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On the role of the Parity Violating Hojman--Holst term in Gravity Theories

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Pith's one-line read Generic f(R,H) metric-affine gravity with the Holst term reduces on shell to a metric scalar-tensor theory containing the graviton and one scalar field.

desk verdict The central kinetic-coupling formula (42) is wrong as printed and fails the paper's own quadratic check, so the advertised equivalence claim needs major revision. read the letter →

arxiv 2506.04738 v1 pith:OQNVHZQM submitted 2025-06-05 gr-qc hep-th

Damianos Iosifidis This is my paper
classification gr-qchep-th
keywords metric-affinegravityHolsttermHojmanparityviolationtorsionscalar-tensorequivalencepseudoscalarcurvaturef(RH)theories
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

This paper asks how many new degrees of freedom the parity-odd Holst (Hojman) curvature pseudoscalar introduces when the gravitational Lagrangian is an arbitrary function $f(R,H)$ of the Ricci scalar and that pseudoscalar. It answers that, in the metric-affine formulation, the theory is on shell equivalent to a torsionless metric scalar-tensor theory carrying the graviton plus one scalar field. The torsion vector and pseudovector become exact gradients built from $f_R$ and $f_H$, the nonmetricity can be gauged away, and the trace equation $Rf_R+Hf_H-2f=0$ ties $H$ to $R$. The result matters because it gives the explicit kinetic coupling and potential of the scalar-tensor theory, making a large class of parity-violating gravity theories accessible to inflationary model building.

What carries the argument

The central mechanism is the exact-gradient form of the torsion pieces. Equation (19) writes the torsion vector as $S_\mu = \frac{3}{8}\partial_\mu \ln(f_R^2 + 4f_H^2)$, and equation (20) writes the torsion pseudovector as $t_\mu = 3\partial_\mu \arctan(2f_H/f_R)$, so all torsion dynamics is carried by gradients of functions of $f_R$ and $f_H$. Combined with the trace equation (29) and the non-degeneracy condition (37), these identities turn the connection equations into a single scalar equation (34) for $\phi = R$, and the auxiliary-field action (35) then produces the scalar-tensor action (41) with $V(\Phi)$ from (40) and $K(\Phi)$ from (42).

What would settle it

A concrete check is to solve the connection field equations for a nontrivial background in any specific $f(R,H)$ with nonvanishing Hessian and a real solution of (29), and test whether the tensor modes $Z_{\mu\nu\lambda}$ and $\Omega_{\alpha\mu\nu}$ are forced to zero as (23) claims; a single solution with nonzero $Z$ or $\Omega$ would falsify the reduction. A complementary test is to linearize the resulting scalar-tensor action (41) around Minkowski space and count the propagating modes: any spectrum beyond the graviton plus one scalar, or a ghostly kinetic sector from $K(\Phi)$, would also falsify it.

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Extended reading notes

Core claim

The paper claims that a generic metric-affine theory with action $S = \frac{1}{2\kappa}\int d^4x \sqrt{-g}\, f(R,H)$, where $H = \varepsilon^{\alpha\beta\mu\nu} R_{\alpha\beta\mu\nu}$ is the parity-odd Holst (Hojman) pseudoscalar, is on shell equivalent to a torsionless metric scalar-tensor theory. From the connection field equations the torsion vector and pseudovector are forced to be exact: $S_\mu = \frac{3}{8}\partial_\mu \ln(f_R^2 + 4f_H^2)$ and $t_\mu = 3\partial_\mu \arctan(2f_H/f_R)$, while nonmetricity can be removed by the projective gauge freedom. The trace equation $Rf_R + Hf_H - 2f = 0$, when it has a real solution $H = F(R)$, leaves a single scalar $\phi = R$; the resulting metric theory is given by action (41), with scalar potential (40) and kinetic coupling (42). In the quadratic special case with $f = a_1R + a_2R^2 + b_1H + b_2H^2 + \gamma RH$ and the parameter constraint (48), the equivalent action is exactly Brans-Dicke with $\omega_0 = -3/2$ and $V = 0$.

Load-bearing premise

The reduction collapses to one scalar only if the trace equation $Rf_R + Hf_H - 2f = 0$ has a real solution $H=F(R)$, the Hessian determinant $f_{\chi\chi}f_{\psi\psi} - f_{\chi\psi}^2$ is nonzero, and the map $\Phi=f_R$ can be inverted; if any of those fails, the auxiliary-field action (35) cannot be reduced to a single-scalar scalar-tensor theory.

