BF model

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

:$mathbf\{F\}equiv\; dmathbf\{A\}+mathbf\{A\}wedge\; mathbf\{A\}$

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).

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.

See also

* Spin foam
* Background field method

External links

* http://math.ucr.edu/home/baez/qg-fall2000/qg2.2.html