nLab
BF-theory

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 forms over a 4-dimensional smooth manifold XX, such that locally on XX the configuration space is given by Lie algebra-valued 1-forms AA with values in some 𝔤 1\mathfrak{g}_1 and 2-forms BB with values in some 𝔤 2\mathfrak{g}_2, together with a homomorphism :𝔤 2𝔤 1\partial : \mathfrak{g}_2 \to \mathfrak{g}_1 and an invariant polynomial ,\langle -,- \rangle, as

S BF:(A,B) XF AB, S_{BF} : (A,B) \mapsto \int_X \langle F_A \wedge \partial B\rangle \,,

where F AF_A is the curvature 2-form of AA.

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 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 indiced 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

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

Revised on May 3, 2014 00:38:55 by Zoran Škoda (109.227.8.217)