BF Theory as a Topological Gauge Theory

When both compact fields are dynamical, an integer-level BF coupling defines a finite topological gauge theory rather than merely a metric-independent background phase. Its local equations make both curvatures flat; its global sector sum leaves finite $\mathbb Z_N$ holonomy; complementary Wilson supports measure a mixed linking phase; and a closed spatial manifold $Y$ carries a finite state space with basis labeled by $H^p(Y;\mathbb Z_N)$. Boundaries are not automatic: a polarization, boundary condition, or boundary theory must specify which variables are fixed and which charged operators may end.

The initial domain is a closed smooth oriented $D$-manifold $M$, with $D\geq3$, $1\leq p\leq D-2$, and $q=D-p-1$. The fields are compact $U(1)$ higher connections $A_p$ and $B_q$, the level is $N\in\mathbb Z_{>0}$, and the Euclidean weight is $e^{-S_E}$. Later sections open a boundary and specialize to three- and four-dimensional controls. Orientation reversal complex-conjugates the linking phases.

Required background. BF Couplings and Discrete Topological Data supplies the compact differential-cohomology fields, level quantization, and positive-linking convention. State Spaces, Cobordisms, and Gluing supplies the cylinder, trace, boundary pairing, and finite-mode gluing tests used below.

Compact BF fields define the finite domain

In a simultaneous local trivialization, the action is

$$S_E[A,B] = \frac{iN}{2\pi} \int_M B_q\wedge\mathrm dA_p, \qquad p+q=D-1.$$

The local gauge transformations are

$$A\longmapsto A+\mathrm d\alpha_{p-1}, \qquad B\longmapsto B+\mathrm d\beta_{q-1}.$$

Large gauge transformations, nontrivial bundles, flat sectors, and gauge-for-gauge data are not contained in these two formulas. Intrinsically,

$$\check A\in\widehat H^{p+1}(M;\mathbb Z), \qquad \check B\in\widehat H^{q+1}(M;\mathbb Z),$$

and the global phase is the differential-cohomology pairing

$$e^{-S_E[A,B]} = \exp\!\left[ -2\pi iN\int_M\check B\smile\check A \right].$$

The fiber integral is $\mathbb R/\mathbb Z$-valued. This is what retains torsion and flat holonomy when the de Rham curvatures vanish. Kapustin and Seiberg construct the compact fields, their Deligne–Beilinson completion, and the integer-level large-gauge check in Kapustin and Seiberg 2014, § 3, arXiv v2, printed pp. 9–13, especially eqs. (3.1)–(3.6), PDF.

Varying the local action on a closed manifold gives

$$\mathrm dA=0, \qquad \mathrm dB=0.$$

These are curvature-flatness equations, not the statement that the compact connections are gauge-trivial. Summing over the global $B$ sectors performs a compact Fourier projection and restricts $A$ holonomy to $\mathbb Z_N$; summing over $A$ does the complementary projection for $B$. Thus three ingredients are inseparable:

• compact $U(1)$ higher connections;
• integer nonzero level; and
• the global path-integral sum modulo the full gauge groupoid.

If either field is noncompact, the holonomy remains continuous. If $N=0$, the pairing is degenerate. Neither case is the finite topological theory discussed on this page.

Complementary Wilson supports carry the linking pairing

Let $C_p$ and $\Sigma_q$ be disjoint oriented closed supports. The basic operators are

$$W_e(C) = \exp\!\left(i e\int_C A\right), \qquad V_m(\Sigma) = \exp\!\left(i m\int_\Sigma B\right),$$

with integer labels $e$ and $m$. In a sector where the ordinary linking number is defined—for example on $S^D$, or when one support bounds—and where the displayed normalizing expectation values are nonzero,

$$\frac{ \left\langle W_e(C)V_m(\Sigma)\right\rangle }{ \left\langle W_e(C)\right\rangle \left\langle V_m(\Sigma)\right\rangle } = \exp\!\left[ \frac{2\pi i\,em}{N} \operatorname{Lk}(C,\Sigma) \right].$$

The sign is the positive-linking convention inherited from the prerequisite. Reversing either support or the spacetime orientation complex-conjugates the phase. Charges are measured modulo $N$: shifting $e$ or $m$ by $N$ leaves every mixed phase unchanged. The compact operator construction and its torsion-sensitive refinement appear in Kapustin and Seiberg 2014, § 3, arXiv v2, printed pp. 11–12, eqs. (3.10)–(3.13), PDF; the charge-$N$ gauge-theory normalization is also developed in Banks and Seiberg 2011, § 2.2, arXiv v2, printed pp. 6–10, especially eqs. (2.9)–(2.13), PDF.

