BF model

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The BF model is a topological field theory, which when quantized, becomes a topological quantum field theory.

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

$S=\int_M K[\mathbf{B}\wedge \mathbf{F}]$

where K is an invariant nondegenerate bilinear form over $\mathfrak{g}$ (if G is semisimple, the Killing form will do) and F is the curvature form

$\mathbf{F}\equiv d\mathbf{A}+\mathbf{A}\wedge \mathbf{A}$

This action is diffeomorphically invariant and gauge invariant. Its Euler-Lagrange equations are

$\mathbf{F}=0$ (no curvature)

and

$d_\mathbf{A}B=0$ (the covariant exterior derivative of B is zero)

Actually, we can always gauge away any local degrees of freedom, which means this model has no local degrees of freedom. That's why it's called a topological field theory.

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