Editorial extensions

If this is right

  • If the trace equation of a given $f(R,H)$ has a real solution and the Hessian condition holds, the theory is on shell indistinguishable from a metric scalar-tensor theory; connection-based observables beyond this scalar sector cannot exist.
  • The explicit kinetic coupling (42) and potential (40) can be used to reconstruct the underlying $f(R,H)$ from an inflationary scalar-tensor model, reversing the usual construction direction.
  • For the quadratic action, the constrained parameter family all map to the same Brans-Dicke action with $\omega_0 = -3/2$ and $V = 0$, so the underlying parameters are not separately observable on shell.
  • In the parity-even $f(R,H^2)$ subclass, $H^2$ reduces to the quadratic curvature combination (54), and the theory still propagates only one additional scalar mode.
  • When additional quadratic torsion and nonmetricity invariants are included, the paper expects further degrees of freedom beyond this single scalar.

Reading between the lines

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

  • The exact-gradient character of the torsion suggests that the scalar mode is a boundary or integration-class degree of freedom of the connection; if so, the on-shell equivalence may lift to a path-integral duality only after careful treatment of boundary terms.
  • Because the equivalence holds for arbitrary $f$, a similar single-scalar reduction may occur for any projective-invariant metric-affine theory built from two curvature scalars, suggesting a general two-scalars-to-one rule.
  • A practical test is to compare the scalar-tensor model (41)-(42) against CMB observables for a chosen $f(R,H)$; any deviation in the scalar spectral index or tensor-to-scalar ratio would constrain the parity-violating parameters.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript studies metric-affine gravity theories with Lagrangian f(R,H), where R is the curvature scalar and H is the parity-odd Holst/Hojman pseudoscalar. It derives the connection field equations, argues that for a non-degenerate Hessian the nonmetricity can be gauged away while torsion reduces to vector and pseudovector parts determined by derivatives of f_R and f_H, and establishes an on-shell equivalence with a metric scalar-tensor theory containing one scalar field. The central claimed results are the explicit potential (40) and kinetic coupling function (42), subject to the trace equation (29) and the Hessian condition (37). A quadratic example is presented and claimed to reduce to Brans-Dicke theory with BD parameter ω0 = -3/2 under condition (48).

Significance. If correct, the paper generalizes earlier special-case results to a broad class of parity-violating metric-affine theories and gives explicit tools for deriving the scalar-tensor description, which is relevant for phenomenological applications such as inflation. The derivation is analytic and includes useful consistency checks against earlier work, and the paper provides a concrete quadratic example. However, the central kinetic-coupling formula appears to be inconsistent with the paper's own example, so the significance can only be assessed after that formula is corrected and verified.

major comments (4)
  1. [Section IV, Eq. (42) and Eq. (49)] Equation (42), the central kinetic-coupling formula, does not reproduce the paper's own quadratic check. Substituting Ω = λΦ and Ω_Φ = λ into (42) gives -K(Φ) = (3 - 4λ^2)/(2Φ(1 + 4λ^2)), which depends on λ, whereas Eq. (49) claims K(Φ) = -3/(2Φ), independent of λ. A direct substitution of the on-shell torsion pieces S_μ = 3/(4Φ) ∂_μΦ and t_μ = 0 into the action (39) together with the post-Riemannian expansions (27)-(28) yields -K = 3/(2Φ), confirming Eq. (50) but contradicting Eq. (42). Since the explicit form of K(Φ) is the central result announced in the abstract, this is a load-bearing algebraic error that must be corrected and re-verified.
  2. [Section III, Eq. (34)] The scalar field equation (34) is stated without derivation; the phrase 'after some trivial algebra' conceals a computation that is central to the claim that φ = R obeys a scalar equation. The notation in (34) is also ambiguous, in particular the term '2t(R)/BoxgR' appears to be a typesetting artifact. The author should provide the intermediate steps or at least a clear statement of the operator appearing in the equation.
  3. [Appendix, Eq. (63)] The proof that the traceless torsion and nonmetricity modes vanish rests on a condensed duality argument. The step from Eq. (63) to the conclusion that Z is either zero or (anti)self-dual with factor ±i/2, and the subsequent exclusion of the self-dual branch because it would require complex-valued f, is not fully shown; in particular, the contractions leading to the system (59)-(62) and the elimination that produces (63) are not exhibited. Since this result is needed to justify the reduction to vector/axial torsion and vanishing nonmetricity, the appendix should be expanded with the explicit algebra.
  4. [Section IV.A, Eqs. (45)-(49)] The quadratic example is not fully self-consistent: Eq. (45) for f is quoted without derivation, the condition (48) is introduced immediately before the claim Ω(Φ) = λΦ without showing the substitution that yields it, and the formula (52) for the generic kinetic function is also presented without derivation. More importantly, because Eq. (42) fails this example as shown above, the example currently serves to expose the error rather than to verify the claimed general formula.
minor comments (4)
  1. [Title and abstract] The title contains a typo: 'Grav ity' should be 'Gravity'. The abstract also writes 'Hojmann' where 'Hojman' is meant.
  2. [Section IV.A, Eq. (52)] The formula (52) for K(Φ) in the generic quadratic case is presented without derivation; a reader cannot see how it follows from (42) without redoing the algebra, so an outline of the computation would improve reproducibility.
  3. [Section III, after Eq. (33)] The text says 'the latter describes the evolution of a new scalar mode φ = R', but the symbol φ was not introduced before this point; it should be defined more carefully to avoid confusion with the auxiliary field Φ used later.
  4. [Section III, Eq. (16)] In Eq. (16), the factor n appears; although the paper is 4-dimensional, it should state explicitly that n = 4 here, or use 4 directly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the scalar-tensor equivalence is derived from stated assumptions and contains no fitted or pre-supposed input.