On a general manifold, ordinary linking must be replaced by the appropriate finite or differential-cohomology pairing. In particular, an order-$\ell$ torsion cycle does not automatically contribute $\gcd(N,\ell)$ independent labels. A cyclic direct summand $\mathbb Z_\ell$ does contribute

$$\operatorname{Hom}(\mathbb Z_\ell,\mathbb Z_N) \cong \mathbb Z_{\gcd(N,\ell)},$$

whereas a particular order-$\ell$ element embedded in a larger group can have a smaller realized evaluation image.

Canonical quantization produces a finite Heisenberg algebra

Take $M=Y^{D-1}\times\mathbb R$ and choose the $A$-polarization. A basis of the topological state space is labeled by the finite flat sectors

$$a\in H^p(Y;\mathbb Z_N), \qquad \mathcal H_A(Y) \cong \operatorname{Fun}\!\left(H^p(Y;\mathbb Z_N),\mathbb C\right).$$

Therefore

$$\boxed{ \dim\mathcal H_A(Y) = \left\lvert H^p(Y;\mathbb Z_N)\right\rvert }.$$

The $B$ holonomies are not a second independent set of basis labels. They act as finite translations of the $A$ label. If $c_p$ and $s_q$ are spatial cycles and $I_Y(c,s)$ is their oriented intersection number, then

$$W_e(c)V_m(s) = \exp\!\left[ \frac{2\pi i\,em}{N}I_Y(c,s) \right] V_m(s)W_e(c).$$

Equivalently, for a wavefunction $\psi(a)$,

$$\begin{aligned} \bigl(W_e(c)\psi\bigr)(a) &= \exp\!\left[ \frac{2\pi i e}{N}\langle a,[c]\rangle \right]\psi(a),\\ \bigl(V_m(s)\psi\bigr)(a) &= \psi\!\left(a-m\,\operatorname{PD}[s]\right), \end{aligned}$$

with the sign of the translation tied to the same intersection convention. This clock–shift algebra is the equal-time form of the spacetime linking phase.

The cardinality formula makes both free and torsion sectors visible. Write

$$H_j(Y;\mathbb Z) \cong \mathbb Z^{b_j} \oplus \bigoplus_a\mathbb Z_{t_{j,a}}.$$

The universal coefficient theorem gives

$$\left\lvert H^p(Y;\mathbb Z_N)\right\rvert = N^{b_p} \prod_a\gcd(N,t_{p,a}) \prod_b\gcd(N,t_{p-1,b}).$$

Three useful controls are

$$\begin{array}{c|c} (Y,p) & \dim\mathcal H_A(Y)\\ \hline (T^2,1) & N^2\\ (T^3,1) & N^3\\ (L(\ell,r),1) & \gcd(N,\ell). \end{array}$$

Closing the identity cylinder gives the TQFT trace check

$$Z\!\left(Y\times S^1\right) = \operatorname{Tr}_{\mathcal H(Y)}\mathbf 1 = \left\lvert H^p(Y;\mathbb Z_N)\right\rvert$$

for the induced product tangential structure. In three dimensions this agrees with the automorphism-weighted finite-bundle sum; omitting its groupoid measure would spoil gluing. Dijkgraaf and Witten give the flat-bundle state and sewing construction in Dijkgraaf and Witten 1990, §§ 6.2–6.3, printed pp. 415–417, eqs. (6.8)–(6.17), while Freed and Quinn give the boundary measure and gluing theorem in Freed and Quinn 1993, § 2, current arXiv v3, internal printed pp. 9–13, especially eq. (2.1), Lemma 2.4, and Theorem 2.13 with eq. (2.17), PDF.

