revealing topological dependencies and deriving a spin-structure-aware bosonized partition function.

Contribution

It provides a detailed analysis of BF models coupled to fermions on higher genus surfaces, including topological effects and a new bosonization approach considering spin structures.

Findings

Partition function depends on topological restrictions of B field.

When B periods are multiples of 4π, the partition function is spin-structure independent.

A bosonized form of the partition function incorporating spin structures is derived.

Abstract

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 $L=dB{\wedge}A$ is written in terms of a globally defined 1-form $A$ and a multi-valued scalar field $B$. Consistency conditions on the periods of $dB$ have to be imposed. It is shown that there exist a non-trivial 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 $4\pi n$, 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 to 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