---
title: Chiral BF Theory Overview
url: https://www.emergentmind.com/topics/chiral-bf-theory
type: topic
---

# Chiral BF Theory Overview

Chiral BF theory denotes a family of BF-type constructions in which chirality is the organizing feature of either the boundary dynamics, the bulk differential structure, or the constrained field content. In the works considered here, the term covers at least three technically distinct settings: \(2+1\)-dimensional BF theories with boundaries supporting counter-propagating chiral currents and Kač–Moody algebras [1106.4641]; mixed three-dimensional theories on \(\mathbb C\times \mathbb R\) that are holomorphic in the complex direction and topological in the real direction, with Chern–Simons theory realized as a chiral deformation [1910.05230]; and higher-dimensional constrained BF/Plebanski-like systems in which self-dual and anti-self-dual sectors are separated by a chiral decomposition [2602.19151]. A related line of work shows that BF edge theories can acquire generalized chirality conditions linking dual boundary fields, with consequences for current algebras, entanglement spectra, and effective edge dynamics [2310.18391].

## 1. Terminological scope and basic BF structures

The common starting point is a BF-type action involving a connection-like field and a multiplier field that enforces a curvature constraint or its analogue. In \(2+1\) dimensions, the Abelian model is written as
\[
S_{BF}=\frac{k}{2\pi}\int d^3x\, \epsilon^{\mu\nu\rho} A_\mu \partial_\nu B_\rho ,
\]
or equivalently \(\frac{k}{2\pi}\int B\wedge dA\), with integer coupling \(k\) required by invariance of the partition function under large gauge transformations [1106.4641]. In the non-Abelian \(2+1\)-dimensional case, the action becomes
\[
S_{BF} = \frac{k}{2\pi}\int d^3x\, \epsilon^{\mu\nu\rho} \left( B_\mu^a F_{\nu\rho}^a +\frac{1}{3}f^{abc}B_\mu^a B_\nu^b B_\rho^c \right),
\]
with
\[
F_{\nu\rho}^a = \partial_\nu A_\rho^a-\partial_\rho A_\nu^a +f^{abc}A_\nu^b A_\rho^c .
\]
These theories are topological at low energy and contain no local propagating bulk degrees of freedom, but they develop nontrivial boundary dynamics when an edge is present [1106.4641].

A different three-dimensional realization is the mixed BF theory on \(\mathbb C\times \mathbb R\), where the fields are
\[
\alpha \in \mathcal A = \Omega^{0,*}(\mathbb C)\widehat\otimes_\pi \Omega^*(\mathbb R)[1]\otimes \mathfrak g, \qquad
\beta \in \mathcal B = \Omega^{1,*}(\mathbb C)\widehat\otimes_\pi \Omega^*(\mathbb R)\otimes \mathfrak g,
\]
with classical action
\[
S_{\mathrm{mix}}(\alpha,\beta) = \int \langle \beta, (\bar\partial + \mathrm d_t)\alpha\rangle +\frac12 \int \langle \beta, [\alpha,\alpha]\rangle .
\]
Here the theory is holomorphic along \(\mathbb C\) and topological along \(\mathbb R\), and this mixed structure is precisely what the corresponding work identifies as the chiral BF perspective [1910.05230].

A broader structural caution is that not every BF theory is chiral. The explicit BV equivalence between three-dimensional triadic gravity and non-Abelian BF theory is formulated entirely in the ordinary BF/gravity setting and does not discuss chiral splitting, self-dual sectors, or left/right decomposition [1707.07764]. This is useful for delimiting the term: chirality is an additional structure imposed on, or emerging from, BF theory rather than a universal property of all BF models.

## 2. Boundary chirality in \(2+1\) dimensions

In \(2+1\)-dimensional BF theory, chirality first appears sharply when the theory is placed on a manifold with boundary. The analysis of the planar boundary problem uses Symanzik’s separability condition, with boundary at \(u=0\), requiring propagators not to connect points on opposite sides of the boundary:
\[
\langle \phi(x)\phi(x')\rangle = \theta(u)\theta(u')\,\Delta_+(x,x') + \theta(-u)\theta(-u')\,\Delta_-(x,x') .
\]
The boundary is encoded by the most general local boundary action compatible with locality, power counting, and helicity conservation. In the Abelian case,
\[
S_{bd} = -\frac{k}{\pi}\int dud^2 z\, \delta(u)\, \big( \alpha_1 A\bar A +\alpha_2 A\bar B +\alpha_3 \bar A B +\alpha_4 B\bar B \big),
\]
up to the light-cone notation of the original presentation [1106.4641].