Three-dimensional BF is off-diagonal Chern–Simons theory

For $D=3$ and $p=q=1$, put $a^1=A$, $a^2=B$ and

$$K_{\mathrm{BF}} = \begin{pmatrix} 0&N\\ N&0 \end{pmatrix}.$$

On a closed manifold the Abelian Chern–Simons presentation is

$$\begin{aligned} S_{E,K} &= \frac{i}{4\pi} \int_M K_{IJ}a^I\wedge\mathrm da^J\\ &= \frac{iN}{2\pi} \int_M B\wedge\mathrm dA. \end{aligned}$$

The matrix is integral, even, nonsingular, and has signature zero. Its line group is

$$\mathcal A_{\mathrm{BF}} \cong \mathbb Z_N\oplus\mathbb Z_N,$$

with labels $\ell=(e,m)$. Since

$$K_{\mathrm{BF}}^{-1} = \begin{pmatrix} 0&1/N\\ 1/N&0 \end{pmatrix},$$

the positive full braid and unit twist are

$$\begin{aligned} M_{(e,m),(e',m')} &= \exp\!\left[ \frac{2\pi i}{N}(em'+me') \right],\\ \theta_{(e,m)} &= \exp\!\left( \frac{2\pi i}{N}em \right). \end{aligned}$$

The pure electric $(e,0)$ and pure magnetic $(0,m)$ lines have trivial self-statistics, while a dyon can have nontrivial spin. Canonical quantization gives

$$\dim\mathcal H(\Sigma_g) = \lvert\det K_{\mathrm{BF}}\rvert^g = N^{2g}.$$

The determinant formula is established in Belov and Moore 2005, § 5.3, arXiv v1, printed p. 26, prose after eq. (5.17), PDF. Because $\operatorname{sign}K_{\mathrm{BF}}=0$, this block has no net Abelian framing phase from the signature. That cancellation does not turn a boundary theory into an oriented absolute theory without the boundary data discussed below.

On an open manifold the symmetric $K$-matrix representative and the $B\wedge\mathrm dA$ representative differ:

$$\frac{iN}{4\pi} \int_M \left(A\wedge\mathrm dB+B\wedge\mathrm dA\right) = \frac{iN}{2\pi} \int_M B\wedge\mathrm dA - \frac{iN}{4\pi} \int_{\partial M}A\wedge B.$$

Thus their bulk equations agree, but their boundary symplectic potentials and natural polarizations differ by a counterterm. One may not transfer a boundary condition between the two presentations without transferring that term as well.

First application: the BF row matches untwisted cyclic gauge theory

Let $N$ be the greatest common divisor of all condensed electric charges. In particular, for a single charge-$N$ condensate assume that no additional lower-charge field also condenses. This Higgs phase supplies a physical route to the compact theory. If

$$\Phi=v e^{i\varphi}, \qquad \varphi\sim\varphi+2\pi,$$

and the compact gauge transformation acts by

$$\varphi\longmapsto\varphi+N\lambda, \qquad A\longmapsto A+\mathrm d\lambda,$$

then the phase stiffness is

$$S_{\mathrm{phase}} = \frac{v^2}{2} \int (\mathrm d\varphi-NA)\wedge* (\mathrm d\varphi-NA).$$

Dualizing the periodic scalar, including its winding sectors, gives a compact $(D-2)$-form connection $B$ and the infrared term

$$S_E^{\mathrm{IR}} = \frac{iN}{2\pi} \int B\wedge\mathrm dA +\text{metric-dependent kinetic terms}.$$

Deep in the gapped regime the kinetic terms are irrelevant to the topological observables. In $2+1$ dimensions, $B$ is another one-form and the basic observables are particle worldlines. In $3+1$ dimensions, $B$ is a two-form and the mixed observable is particle–string linking. The periodicity of $\varphi$ is essential: dualizing a noncompact scalar would not produce the finite sector sum. Banks and Seiberg give the charge-$N$ derivation in Banks and Seiberg 2011, §§ 2.2–2.3, arXiv v2, printed pp. 6–10, eqs. (2.3)–(2.15), PDF; a pedagogical derivation of the compact Fourier projection, operators, and Higgs reduction appears in Brennan and Hong 2023, § 3, arXiv v2, printed pp. 32–36, eqs. (3.22)–(3.45), PDF.

