Pith. sign in

REVIEW 5 major objections 4 minor 24 references

Four-dimensional Palatini–Cartan gravity admits a genuine Poisson corner phase space after reduction by the kernel of residual corner one-forms.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

The reduced corner phase space of Palatini–Cartan gravity is a Dirac structure—the graph of a Poisson bivector—yielding a strict BF2V theory.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection The corner reduction to a Poisson phase space is a real advance, but the central proof leans on infinite-dimensional quotient regularity and a bivector formula whose well-definedness is not fully nailed down—still worthy of a serious referee. the 5 major comments →

arxiv 2607.28262 v2 pith:QENT6ARK submitted 2026-07-30 math-ph math.DGmath.MPmath.SG

The reduced Dirac structure of General Relativity on manifolds with corners

Alberto S. Cattaneo , Filippo Fila-Robattino , Manuel Tecchiolli This is my paper
classification math-ph math.DGmath.MPmath.SG MSC 53D1770S0583C05
keywords Dirac structurecorner phase spacePalatini–Cartan gravityBF2V formalismPoisson bivectorCourant algebroidcoisotropic reductionmanifolds with corners
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper aims to prove that the codimension-two corner structure of four-dimensional Palatini–Cartan gravity, previously shown to be singular and only pre-BF2V, is in fact a genuine Poisson structure after a classical reduction. Starting from the boundary constraint algebra, the authors build a pre-Dirac structure on the space of corner fields, verify its isotropy and involutivity, and then quotient by the kernel of the residual corner one-forms. The result is a maximal Dirac structure that is the graph of a Poisson bivector on the reduced corner phase space. From this Poisson structure a strict BF2V theory follows directly, and an equivalent affine Poisson description shows that the reduced corner theory is a relative of 4D BF theory. A sympathetic reader would care because this gives the corner degrees of freedom of gravity a well-defined geometric phase space, opening a concrete route toward corner quantization and a unified bulk–boundary–corner framework.

Core claim

The central claim is Theorem 5.29: the generalized distribution D defined by Eq. (57) on the reduced corner field space PΓ is a Dirac structure, and it is obtained as the graph of the Poisson bivector π given in Eq. (58). In the paper's own terms, the previously obstructed codimension-two structure of Palatini–Cartan gravity becomes, after reduction, an honest Poisson corner phase space. The paper further proves that this Poisson structure admits an affine Poisson description, which yields the BF2V action S^PC_Γ = ∫_Γ (1/2)E[c,c] + µ e d_{ω0} c + (1/2)µ² F_{ω0}, and shows that the resulting BF2V data is equivalent to a four-dimensional BF theory on a constrained submanifold. If the theorem h

What carries the argument

The load-bearing object is the standard Courant algebroid T⊕T* over the corner field space, equipped with its Dorfman bracket and symmetric pairing. The pre-Dirac structure is the span of generalized sections Xα + Xα, where Xα are the push-forwards of the Hamiltonian vector fields of the boundary constraints (Lorentz, tangential diffeomorphism, normal deformation) and Xα are the residual one-forms supported on the corner. The reduction proceeds by quotienting by the kernel of the one-form components, and the reduced space PΓ is parameterised by the field redefinitions E = e²/2 and Ω = e(ω−ω0), with representatives fixed by the conditions ϵ_m ∈ Ω^{0,1}_Γ, ω_m = 0, and all higher transverse je

Load-bearing premise

The reduction is legitimate only if the quotient PΓ = FΓ/K is a smooth manifold and if the chosen representatives (ϵ_m, ω_m = 0, and zero higher transverse jets) do not change the induced generalized distribution; the paper proves the kernel distribution W̃ is regular and involutive but does not prove smoothness of the infinite-dimensional quotient or invariance of the Dirac structure under these choices.

