nLab BF-theory



\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory




Quantum field theory


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Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



What is called BF-theory is a topological quantum field theory defined by an action functional S BFS_{BF} on a space of certain connections and differential forms over a 4-manifold XX, such that locally on XX the space of field histories is given by

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

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

  3. together with

    1. a homomorphism :𝔤 2𝔤 1\partial \colon \mathfrak{g}_2 \to \mathfrak{g}_1

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


S BF:(A,B) XF AB. 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 𝔤=(𝔤 2𝔤 1)\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 GG-principal 2-bundles with connection on a 2-bundle, where G=(G 2G 1)G = (G_2 \to G_1) is a Lie 2-group integrating 𝔤\mathfrak{g}.

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

X(F AB12F AF A12BB) \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 𝔤\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

  • 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

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

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

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 Martins, Aleksandar Miković, Lie crossed modules and gauge-invariant actions for 2-BF theories, Adv. Theor. Math. Phys. 15:4 (2011), 913-1199 euclid

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

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

Last revised on November 27, 2023 at 15:09:47. See the history of this page for a list of all contributions to it.