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On Covariant Actions for Chiral $p-$Forms

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper constructs a Lorentz- and diffeomorphism-covariant polynomial action for free chiral p-forms, classically equivalent to the Pasti–Sorokin–Tonin formulation.

desk verdict A genuinely new polynomial covariant action for chiral p-forms that reduces to PST on generic configurations, but the proof of equivalence leaves the null-locus c^2=0 unaddressed, so the single-chiral-DOF claim is not fully demonstrated. read the letter →

arxiv 1908.01789 v2 pith:FRSY2X5R submitted 2019-08-05 hep-th

Karapet Mkrtchyan This is my paper · ORCID
classification hep-th
keywords chiralp-formsself-dualfieldsPSTformulationduality-symmetricactionspolynomialactionauxiliaryp-formmanifestLorentzcovariancegaugepotentials
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

Chiral $p$-forms—fields whose self-dual part alone propagates—have resisted a manifestly Lorentz- and diffeomorphism-invariant action that is polynomial in the fields. This paper claims to remove that obstruction: it writes down a polynomial action, built from the physical gauge $p$-form, an auxiliary $p$-form, and a scalar, that is classically equivalent to the Pasti–Sorokin–Tonin (PST) formulation and describes exactly one chiral degree of freedom in $2p+2$ dimensions. If true, this matters because the new formulation is written in gauge potentials rather than field strengths, giving a natural starting point for non-abelian and gravitational interactions of chiral forms. The same mechanism is shown to work for duality-symmetric electromagnetism in four dimensions.

What carries the argument

Central object is the Lagrangian (2.1). For a physical $p$-form potential $\phi$ with field strength $F_{\mu_1\dots\mu_{p+1}}$ in $d=2p+2$ dimensions, the action is $$L = -\frac{1}{2(p+1)}$F^{2}$ - \frac{1}{2(p+1)}\bigl(\tilde F - (p+1)c_{[\mu_1}R_{\mu_2\dots\mu_{p+1}]}\bigr)^2 + $G^{{\mu\nu}}$\partial_{[\mu}c_{\nu]},$$ where $\tilde F$ includes the Hodge-dual piece so that the second term sees the chirality, $R$ is the auxiliary $p$-form, $c$ is the gradient of the PST scalar, and $G$ is a Lagrange multiplier enforcing $c=da$. The essential mechanism is the algebraic shift symmetry $\delta R_{\mu_1\dots\mu_p}=c_{[\mu_1}\lambda_{\mu_2\dots\mu_p]}$; the $(c\cdot R)^2$ term present in (2.1) is what promotes $c\cdot R=0$ from a second-class constraint to a first-class gauge condition, making the elimination of $R$ consistent. Substituting the gauge-fixed solution for $R$ and integrating out $G$ reproduces the PST Lagrangian, while the matrix form (2.18) makes the duality rotations $\phi\leftrightarrow R$ manifest.

What would settle it

A complete Hamiltonian or Dirac analysis of the action (2.8) should find exactly one chiral degree of freedom in $2p+2$ dimensions. If the constraint algebra yields additional propagating modes, or if negative-norm states appear on the locus $(\partial a)^2=0$, the central claim is false.

Watch

Extended reading notes

Core claim

The paper's claim is that a free chiral $p$-form in $2p+2$-dimensional Minkowski space admits a Lagrangian that is simultaneously polynomial, Lorentz covariant, and generally covariant, with a finite number of auxiliary fields. The construction adds an auxiliary $p$-form $R$ and a PST scalar $a$ to the physical $p$-form $\phi$; the resulting action (2.8) carries exactly one chiral degree of freedom and is classically equivalent to the non-polynomial PST action. In four dimensions the same mechanism gives a duality-symmetric formulation of Maxwell theory in which a single photon is described by four vector gauge fields plus a scalar. The paper presents this as a direct counterexample to the folklore that manifest covariance forces non-polynomiality or extra ghosts in chiral-form actions.

Load-bearing premise

The reduction to PST requires solving a differential equation for the Lagrange multiplier and is performed only where $(\partial a)^2\neq 0$; if the polynomial action fails to describe exactly one chiral mode on that singular locus, the classical equivalence, and with it the central claim, collapses.

