The BF model or BF theory is a topological field, which when quantized, becomes a topological quantum field theory.
BF stands for background field B and F, as can be seen below, are also the variables appearing in the Lagrangian of the theory, which is helpful as a mnemonic device.
We have a 4-dimensional differentiable manifold M, a gauge group G, which has as "dynamical" fields a 2-form B taking values in the adjoint representation of G, and a connection form A for G. The action is given by where K is an invariant nondegenerate bilinear form over
Its Euler–Lagrange equations are and In fact, it is always possible to gauge away any local degrees of freedom, which is why it is called a topological field theory.
One can add additional twist terms allowed by group cohomology theory such as Dijkgraaf–Witten topological gauge theory.