The boundary modifies the equations of motion by \(\delta(u)\)-localized terms, leaves residual gauge symmetry after axial gauge fixing, and produces Ward identities with boundary breakings. Those Ward identities imply conserved boundary currents and their Kač–Moody algebra [1106.4641]. In the Abelian theory, nontrivial time-reversal-invariant solutions require
\[
\alpha_1 \alpha_4 - (1-\alpha_2)^2 = 0, \qquad \alpha_2 = -\alpha_3.
\]
The boundary currents satisfy opposite chirality conditions,
\[
\bar\partial R_+ = 0, \qquad \partial S_+ = 0,
\]
and their equal-time algebra is
\[
[R_+(z),R_+(z')] = \frac{2\pi}{k}\,\alpha_4(1-\alpha_2)\,\partial_z \delta(z-z'),
\]
\[
[S_+(z),S_+(z')] = -\frac{2\pi}{k}\,\alpha_4(1-\alpha_2)\,\partial_z \delta(z-z'),
\]
\[
[R_+(z),S_+(z')] = 0.
\]
The resulting structure is a pair of Abelian Kač–Moody algebras with opposite chiralities [1106.4641].

The non-Abelian model is more rigid. Time-reversal invariance at the boundary fixes the parameters to
\[
\alpha_1 = \alpha_4 = 1,\qquad \alpha_2 = \alpha_3 = 0,
\]
and the natural combinations
\[
R^a = A_+^a + B_+^a, \qquad S^a = -A_+^a + B_+^a
\]
obey
\[
\bar\partial R^a = 0, \qquad \partial S^a = 0.
\]
Their current algebra is the direct sum of two non-Abelian Kač–Moody algebras of opposite chirality,
\[
[R^a(z),R^b(z')] = f^{abc} R^c(z)\,\delta(z-z') +\frac{k}{2\pi}\,\delta^{ab}\,\partial_z\delta(z-z'),
\]
\[
[S^a(z),S^b(z')] = f^{abc} S^c(z)\,\delta(z-z') -\frac{k}{2\pi}\,\delta^{ab}\,\partial_z\delta(z-z'),
\]
\[
[R^a(z),S^b(z')] = 0.
\]
A central contrast with the Abelian case is that the coefficient of the central extension is fixed by the bulk coupling \(k\) in the non-Abelian theory, whereas the Abelian boundary central term depends on boundary parameters unless further decoupling conditions are imposed [1106.4641].

This boundary construction is directly tied to time-reversal-invariant topological matter. The bulk is gapped and topological, while the boundary supports counter-propagating chiral currents, matching the expected helical edge structure of quantum spin Hall systems and related topological insulators [1106.4641].

## 3. Curved boundaries, local velocities, and accelerated chiral edge modes

A second boundary realization of chirality arises in Abelian BF theory on a generic three-dimensional manifold with a radial boundary at \(r=R\). The bulk action is
\[
S_{\text{BF}}=\kappa \int d^3x\, \theta(R-r)\,\varepsilon^{\mu\nu\rho} A_\mu \partial_\nu B_\rho ,
\]
and the full setup includes radial gauge fixing, sources, and a general boundary term
\[
S_{\text{bd}}=\int d^3x\, \delta(r-R)\,\Big(a_{ij}A^iA^j+b_{ij}B^iB^j+c_{ij}A^iB^j\Big),
\]
with coefficients that may depend on the induced metric \(Y_{ij}\) and its determinant \(y\) [2203.13520].

The broken Ward identities imply conserved boundary currents, and on the boundary those currents can be expressed through two scalar fields,
\[
B_i(X)=\partial_i \varphi(X), \qquad A_i(X)=\partial_i \zeta(X).
\]
The induced equal-time commutators generate a semidirect sum of two Kač–Moody algebras with central charge
\[
c=\frac{1}{\kappa},
\]
so positivity of the central charge requires \(\kappa>0\) [2203.13520].

