BF model

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The BF model is a topological field theory, 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

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)


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

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.

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