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
functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
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 differential forms over a 4-manifold $X$, such that locally on $X$ the space of field histories is given by
Lie algebra-valued 1-forms$\;A$ with values in some Lie algebra $\mathfrak{g}_1$ and with field strength/curvature 2-form $F_A$;
differential 2-forms$\;B$ with values in some Lie algebra $\mathfrak{g}_2$,
together with
a homomorphism $\partial \colon \mathfrak{g}_2 \to \mathfrak{g}_1$
an invariant polynomial $\langle -,- \rangle$
by
There is not much of a proposal in the literature for how to make sense of this expression 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 induced 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
BF gravity
For the use of a version of BF-theory in arithmetic to deal with the non-orientability of arithmetic schemes see
Last revised on May 4, 2020 at 08:59:03. See the history of this page for a list of all contributions to it.