The holographically induced boundary action depends on \(\varphi\) and \(\zeta\) and, after the linear redefinition
\[
\Phi_\pm = \zeta \pm \varphi,
\]
can be decoupled into two Luttinger actions, one for each chirality, when the mixed term vanishes [2203.13520]. The chiral equations of motion are
\[
\partial_t \Phi_\pm + v_\pm\, \partial_\theta \Phi_\pm = 0.
\]
A distinctive result is that on a generic curved bulk background the velocities are local,
\[
v_\pm = v_\pm(t,\theta),
\]
with their dependence transmitted from the bulk geometry through the induced boundary metric, in particular through the determinant \(y\). The edge modes are therefore accelerated chiral modes rather than constant-velocity ones [2203.13520].

The induced theory admits three qualitative patterns: opposite directions, same direction, and one moving mode. However, boundedness of the Hamiltonian rules out the same-direction case, because positivity implies
\[
v_+\ge 0,\qquad v_-\le 0.
\]
Time-reversal symmetry is even more restrictive and uniquely selects
\[
v_+ = -v_-,
\]
that is, equal-and-opposite counter-propagating modes, identified with the edge structure of topological insulators [2203.13520]. This provides a curved-space extension of the boundary chirality found in planar analyses: chirality survives, but its kinematics becomes locally metric-dependent.

## 4. Holomorphic-topological chiral BF theory and its Chern–Simons deformation

In the mixed theory on \(\mathbb C\times \mathbb R\), chirality is not primarily a boundary property but a property of the differential operators appearing in the action. The equations of motion are
\[
\bar\partial \alpha + \mathrm d_t \alpha + \frac12[\alpha,\alpha]=0, \qquad
\bar\partial \beta + \mathrm d_t \beta + [\alpha,\beta]=0,
\]
so at degree zero the connection defined by \(\alpha\) is holomorphic in the \(\mathbb C\)-direction and flat in the \(\mathbb R\)-direction [1910.05230].

Chern–Simons theory appears as a chiral deformation of this mixed BF system through the purely holomorphic interaction
\[
I_{\mathrm{CS}}(\alpha)=\frac12 \int \langle \alpha, \partial \alpha\rangle,
\]
which yields the deformed action
\[
S_{\mathrm{CS}} = S_{\mathrm{mix}} + I_{\mathrm{CS}}.
\]
The deformed equation of motion for \(\alpha\) becomes
\[
\bar\partial \alpha + \mathrm d_t \alpha + \frac12[\alpha,\alpha] + \partial \alpha = 0.
\]
The deformation is termed chiral because it uses only the holomorphic derivative \(\partial\), not \(\bar\partial\) [1910.05230].

The main technical innovation is a holomorphic gauge fixing. The kinetic operator is
\[
Q = (\bar\partial + \mathrm d)\otimes \mathrm{id}_{\mathfrak g},
\]
and the gauge-fixing operator is
\[
Q^* = (2\bar\partial^* + \mathrm d^*)\otimes \mathrm{id}_{\mathfrak g}.
\]
On flat \(\mathbb C\times \mathbb R\),
\[
\bar\partial^*(f\,\mathrm d\bar z)=2\frac{\partial f}{\partial z}, \qquad
\mathrm d^*(f\,\mathrm d t)=\frac{\partial f}{\partial t},
\]
and
\[
\triangle = [Q,Q^*] = 2[\bar\partial,\bar\partial^*] + [\mathrm d,\mathrm d^*].
\]
Because chiral vertices have at most one \(\beta\)-leg, while the propagator carries one \(\alpha\)-leg and one \(\beta\)-leg, the perturbation theory is constrained to trees with one \(\beta\)-root, single vertices with no \(\beta\)-legs, and one-loop wheel graphs with no \(\beta\)-legs. This yields the theorem that any chiral deformation of mixed BF theory admits a one-loop exact quantization [1910.05230].