What would settle it

Compute the Schouten bracket [π,π] of the bivector in Eq. (58) on a concrete corner (e.g., Γ = S² with a round coframe and a fixed reference connection ω0) without imposing the constraints, and check whether it vanishes; if it does not, Theorem 5.29 is false. Equivalently, verify the identity ι_α π = X_α for every generating one-form α = aδE + bδΩ on PΓ; a single failure of these defining equations would refute the claim that D = Graph(π).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The corner phase space of Palatini–Cartan gravity is Poisson, so the standard Poisson-geometric toolkit—moment maps, deformation quantization, BRST/BFV cohomology—becomes applicable to the corner degrees of freedom.
  • The BF2V action is obtained directly from the Poisson bivector, making the codimension-two theory strict rather than pre-BF2V; this provides a concrete starting point for a BV-BFV-type quantization of gravity on manifolds with corners.
  • The affine Poisson form identifies the term µ² F_{ω0} as a classical seed of central-extension data, so a gravitational corner algebra can be constructed along the lines of the BF corner algebra, with possible non-trivial representations.
  • The reduction procedure—quotient by the kernel of the residual one-forms—is a general mechanism that unifies the corner structures of Yang–Mills, BF and gravity, suggesting that any first-class boundary constraint algebra yields a Poisson corner phase space after this reduction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A widely applicable corollary is that the obstruction to maximality of corner Dirac structures in gauge theories is precisely the existence of directions invisible to all corner charges; quotienting by them should convert any pre-Dirac structure from first-class constraints into a Poisson structure, not just in gravity.
  • The affine Poisson form suggests that the gravitational corner algebra is a central extension of the Lorentz–diffeomorphism algebra, with the central charge determined by F_{ω0}; computing the natural Lie–Poisson bracket on PΓ would yield a concrete prediction for the corner algebra.
  • The representative choices (ϵ_m, ω_m=0, vanishing jets) amount to a gauge-fixing of the reduction. If the Dirac structure turns out to depend on these choices, different corners of the same spacetime could carry inequivalent Poisson structures, which would be a physically important subtlety; the paper does not address this invariance.
  • The similarity to BF theory suggests that, for a suitable embedding, the gravitational corner algebra may be a subalgebra or quotient of the BF corner algebra, potentially making the quantum corner state space more tractable.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper proposes a reduction procedure for the codimension-two structure of four-dimensional Palatini–Cartan gravity. Starting from the boundary constraint algebra reviewed in Section 4, it constructs a generalized distribution on the space of corner fields (Definition 5.6, Eq. (57)), proves its isotropy and involutivity (Proposition 5.7, Appendix B.1), identifies a reduction by the kernel of the corner one-forms, and claims that on the reduced space PΓ the induced maximal distribution is the graph of an explicit Poisson bivector π (Theorem 5.29, Eq. (58)). The paper further promotes this Poisson structure to a BF^2V action (Section 6), proposing an affine Poisson reformulation resembling 4D BF theory. The central assertion is well motivated and the explicit computations in the appendices are substantial; however, several load-bearing definitions and reduction steps are asserted or only sketched, leaving the main theorem not fully established in its stated form.

Significance. If the main theorem is correct, the paper would remove a known obstruction at codimension two in Palatini–Cartan gravity and provide a genuine Poisson corner phase space together with a strict BF^2V description; this would be a significant advance over the pre-BF^2V and local P∞ structures of [CC24]. The paper is strong in its explicit presentation: the distribution D is defined concretely, the isotropy/involutivity proof is carried out in detail in Appendix B.1, the candidate bivector π is written out, and the equivalence check π(X_α)=X_α is spelled out in Appendix B.2. The use of prior published results [CCS21a], [Can24] and [CC24] is appropriate and not circular. The main weakness is not the internal algebra but the lack of a proof that the reduced space PΓ is a well-defined smooth manifold and that the bivector π is a well-defined section of ∧²TPΓ; these are necessary for the graph interpretation to be meaningful.

