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On Lorentz Invariant Actions for Chiral P-Forms

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Paolo Pasti, Dmitri Sorokin, Mario Tonin

High Energy Physics - Theory · Submitted 13 Nov 1996 · 27 references

Pith's verdict: The new content is the truncation map and the Dirac analysis of the single-scalar chiral p-form action; both hold up, and the anomaly claim is properly conjectural.

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arXiv IDhep-th/9611100 (version 1)
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CitationDmitri Sorokin, Mario Tonin, Paolo Pasti. "On Lorentz Invariant Actions for Chiral P-Forms." arXiv:hep-th/9611100 (1996). https://pith.science/paper/IIHJHXSJ
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The paper

The paper claims

Two Lorentz-covariant actions for chiral p-forms are shown to describe the same theory: one an infinite tower of auxiliary forms, the other a single nonpolynomial auxiliary scalar.

The reading

A chiral $p$-form is an antisymmetric field whose $(p+1)$-form field strength equals its own Hodge dual. Because self-duality is a first-order condition, the theory is already on shell before any action is written, and building a manifestly Lorentz-invariant action for it has been an open problem since the 1980s. Two covariant strategies existed: an infinite tower of auxiliary (anti-)self-dual forms, and a finite action in which one auxiliary scalar $a(x)$ enters nonpolynomially through $u_m = partial_m a$. The paper's central claim is that the second is a consistent truncation of the first: the scalar term can be pushed arbitrarily far down the tower and the infinite action recovered in the limit. The authors then show that the one-scalar model has an unusually clean Hamiltonian structure, and that in the two-dimensional case its constraint can be twisted so that the would-be quantum central charge vanishes.

What carries the argument

The load-bearing object is the auxiliary scalar $a(x)$ entering through its gradient $u_m = partial_m a$ in the term $frac{1}{2(u^q u_q)} u^m F_{mnl} F^{nlr} u_r$, added to the ordinary Maxwell-like kinetic term $-frac{1}{6}F_{lmn}F^{lmn}$ for the anti-self-dual part of the field strength. The norm $u^2$ in the denominator is what makes the action nonpolynomial and what makes the gauge $u_m = 0$ inadmissible; the paper requires $u^m u_m$ nonzero, in analogy with requiring an invertible metric in gravity. The second piece of machinery is the tower-rewriting identity: each auxiliary form converts a derivative of $a$ into an independent field, so that the scalar can be moved to any depth in an alternating chain of auxiliary forms without changing the classical content. The third is the twisting improvement $lambda partial_1(P_a a)$, borrowed from the ghost-twisting used in conformal field theory, which shifts the central charge while preserving the first-class property.

Extended reading notes

Core claim

The paper establishes a bridge between two formulations of chiral $p$-forms in $D = 2(p+1)$. Starting from the nonpolynomial action with one auxiliary scalar, the authors introduce auxiliary three-form fields step by step, each new field replacing a derivative of $a(x)$ by an independent field constrained to equal it, and each step adding a term of the form $(-1)^n hat{Lambda}_{(n+1)} hat{Lambda}_{(n+2)}$. After $N$ such steps the scalar sits at the far end of a chain of $2N+2$ auxiliary forms; sending $N$ to infinity and splitting the fields into self-dual and anti-self-dual parts reproduces the infinite-tower action of McClain, Wu and Yu. Read backwards, this is a prescription for truncating the infinite series without destroying its physical content: replace the sum of the last retained self-dual and anti-self-dual forms by a term $partial_p a, hat{Lambda}^{(0)p[lm} partial_{n]} a$. The authors then perform the Dirac analysis of the one-scalar action. In $d=2$ the primary constraint is $C = frac{1}{4}(P_varphi - varphi')^2 - P_a a' = 0$, which is first class and obeys a Virasoro-like algebra with itself; in $D=6$ the analogous constraints are $Pi^{alpha eta}partial_eta a = 0$ and $frac{1}{8}arepsilon^{alpha eta gamma delta ho}Pi_{alpha eta}Pi_{gamma delta}partial_ho a + P_a (partial a)^2 = 0$, again first class. In neither case is the constraint the square of a second-class constraint, which is what distinguishes this construction from Siegel's. The canonical Hamiltonian is quadratic and contains only the physical chiral mode. Finally, in $d=2$ the constraint admits the improvement $C(lambda) = frac{1}{4}(P_varphi - varphi')^2 - (1-lambda)P_a a' + lambda P_a' a$, still first class, whose quantum central charge is $c = 1 + 2(6lambda^2 - 6lambda + 1)$ and vanishes at $lambda = frac{1}{2}$.

Load-bearing premise

The equivalence rests on the assumption, stated in Section II.A, that only solutions of the infinite-tower equations containing a finite number of nonzero auxiliary fields are admissible, a restriction the authors themselves call somewhat artificial, together with the claim that the truncation preserves the physical content at least at the classical level.