The same framework extends to the half-space \(\mathbb C\times \mathbb R_{\ge 0}\), where the chiral boundary condition is imposed by requiring
\[
\alpha \in \mathcal A_{\mathrm{odd}}\otimes \mathfrak g[1].
\]
The reflected heat kernel
\[
K^\chi_T(z-w,t-s) = K^{an}_T(z-w,t-s) - K^{an}_T(z-w,t+s)
\]
implements the boundary condition, and the quantization remains one-loop exact and anomaly-free on the half-space. The boundary quantum correction shifts the level by
\[
\hbar \int_{\mathbb C \times \mathbb R_{\ge0}} \operatorname{Tr}_{\mathfrak g^{ad}}(\alpha\partial\alpha),
\]
which is the expected level shift from the Chern–Simons/WZW correspondence. Near the boundary, the observables recover the chiral current algebra factorization algebra \(\mathcal F^\kappa\), while in the bulk they recover the Chern–Simons factorization algebra [1910.05230].

## 5. Chiral decomposition in constrained BF and Plebanski-like theories

In higher-dimensional constrained BF systems, chirality refers to self-dual and anti-self-dual sectors rather than to boundary propagation. A six-dimensional \(SO(3,3)\) BF-type gauge theory on a split-signature manifold \(\mathcal M_6\) is formulated with an \(SO(3,3)\) connection \(A\), curvature \(R=dA+A\wedge A\), a Lie-algebra-valued \(4\)-form \(\Sigma\), auxiliary tensor fields \(\Phi\), and optionally a curvature-squared completion. The general action is
\[
S(\Sigma,A)=\int_{\mathcal M_6}\Big[ k_G\,\epsilon_{IJKLMN}\,(\Sigma^{IJKL}\wedge R^{MN}) +\Phi_{IJKLMN}\,(\Sigma^{IJKL}\wedge \star \Sigma^{MN}) +\epsilon_{ABCDEF}\big(\Lambda_G+g\,\Phi^{IJKLMN}\Phi_{IJKLMN}\big)(\Sigma^{ABCD}\wedge\star\Sigma^{EF}) +\alpha\,\epsilon_{IJKLMN}(R^{IJ}\wedge\star R^{KLMN}) \Big].
\]
In compact form,
\[
S(\Sigma,A)=\int_{\mathcal M_6}\Big[ k_G(\Sigma\wedge R)+\Phi(\Sigma\wedge\star\Sigma)+(\Lambda_G+g\Phi^2)(\Sigma\wedge\star\Sigma)+\alpha(R\wedge\star R) \Big].
\]
The BF term is topological before constraints, while the \(\Phi\)-terms impose simplicity-type relations [2602.19151].

After a MacDowell–Mansouri-type breaking
\[
SO(3,3)\to SU(2)_L\times SU(2)_R,
\]
the connection splits into stabilizer and coset pieces,
\[
A=\frac12\,\omega^a J_a+\frac12\,\kappa^i K_i+\frac1\ell\,\xi^{ai}E_{ai},
\]
and the curvature decomposes into chiral sectors [2602.19151]. The coset one-forms \(\xi^{ai}\) act as tetrad-like variables and are used to build Plebanski \(\Sigma\)-fields.

Two four-dimensional sectors then emerge. In Region \(\mathcal I\), the self-dual two-forms are
\[
\Sigma^{(+)i}=e^{(\mathcal I)0}\wedge e^{(\mathcal I)i} +\frac12\sqrt{\sigma}\,\epsilon^{ijk}e^{(\mathcal I)}_j\wedge e^{(\mathcal I)}_k,
\]
with constrained BF action
\[
S_{\mathcal I}=\int_{M_{\mathcal I}} \left[ \Sigma^{(+)i}\wedge F^i(A) -\frac{\lambda}{2}\Sigma^{(+)i}\wedge\Sigma^{(+)i} +\frac12\varphi^{(+)}_{ij}\,\Sigma^{(+)i}\wedge\Sigma^{(+)j} \right].
\]
In Region \(\mathcal{II}\), the anti-self-dual two-forms are
\[
\tilde\Sigma^{(-)i}=e^{(\mathcal{II})0}\wedge e^{(\mathcal{II})i} -\frac12\sqrt{\sigma}\,\epsilon^{ijk}e^{(\mathcal{II})}_j\wedge e^{(\mathcal{II})}_k,
\]
with action
\[
S_{\mathcal{II}}=\int_{M_{\mathcal{II}}} \left[ \tilde\Sigma^{(-)i}\wedge F^i(\widetilde A) -\frac{\tilde\lambda}{2}\tilde\Sigma^{(-)i}\wedge\tilde\Sigma^{(-)i} +\frac12\varphi^{(-)}_{ij}\,\tilde\Sigma^{(-)i}\wedge\tilde\Sigma^{(-)j} \right].
\]
Thus both sectors share the BF/Plebanski structure, but one is self-dual and the other anti-self-dual [2602.19151].