major comments (5)
  1. [Section 5.4, Eq. (58)] The bivector π is not shown to be a well-defined section of ∧²TPΓ. The reduced space PΓ (Theorem 5.26) fixes ϵ_m, Ω_m, and all higher jets by Eqs. (46), (50) and Table 1, yet Eq. (58) contains explicit functional derivatives δ/δϵ_m and δ/δΩ_m. The proof in Appendix B.2 checks π(X_α)=X_α for the generators X_α, but it never proves that the resulting vector fields are tangent to the submanifold PΓ or that the transverse derivatives are not independent coordinates. Without this, the equality D = Graph(π) is not established on the stated reduced space.
  2. [Eq. (58), last term] The expression W_e^{-1}[δ/δΩ ϵ_m ϵ_n b, ϵ_m] requires the argument of W_e^{-1} to lie in the image of the wedge map W_e. No such argument is given. Since W_e is not invertible on the full space (Theorem A.1), the notation W_e^{-1} is only defined on Im(W_e); the paper does not justify that the functional derivative with respect to Ω of the bracket lands there. The same issue appears in Section 6 in the heuristic use of W_e^{-1} in Eq. (61).
  3. [Section 5.2.1, Theorem 5.26] The definition of PΓ relies on three representative-fixing choices: ϵ_m via Eq. (46), ω_m=0 via Eq. (50), and vanishing of all higher transversal jets (Theorem 5.26). Proposition 5.9 proves regularity and involutivity of the kernel W-tilde on F̃Γ, but it does not prove that the quotient F̃Γ/W-tilde is a smooth infinite-dimensional manifold, nor that the induced generalized distribution is independent of the chosen representatives. The paper also does not verify that the constraints (44), (45), (41) are preserved by the quotient, which is necessary for the submanifold PΓ to inherit the Dirac structure.
  4. [Prop. 5.24 and Table 1] The transversal (n-direction) Bianchi/Einstein propagation is only sketched. Proposition 5.24 says 'We give some hints of proof here and leave the full one for the reader', and the analogous corner statement (Prop. 5.25) is stated with no computation. Table 1, which fixes F_{ωn} and ∂_nω, is the basis for the structural constraints used in the main theorem, including the definition of Θ and Θ_m in Theorem 5.29. An unproved fix of 18 components of ∂_nω is load-bearing for PΓ and for the bivector formula, so it must be supplied or replaced by a reference to a full derivation.
  5. [Section 6, Prop. 6.4 and Prop. 6.6] The BF^2V pullback steps are asserted rather than demonstrated. In Prop. 6.4 the text says 'the full computation of the pullback of S_Γ along Ψ is quite involved, we prefer to omit it', and Prop. 6.6 similarly omits the computation. Since the affine Poisson reformulation in Eq. (71) is one of the paper's advertised outcomes, this omission is significant, even if the final CME check in Prop. 6.4 is performed. The reader cannot verify that the symplectomorphism Ψ and Φ indeed yield Eqs. (63)–(64) and Eq. (69).
minor comments (4)
  1. [Throughout] The references to 'Table 5.2.2' (e.g., in Theorem 5.26 and Section 5.3) should be 'Table 1'; the table has no caption number and is introduced as 'Table 1'.
  2. [Remark 3.17] Typo: 'Ww will see' should be 'We will see'.
  3. [Eq. (58)] The bivector π is written with both E (the field e²/2) and derivatives δ/δE; the notation is compressed and not defined for the multi-vector wedge products of odd/even functional derivatives. A paragraph explaining the convention for the graded wedge product of δ/δE, δ/δΩ, δ/δΩ_m and δ/δϵ_m would improve readability.
  4. [Section 3.1.2] The commutative diagram after Definition 3.13 is not referenced in the text and some arrows are not labeled; adding a short explanation would help the reader navigate the spaces F̃Σ, FΣ, FΓ and PΓ.

Circularity Check

0 steps flagged

No significant circularity: the corner Poisson/Dirac construction is an independent derivation; self-citations are prior published background, and the main gaps are well-definedness and omitted computations, not circular steps.

full rationale

The derivation chain runs from the boundary constraint algebra (Theorem 4.15, quoted from [CCS21a]) to the pre-Dirac distribution D (Eq. (57)), then reduces by the kernel of the residual one-forms and exhibits a bivector π (Eq. (58)) satisfying π(X_α)=X_α for the generators (Appendix B.2). No fitted parameter is renamed as a prediction, and no equation in the chain is equal to its input by construction: the boundary algebra of [CCS21a] and the coframe facts of [CC24]/[Can24] (e.g., Theorem A.1, E=e^2/2) are prior published results that do not contain Theorem 5.29, so citing them is not a circular reduction. The final BF2V action (69) is obtained by explicit field redefinitions from π, not assumed. Two caveats are correctness risks rather than circularity: Prop. 5.24 says "We give some hints of proof here and leave the full one for the reader," and Prop. 6.4 says "the full computation ... is quite involved, we prefer to omit it." Moreover, π in Eq. (58) contains δ/δϵ_m and δ/δΩ_m even though Theorem 5.26 fixes ϵ_m and (ω−ω_0)_m=0; the paper does not prove these are well-defined tangent directions on PΓ. That is a gap in the proof of Theorem 5.29, but not a reduction of the theorem to its own assumptions.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The construction relies on stated structural assumptions (non-differentiated multipliers, isotropy/involutivity, non-degenerate metrics) and on prior external results for coframe decompositions; it introduces representative choices and assumes the reduced quotient is smooth. No new physical entity such as a new particle, force, or dimension is postulated.

