Non-Propagating 3-Form Gauge Potentials

• Non-propagating 3-form gauge potentials are higher-degree differential forms with gauge redundancy that encode global topological effects and constraint conditions in various field theories.
• Algebraic structures such as BRST symmetry and Curci-Ferrari-type restrictions, alongside FFBRST and BV quantization, systematically remove nonphysical degrees of freedom.
• These potentials provide dual descriptions in axion physics, string/M-theory, and confinement scenarios, underpinning non-perturbative and topological mechanisms.

Non-propagating 3-form gauge potentials are higher-degree differential forms whose gauge dynamics and physical implications fundamentally differ from those of standard propagating fields. Typically arising in field theories, string theory, M-theory, and quantum gravity, these objects are characterized by the absence of local degrees of freedom, playing crucial roles in encoding topological, global, and constraint structures, implementing non-perturbative effects, and providing dual descriptions of key physical phenomena such as axion potentials, brane fluxes, and confinement.

1. Algebraic and Geometrical Structure

A 3-form gauge potential $C_3$ is a totally antisymmetric tensor field with local gauge redundancy: $C_3 \rightarrow C_3 + d\Lambda_2$ where $\Lambda_2$ is a 2-form. The associated field strength $F_4 = dC_3$ is invariant under this transformation. The gauge invariance and redundancy mean that, like all higher $p$-form gauge theories for $p\geq 2$, many components do not correspond to propagating physical degrees of freedom.

The structure and constraints of these potentials are naturally formulated in terms of BRST and anti-BRST symmetry. In $D$-dimensional Abelian 3-form gauge theory, the Bonora–Tonin superfield formalism replaces the gauge fields and ghosts by superfields on a $(D,2)$-dimensional supermanifold, with two Grassmann directions $(\theta, \bar\theta)$. The so-called "horizontality condition" enforces independence of the curvature form on these extra coordinates, automatically generating off-shell nilpotent and absolutely anticommuting BRST ($s_b$) and anti-BRST ($s_{ab}$) transformations. The superfield expansion produces towers of ghost and auxiliary fields, their structure uniquely fixed by the horizontality condition (Malik, 2011).

2. Constraint Structure and Curci-Ferrari-Type Restrictions

For $p\geq 2$-form gauge fields, the complete removal of nonphysical (gauge) degrees of freedom in the quantum theory requires additional algebraic constraints. For 3-form theories, these take the form of Curci-Ferrari (CF) type restrictions. Schematic examples include: $f^{(2)} + f^{(3)} = \partial_\mu C_1$ where $f^{(2)}, f^{(3)}$ are secondary fields from the superfield expansion and $C_1$ is a scalar ghost. These (anti-)BRST-invariant constraints are essential for ensuring that the BRST and anti-BRST algebras close off-shell and absolutely anticommute: $$s_b^2 = 0, \quad s_{ab}^2 = 0, \quad \{s_b, s_{ab}\} = 0$$ They are the haLLMark of all higher $p$-form gauge theories in the BRST framework (Malik, 2011). The CF conditions embody the first-class constraints and encode the removal of nonphysical, non-propagating components.

3. Gauge Fixing, BRST/BV Quantization, and Non-propagating Degrees

The elimination of gauge-redundant and thus non-propagating components is operationalized by gauge fixing and the introduction of ghost/auxiliary sectors. Covariant and noncovariant gauges can be systematically related via finite field-dependent BRST (FFBRST) transformations, with the generating functional mapping between different gauge selections through a controlled Jacobian in the path integral (Upadhyay et al., 2013). When gauge fixing is completely implemented, only the physical degrees of freedom propagate, with the nonphysical parts remaining as non-dynamical Lagrange multipliers or auxiliary variables.

In the Batalin-Vilkovisky (BV) formalism, the extended action introduces antifields for every field and is subject to the quantum master equation $(W, W) = 0$. The FFBRST and BV methods ensure that all physical observables and propagating modes are gauge choice-independent and that in particular, non-propagating modes are not present in the spectrum.

4. Topological and Non-Propagating 3-Form Theories

A canonical example of a non-propagating 3-form is the "BdC" theory in six dimensions: $S = \int B \wedge dC$ with $B$ a 2-form and $C$ a 3-form. This action leads only to constraint equations (e.g., $dB = 0$, $dC = 0$) and, even under deformations by potential terms such as $B\wedge B\wedge B$ ("symplectic" case) or $C\wedge \widehat{C}$ ("Hitchin/complex" case), remains strictly topological, with no propagating degrees of freedom (Herfray et al., 2017).

