The BF model is a topological field theory, which when quantized, becomes a topological quantum field theory. In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... Topological quantum field theory in theoretical physics provides an example of background-independent quantum theory, but with no local degrees of freedom, and only finitely many degrees of freedom globally. ...

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. This article is in need of attention. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... The adjoint representation of a Lie group G is the linearized version of the action of G on itself by conjugation. ... In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ...

The action In physics, the action principle is an assertion about the nature of motion, from which the trajectory of an object subject to forces can be determined. ...

where K is an invariant nondegenerate bilinear form over (if G is semisimple, the Killing form will do) and F is the curvature form In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... In mathematics, the Killing form, named for Wilhelm Killing (1847-1923), is a bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. ... In differential geometry, the curvature form describes curvature of principal bundle with connection. ...

This action is diffeomorphically invariant and gauge invariant. Its Euler-Lagrange equations are In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... The Euler-Lagrange Equation is the major formula of the Calculus of variations. ...

(no curvature)

and

(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. In differential geometry, the connection form describes connection on principal bundles (or vector bundles). ...

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