free parameters (3)
  • Reference connection ω0 = arbitrary, chosen by hand
    The final affine BF2V action (69) and the affine Poisson bivector (71) depend explicitly on ω0 via dω0 and Fω0. The paper does not prove independence of the reduced corner structure from this choice.
  • Normal field ϵm representative = defined by Equation (46), with sign ν = ±1
    Definition 5.10 fixes ϵm by [e,ϵm] = 0, [ϵn,ϵm] = 0, [ϵm,ϵm] = ν. This is a choice of representative of [em] in the quotient; it enters the Poisson bivector and BF2V data, and no invariance under alternative representatives is shown.
  • Higher transversal jets = 0
    Theorem 5.26 sets all higher transversal jets to zero. This is a gauge/representative choice that the quotient construction relies on, with no proof that the Dirac structure is independent of it.
axioms (6)
  • domain assumption Assumption 3.4: constraint functionals do not differentiate the Lagrange multipliers.
    Invoked to ensure residual one-forms appear only at the corner and to justify the pre-Dirac construction; satisfied by the PC constraints, but restricts the class of theories.
  • domain assumption Assumption 3.8 / Proposition 5.7: the generalized distribution D is isotropic and involutive on FΓ.
    This is proven for PC gravity, but it is an additional structural condition used to define a pre-Dirac structure rather than a consequence of standard variational calculus alone.
  • domain assumption Non-degeneracy of the induced boundary and corner metrics.
    Used throughout Lemma 4.9 / Theorem 4.12 and Lemma A.7 to guarantee unique decompositions; excludes null boundary/corner cases, which are only referenced to [CCT21].
  • ad hoc to paper The infinite-dimensional quotient PΓ = FΓ/K is a smooth manifold with the expected tangent space.
    Proposition 5.9 proves W-tilde is regular and involutive, but the paper does not prove the quotient is smooth nor that the induced distribution descends to a genuine subbundle; this is assumed in Theorem 5.26 and 5.29.
  • ad hoc to paper The propagation table of torsion/Einstein/Bianchi identities (Table 1) fixes the transversal fields and jets as stated.
    Propositions 5.24 and 5.25 are only sketched, yet they are used to define PΓ and to rewrite the vector fields in terms of E and Ω.
  • domain assumption Coframe decomposition and dimension results for the wedge maps W^{i,j}_k on Γ, imported from [Can24] and [CC24].
    Theorem A.1, Lemma A.7 and related injectivity/surjectivity statements are external results on the linear algebra of coframes; the corner construction relies on them without re-proving them.

reviewed 2026-08-04 · how reviews work

0 comments
Cite this review

Pith. "Pith review of The reduced Dirac structure of General Relativity on manifolds with corners." pith.science (2026). https://pith.science/paper/QENT6ARK

@misc{pith2026260728262,
  author       = {Pith},
  title        = {Pith review of: The reduced Dirac structure of General Relativity on manifolds with corners},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QENT6ARK}},
  note         = {Machine review of arXiv:2607.28262}
}
Share X Bluesky LinkedIn Reddit HN
abstract

In this paper, the corner Poisson structure of four-dimensional Palatini-Cartan gravity is derived. Building on the classical description of gravity on manifolds with boundary, specifically on the boundary constraint algebra, a pre-Dirac structure on the space of corner fields is obtained together with a reduction procedure that yields a maximal Dirac structure, identified as the graph of a Poisson bivector field, on the reduced space of corner fields. It is further shown that this Poisson structure admits an equivalent affine Poisson description, which naturally exhibits the reduced corner theory as a $BF$-like theory and leads to a BF$^2$V formulation. This provides the basis for a unified framework for the bulk, boundary, and corner structures of Palatini-Cartan gravity.