Upon Hamiltonian reduction, the phase space is finite-dimensional, parameterized only by global topological invariants (e.g., moduli of closed forms, cohomology classes). The absence of any local propagating degrees of freedom is thus robust under standard deformations and carries over to the dimensionally reduced theory (for instance, reproducing 3D gravity upon compactifying on $S^3$).

5. Physical Mechanisms: Potentials, Duality, and Axion Physics

Non-propagating 3-form gauge potentials have a central role in the generation of potentials for pseudo-scalar (axion-like) fields. The archetypal coupling is of the form: $$\mathcal{L}_\text{top} = G(F) - \frac{1}{2}\partial^\mu\phi\partial_\mu\phi + m\phi F + \text{(boundary terms)}$$ where $F$ is (the Hodge dual of) the 3-form field strength, $G(F)$ is an arbitrary (ghost/tachyon-free) function. The algebraic equation of motion for $F$ allows it to be integrated out, yielding an effective potential for $\phi$, such as quadratic or cosine-types depending on $G$: $$V(\phi) = -G(F) + FG'(F)\quad\text{with}\quad -G'(F) = m\phi + c$$ In supersymmetric contexts, this translates into superpotential terms $m\Phi Y$ with $Y$ the chiral superfield containing the 3-form, allowing higher-derivative deformations and multiple vacua (Nitta et al., 2018).

In axion physics and string compactifications, this dual description of the axion mass via a non-propagating 3-form is central. For example, the axion potential generated by instantons can be recast as a 3-form "eating" a 2-form, i.e., $|db_2 + nc_3|^2 + |F_4|^2$, enforcing discrete shift symmetry. In string theory, D-brane instantons can generate such potentials, but the required 3-form is not present as a harmonic form in Calabi–Yau compactifications; it arises in the backreacted, generalized geometry upon including nonperturbative effects. The Kaluza–Klein reduction along non-harmonic forms then yields a non-propagating 3-form $c_3$ in four dimensions whose only dynamical effect is in enforcing the global axion potential structure with multi-branch behavior (García-Valdecasas et al., 2016).

6. Non-propagation in QCD and Confinement

In certain Yang–Mills-type models, non-propagating 3-form gauge potentials realize physical confinement. The generalized Yang–Mills SU(3) model introduces a "phase field" $H_\mu^a$ ("confion") satisfying a fourth-order differential equation: $L_s^2 \partial^2 \partial^2 H^a_\nu = J^a_\nu$ The Green function $D(k) = 1/(k^2)^2$ does not correspond to propagating physical quanta; the solutions have support only inside and on the light cone, with constant, non-radiating fields and discontinuities at the light cone. The resulting static potentials display both linearly rising (confining) and Coulomb components. The confion quanta never appear as physical, propagating states and are permanently confined to virtual intermediate states (Hsu, 2019).

7. Global Topological and Cohomological Origin

The global definition of 3-form gauge potentials, particularly in the context of M-theory and on M5-branes, necessitates a formulation in terms of nonabelian cohomotopy. "Hypothesis H" posits the 4-flux in M-theory is flux quantized in 4-cohomotopy, and the 3-flux on the M5 brane in (twisted) 3-cohomotopy. The local 3-form potential is obtained by integrating a "null concordance" of the flux density. The physical data are globally specified not by local form fields but by their total cohomotopical class and transition data on open overlap patches. This means that the 3-form gauge potential is, in a non-perturbative sense, non-propagating: its degrees of freedom are encoded in the global topology (or, in the modern sense, in the differential, twisted, equivariant cohomotopy class) and all local fluctuations correspond to gauge transformations or higher gauge equivalences (Banerjee, 9 Jul 2025).

The traditional Bianchi identities are reproduced as consequences of this global construction, with implications for anomaly cancellation, coupling to gravity, and other topological features of string/M-theory.

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In sum, non-propagating 3-form gauge potentials constitute a rich and unifying framework appearing across quantum field theory, topological field theory, string compactification, axion dynamics, brane physics, and confinement. Their defining attributes are (i) constraint-dominated, topological, or auxiliary character, (ii) role in preserving gauge invariance or duality structures, (iii) emergence as global or homotopical objects upon imposing appropriate flux quantization, and (iv) non-appearance in the physical spectrum as propagating quantum degrees of freedom. These properties enable them to encode critical global or non-perturbative effects that are inaccessible to local propagating fields.