The following table keeps the three-model thread in one dimension, $2+1$, and makes the field role explicit. Its rows share finite data after global completion; the table is not, by itself, a proof of equivalence.

Three dynamical topological models under one finite-state and operator test; matching one row of data does not prove a fully extended equivalence

Dynamical model	Global fields summed	Finite line labels	Decisive topological datum	Genus-g state check
Compact U(1) Chern–Simons, nonzero level k	One compact one-form connection	Cyclic group of order |k|	One self-pairing controls mutual braiding and spin	|k| to the power g; odd k is spin-dependent
Untwisted compact BF, positive level N	Two compact one-form connections	Electric–magnetic pair in ZN × ZN	Off-diagonal pairing; pure electric and magnetic spins are trivial	N to the power 2g
Untwisted finite ZN gauge theory	Finite flat bundles with automorphism weights	Electric charges and magnetic fluxes	Finite holonomy pairing and groupoid gluing measure	N to the power 2g

After the compact sectors and path-integral measure are matched, the three-dimensional BF row is the continuum presentation of untwisted $\mathbb Z_N$ gauge theory. The agreement

$$Z_{\mathrm{BF}}\!\left(\Sigma_g\times S^1\right) = N^{2g} = \frac{ \left\lvert \operatorname{Hom}\!\left( \pi_1(\Sigma_g\times S^1),\mathbb Z_N \right) \right\rvert }{N}$$

checks the state-space trace and the connected-spacetime automorphism factor. It does not yet prove equality of every bordism map, boundary condition, or extended operator. A nonzero Dijkgraaf–Witten cocycle can change spins and associative data, and a generic finite group need not admit this Abelian BF presentation.

Boundary polarizations decide which operators can end

Now let $\partial M\ne\varnothing$. Varying the local Euclidean action gives

$$\begin{aligned} \delta S_E = \frac{iN}{2\pi} \Bigg\{& \int_M \left[ \delta B\wedge\mathrm dA - (-1)^q\mathrm dB\wedge\delta A \right]\\ &+ (-1)^q \int_{\partial M} B\wedge\delta A \Bigg\}. \end{aligned}$$

The boundary term is the canonical pairing between the two fields. After removing the overall factor of $i$, a boundary symplectic potential is

$$\Theta_{\partial} = (-1)^q\frac{N}{2\pi} \int_{\partial M}B\wedge\delta A, \qquad \Omega_{\partial} = \delta\Theta_{\partial}.$$

Fixing the pullback of $A$ sets $\delta A|_{\partial M}=0$ and defines an $A$-polarization. Adding

$$S_{\partial}^{\mathrm{swap}} = - (-1)^q \frac{iN}{2\pi} \int_{\partial M}B\wedge A$$

exchanges the boundary variation for one proportional to $\delta B\wedge A$, giving the complementary $B$-polarization. These are choices of relative presentation, not two new bulk theories.

Gauge invariance is a separate boundary question. Under $B\mapsto B+\mathrm d\beta$,

$$\Delta_\beta S_E = \frac{iN}{2\pi} \int_{\partial M}\beta\wedge\mathrm dA.$$

One must restrict the boundary gauge transformations, impose a compatible boundary condition, or add boundary degrees of freedom whose variation cancels this term. The bulk density does not select a unique edge theory.

In three dimensions, the pure electric subgroup $\{(e,0)\}$ and pure magnetic subgroup $\{(0,m)\}$ are elementary maximal isotropic boundary choices. An electric boundary can absorb the corresponding electric lines; a magnetic boundary can absorb the magnetic lines. For composite $N$ there are additional possibilities, and a full classification requires the boundary algebra and condensation data. Kapustin and Saulina analyze Abelian Chern–Simons boundary conditions through isotropic and Lagrangian data in Kapustin and Saulina 2011, § 4, corrected arXiv v2, printed pp. 10–16, PDF. This page uses only the two basic BF polarizations.

