For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
for higher abelian targets
for symplectic Lie n-algebroid targets
FQFT and cohomology
Types of quantum field thories
What is called BF-theory is a topological quantum field theory defined by an action functional on a space of certain connections and forms over a 4-dimensional smooth manifold , such that locally on the configuration space is given by Lie algebra-valued 1-forms with values in some and 2-forms with values in some , together with a homomorphism and an invariant polynomial , as
where is the curvature 2-form of .
There is not much of a proposal in the literature for what exactly that would or should mean globally. It has been observed that it looks like the action functional is one on ∞-Lie algebra-valued forms with values in a strict Lie 2-algebra .
This would suggest that the BF-action functional is to be regarded as a functional on the space (2-groupoid) of -principal 2-bundles with connection on a 2-bundle, where is a Lie 2-group integrating .
Much of the interest in BF-theory results from the fact that on a 4-dimensional manifold, to some extent the Einstein-Hilbert action for gravity may be encoded in BF-theory form. See gravity as a BF-theory.
BF theory was maybe first considered in
The observation that the BF-theory action functional looks like it should be read as a functional on a space of ∞-Lie algebra valued forms with values in a strict Lie 2-algebra possibly appears in print first in section 3.9 of
and a more comprehensive discussion is in section 4.3 of
There is a more general BFCG action introduced by Girelli, Pfeiffer and Popescu which has been shown to be a special case of the categorified BF-theory in