The simplicity constraints
\[
C^{(+)ij}_{\text{quad}}:=\Sigma^{(+)i}\wedge\Sigma^{(+)j} -\frac13\delta^{ij}\Sigma^{(+)k}\wedge\Sigma^{(+)k}=0,
\]
\[
C^{(-)ij}_{\text{quad}}:=\tilde\Sigma^{(-)i}\wedge\tilde\Sigma^{(-)j} -\frac13\delta^{ij}\tilde\Sigma^{(-)k}\wedge\tilde\Sigma^{(-)k}=0
\]
and the non-degeneracy conditions
\[
V^{(+)ij}_{\text{quad}}:=\frac13\delta^{ij}\Sigma^{(+)k}\wedge\Sigma^{(+)k}\neq 0, \qquad
V^{(-)ij}_{\text{quad}}:=\frac13\delta^{ij}\tilde\Sigma^{(-)k}\wedge\tilde\Sigma^{(-)k}\neq 0
\]
select the non-degenerate Einstein-gravity branch. Reality conditions are imposed because the chiral variables are complex in Lorentzian signature [2602.19151].

This usage of chiral BF theory differs conceptually from boundary chirality. Here chirality separates gravitational and weak-like sectors, self-dual and anti-self-dual two-forms, and opposite-signature four-dimensional leaves. The interface between these sectors carries its own gluing condition,
\[
\mathcal G[\phi]=\Sigma^{(+)}|_S - U_\phi\cdot\Sigma^{(-)}|_S \approx 0,
\]
implemented by
\[
S_{\text{glue}}=\int_S \mu^i(x)\left[\Sigma^{(+)i}|_S-U_\phi(x)\cdot\Sigma^{(-)i}|_S\right],
\]
so chirality is tied to a corner or edge structure even in this higher-dimensional setting [2602.19151].

## 6. Edge modes, generalized chirality conditions, and current algebras

A further extension of the chiral BF theme appears in entanglement and edge-mode analyses of Abelian \(p\)-form BF theory in \(d\) dimensions. The bulk action is
\[
S_{BF}[A,B] = \frac{\mathbb K^{IJ}}{2\pi}\int_X B_I\wedge dA_J,
\]
with \(A\) a vector of \(p\)-form gauge fields, \(B\) a vector of \((d-p-1)\)-form gauge fields, and \(\mathbb K\) an integer, symmetric, nondegenerate level matrix of rank \(\kappa\) [2310.18391].

On a manifold with boundary, the path integral splits as
\[
Z_{BF+edge}[X,\partial X] = Z_{bulk}[X]\,Z_{edge}[\partial X],
\]
and the relevant edge theory arises on the regulated entangling surface. The boundary term is
\[
S_{\partial}[A_\partial,B_\partial] = \frac{\mathbb K}{4\pi}\mu^\Delta \int A_\partial\wedge *A_\partial, \qquad \Delta=d-1-2p,
\]
together with the generalized chiral boundary condition
\[
B_\partial + \mu^\Delta (-1)^{(d-p-1)(p+1)} *A_\partial = 0.
\]
This links the boundary values of the \(p\)-form and \((d-p-1)\)-form in a self-duality-like way [2310.18391].

