The BF model or BF theory is a topological field (physics), field, which when quantization (physics), 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 (field theory), 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 a Lie group, adjoint representation of G, and a connection form A for G.
The action (physics), action is given by
:
where K is an invariant nondegenerate bilinear form over
(if G is semisimple Lie algebra, semisimple, the Killing form will do) and F is the curvature form
:
This action is diffeomorphism, diffeomorphically invariant and gauge invariance, gauge invariant. Its Euler–Lagrange equations are
:
(no curvature)
and
:
(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.
In fact, BF theory can be used to formulate discrete gauge theory. One can add additional twist terms allowed by group cohomology theory such as Robbert Dijkgraaf, Dijkgraaf–Edward Witten, Witten topological gauge theory.
There are many kinds of modified BF theories as topological quantum field theory, topological field theories, which give rise to Linking number, link invariants in 3 dimensions, 4 dimensions, and other general dimensions.
See also
* Background field method
* Barrett–Crane model
* Dual graviton
* Plebanski action
* Spin foam
* Tetradic Palatini action
References
External links
* http://math.ucr.edu/home/baez/qg-fall2000/qg2.2.html
Quantum field theory
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