Relating electrodynamics and gravity in two Euclidean dimensions

Rodrigo F. Sobreiro; Thales F. Bittencourt

REVIEW 3 major objections 5 minor 31 references

Relating electrodynamics and gravity in two Euclidean dimensions

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Two-dimensional electrodynamics and two-dimensional gravity are physically equivalent: the photon maps to the spin connection, and the two quantum path integrals agree up to harmless Jacobian factors.

desk verdict Classical map is fine; the quantum equivalence claim does not survive contact with the paper's own BRST transformations. read the letter →

arxiv 2502.09272 v1 pith:UAXFOJ77 submitted 2025-02-13 hep-th

Thales F. Bittencourt , Rodrigo F. Sobreiro This is my paper · ORCID

classification hep-th MSC 81T1381T2081T4081T7083C45 PACS 11.15.-q04.60.-m

keywords two-dimensionalelectrodynamicsgravitygauge-gravitycorrespondenceBRSTsymmetryfirst-orderformalismparityviolationzweibeinpathintegral

topics Yang-Mills Mass Gap

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 sets out to prove that two-dimensional electrodynamics with Dirac fermions and two-dimensional gravity in the first-order formalism are two descriptions of the same physics. A field map built on the $U(1) \to SO(2)$ group isomorphism, valid in Euclidean spacetime, turns the photon into the spin connection, the auxiliary Maxwell field into the gravitational zweibein (the local-frame 1-form that carries the metric), and the Dirac electron into a fermion with an unusual coupling to geometry. The authors then argue the equivalence survives quantization: the path integrals $Z_{\mathrm{QED}}$ and $Z_{\mathrm{QG}}$ are equal, because every Jacobian factor generated by the map is re-expressed as a BRST-exact term, which is physically inert. If this is right, every correlation function of two-dimensional QED has an identical gravitational counterpart, and the gravitational constants — Newton's constant and the cosmological constant — are fixed by the electric charge and the Chern-Pontryagin coefficient of the electrodynamic action. A by-product is a gravity theory whose fermion interaction violates parity and time reversal, with left- and right-handed fermion parts rotating in opposite senses under $SO(2)$.

What carries the argument

The engine of the argument is the $U(1) \to SO(2)$ group isomorphism, which exists only in Euclidean signature and allows an Abelian gauge field to be identified with the $SO(2)$ spin connection of a first-order gravity theory. On that isomorphism rests an explicit field map between the two actions, enlarged with auxiliary boundary fields so that the number of independent field components matches on both sides; these auxiliary fields are organized into BRST doublets so that they cannot contribute to physical content. The quantum step rests on a second mechanism: the Jacobian of the map, $J \propto \det^{3/2}(\varphi^a{}_b)\, e^3$, is represented as an integral over newly introduced auxiliary fields $(\bar Y^a, Y^a, Z^a, \bar W^a, W^a, K^a)$, and the combined term $S_{\mathrm{det}}$ is shown to be BRST-exact, hence a trivial co-cycle that cannot alter physical observables. Finally, the map fixes the gravitational constants to $G = e^2/[8\pi(\kappa + e^2)]$ and $\Lambda^2 = e^4/(\kappa + e^2)$, tying the strength of gravity to the charge and topological coefficient of electrodynamics.

What would settle it

Compute the fully gauge-fixed partition functions of both theories on a compact surface such as a torus — U(1) gauge fixing for the QED side, $SO(2)$ plus diffeomorphism gauge fixing for the gravity side — and compare them, or compare a gauge-invariant correlator such as the fermion two-point function. A mismatch, for instance from a diffeomorphism ghost contribution or an anomaly with no counterpart in the U(1) ghost sector, would falsify the claimed quantum equivalence.

