BF Theory

Definition

BF theory is the topological field theory of a 1-form gauge field $A$ and a $(d-2)$-form $B$ in $d$ spacetime dimensions, with the action built from the wedge product alone,

$$S=\frac{iN}{2\pi }\int B\wedge dA,$$

in the Euclidean normalization of the generalized-symmetries course, where $N$ is an integer level. No metric enters, so the theory is topological. The two gauge redundancies are $A\to A+d\lambda$ and $B\to B+d\mu$, the equations of motion are $dA=0$ and $dB=0$, and the solutions are flat connections: there are no local degrees of freedom.

What the theory does carry is a set of extended operators and the phases they see. The Wilson line $e^{iq\oint A}$ and the surface operator $e^{i\oint B}$ link, and the linking phase is $e^{2\pi iq/N}$, which is the Aharonov–Bohm phase of a charge $q$ around a unit of discrete flux. That phase is the whole content: level-$N$ BF theory is the continuum form of ${\mathbb{Z}}_N$ gauge theory, with an electric ${\mathbb{Z}}_N^{(1)}$ symmetry generated by the surfaces and acting on the lines, and a magnetic ${\mathbb{Z}}_N^{(2)}$ acting on the surfaces (in four dimensions; see higher-form-symmetries for the degree counting in general $d$).

Two facts about the level are worth keeping in front:

• The level counts the ground states. On a spatial three-torus the four-dimensional theory has $N^3$ ground states, one for each class in $H^1(T^3,{\mathbb{Z}}_N)$; in three dimensions the count on a genus-$g$ surface is $N^{2g}$. This degeneracy is topological order, and it is what the toric code realizes at $N=2$ — the toric code is ${\mathbb{Z}}_2$ BF theory, the $K$-matrix $\left(\begin{array}{cc}0& 2\\ 2& 0\end{array}\right)$ of the Chern–Simons description.
• Level one is trivial. At $N=1$ the degeneracy is $1$ on any spatial topology and no line or surface carries a nontrivial phase. A level-one BF term is therefore not a topologically ordered phase; it is an invertible theory, and reading it as “topological order” is a mistake that costs the distinction between a gapped phase with a degeneracy and one without.

In three dimensions BF theory is the doubled Chern–Simons theory $U(1{)}_N\times U(1{)}_{-N}$, which is why it is non-chiral and why string-net constructions produce it rather than the chiral states of the quantum Hall effect.

Role in Research

BF theory is where the group’s confinement-duality line meets the modern language of generalized symmetries.

• The electric Julia–Toulouse condensate. Condensing electric charges and applying the Julia–Toulouse prescription produces an abelian BF term built on the original photon, at level one and accompanied by the Maxwell and $|dB{|}^2$ kinetic terms (Grigorio, Guimaraes, Rougemont, Wotzasek, Zarro, Phys. Rev. D 86 (2012) 027705). The same paper includes the magnetic defects consistently and obtains Dirac’s veto from the brane symmetry. Because the level is one, what the construction delivers is the massive sector — integrating out $B$ gives the Proca theory of Phys. Lett. B 723 (2013) 422 — and not topological order.
• What a charge-$N$ condensate would give. Condensing matter of charge $N$ instead should leave a level-$N$ BF term behind, and with it a ${\mathbb{Z}}_N^{(1)}$ remnant of the electric symmetry. Working out how the level is fixed by the charge of the condensate is an open problem of the generalized-symmetries course (Sem I Week 11, problem 6⋆⋆) and the natural sharpening of the 2012 result.
• The magnetic condensate is the mirror image. There the Gaussian dual of the massive Kalb–Ramond form is a level-one BF term built on the dual photon. Which photon the $B$ field couples to is what distinguishes the two condensates.
• Higher gauging. The surfaces of BF theory are the symmetry operators that higher gauging sums over, so condensation defects and the group’s manuscript on higher-gauging walls live directly on top of this structure.

Relations

• julia-toulouse-mechanism — the electric condensate produces a level-one BF term; the magnetic one produces it for the dual photon
• topological-order — level-$N$ BF theory is the continuum description of abelian topological order; level one carries none
• toric-code — the ${\mathbb{Z}}_2$ case, solved on the lattice
• chern-simons-theory — in three dimensions BF is the doubled theory $U(1{)}_N\times U(1{)}_{-N}$; the $K$-matrix language is shared
• higher-form-symmetries — the electric ${\mathbb{Z}}_N^{(1)}$ and magnetic symmetries of BF theory, and the degree counting in general $d$
• condensation-defects — the surfaces of BF theory are what higher gauging sums over
• lattice-gauge-theory, villain-action — ${\mathbb{Z}}_N$ gauge theory on the lattice, of which this is the continuum limit
• axionic-electrodynamics — integrating out $B$ or $A$ in the Maxwell–BF system relates the massive and the axionic descriptions
• confinement — a cautionary relation: BF theory describes a deconfined phase, with perimeter-law Wilson lines, and not a confining one

Papers

• A. Schwarz, Commun. Math. Phys. 67 (1979) 1 — the abelian topological action.
• G. Horowitz, Commun. Math. Phys. 125 (1989) 417 — exactly soluble diffeomorphism-invariant theories.
• D. Birmingham, M. Blau, M. Rakowski, G. Thompson, Phys. Rept. 209 (1991) 129 — the standard review.
• Grigorio, Guimaraes, Rougemont, Wotzasek, Zarro, Phys. Rev. D 86 (2012) 027705 — BF theory as an electric Julia–Toulouse condensate.
• See confinement-duality and the generalized-symmetries course conventions (§7) for the normalization used here.

Notes

• The level is the only datum: two BF theories with the same $N$ are the same theory, and $N$ is quantized because the linking phase must be single-valued.
• BF theory has a local Lagrangian but is not a local net of type III₁ algebras in the sense of the Haag–Kastler axioms, since it has no local observables to speak of. That is the distinction the magic paper uses to separate local QFT from topological field theory, and the reason a BF ground state can be a stabilizer state while a QFT vacuum cannot.
• This page was created on 2026-10-15; six pages had been linking to it since the Julia–Toulouse and condensed-matter material was written.