nLab
BF-theory

Idea

What is called BF-theory is a topological quantum field theory defined by an action functional $S_{\mathrm{BF}}$ 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 ${𝔤}_1$ and 2-forms $B$ with values in some ${𝔤}_2$, together with a homomorphism $\partial :{𝔤}_2\to {𝔤}_1$ and an invariant polynomial $⟨-,-⟩$, as

$$S_{\mathrm{BF}}:(A,B)\mapsto {\int }_X⟨F_A\wedge \partial B⟩,$$S_{BF} : (A,B) \mapsto \int_X \langle F_A \wedge \partial B\rangle \,,

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 $𝔤=({𝔤}_2\stackrel{\partial }{\to }{𝔤}_1)$.

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 $𝔤$.

If one couples to the above action functional that for topological Yang-Mills theory and a cosmological constant with coefficients as in

$${\int }_X(⟨F_A\wedge B⟩-\frac{1}{2}⟨F_A\wedge F_A⟩-\frac{1}{2}⟨\partial B\wedge \partial B⟩)$$\int_X( \langle F_A \wedge B\rangle - \frac{1}{2} \langle F_A \wedge F_A\rangle - \frac{1}{2}\langle \partial B \wedge \partial B\rangle)

then this is the generalized Chern-Simons theory action functional indiced from the canonical Chern-Simons element on the strict Lie 2-algebra $𝔤$. See Chern-Simons element for details.

Applications

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.

References

BF theory was maybe first considered in

• Gary Horowitz, Exactly soluable diffeomorphism invariant theories Commun. Math. Phys. 125, 417-437 (1989)

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

• Florian Girelli, Hendryk Pfeiffer, Higher gauge theory – differential versus integral formulation, arXiv:hep-th/0309173

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

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_{\mathrm{\infty }}$-conections (web))

and a more comprehensive discussion is in section 4.3 of

• Urs Schreiber, differential cohomology in a cohesive topos .

See also

• Aristide Baratin, Florian Girelli, Daniele Oriti, Diffeomorphisms in group field theories, Physical Review D, vol. 83, Issue 10, id. 104051, doi, arxiv/1101.0590