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Extended reading notes

Core claim

The central claim is that action (2.1), two-dimensional Maxwell–Dirac theory with an auxiliary field $\Theta$ and a Chern–Pontryagin term, and action (2.12), two-dimensional gravity in the first-order formalism built from a zweibein $e^a$ and a spin connection $\omega^{ab}$ together with the same fermions and auxiliary boundary fields, are physically equivalent. The correspondence sends $A \mapsto \epsilon_{ab}\,\omega^{ab}$, $\Theta \mapsto \mu^2 \epsilon_{ab} e^a e^b$, and $\gamma^i dx^i \mapsto \gamma^a e_a$, with 17 independent field components on each side matched by auxiliary fields that form BRST doublets and therefore carry no physical degrees of freedom. The gravitational parameters are not free: $G = e^2/[8\pi(\kappa + e^2)]$ and $\Lambda^2 = e^4/(\kappa + e^2)$, so when the topological coefficient $\kappa$ vanishes, Newton's constant is fixed at $1/8\pi$. At the quantum level, the non-trivial Jacobian $J \propto \det^{3/2}(\varphi^a{}_b)\, e^3$ of the map is written as a Gaussian integral over auxiliary fields whose combined action $S_{\mathrm{det}}$ is BRST-exact, a trivial element of BRST cohomology, and the paper concludes that $Z_{\mathrm{QED}} = Z_{\mathrm{QG}}$: 'two-dimensional QED and two-dimensional gravity are, essentially, two descriptions of the same physical phenomenon.' The mapped theory also exhibits a new fermion–gravity interaction, $L_I = e^a{}_\mu \epsilon_{bc}\,\omega^{bc}{}_\mu\, \bar\psi\gamma^a\psi$, which breaks parity and time reversal even when the original action has no topological term, and the fermions transform under $SO(2)$ with an extra $\gamma^3$ factor so that right- and left-handed chiral components rotate with opposite signs.

Load-bearing premise

The quantum equivalence rests on the assumption that the only thing separating the two path integrals is a change-of-variables factor that is provably harmless, and this is asserted before either theory has been fully gauge-fixed and before the extra terms that gauge fixing introduces on each side have been compared.

Editorial extensions

If this is right

Every physical observable of two-dimensional QED, including fermion and gauge-field correlation functions, has an identical counterpart in the mapped gravity theory, so computing either side computes both.
The Newton and cosmological constants are determined, not adjustable: with $\kappa = 0$, the dimensionless Newton constant takes the parameter-free value $G = 1/8\pi$.
Because the starting electrodynamics is renormalizable, the mapped gravity theory is expected to be renormalizable, offering a tractable two-dimensional model of quantum gravity.
The parity- and time-reversal-violating fermion interaction emerges from the geometric map itself rather than from the topological term, so even P- and T-symmetric QED maps to a P- and T-violating gravity.
The fermions' unusual $SO(2)$ behavior — chiral components rotating in opposite senses — constitutes a new type of fermion dynamics that the authors propose to study further, including its charge-conjugation properties.

Reading between the lines

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

The deferred gauge-fixing program is the natural stress test: two-dimensional theories carry diffeomorphism and trace anomalies, and nothing in the paper shows the ghost sectors of the two sides produce matching anomaly content, so the equivalence could in principle fail there.
A concrete quantitative check would be the partition function on a compact surface such as a torus: the fully gauge-fixed gravity side, diffeomorphism ghosts included, would have to reproduce the known torus partition function of two-dimensional QED exactly.
The parameter-free value $G = 1/8\pi$ for $\kappa = 0$ is a sharp prediction: any concrete realization of the correspondence, such as the graphene-sheet setting the authors sketch, would be constrained to exhibit this effective Newton constant.
Because the Einstein–Hilbert term in two dimensions is purely topological, the mapped theory is effectively a topological gravity sector plus propagating fermions; one reasonable conjecture, beyond the paper's claims, is that the correspondence is a duality between a gauge theory and a topological gravity theory.

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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

Dear colleague,

Quick take: the paper extends Sobreiro's earlier pure-gauge map to include Dirac fermions, and the resulting parity-violating fermion–spin connection coupling (3.2) is a real, if modest, new result. The classical substitution of the field map into the action is straightforward and seems self-consistent. But the headline claim—that QED and this gravity theory are equivalent at the quantum level—is not supported. The stress-test note is right: from the paper's own transformations (4.13), s^2 \bar{Y}^a = -1/2 Y^a ≠ 0, and similarly for W, so the BRST operator is not nilpotent. The entire Jacobian-decoupling argument in Section 4.2 relies on Sdet being a BRST-trivial cocycle, and without nilpotency that notion is undefined. The claim 'we may safely conclude ... consistent at the quantum level' is therefore premature.

