Conformal Gravity as a Deformed Topological Field Theory

James A. Reid

REVIEW 4 major objections 5 minor 25 references

Conformal Gravity as a Deformed Topological Field Theory

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Promoting the fixed length scale of the MacDowell-Mansouri connection to a dynamical scalar field turns the deformed BF action for general relativity into an action for conformal gravity, whose on-shell field equations are the conformal…

desk verdict A plausible but under-derived reformulation of conformal gravity as a deformed BF theory; the variation in Section 3 has a concrete error and the reduction to the primary sector is assumed, so the central equivalence is not established as written. read the letter →

arxiv 2508.05683 v1 pith:YKW6P7I4 submitted 2025-08-05 gr-qc

James A. Reid This is my paper · ORCID

classification gr-qc PACS 04.50.Kd04.20.-q02.40.-k

keywords conformalgravityMacDowell-MansourideformedBFtheoryCartangeometrycosmologicalscalarfieldscale-invariantPalatinivariationtopological

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that conformal gravity is the deformed topological field theory obtained when the fixed length scale $l$ in the MacDowell-Mansouri connection is promoted to a dynamical scalar field $\phi$. The central result is that the deformed BF action $S^\phi_{BF}=-\int(\phi^2 B\wedge F - (\epsilon G_N\Lambda/12)\phi^4 \tilde B\wedge \tilde B \epsilon)$ has on-shell field equations identical to the conformal Einstein equations, and reduces to a Palatini action with a quartic potential. If true, this gives the cosmological scalar field a geometric origin: spacetime points are infinitesimally modeled by homogeneous spaces whose radius is set by $\phi$, rather than by a fixed length. It also connects scale-invariant gravity to the BF-theoretic route toward quantum gravity.

What carries the argument

The central object is the modified MacDowell-Mansouri connection $\omega = A + \sqrt{\Lambda/3}\,\phi\, e$, in which the fixed length scale has become a scalar-field-dependent inverse length. The argument is carried by substituting the on-shell solution of the $B$ field equation, $B^{ij}=-(3/2\epsilon G_N\Lambda\phi^2)\epsilon^{ijkl}F_{kl}$, back into the deformed BF action, which reduces it first to a MacDowell-Mansouri-type action and then to the Palatini action with $\phi^2$ coupled to the curvature and a $\phi^4$ potential. The final step uses the general Palatini variation formulas of [5] with $D(\phi)=\phi^2$ and $C(\phi)=0$, which yield precisely the conformal Einstein equations (47). The Hodge-star identity $\star\star\sigma=-\sigma$ and the antisymmetry of the alternating tensor are what make the $\phi^4$ deformation conspire with the $\phi^2 B\wedge F$ term to produce the quartic potential.

What would settle it

Vary the action $S^\phi_{BF}$ directly with the coframe $e$ and spin connection $A$ as independent gravitational variables (with $\phi$ kept separate), retaining the $\partial\phi$ contributions in the variation of $F$; then check whether the resulting equations still imply the conformal Einstein equations (47). If extra $\partial\phi$ terms survive in the equations of motion, the claimed on-shell equivalence does not hold.

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Extended reading notes

Core claim

The paper's claim is that replacing the fixed inverse length $1/l$ in the MacDowell-Mansouri connection by $\sqrt{\Lambda/3}\,\phi$ converts general relativity in MacDowell-Mansouri form into conformal gravity. The resulting action $S^\phi_{BF}$ is shown to be equivalent on shell to the conformal Einstein-Hilbert action: the $B$ equation fixes $B$ in terms of $F$, the $\omega$ and $\phi$ equations then reduce to the system (47), which is exactly the conformal Einstein equations. The same reduction takes $S^\phi_{BF}$ to the MacDowell-Mansouri-type action (36) and then to the Palatini form $(1/G_N)\int\sqrt{-g}(\phi^2 \mathrm{scal}(A)-2\epsilon\Lambda\phi^4)$. Geometrically, the paper argues that in this theory each spacetime point is infinitesimally approximated by a homogeneous space $G/H$ whose radius is parameterised by the value of $\phi$, in contrast to general relativity where that radius is the fixed length $l$.

Load-bearing premise

The derivation assumes that $\phi$ and the connection $\omega$ can be varied independently, even though $\omega$ itself contains $\sqrt{\Lambda/3}\,\phi\, e$, so the $\phi$-derivative terms that a fully independent variation of the coframe would produce in the $\omega$ equation are not included.