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 13 linked inside Pith

  1. [1]

    The geometry of the master equation and topological quantum field theory

    [Ale+97] M. Aleksandrov, M. Kontsevich, A. Schwarz, and O. Zaboronsky. “The geometry of the master equation and topological quantum field theory”.Int. J. Mod. Phys. A(12 1997), pp. 1405–1430.doi:10.1142/S0217751X97001031. [AS15] S. Alexandrov and S. Speziale. “First order gravity on the light front”.Physical Review D 91.6 (Mar. 2015).issn: 1550-2368.doi:1...

    doi:10.1142/s0217751x97001031 1997

  2. [8]

    Boundary structure of the standard model coupled to gravity

    arXiv:2605.30006 [math-ph].url:https://arxiv.org/abs/2605. 30006. [Can+23] G. Canepa, A. S. Cattaneo, F. Fila-Robattino, and M. Tecchiolli. “Boundary structure of the standard model coupled to gravity” (July 2023). arXiv:2307.14955 [math-ph]. [CCS21a] G. Canepa, A. S. Cattaneo, and M. Schiavina. “Boundary structure of general relativity in tetrad variable...

    Pith/arXiv arXiv 2023

  3. [9]

    Dirac manifolds

    Cambridge University Press, 2016.doi:10.1017/9781316678626. [Cou90] T. Courant. “Dirac manifolds”.Transactions of the American Mathematical Society319.2 (1990), pp. 631–661. [CHv95] O. Coussaert, M. Henneaux, and P. van Driel. “The asymptotic dynamics of three- dimensional Einstein gravity with a negative cosmological constant”.Classical and Quantum Gravi...

    Pith/arXiv arXiv 2016

  4. [19]

    Cosmology with Torsion: An alternative to cosmic inflation

    [Pop10] N. J. Poplawski. “Cosmology with Torsion: An alternative to cosmic inflation”.Phys. Lett. B694 (2010), p

    2010

  5. [64]

    On synthetic interpretation of quantum principal bundles

    [Brz09] T. Brzezi´ nski. “On synthetic interpretation of quantum principal bundles” (2009). arXiv: 0912.0213 [math.QA]. [Bur] M. Burkardt. “Light Front Quantization”.Advances in Nuclear Physics(), pp. 1–74.doi: 10.1007/0-306-47067-5_1. [CLR19] F. Cabral, F. S. N. Lobo, and D. Rubiera-Garcia. “Einstein-Cartan-Dirac gravity withU(1) symmetry breaking” (2019...

    Pith/arXiv arXiv 2009

  6. [96]

    The Poisson brackets of free null initial data for vacuum general relativity

    [Rei18] M. P. Reisenberger. “The Poisson brackets of free null initial data for vacuum general relativity”.Classical and Quantum Gravity35.18 (Aug. 2018), p. 185012.issn: 1361-6382. doi:10.1088/1361-6382/aad569. [Rei13] M. P. Reisenberger. “The symplectic 2-form for gravity in terms of free null initial data”. Classical and Quantum Gravity30.15 (July 2013...

    Pith/arXiv arXiv 2018

  7. [122]

    Hamiltonian BRST and Batalin-Vilkovisky Formalisms for Second Quantization of Gauge Theories

    [BG05] G. Barnich and M. Grigoriev. “Hamiltonian BRST and Batalin-Vilkovisky Formalisms for Second Quantization of Gauge Theories”.Communications in Mathematical Physics254.3 (2005), pp. 581–601.doi:10.1007/s00220-004-1275-4. [BF83] I. A. Batalin and E. S. Fradkin. “A generalized canonical formalism and quantization of reducible gauge theories”.Physics Le...

