nLab BF-theory

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Contents

Context

\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

Quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

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

by

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.

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

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

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

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

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

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]

Last revised on August 21, 2024 at 09:06:47. See the history of this page for a list of all contributions to it.