Editorial extensions

If this is right

Conformal gravity inherits the deformed-BF formulation, so its quantization can be approached by deforming a topological field theory rather than by quantizing the metric action directly.
The cosmological scalar field $\phi$ is not an external matter field but the dynamical avatar of the MacDowell-Mansouri length scale, giving it a geometric origin.
The Palatini variation fixes the connection to be the conformally transformed Levi-Civita connection with conformal factor $\phi$, so the metric and Palatini formulations of conformal gravity coincide.
In this Cartan-geometric picture, conformal transformations change the radius of the homogeneous spaces that infinitesimally model spacetime, so conformal symmetry acts on the local model geometry itself.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the same promotion of $l$ to a scalar field in the higher-dimensional MacDowell-Mansouri action (20) would likely produce conformally invariant scalar-tensor theories in other dimensions; the paper treats only $n=4$.
Beyond the paper: because the undeformed BF action is topological and the deformation is controlled by $\phi$, the equivalence suggests a quantization route in which conformal gravity is a deformation of a topological field theory; the paper does not pursue quantization.
Beyond the paper: repeating the derivation with the coframe and spin connection as independent variables, rather than varying $\omega$ and $\phi$ jointly, would pin down exactly which field content the on-shell equivalence covers; this check is a natural next step not carried out in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The genuinely new thing here is the BF-type action (26) with the MacDowell-Mansouri length scale promoted to a scalar field. That construction is not in the cited literature, and it is a natural enough idea that someone working on gauge-theoretic gravity will want to know about it. The on-shell reduction of that action to the Palatini form (41) is also plausible: if you eliminate B using its own equation, the algebra leading to the scalar-tensor action is correct, and the Burton-Mann formulas do produce the conformal Einstein equations (47). The paper is clearly written and honest about its sources. Credit where it is due: the new BF formulation itself is a real contribution, and the geometrical interpretation of the scalar field as a varying Cartan radius in Section 4 is a nice observation.

The soft spots are not minor. In Section 3, the variation with respect to ω gives equation (31), d_ωB = 0, but that only follows if φ² is treated as constant when integrating by parts. The correct Euler-Lagrange equation is d_ω(φ²B) = 0, which contains a dφ² ∧ B term that is absent from the paper. That is a load-bearing error because the full set of BF equations is supposed to imply the conformal Einstein equations. The second issue is the silent truncation to primary components around equations (33)–(34). The statement “B = bB is imposed by consistency of the index structure” is not derived. A full variation with respect to B would enforce the mixed components of F to vanish, and those mixed components contain precisely the torsion condition T = −d ln φ ∧ e that one needs to recover the Palatini connection (46). The paper assumes that condition instead of deriving it from the BF equations. Both problems are fixable, and I suspect the construction can be repaired by varying with respect to A, e, and φ as independent fields, or by keeping the mixed B components and showing that they impose the Palatini torsion relation. As written, though, the central claim is not rigorously supported.

The citation pattern looks fine, and there is no circularity beyond the fact that the paper takes the conformal Einstein equations as the known target and finds an action that reproduces them; that is a reformulation, not a new prediction, and the paper does not oversell it. Who is this for? Researchers working on MacDowell-Mansouri, BF-theoretic gravity, or scalar-tensor reformulations. They should read it with caution. I would send it to peer review, because the new BF action deserves referee time, but I would expect major revision before publication.

Recommendation: engage with it, but require the derivation to be redone properly.

Referee Report

4 major / 5 minor

Summary. This paper proposes that promoting the constant length scale l of the MacDowell-Mansouri connection to a dynamical scalar field φ yields a deformed BF action whose field equations are equivalent on shell to the conformal Einstein equations. The author writes the modified Cartan connection (24), computes its curvature (25), defines the deformed BF action (26), and derives the purported equations of motion (30)-(32). The action is then reduced to a MacDowell-Mansouri-type action and a Palatini-type action (41), and the Burton-Mann formulas are used to obtain the conformal Einstein equations (47). Section 4 offers a Cartan-geometric interpretation in which the scalar field parameterizes the radius of the infinitesimal homogeneous-space approximation to spacetime.