There is also the acknowledged gap: gauge fixing is never implemented. The path integrals (4.5) and (4.17) are over gauge orbits (and diffeomorphisms on the gravity side), so the equality of Z_qed and Z_qg is not even well-defined as stated. The authors admit this in the conclusions, but that admission undercuts the central result.

Two smaller complaints. First, the 'unconventional' fermion transformation (3.4) is not actually unconventional: using (A.5), γ3 σab = -εab, so (3.4) is identical to the standard SO(2) rotation in (3.3). The novelty claim about a new type of gauge symmetry is overstated. Second, the relations (2.13) fixing G and Λ are just parameter redefinitions; they are not independent checks of the map.

What the paper does well: the classical mapping is clean, the derivation of the fermion-gravity term is explicit, and the authors are candid about the missing gauge fixing. The extension to fermions is a legitimate step beyond [24].

Who this is for: readers working on formal toy models of gauge-gravity duality in 2D. The classical part is worth a look; the quantum part needs serious repair. I'd send it to a referee—the concrete BRST nilpotency issue is checkable and the authors should have caught it, but the paper is not vacuous and a referee report could help them fix it.

Recommendation: send to peer review, but the referee should be told to focus on the nilpotency of s and on gauge-fixing before any claims of quantum equivalence can be taken seriously.

Referee Report

3 major / 5 minor

Summary. This paper presents a classical map from two-dimensional Euclidean Maxwell-Dirac theory to a first-order two-dimensional gravity theory with fermions, using the U(1) to SO(2) isomorphism and an auxiliary boundary term to balance field content. The mapped gravity action has a parity- and time-reversal-violating fermion-spin-connection coupling and fermions with a modified SO(2) transformation law. The authors then attempt to extend the map to the quantum level by comparing path integrals and using BRST techniques, concluding that the Jacobian of the field map is a trivial BRST cocycle and therefore that two-dimensional QED and two-dimensional quantum gravity are physically equivalent.

Significance. The classical computation leading to the gravity action (2.12) is explicit and straightforward, and the resulting unconventional fermion-gravity coupling is an interesting observation. If a full quantum equivalence were proven, it would be a notable result connecting two-dimensional QED with first-order gravity. However, the quantum equivalence is not established by the present manuscript: the BRST operator used for the Jacobian is not nilpotent, and the path integrals are not gauge-fixed. These are central to the paper's main claim rather than presentation issues.

major comments (3)

[Section 4.2, Eqs. (4.13)-(4.16)] The operator s is not nilpotent. From (4.13), s^2 \bar{Y}^a = s(1/2 Z^a) = -1/2 Y^a \neq 0, and from (4.14), s^2 \bar{W}^a = -1/2 W^a \neq 0, assuming c^2=0. Since the notion of a trivial BRST cocycle requires s^2=0, Eq. (4.16) does not establish that Sdet decouples. Consequently the Jacobian (2.11) is not shown to be unphysical, and the statement after Eq. (4.19) that the map is consistent at the quantum level does not follow.
[Section 4.1, Section 4.2, Section 5] The partition functions Z_qed and Z_qg in Eqs. (4.5) and (4.17) are written without gauge fixing. In Section 4.1 the authors say that gauge fixing is not required, but in the Conclusions they concede that 'gauge fixing is still needed' and that diffeomorphisms must also be treated. Without gauge fixing, the integrals over the U(1) or SO(2) and diffeomorphism orbits are not well defined, so the formal equality of the two path integrals is not a well-defined statement. A valid quantum comparison would require gauge fixing on both sides, including Faddeev-Popov determinants and a discussion of possible anomalies.
[Section 4.2, after Eq. (4.12)] The decisive step in which the nontrivial Jacobian (2.11) is represented by the Gaussian integrals (4.12) and then shown to be a BRST variation is introduced with 'one can confirm' and 'Following the steps developed in [24]'. This step is load-bearing for the quantum equivalence and is not reproduced in the present paper. The manuscript should contain a self-contained derivation of (4.12), (4.15), and (4.16), rather than relying on a prior paper by one of the authors.

minor comments (5)