Editorial extensions

If this is right

  • For each even $p$, the action (2.8) carries a single chiral $p$-form degree of freedom in $2p+2$ dimensions while keeping the action polynomial, Lorentz covariant, and diffeomorphism invariant.
  • The same mechanism yields a duality-symmetric Maxwell action in four dimensions in which one photon is described by four vector gauge fields and one scalar.
  • Unlike field-strength-based covariant actions, this formulation is written in gauge potentials, so it offers a concrete starting point for Yang-Mills-type and Born-Infeld-type non-abelian interactions.
  • Integrating out auxiliary fields recovers the PST action, and the paper shows that further reduction reproduces known non-covariant chiral- and duality-symmetric formulations.
  • The matrix form (2.18) suggests a straightforward $2N\times 2N$ extension to $N$ chiral $p$-forms, with duality rotations exchanging physical and auxiliary fields.

Reading between the lines

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

  • The paper leaves implicit that the same rank-one background-matrix trick may produce a whole family of duality-symmetric theories for larger $M$ and $K$; a concrete test is whether the $p=0$ scalar action (4.12) is the first member of that family.
  • If the singular locus $(\partial a)^2=0$ is only an artifact of the PST reduction and not of the polynomial action itself, the new formulation could support quantization or backgrounds on which PST fails; checking the constraint algebra there would settle this.
  • A minimal non-abelian deformation of the six-dimensional action that preserves the reducible gauge symmetries listed in Section 3 is a direct probe of whether the polynomial form is more than a rewriting of PST.
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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

2 major / 5 minor

Summary. The paper constructs a Lorentz- and generally-covariant, polynomial action for free chiral p-forms in d=2p+2 dimensions. The action (2.1) couples the physical (p+1)-form field strength F to an auxiliary p-form R through a vector c, with a Lagrange multiplier G enforcing ∂[μcν]=0. After algebraically eliminating R, the action reduces to the Pasti-Sorokin-Tonin (PST) action (2.7); after integrating out G, it takes the polynomial form (2.8) with an auxiliary p-form R and a PST scalar a. The construction is illustrated for the chiral two-form in six dimensions and the chiral boson in two dimensions, and a duality-symmetric Maxwell analogue in four dimensions is given using four vector fields and a scalar. The paper also exhibits several gauge symmetries and duality symmetries and argues that the formulation may be useful for non-abelian interactions and gravitational coupling.

Significance. If the equivalence to PST holds, the construction is a useful contribution: it provides a polynomial, manifestly Lorentz- and diffeomorphism-covariant action for free chiral p-forms with finitely many auxiliary fields, avoiding the non-polynomial 1/(∂a)^2 terms of PST. The explicit reductions in d=2 and d=6 and the worked Maxwell example make the construction concrete, and the symmetries in Appendix A are a useful resource. The derivation is self-contained and has no fitted parameters; the coefficients are fixed by gauge invariance. However, the equivalence and the single-degree-of-freedom count are proven only on the open subspace c^2≠0, and the paper does not settle what happens on the null locus, where the polynomial action remains smooth but the PST reduction degenerates. This limits the unqualified claims in the abstract.

major comments (2)
  1. [Section 2.1, Eqs. (2.4)-(2.7)] The proof of equivalence to PST solves for R by dividing by c^2 in Eq. (2.6). When c^2=0, Eq. (2.4) degenerates: it no longer determines R but instead imposes a constraint on F·c, and the auxiliary-field gauge symmetry (2.5) becomes singular. The gauge transformations quoted later, e.g. (4.7), (A.16), and (A.19), contain explicit 1/c^2 factors, so null configurations are not connected to non-null configurations by regular gauge transformations. The paper acknowledges the singularity of the PST form after Eq. (2.12), but it does not analyze the polynomial action (2.1) on the null locus. Therefore the central claim—classical equivalence to PST and the count of a single chiral degree of freedom—is demonstrated only on the open dense subspace c^2≠0, not on the full field space. This is a load-bearing point because the abstract states the equivalence without qualification. The author should either prove a regular extension to the null locus or state the main theorem with this restriction and discuss whether the degree-of-freedom count is affected.
  2. [Section 2.2, Eqs. (2.13)-(2.16)] The on-shell analysis that leads to the self-duality condition P=0 uses the contraction ∂μa P^{μ...}=0 and concludes P=0. This implication is valid only when ∂a is non-null; for null ∂a the contraction does not force P to vanish. Thus the single-chiral-degree-of-freedom argument relies on the same c^2≠0 condition that appears in the PST reduction. The paper should state this condition explicitly in the derivation, not only as a remark about the discontinuity of the PST form after Eq. (2.12).
minor comments (5)
  1. [Section 2.1] After Eq. (2.7), the sentence 'It is a trivial exercise to show the equivalence of the Lagrangian (2.7) with that of [20]' is too terse for a central step. Please write out the explicit PST action with c=∂a and identify the non-polynomial term.
  2. [Eq. (2.14)] The index placement in P_{μ1...μp+1} ≡ F^{μ1...μp+1} + aQ^+_{μ1...μp+1} is inconsistent; the first term has raised indices while the other terms have lowered indices. Please correct the notation.
  3. [Section 4, Eq. (4.9)] The solution for cμ divides by R, so the locus R=0 is not discussed. If this is a formal manipulation, please state the domain or use a gauge where the expression is regular.
  4. [Section 5, Eqs. (5.7)-(5.8)] The 4x4 matrices M and K are presented without derivation. A short explanation of the block structure and the field-redefinition freedom would make the duality-symmetric Maxwell example easier to follow.
  5. [References] Reference [35] is listed as 'Work in progress' with no arXiv number; if the companion paper becomes available, please update the reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new polynomial action is reduced by direct algebraic elimination to the independent PST action, and the central claim does not reduce to its own inputs.