Significance. If the derivation were correct, the paper would give a gauge-theoretic embedding of conformal gravity in which the cosmological scalar field is not introduced by hand but is the dynamical length scale of the MacDowell-Mansouri connection. The reduction to the Palatini-type action and the use of Burton-Mann's formulas is a clean route to the conformal Einstein equations, and the Cartan-geometric interpretation of φ as a variable model-geometry radius is conceptually appealing. The paper is a structural reformulation rather than a source of new observational predictions; its value lies in the proposed origin of the scalar field and in the BF/deformed-topological-field-theory packaging. The central claim is currently not rigorously established because the variational steps in Section 3 contain load-bearing gaps.

major comments (4)

[§3, Eqs. (29)-(31)] The integration by parts leading to equation (31) treats φ² as a spacetime constant. Since φ is a scalar field, the correct Euler-Lagrange equation is d_ω(φ²B)=0, i.e. φ²d_ωB + dφ²∧B = 0, not d_ωB = 0. The omitted dφ²∧B term couples the connection equation to gradients of φ and is essential to the claimed on-shell content, so equation (31) cannot be used as the connection equation of the theory.
[§3, Eq. (24) and Eq. (29)] The fields ω and φ are not independent, because ω is defined in terms of φ through ω = A + √(Λ/3) φ e in equation (24). Varying ω and φ as independent fields double-counts φ. The variation should be performed with respect to the actual independent fields A, e, and φ, or with a constraint relating δω to δφ and δe; otherwise the variation leading to (31) is not the variation of the theory defined by (24).
[§3, Eqs. (33)-(35)] The passage from equations (30)-(32) to (33)-(34) asserts that 'B = bB is imposed by consistency of the index structure' without deriving the mixed components. The full B-field equation (30) also constrains the mixed components: because the right-hand side contains only the primary part of B, the mixed components of F must vanish. In particular, F_{i5}=0 imposes the torsion relation T = -d ln φ ∧ e (up to sign conventions), which is precisely the Palatini connection condition (46) needed to obtain conformal gravity. This condition is silently assumed rather than derived.
[§3, Eq. (36)] The on-shell reduction S^φ_BF = S^φ_MM uses only the B equation (30) to eliminate B, after which the φ equation (32) is shown to be redundant. The author should verify that, with the corrected connection equation and with the mixed torsion constraint derived rather than assumed, the BF system (30)-(32) is still equivalent on shell to the Palatini variation of (41). The current text does not establish this equivalence because the connection equation used in the argument is not the correct one.

minor comments (5)

[Eq. (30)] The symbol ε_IJKL is used with indices I,J,K,L that range from 0 to 4, but the Levi-Civita tensor ε is only defined for the four-dimensional internal indices; please clarify the index conventions and the projection to the primary block.
[Eq. (38)] In the sentence before (38), 'summetric' should be 'symmetric', and in (38) the index label 'ε_abcd' should be 'ε_ijkl' for consistency with the surrounding equations.
[Eqs. (41) and (48)] The reduced action (41) has no kinetic term for φ, whereas the conformal Einstein-Hilbert action (48) explicitly contains 6∂φ∂φ; the text should explain that the kinetic term in the final field equations arises from the Palatini connection condition (46), not from an explicit kinetic term in (41).
[Section 4] The phrase 'infinitesimally approximated by homogeneous spaces (restricted to a point)' should be made precise; the relevant notion is that the tangent space is modeled by the homogeneous space G/H, and the specific sense in which the radius varies with φ should be stated.
[Section 2] The matrix B is described as 'entirely unconstrained', but Section 3 later restricts B to its primary part; the apparent inconsistency should be resolved by explaining that the mixed components are Lagrange multipliers whose equations impose the torsion constraint.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the conformal Einstein equations are derived via an independent Palatini theorem, not assumed as an input.

full rationale

The paper's central result is an equivalence proof. It explicitly introduces the dynamical-length BF action (26), eliminates the auxiliary B field using equation (33), reduces the action to the MacDowell-Mansouri-type expression (41), and then invokes the external, parameter-free Palatini variation formulas of Burton and Mann, equations (43)-(46), with D(phi)=phi^2 and C(phi)=0, to obtain equations (47), which coincide with the conformal Einstein equations (3). Nothing is fitted to the target field equations: the action coefficients are fixed once the phi^2 and phi^4 weights are chosen, and the target equations appear at the end of the calculation rather than as assumptions inside it. There are no self-citations, and the cited Burton-Mann result is independent of the present paper. The geometric interpretation in Section 4 is an interpretive commentary rather than a load-bearing step in the derivation. The main weakness in the paper is a derivation gap rather than circularity: equation (31) is obtained by integrating phi^2 B wedge delta F by parts as though phi^2 were constant, whereas the correct connection equation should involve d_omega(phi^2 B), and the subsequent reduction to B = bB drops mixed components without deriving them. These are correctness issues in the BF-level derivation, not cases of a prediction being equivalent to its inputs by construction. No circular step can be exhibited, so the circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The construction uses a known equivalence: Palatini φ^2 R theories reduce to conformal scalar-tensor gravity. The only genuinely new element is the BF action (26), whose form is chosen so that its on-shell reduction gives the Palatini action. No new particles or entities are introduced; the scalar field φ is the standard conformal scalar.

