BF model

The BF model 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 two-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 g (if G is semisimple, the Killing form will do) and F is the curvature form

This action is diffeomorphically invariant and gauge invariant. 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.

However, if M is topologically nontrivial, A and B can have nontrivial solutions globally.