full rationale

The derivation chain is self-contained. The field equations (10)-(15), the trace condition (29), the auxiliary-field action (35), and the post-Riemannian expansions (27)-(28) are obtained in the paper from stated definitions and variational principles; the potential (40) and kinetic coupling (42) are computed from these expressions rather than imposed. The Hessian condition (37), existence of a real solution to the trace equation, and invertibility of Phi(R) are explicitly stated assumptions for the equivalence, and they are conditions on the input function f, not encoded output claims. The self-citations ([25,26] for projective invariance and [27] for the fH=constant case) are used for background, conventions, and consistency checks and are not load-bearing for the generic equivalence result; the appendix independently proves the vanishing of the tensor modes. A possible algebraic mismatch between Eq. (42) and the quadratic example in Sec. IV.A would be a correctness concern, not circularity, because it does not arise from fitting a parameter or defining the result into the input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard metric-affine geometry, a gauge choice that is legitimate because R and H are projectively invariant, and the explicit solvability conditions stated by the author. No numbers are fitted to data, and no new physical entities are introduced.

assumptions (5)
  • domain assumption 4-dimensional metric-affine spacetime with torsion and non-metricity
    All definitions in Section II, including the pseudoscalar H, are specific to 4 dimensions and to a general connection.
  • domain assumption Projective invariance allows gauge fixing q_mu = 0
    The paper uses projective freedom after Eq. (16) to set one non-metricity vector to zero; if this gauge choice were illegitimate, the non-metricity would not vanish and the torsion solution would change.
  • domain assumption Trace equation (29) has at least one real solution and the Hessian condition (37) is non-degenerate
    These are the stated conditions for the equivalence: the conformally invariant case f = R^2 G(R/H) and cases with no real roots are explicitly excluded.
  • domain assumption f is real-valued
    The appendix uses reality of f to exclude complex self-dual torsion modes when deriving Z = 0 and Omega = 0.
  • standard math Standard curvature, torsion, non-metricity decompositions and epsilon-tensor identities
    The paper relies on standard identities from references [21-24] throughout the derivation.

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Cite this review

Pith. "Pith review of On the role of the Parity Violating Hojman--Holst term in Gravity Theories." pith.science (2026). https://pith.science/paper/OQNVHZQM

@misc{pith2026250604738,
  author       = {Pith},
  title        = {Pith review of: On the role of the Parity Violating Hojman--Holst term in Gravity Theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OQNVHZQM}},
  note         = {Machine review of arXiv:2506.04738}
}
read the original abstract

We study Parity Violating Gravity Theories whose gravitational Lagrangian is a generic function of the scalar curvature and the parity odd curvature pseudoscalar, commonly known as the Holst (or Hojmann) term. Generalizing some previous results in the literature, we explicitly show that if the Hessian of this function is non-degenerate, the initial non-Riemannian Theory is on-shell equivalent to a metric Scalar-Tensor Theory. The generic form of the kinetic coupling function and the scalar potential of the resulting Theory are explicitly found and reported.

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Forward citations

Cited by 1 Pith paper

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