Abelian BF is a bounded AKSZ and BV–BFV example

There is a useful local classical construction behind the preceding boundary pairing. It must be kept separate from the compact quantum theory. Take an oriented compact three-manifold $M$ with boundary $\Sigma=\partial M$ and a real vector space $V$. The AKSZ source is $T[1]M$ with degree-one differential $\mathsf D=\mathrm d_M$. The target is

$$\mathcal X = T^*[2]\!\left(V[1]\right) \cong V[1]\oplus V^*[1],$$

with degree-one coordinates $a^i,b_i$, primitive and symplectic form

$$\alpha_{\mathcal X} = b_i\,\delta a^i, \qquad \omega_{\mathcal X} = \delta b_i\,\delta a^i.$$

For Abelian BF theory the target Hamiltonian is $\Theta=0$, so $\{\Theta,\Theta\}=0$. A map $T[1]M\to\mathcal X$ is a pair of superfields $\mathbf A\in\Omega^\bullet(M,V)[1]$ and $\mathbf B\in\Omega^\bullet(M,V^*)[1]$; their ghost-number-zero components are the physical one-forms $A$ and $B$. With $c=N/(2\pi)$, transgression gives

$$\begin{aligned} \Omega_M &= c\int_{T[1]M} \langle\delta\mathbf B,\delta\mathbf A\rangle,\\ \mathcal S_M &= c\int_{T[1]M} \langle\mathbf B,\mathsf D\mathbf A\rangle. \end{aligned}$$

Choose the Cattaneo–Mnev–Reshetikhin sign convention

$$Q_M\mathbf A=-\mathsf D\mathbf A, \qquad Q_M\mathbf B=-\mathsf D\mathbf B.$$

On a closed $M$,

$$Q_M^2=0 \quad\Longrightarrow\quad \frac12(\mathcal S_M,\mathcal S_M)=0.$$

This is the classical master equation for the linear model. With a boundary, the exact statement is relative. The transgressed boundary data are

$$\begin{aligned} \alpha_\Sigma &= -c\int_{T[1]\Sigma} \langle\mathbf B,\delta\mathbf A\rangle,\\ \omega_\Sigma &= \delta\alpha_\Sigma,\\ \mathcal S_\Sigma &= c\int_{T[1]\Sigma} \langle\mathbf B,\mathsf D\mathbf A\rangle, \end{aligned}$$

and the BV–BFV identities read

$$\iota_{Q_M}\Omega_M = \delta\mathcal S_M+\pi^*\alpha_\Sigma, \qquad \frac12\iota_{Q_M}\iota_{Q_M}\Omega_M = \pi^*\mathcal S_\Sigma.$$

The condition

$$L_A = \left\{ \mathbf A|_\Sigma=0 \right\}$$

is Lagrangian in the boundary field space, $\alpha_\Sigma|_{L_A}=0$, and is preserved by $Q_M$; $\mathbf B|_\Sigma=0$ is the dual elementary choice. Thus the same local construction supplies the bulk action, the boundary symplectic pairing, the closed-manifold master equation, and a concrete Lagrangian boundary condition.

Two adversarial controls expose the hypotheses. First, leaving all boundary data free is not a Lagrangian condition because $\omega_\Sigma$ remains nonzero; the boundary term and modified master equation cannot be discarded. Second, try the non-Abelian-looking target Hamiltonian

$$\Theta_f = \frac12 b_i f^i{}_{jk}a^ja^k.$$

Its self-bracket is proportional to the Jacobiator. For example, if

$$[e_1,e_2]=e_1, \qquad [e_2,e_3]=e_2, \qquad [e_3,e_1]=e_3,$$

then

$$J(e_1,e_2,e_3) = e_1+e_2+e_3 \ne0.$$

Consequently $\{\Theta_f,\Theta_f\}\ne0$, and its transgression produces a nonzero classical-master-equation defect even on closed $M$. The AKSZ condition is doing real work.

