nLab BF-theory

Idea

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

1. Lie algebra-valued 1-forms$\;A$ with values in some Lie algebra $\mathfrak{g}_1$ and with field strength/curvature 2-form $F_A$;

2. differential 2-forms$\;B$ with values in some Lie algebra $\mathfrak{g}_2$,

3. together with

   1. a homomorphism $\partial \colon \mathfrak{g}_2 \to \mathfrak{g}_1$

   2. an invariant polynomial $\langle -,- \rangle$

by

$$S_{BF} \;\colon\; (A,B) \;\mapsto\; \int_X \langle F_A \wedge \partial B\rangle \,.$$

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

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

then this is the generalized Chern-Simons theory action functional induced from the canonical Chern-Simons element on the strict Lie 2-algebra $\mathfrak{g}$. 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

General

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, $L_\infty$-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

Relation to Einstein gravity (in first-order formulation):

• Mariano Celada, Diego González, Merced Montesinos, BF gravity [arxiv/1610.02020]

• Alberto S. Cattaneo, Leon Menger, Michele Schiavina, Gravity with torsion as deformed BF theory [arXiv:2310.01877]

On a version of BF-theory in arithmetic related to non-orientability of arithmetic schemes:

• Magnus Carlson, Minhyong Kim, A note on abelian arithmetic BF-theory, (arXiv:1911.02236)

BFCG theory

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

• João Faria Martins, Aleksandar Miković: Lie crossed modules and gauge-invariant actions for 2-BF theories, Adv. Theor. Math. Phys. 15 4 (2011) 913-1199 [arXiv:1006.0903, euclid:atmp/1339438351]

Discussion of BFCG theory via connections on 2-bundles and higher parallel transport:

• A.D. López-Hernández, Graciela Reyes-Ahumada, Javier Chagoya: Categorical generalization of BF theory coupled to gravity [arXiv:2408.02889]