full rationale

The paper's central claim is that the polynomial action (2.1) is classically equivalent to the Pasti-Sorokin-Tonin formulation. The derivation is self-contained and proceeds by explicit computation: the auxiliary field R is eliminated using its algebraic equation of motion (2.4), giving (2.6), and substitution into (2.1) yields the non-polynomial Lagrangian (2.7), which is then identified with the PST action after integrating out the Lagrange multiplier G. This is a direct algebraic reduction to an external, independently established benchmark, not a definition of the target result in terms of the new action. The coefficients of the action are fixed by the requirement of gauge symmetry and are not fitted to any data or to the PST result. The cited PST work is external and not a self-citation; moreover, the equivalence is demonstrated by the paper's own equations rather than imported by citation. The admitted caveat that the PST form is singular when c^2 = 0 is a domain-of-validity restriction, explicitly acknowledged in the paper, and does not make the derivation circular. No fitted parameter is renamed as a prediction, and no load-bearing uniqueness theorem is invoked from the author's prior work. The action is compared with PST as a consistency check and as evidence for the chiral degree-of-freedom count, which is a legitimate external check. Overall, no circular step satisfying the required standard was found.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The construction relies on standard mathematical tools and the known PST formulation. It introduces one new auxiliary field (R) but no physical entities or fitted parameters. The main non-trivial assumption is the requirement that the PST scalar has a non-null gradient for the equivalence to hold.

assumptions (3)
  • standard math Standard differential geometry and exterior calculus are used, including Hodge duality and antisymmetric tensors.
    The paper freely employs these tools without proof; they are standard background.
  • domain assumption The Pasti-Sorokin-Tonin action correctly describes a single free chiral p-form.
    The paper's central claim is equivalence to PST; it relies on PST's known property as the reference formulation. This is an external well-established result, not a postulate of this paper.
  • domain assumption The classical equivalence holds only when (∂a)^2 ≠ 0.
    The reduction from the polynomial action to PST requires inverting c^2 = (∂a)^2; the PST action is singular when this vanishes. The paper acknowledges this in Section 2.1.
invented entities (1)
  • Auxiliary p-form field R
    purpose: Introduced to make the action polynomial while preserving covariance.
    The paper introduces a new auxiliary p-form field R. It has no physical degrees of freedom and is eliminated by its algebraic equation of motion, leading to PST. No independent evidence is provided, but it is not claimed to be physical.

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Pith. "Pith review of On Covariant Actions for Chiral $p-$Forms." pith.science (2026). https://pith.science/paper/FRSY2X5R

@misc{pith2026190801789,
  author       = {Pith},
  title        = {Pith review of: On Covariant Actions for Chiral $p-$Forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FRSY2X5R}},
  note         = {Machine review of arXiv:1908.01789}
}
abstract

We construct a Lorentz and generally covariant, polynomial action for free chiral $p-$forms, classically equivalent to the Pasti-Sorokin-Tonin (PST) formulation. The minimal set up requires introducing an auxiliary $p-$form on top of the physical gauge $p-$form and the PST scalar. The action enjoys multiple duality symmetries, including those that exchange the roles of physical and auxiliary $p-$form fields. Actions of the same type are available for duality-symmetric formulations, which is demonstrated on the example of the electromagnetic field in four dimensions. There, the degrees of freedom of a single Maxwell field are described employing four distinct vector gauge fields and a scalar field.

Discussion (0). Continue with ORCID to comment.

Forward citations

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