After Hodge decomposition, the edge theory becomes what the authors explicitly call a chiral mixed Maxwell theory: a coupled system of a \((p-1)\)-form Maxwell theory and a \((d-p-2)\)-form Maxwell theory constrained by a chirality relation. Its partition function is
\[
Z_{edge}[S^1_\beta\times R] = \frac{\Theta[q;\mathbb K\otimes \widetilde{\mathbb G}_p^R]} {\big(\eta_R^{(p-1)}[q]\big)^\kappa} \, \frac{\Theta[q;\mathbb K\otimes \widetilde{\mathbb G}_{d-p-1}^R]} {\big(\eta_R^{(d-p-2)}[q]\big)^\kappa}, \qquad q=e^{-\beta\mu}.
\]
The universal entropy contribution is
\[
TEE = -\frac12\big(b_p(\partial R)+b_{d-p-1}(\partial R)\big)\log|\det\mathbb K|
\]
in odd dimensions, after separating the nonuniversal geometric terms [2310.18391].

The extended Hilbert space is organized by an infinite-dimensional, centrally extended current algebra,
\[
[\hat Q[\alpha],\hat Q[\beta]] = i\frac{\mathbb K}{2\pi}\int_R d\beta\wedge d\alpha,
\]
which generalizes \(2\)-dimensional Kac–Moody algebras to arbitrary dimension and topology. In Laplacian eigenmodes,
\[
[\hat b_n,\hat a_m] = i\frac{\mathbb K}{2\pi}\sqrt{\lambda_n}\,\delta_{nm}.
\]
The thermal partition function of the chiral mixed Maxwell theory coincides exactly with an extended representation character of this current algebra,
\[
\chi_0[q] = Z_{edge}[S^1_\varepsilon\times R].
\]
This establishes an exact correspondence between the edge-mode theory and the entanglement spectrum [2310.18391].

A recurrent structural theme therefore spans the boundary, mixed, and constrained formulations: chirality in BF theory is repeatedly tied to residual gauge symmetry becoming physical at boundaries, interfaces, or entangling surfaces. A plausible implication is that the phrase “chiral BF theory” should be read less as the name of a unique Lagrangian and more as a class of BF-based constructions in which chiral decomposition controls the observable sector.

## 7. Conceptual distinctions and recurrent themes

Several distinctions recur across the literature.

First, boundary chirality and bulk chirality are not the same notion. In \(2+1\)-dimensional topological BF theory with an edge, chirality means counter-propagating boundary currents satisfying holomorphic and antiholomorphic conservation laws and generating Kač–Moody algebras [1106.4641]. In the mixed theory on \(\mathbb C\times\mathbb R\), chirality means dependence on the holomorphic derivative \(\partial\) and a holomorphic-topological splitting of the differential complex [1910.05230]. In the constrained \(SO(3,3)\) theory, chirality means the separation into self-dual and anti-self-dual BF/Plebanski sectors [2602.19151].

Second, time-reversal symmetry plays sharply different roles in boundary realizations. In planar and radial boundary theories, time reversal selects counter-propagating sectors; in the curved radial case it uniquely enforces equal-and-opposite local velocities, \(v_+=-v_-\) [2203.13520], while in the non-Abelian planar case it fixes the boundary physics much more rigidly than in the Abelian case [1106.4641]. This contrasts with the mixed holomorphic-topological construction, where the decisive structure is the chiral deformation and holomorphic gauge fixing rather than time reversal [1910.05230].

Third, ordinary BF theory should not be conflated with chiral BF theory. The BV-equivalence between three-dimensional gravity and BF theory provides a strong off-shell equivalence of BV theories, including ghosts and antifields, but explicitly does not discuss a chiral BF theory in the sense of left/right decomposition or self-dual splitting [1707.07764]. This limitation is conceptually important because it prevents the term “chiral” from being absorbed into the generic definition of BF theory.

Finally, the recurring mathematical output is the appearance of current algebras, edge degrees of freedom, and induced lower-dimensional theories. In \(2+1\)-dimensional topological matter, those are Kač–Moody currents of opposite chirality [1106.4641]. In the mixed theory, they appear as the chiral current algebra factorization algebra near the boundary [1910.05230]. In higher-dimensional entanglement problems, they become a centrally extended current algebra indexed by Laplacian modes on the entangling surface [2310.18391]. Across these settings, chiral BF theory serves as a framework in which topological bulk data produce chiral observable sectors on boundaries, interfaces, or reduced leaves.

Source: https://www.emergentmind.com/topics/chiral-bf-theory