[Eq. (4.12)] The notation 'ǫcdeced' appears to be malformed; it should presumably read 'ǫ_{cd} e^c e^d' or similar. Please correct the notation and verify the factors in the exponentials.
[Section 2.1] The claim that the boundary term allows one to map '17 independent fields' into '17 independent fields' is not transparent. Please spell out the component counting using Tables 1, 2, and 3, since the ranks and form degrees make the count non-obvious.
[Section 3.2, around Eq. (3.9)] The two-observer example is confusing: two particles at rest with respect to O1 would both move to the left with speed V in O2's frame, so v_{R,L} = \mp V does not follow from a boost alone. The relation between the modified SO(2) transformations and this velocity statement needs a clearer explanation.
[References] Reference [15] is missing a journal or preprint identifier, and reference [23] is cited as 'Accepted for publication' without complete publication data. Please complete these references.
[Section 5] There is a typo in the Conclusions: 'analisys' should be 'analysis'.

Circularity Check

1 steps flagged · score 4.0 of 10

Quantum equivalence relies on a load-bearing self-citation to [24]; the classical dictionary is explicit but includes a hand-picked normalization μ=e.

self citation load bearing [Section 4.2, Eqs. (4.12)-(4.16) and the following paragraph]
"Following the steps developed in [24], one can confirm that we can write e^3 = ∫ [dȲdYdZ] exp[−∫(Ȳ^a Y_a + 1/2 Z^a Z_a) ε_cd e^c e^d], det^{3/2}(φ^a_b) = ... (4.12). ... Then, defining Sdet = ... we have that Sdet = s∫(Ȳ^a Z^a + W̄^a φ_ab K^b) ε_cd e^c e^d, (4.16). Thence, although the fields originating from the Jacobian are not BRST doublets, Sdet is a trivial BRST co-cycle. Therefore, the Jacobian does not interfere at the physical content of the model."

The central quantum equivalence claim is that the non-trivial Jacobian (2.11) does not affect the path integral. The key technical steps - representing e^3 and det^{3/2}(φ) by Gaussian integrals and writing Sdet as a BRST-exact term - are not derived in this paper but are explicitly imported from [24], a paper whose sole author is one of the present authors. Thus the load-bearing step of the quantum conclusion reduces to a self-citation. The paper also states in the Conclusions that 'gauge fixing is still needed' and that diffeomorphisms must be added, so the equality of Z_qed and Z_qg is not independently established. The self-citation is therefore not a minor bibliographic point; it carries the central argument.

full rationale

The paper is a constructive dictionary: Eq. (2.6) explicitly imposes the field identification Θ→μ²ε e e and A→εω, and the choice μ=e is made 'for simplicity'. The resulting relations (2.13) for G and Λ are consequences of this normalization rather than independent predictions; this is a transparency issue, not a hidden circularity, because the map is openly defined. The fermionic extension and the parity/time-reversal violating interactions are new content obtained by substitution, not assumed. The main circularity concern is in Section 4.2: the proof that the Jacobian (2.11) is BRST-trivial leans on [24] by one of the present authors. Since the BRST transformations in (4.13)-(4.14) are introduced so that Sdet is s-exact, and the paper does not verify nilpotency (indeed s²Ȳ = -1/2Y from those equations), the cited treatment is load-bearing and the equality Z_qed = Z_qg is not independently demonstrated. The Conclusions also concede that gauge fixing, including diffeomorphisms, is still needed, so the quantum claim is incomplete. On balance, the classical mapping is self-contained and transparent, while the quantum equivalence has a partial circularity via self-citation and an omitted proof; the central fermionic map retains independent content.

Assumptions & free parameters 1 free parameters · 4 assumptions · 2 invented entities

The central construction is a field redefinition, so the main ledger items are the postulated map, the formal path-integral treatment, and the ad hoc scale μ.

free parameters (1)

μ (mapping scale) = μ = e
In (2.6), Θ maps to μ² ε_ab e^a e^b; the authors set μ=e 'for simplicity'. This choice fixes the derived relations G = e²/(8π(κ+e²)) and Λ² = e⁴/(κ+e²) in (2.13); other choices would change them.

