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
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
What is called BF-theory is a topological quantum field theory defined by an action functional $S_{BF}$ on a space of certain connections and forms over a 4-dimensional smooth manifold $X$, such that locally on $X$ the configuration space is given by Lie algebra-valued 1-forms $A$ with values in some $\mathfrak{g}_1$ and 2-forms $B$ with values in some $\mathfrak{g}_2$, together with a homomorphism $\partial : \mathfrak{g}_2 \to \mathfrak{g}_1$ and an invariant polynomial $\langle -,- \rangle$, as
where $F_A$ is the curvature 2-form of $A$.
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 $\mathfrak{g} = (\mathfrak{g}_2 \stackrel{\partial}{\to} \mathfrak{g}_1)$.
This would suggest that the BF-action functional is to be regarded as a functional on the space (2-groupoid) of $G$-principal 2-bundles with connection on a 2-bundle, where $G = (G_2 \to G_1)$ is a Lie 2-group integrating $\mathfrak{g}$.
If one couples to the above action functional that for topological Yang-Mills theory and a cosmological constant with coefficients as in
then this is the generalized Chern-Simons theory action functional indiced from the canonical Chern-Simons element on the strict Lie 2-algebra $\mathfrak{g}$. See Chern-Simons element for details.
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
The observation that coupled to topological Yang-Mills theory it can be read as the ∞-Chern-Simons theory action functional on connections on 2-bundles is in
and a more comprehensive discussion is in section 4.3 of
See also
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