    Pith/arXiv arXiv 2005

  8. [132]

    Some reference formulas for the generating functions of canonical transforma- tions

    Con- temporary Mathematics. Providence, RI: American Mathematical Society, 1992, pp. 51–73. doi:10.1090/conm/132/1188434. [Ans16] D. Anselmi. “Some reference formulas for the generating functions of canonical transforma- tions”.European Physical Journal C76, 49 (Feb. 2016), p. 49.doi:10.1140/epjc/s10052- 015-3874-y. [ADM59] R. Arnowitt, S. Deser, and C. W...

    doi:10.1090/conm/132/1188434 1992

  9. [181]

    Universe in a Black Hole with Spin and Torsion

    [Pop16] N. J. Poplawski. “Universe in a Black Hole with Spin and Torsion”.ApJ832 (2016), p

    2016

  10. [1545]

    Reduction of symplectic manifolds with symmetry

    [MW74] J. Marsden and A. Weinstein. “Reduction of symplectic manifolds with symmetry”.Reports on Mathematical Physics5.1 (1974), pp. 121–130.issn: 0034-4877.doi:10.1016/0034- 4877(74)90021-4. [Mat75] J. Mather.Notes on Topological Stability. Tech. rep. Harvard University,

    doi:10.1016/0034- 1974

  11. [1969]

    Symplectic reduction, BRS cohomology, and infinite- dimensional Clifford algebras

    [KS87] B. Kostant and S. Sternberg. “Symplectic reduction, BRS cohomology, and infinite- dimensional Clifford algebras”.Annals of Physics176.1 (1987), p. 49.issn: 0003-4916. doi:10.1016/0003-4916(87)90178-3. [Kra+19] D. Kranas, C. G. Tsagas, J. D. Barrow, and D. Iosifidis. “Friedmann-like universes with torsion”.Eur. Phys. J. C79 (2019), p

    doi:10.1016/0003-4916(87)90178-3 1987

  12. [1975]

    Einstein-Cartan Wormhole Solutions

    56 REFERENCES [MZ17] M. R. Mehdizadeh and A. H. Ziaie. “Einstein-Cartan Wormhole Solutions”.Phys. Rev. D 95 (2017), p. 064049. [Mei25] E. Meinrenken. “Introduction to Moduli Spaces and Dirac Geometry”.arXiv preprint arXiv:2506.04150(2025).doi:10.48550/arXiv.2506.04150. [Mel] R. B. Melrose.Differential Analysis on Manifolds with Corners. Lecture notes.url:...

    Pith/arXiv arXiv doi:10.48550/arxiv.2506.04150 2017

  13. [1988]

    Superfield BRST charge and the master action

    [GD00] M. Grigoriev and P. Damgaard. “Superfield BRST charge and the master action”.Physics Letters B474.3 (2000), pp. 323–330. [Gri11] M. Grigoriev. “Parent formulation at the Lagrangian level”.Journal of High Energy Physics 2011.7 (2011), p. 61.doi:10.1007/JHEP07(2011)061. [Gri16] M. Grigoriev.Presymplectic structures and intrinsic Lagrangians

    doi:10.1007/jhep07(2011)061 2000

  14. [1993]

    Lectures on Batalin-Vilkovisky formalism and its applications in topological quantum field theory

    [Mne17] P. Mnev. “Lectures on Batalin-Vilkovisky formalism and its applications in topological quantum field theory” (July 2017). arXiv:1707.08096 [math-ph]. [MSW19] P. Mnev, M. Schiavina, and K. Wernli. “Towards holography in the BV-BFV setting”. Annales Henri Poincare21.3 (2019), pp. 993–1044.doi:10.1007/s00023-019-00862-8. [Mon+20] M. Montesinos, R. Es...

    Pith/arXiv arXiv 2017

  15. [1994]

    Local BRST cohomology in the antifield for- malism. I. General theorems

    [BBH95] G. Barnich, F. Brandt, and M. Henneaux. “Local BRST cohomology in the antifield for- malism. I. General theorems”.Comm. Math. Phys.174.1 (1995), pp. 57–91. [BG11] G. Barnich and M. Grigoriev. “First order parent formulation for generic gauge field theo- ries”.Journal of High Energy Physics2011.1 (2011), p. 122.doi:10.1007/JHEP01(2011)

    doi:10.1007/jhep01(2011 1995

  16. [1997]

    Homological reduction of constrained Poisson algebras

    [Sta97] J. Stasheff. “Homological reduction of constrained Poisson algebras”.J. Differential Geom. 45.1 (1997), pp. 221–240.doi:10.4310/jdg/1214459757. [Str12] N. Straumann.General Relativity. Springer,

    arXiv 1997

  17. [2001]

    Symmetric teleparallel general relativity

    [NY99] J. M. Nester and H. J. Yo. “Symmetric teleparallel general relativity”. 37 (1999). eprint: gr-qc/9809049. [Pal19a] A. Palatini. “Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamil- ton”.Rendiconti del Circolo Matematico di Palermo (1884-1940)43.1 (1919), pp. 203–212. issn: 0009-725X.doi:10.1007/BF03014670. [Pal19b] A. Pala...