assumptions (4)

standard math U(1) is isomorphic to SO(2) in two Euclidean dimensions
Used throughout Section 2.2 to justify the field map between the QED connection and the spin connection.
ad hoc to paper The field redefinitions (2.6)-(2.10) are a valid bijection between field spaces after adding boundary auxiliary fields
This is the central construction; it is postulated to make the mapping one-to-one and is not derived from a more fundamental principle.
domain assumption BRST-exact actions and BRST doublet fields do not affect physical observables
Used in Section 4 to claim that the boundary term and the Jacobian contribution are harmless. Standard in BRST quantization, but requires a proper gauge-fixed setting.
domain assumption The path integral over all fields (without gauge fixing) is meaningful for comparing Z_qed and Z_qg
Section 4.1 sets up Z_qed without gauge fixing; Section 4.2 compares to Z_qg without fixing diffeomorphisms. The authors later concede these are still needed.

invented entities (2)

Auxiliary BRST-trivial fields (ξ^a, ρ^a, Y, Ybar, W, Wbar, Z, K)
purpose: To absorb the Jacobian of the field map and to keep BRST doublet structure in the quantum action (4.12)-(4.19).
These fields are introduced for the path integral representation and have no independent experimental handle; they appear only in BRST-exact combinations.
Unconventional SO(2) fermions with γ3-modified gauge transformations
purpose: The paper presents these as a new kind of fermion whose left- and right-handed parts rotate oppositely under SO(2), forming the basis for the claimed new gauge theory.
No experimental or independent theoretical evidence is provided. Using (A.5), the transformation (3.4) is algebraically equivalent to the mapped transformation (3.3), so the entity may be a convention rather than a new object.

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Cite this review

Pith. "Pith review of Relating electrodynamics and gravity in two Euclidean dimensions." pith.science (2026). https://pith.science/paper/UAXFOJ77

@misc{pith2026250209272,
  author       = {Pith},
  title        = {Pith review of: Relating electrodynamics and gravity in two Euclidean dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UAXFOJ77}},
  note         = {Machine review of arXiv:2502.09272}
}

abstract

Two-dimensional electrodynamics coupled to Dirac fermions is mapped onto two-dimensional gravity in the first-order formalism, also including fermions. However, the resulting fermion-gravity coupling deviates from the conventional form, explicitly violating parity and time-reversal symmetries. Additionally, these fermions exhibit an unconventional transformation behavior under $SO(2)$ transformations. Furthermore, we analyze the consistency of this mapping at the quantum level using the path integral formalism and Becchi-Rouet-Stora-Tyutin techniques. Our findings demonstrate that quantum electrodynamics and quantum gravity remain equivalent at the quantum level.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [24]

Geometrodynamical description of two -dimensional electrodynamics

R. F. Sobreiro, “Geometrodynamical description of two -dimensional electrodynamics” . Europhysics Letters 139 (2022) 64002

work page 2022

[2] [1]

The electronic properties of graphene

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselo v, and A. K. Geim, “The electronic properties of graphene” . Rev. Mod. Phys. 81 (2009) 109–162 . 8This representation diﬀers from [ 31] because we are in Euclidean spacetime. 12

work page 2009

[3] [2]

Solid-s tate platform for space-time engineering: The 8Pmmn borophene sheet

T. Farajollahpour, Z. Faraei, and S. A. Jafari, “Solid-s tate platform for space-time engineering: The 8Pmmn borophene sheet” . Phys. Rev. B 99 no. 23, (2019) 235150

work page 2019

[4] [3]

Two- and One-Dimensional Honeycomb Structures of Silicon and Germanium

S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and C. S. , “Two- and One-Dimensional Honeycomb Structures of Silicon and Germanium” . Phys. Rev. Lett. 102 (2009) 236804

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A. Staruszkewicz, “Fokker Action Principle and the Hype rbolic Motion” . Am. J. Phys. 35 (1967) 437

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I. Bialynicki-Birula, “Classical electrodynamics in t wo dimensions - exact solution” . Phys. Rev. D 3 (1971) 864–866

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B. P. Kosyakov, “Exact solutions of classical electrody namics and the Yang-Mills-Wong theory in even-dimensional space-time” . Theor. Math. Phys. 119 (1999) 493–505

work page 1999

[8] [7]

B. P. Kosyakov, Introduction to the classical theory of particles and ﬁelds . Springer, Berlin, Heidelberg, 2007. http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521545889

work page 2007