free parameters (1)

Λ
Introduced as a dimensionless parameter controlling the length scale; its value is not fitted here but is an input of the theory. The paper changes the usual dimensionful Λ into a dimensionless one, which is a modeling choice.

assumptions (4)

standard math The Hodge star on Lorentzian 2-forms satisfies ⋆⋆σ = -σ
Used to solve the B equation of motion (30) for B in terms of F, just below equation (33).
domain assumption The deformed BF action (26) is a valid gravitational action whose dynamics is equivalent to the on-shell reduction to the Palatini action (41)
The paper assumes the auxiliary field B can be integrated out and that this substitution preserves the dynamics, which is standard for BF theories but is not proved in detail.
domain assumption Burton-Mann's Palatini variation formulas (43)-(46) for scalar-tensor actions are correct and apply to action (41)
The derivation of the conformal Einstein equations (47) relies entirely on equations quoted from reference [5].
domain assumption The conformal Einstein equations (3) are the target field equations of conformal gravity
The paper takes the conformal Einstein-Hilbert action and its equations as the definition of conformal gravity, which is the standard but not universal definition.

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Cite this review

Pith. "Pith review of Conformal Gravity as a Deformed Topological Field Theory." pith.science (2026). https://pith.science/paper/YKW6P7I4

@misc{pith2026250805683,
  author       = {Pith},
  title        = {Pith review of: Conformal Gravity as a Deformed Topological Field Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YKW6P7I4}},
  note         = {Machine review of arXiv:2508.05683}
}

abstract

In the MacDowell-Mansouri formulation of general relativity, the spin connection and coframe variables are incorporated into a single Lie algebra-valued connection called the MacDowell-Mansouri connection, $\omega$. From the curvature form $F$ of $\omega$ and an auxiliary field, $B$, one may formulate general relativity as a deformed topological field theory by constructing an action functional whose variation yields a set of field equations that are equivalent to the Einstein equations on shell. In this article, we show that when the fundamental length scale of the MacDowell-Mansouri connection is regarded as a dynamical variable -- a cosmological scalar field -- the field equations obtained from the variation of the resulting action are equivalent to the conformal Einstein equations on shell. Through the lens of Cartan geometry, we then discuss a notable geometrical difference between general relativity and its conformally transformed counterpart. Specifically, for the latter, we show that points in spacetime are infinitesimally approximated by homogeneous spaces (restricted to a point) whose radii are parameterised by the value of the cosmological scalar field.

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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R. Arnowitt, S. Deser and C. W. Misner, “The Dynamics of General Relativity” in Gravitation: an intro- duction to current research (Wiley, 1962), e-print: arXiv:gr-qc/0405109

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work page 1962

[4] [4]

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L. Bombelli, “Actions for General Relativity” webpage: http://www.phy.olemiss.edu/~luca/Topics/ gr/action.html (2014)

work page 2014

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Palatini Variational Principle for N-Dimensional Dilaton Gravity

H. Burton and R. B. Mann, “Palatini Variational Principle for N-Dimensional Dilaton Gravity” Class. Quantum Grav. 15, 1375-1385 (1998)

work page 1998

[6] [6]

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R. Capovilla, J. Dell, T. Jacobson and L. Mason, “Self-dual 2-forms and gravity” Class. Quantum Grav. 8, 41 (1991)

work page 1991

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P. A. M. Dirac, “Long ranges force and broken symmetries” Proc. R. Soc. Lond. A. 333, 1595 403-416 (1973)

work page 1973

[8] [8]

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R. Durka, “Deformed BF theory as a theory of gravity and supergravity” e-print arXiv:1208.5185 [gr-qc] (2012) 12 JAMES A. REID

arXiv 2012