    Pith/arXiv arXiv 1999

  18. [2006]

    Palatini formalism and new canonical variables for GL(4) invariant gravity

    [FP90] R. Floreanini and R. Percacci. “Palatini formalism and new canonical variables for GL(4) invariant gravity”.Class. Quant. Grav.7 (1990), pp. 1805–1818.doi:10 . 1088 / 0264 - 9381/7/10/011. [FR12] K. Fredenhagen and K. Rejzner. “Batalin-Vilkovisky Formalism in Perturbative Algebraic Quantum Field Theory”.Communications in Mathematical Physics317.3 (...

    Pith/arXiv arXiv 1990

  19. [2012]

    Algebra of Constraints for the Linearized Palatini-Cartan Theory on a Light- Like Boundary

    REFERENCES 57 [Tec19a] M. Tecchiolli. “Algebra of Constraints for the Linearized Palatini-Cartan Theory on a Light- Like Boundary”. Master thesis. Nov. 2019.url:http://user.math.uzh.ch/cattaneo/ tecchiolli.pdf. [Tec19b] M. Tecchiolli. “On the Mathematics of Coframe Formalism and Einstein–Cartan The- ory—A Brief Review”.Universe5.10 (Sept. 2019), p. 206.is...

    2019

  20. [2016]

    Generalized Complex Geometry

    arXiv:1606.07532 [hep-th]. [Gua04] M. Gualtieri. “Generalized Complex Geometry”. PhD Thesis, supervised by Nigel Hitchin. PhD thesis. University of Oxford, 2004.url:https://arxiv.org/abs/math/0401221. [Heh+76a] F. W. Hehl, P. V. D. Heyde, G. D. Kerlick, and J. M. Nester. “General Relativity with Spin and Torsion: Foundations and Prospects”.Rev. Mod. Phys....

    Pith/arXiv arXiv 2004

  21. [2018]

    Lovelock-Cartan theory of gravity

    arXiv:1805.10317 [math-ph].url:https://hdl.handle.net/20.500.11811/7534. [Lot] J. Lott.The Geometry of Supergravity Torsion Constraints. arXiv:math/0108125. [Mac87] K. Mackenzie.Lie Groupoids and Lie Algebroids in Differential Geometry. London Math- ematical Society Lecture Note Series. Cambridge University Press, 1987.doi:10.1017/ CBO9780511661839. [MZ91...

    Pith/arXiv arXiv 1987

  22. [2019]

    Symmetric Space Cartan Connections and Gravity in Three and Four Dimen- sions

    arXiv:1911.09766 [math.DG]. [Wis09] D. K. Wise. “Symmetric Space Cartan Connections and Gravity in Three and Four Dimen- sions”.SIGMA5, 080 (Aug. 2009), p. 080.doi:10.3842/SIGMA.2009.080. [Wit89] E. Witten. “2+1 dimensional gravity as an exactly soluble system”.Nuclear Physics B(311 1988/89), pp. 46–78. [Wit88] E. Witten. “Topological sigma models”.Comm. ...

    Pith/arXiv arXiv 1911

  23. [2025]

    Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism

    arXiv: 2507.06279 [math-ph].url:https://arxiv.org/abs/2507.06279. [CC24] G. Canepa and A. S. Cattaneo. “Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism”.Annales Henri Poincar´ e25.5 (2024), pp. 2585–2639.doi:10. 1007/s00023-023-01360-8. [CCF22] G. Canepa, A. S. Cattaneo, and F. Fila-Robattino. “Boundary structure of gauge ...

    Pith/arXiv arXiv 2024

  24. [2026]

    Einstein-Cartan Cosmologies

    [BNB19] S. Bravo Medina, M. Nowakowski, and D. Batic. “Einstein-Cartan Cosmologies”.Ann. Phys.400 (2019), p

    2019

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.