This fixture is the formal real, linear classical model. Multiplying its local data by $N/(2\pi)$ preserves the classical identities but does not produce compact differential cocycles, large gauge transformations, integer level quantization, torsion sectors, the zero-mode measure, or the finite $\mathbb Z_N$ quantum theory. Those are the global inputs developed in the earlier sections. The formal construction and boundary identities are derived in Cattaneo, Mnev, and Reshetikhin 2014, §§ 3.7, 5.4.1, and 6.1–6.3, arXiv v3, printed pp. 22, 36–37, and 41–45, especially eq. (52), PDF; a compact account of the target master condition and BF transgression appears in Mnev 2012, §§ 2.1–2.3, arXiv v1, printed pp. 6–9, especially eqs. (7) and (23)–(26), PDF.

The displayed modified master equation follows the convention and boundary projection used in Cattaneo, Mnev, and Reshetikhin 2018, § 2.1.1, arXiv v2, printed pp. 9–10, eqs. (2.1)–(2.2), and § 3, printed pp. 26–27, eqs. (3.1)–(3.3), PDF. Changing the BV symplectic-form sign changes $Q$, $\alpha_\Sigma$, and $\mathcal S_\Sigma$ together; mixing those convention packages would invalidate the identities.

What compact BF theory does not determine

The compact level-$N$ theory fixes a strong but finite package:

• curvature flatness and finite $\mathbb Z_N$ holonomy;
• the electric–magnetic linking pairing;
• the finite Heisenberg algebra of complementary operators;
• the state-space dimension in a chosen polarization; and
• the bulk boundary symplectic pairing.

It does not determine every completion of that package.

A local density is not the global theory. Differential forms alone miss bundle sectors, torsion, the gauge-groupoid measure, and higher gauge-for-gauge normalization.

The untwisted pairing is not every finite gauge theory. Cocycle twists can change spins, associators, and bordism amplitudes without changing the underlying finite group. Non-Abelian finite groups require data that no two-field Abelian BF action captures.

A bulk action does not choose a boundary. Polarization, allowed boundary gauge transformations, counterterms, condensed operators, and boundary dynamics are additional data.

Topological infrared data do not determine phase dynamics. Maxwell or stiffness terms, the hierarchy of gaps, defects with finite tension, phase transitions, and lattice realizations lie outside the strict TQFT limit.

The classical AKSZ model is not the compact quantum completion. Its master equation organizes local fields, ghosts, antifields, and boundary symplectic data. It neither quantizes $N$ nor supplies the finite sector sum.

For $N>1$, the theory is noninvertible: already $\dim\mathcal H(T^{D-1})>1$. At $N=1$ the finite holonomy and mixed phase are trivial. This limiting check is one reason to state $N>0$ and to separate the nontrivial finite theory from the degenerate $N=0$ action.

Check your understanding

1. Recover the clock–shift sign

Use the wavefunction actions above to compute $W_e(c)V_m(s)\psi$ and $V_m(s)W_e(c)\psi$.

Solution

The shift operator evaluates the Wilson phase at $a-m\,\operatorname{PD}[s]$. Their ratio is

$$\exp\!\left[ \frac{2\pi i\,em}{N}I_Y(c,s) \right],$$

which gives the displayed positive-intersection Heisenberg relation. A plus sign in the wavefunction shift would conjugate the commutator and would have to be accompanied by the opposite linking convention.

2. Count a torsion state space

Let $Y=L(12,r)$ and $N=8$. Find $\dim\mathcal H_A(Y)$ for $p=1$.

Solution

Since $H_1(Y;\mathbb Z)\cong\mathbb Z_{12}$,

$$H^1(Y;\mathbb Z_8) \cong \operatorname{Hom}(\mathbb Z_{12},\mathbb Z_8) \cong \mathbb Z_4.$$

Therefore $\dim\mathcal H_A(Y)=\gcd(12,8)=4$.

3. Test the off-diagonal K matrix

For $N=3$, compute the full braid of $(1,0)$ with $(0,1)$ and the twist of $(1,1)$.

Solution

The mutual phase is

$$M_{(1,0),(0,1)} = e^{2\pi i/3}.$$

The dyon twist is also

$$\theta_{(1,1)} = e^{2\pi i/3}.$$

Each pure electric or pure magnetic line has unit twist.

4. Swap the boundary polarization

Show that adding $S_{\partial}^{\mathrm{swap}}$ cancels the $B\wedge\delta A$ term and leaves a variation proportional to $\delta B\wedge A$.

Solution

Varying the counterterm gives

$$\delta S_{\partial}^{\mathrm{swap}} = - (-1)^q\frac{iN}{2\pi} \int_{\partial M} \left( \delta B\wedge A+B\wedge\delta A \right).$$

The second term cancels the bulk boundary variation. The surviving first term vanishes when the pullback of $B$ is fixed, so the counterterm exchanges the $A$- and $B$-polarizations.

5. Make the AKSZ failure visible

Why does the cubic target Hamiltonian built from the displayed non-Jacobi bracket fail even when $\partial M=\varnothing$?

Solution

Its target self-bracket contains the nonzero Jacobiator:

$$\{\Theta_f,\Theta_f\} \propto J^i{}_{jkl}\,b_i a^j a^k a^l \ne0.$$

Transgression carries this term into $\tfrac12(\mathcal S_M,\mathcal S_M)$, so the closed-manifold classical master equation fails. No boundary condition can repair a bulk target that does not satisfy its own master equation.

Continue to finite gauge theory and relative boundaries

Finite Gauge Theory and Dijkgraaf–Witten Twists will replace the continuum BF presentation by the finite bundle groupoid and develop cocycle twists. Operators, Boundaries, and Relative Topological Theories will develop condensation, boundary operator algebras, and relative amplitudes beyond the two elementary polarizations used here.

For the theorem-level construction, AKSZ Sigma Models as BV–BFV Examples will develop graded targets, transgression, master equations, compatible boundary conditions, and their exact equivalence notions. The already developed Abelian Chern–Simons Theory supplies the self-pairing and spin/framing side of the three-model comparison.

References

• Banks, Tom, and Nathan Seiberg. “Symmetries and Strings in Field Theory and Gravity.” Physical Review D 83 (2011): 084019. DOI. Open PDF, arXiv:1011.5120v2.
• Belov, Dmitriy M., and Gregory W. Moore. “Classification of Abelian Spin Chern–Simons Theories.” arXiv:hep-th/0505235v1 (2005). Stable record. Open PDF.
• Brennan, T. Daniel, and Sungwoo Hong. “Introduction to Generalized Global Symmetries in QFT and Particle Physics.” arXiv:2306.00912v2 (2023). DOI. Open PDF.
• Cattaneo, Alberto S., Pavel Mnev, and Nicolai Reshetikhin. “Classical BV Theories on Manifolds with Boundary.” Communications in Mathematical Physics 332 (2014): 535–603. DOI. Open PDF, arXiv:1201.0290v3.
• Cattaneo, Alberto S., Pavel Mnev, and Nicolai Reshetikhin. “Perturbative Quantum Gauge Theories on Manifolds with Boundary.” Communications in Mathematical Physics 357 (2018): 631–730. DOI. Open PDF, arXiv:1507.01221v2.
• Dijkgraaf, Robbert, and Edward Witten. “Topological Gauge Theories and Group Cohomology.” Communications in Mathematical Physics 129 (1990): 393–429. DOI.
• Freed, Daniel S., and Frank Quinn. “Chern–Simons Theory with Finite Gauge Group.” Communications in Mathematical Physics 156 (1993): 435–472. DOI. Open PDF, current arXiv:hep-th/9111004v3; v1 and v2 are withdrawn.
• Kapustin, Anton, and Natalia Saulina. “Topological Boundary Conditions in Abelian Chern–Simons Theory.” Nuclear Physics B 845 (2011): 393–435. DOI. Open PDF, arXiv:1008.0654v2.
• Kapustin, Anton, and Nathan Seiberg. “Coupling a QFT to a TQFT and Duality.” Journal of High Energy Physics 2014, no. 4 (2014): 001. DOI. Open PDF, arXiv:1401.0740v2.
• Mnev, Pavel. “A Construction of Observables for AKSZ Sigma Models.” Letters in Mathematical Physics 105, no. 12 (2015): 1735–1783. DOI. Open PDF, arXiv:1212.5751v1.