Editorial extensions

If this is right

  • The one-scalar action can be regarded as the minimal covariant description of a chiral $p$-form, replacing an infinite auxiliary sector with a single field and a finite number of local symmetries.
  • Because the constraints are all first class and the Hamiltonian is quadratic, the classical energy of the model is that of exactly one chiral $p$-form, with no propagating anti-chiral partner.
  • In $d=2$ the twisted constraint at $lambda = 1/2$ has vanishing central charge, so an operator ordering may exist in which the local symmetry is not anomalous.
  • The action couples to gravity in a natural covariant way, opening a route to studying gravitational anomalies caused by chiral forms in this formulation.
  • The construction admits supersymmetrization, and the paper suggests truncating an existing superfield formulation with infinitely many fields down to a superfield version of the one-scalar action.

Reading between the lines

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

  • If the truncation really commutes with quantization, the partition function of the one-scalar model should equal that of the infinite-tower model after gauge fixing, which would give a finite-dimensional check on the regularization prescriptions previously needed for the infinite series.
  • The twisting freedom suggests that the anomaly question for chiral forms is not a property of the field content but of how the constraint is presented, which would parallel the way ghost twisting is used to move between equivalent conformal field theories.
  • The requirement that $u^m u_m$ be nonzero means the auxiliary scalar defines a preferred direction everywhere; this looks like a Stueckelberg or Goldstone structure, and one might expect the model to admit a reformulation as a gauge-fixed version of a theory with a local shift symmetry for $a$.
  • The same tower-truncation identity should apply to duality-symmetric Maxwell theory in $D=4$, where the auxiliary scalar has been interpreted as an axion, giving a concrete link between the axion interpretation and the infinite-field formulation.
Original abstract

We demonstrate how a Lorentz covariant formulation of the chiral p-form model in D=2(p+1) containing infinitely many auxiliary fields is related to a Lorentz covariant formulation with only one auxiliary scalar field entering a chiral p-form action in a nonpolynomial way. The latter can be regarded as a consistent Lorentz-covariant truncation of the former. We make the Hamiltonian analysis of the model based on the nonpolynomial action and show that the Dirac constraints have a simple form and are all of the first class. In contrast to the Siegel model the constraints are not the square of second-class constraints. The canonical Hamiltonian is quadratic and determines energy of a single chiral p-form. In the case of d=2 chiral scalars the constraint can be improved by use of `twisting' procedure (without the loss of the property to be of the first class) in such a way that the central charge of the quantum constraint algebra is zero. This points to possible absence of anomaly in an appropriate quantum version of the model.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,620 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.45 · compares with prior results 0.91 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We demonstrate how a Lorentz covariant formulation of the chiral p-form model in D=2(p+1) containing infinitely many auxiliary fields is related to a Lorentz covariant formulation with only one auxiliary scalar field entering a chiral p-form action in a nonpolynomial way. The latter can be regarded as a consistent Lorentz-covariant truncation of the former. We make the Hamiltonian analysis of the model based on the nonpolynomial action and show that the Dirac constraints have a simple form and are all of the first class. In contrast to the Siegel model the constraints are not the square of second-class constraints. The canonical Hamiltonian is quadratic and determines energy of a single chiral p-form. In the case of d=2 chiral scalars the constraint can be improved by use of `twisting' procedure (without the loss of the property to be of the first class) in such a way that the central charge of the quantum constraint algebra is zero. This points to possible absence of anomaly in an appropriate quantum version of the model.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We demonstrate how a Lorentz covariant formulation of the chiral p-form model in D=2(p+1) containing infinitely many auxiliary fields is related to a Lorentz covariant formulation with only one auxiliary scalar field entering a chiral p-form action in a nonpolynomial way. contribution 0.77 0.93 0.52 asserted 1.00
s1 The latter can be regarded as a consistent Lorentz-covariant truncation of the former. interpretation 0.69 0.70 0.19 asserted 0.77
s2 We make the Hamiltonian analysis of the model based on the nonpolynomial action and show that the Dirac constraints have a simple form and are all of the first class. method 0.62 0.96 0.13 asserted 1.00
s3 In contrast to the Siegel model the constraints are not the square of second-class constraints. result 0.99 0.79 0.09 asserted 1.00
s4 The canonical Hamiltonian is quadratic and determines energy of a single chiral p-form. result 1.00 0.88 0.12 asserted 1.00
s5 In the case of d=2 chiral scalars the constraint can be improved by use of `twisting' procedure (without the loss of the property to be of the first class) in such a way that the central charge of the quantum constraint algebra is zero. result 0.78 0.81 0.10 asserted 0.90
s6 This points to possible absence of anomaly in an appropriate quantum version of the model. interpretation 1.00 0.47 0.21 hedged possibility 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References
How the literature uses it

93 arXiv papers cite this in 144 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 82 sentences without a confident label

Builds on it 2 papers
  • In the next section we will construct an action that reproduces the first-order (self-)duality equations ([ref]), ([ref]), ([ref]) by extending the construction of [ref] to the non-Abelian case.

    Duality-symmetric actions for non-Abelian tensor fields · Bosonic part of the $(1,0)$ superconformal field equations · 0.99

  • The main constructions of this paper can be extended to the conformal chiral two-form field theory in six dimensions [ref] within the approach of [ref] , which combined the virtues of the six-dimensional Pasti-Sorokin-Tonin formulation [ref] with the four-dimensional formalism developed by Ivanov, Nurmagambetov and Zupnik [ref] .

    Higher-derivative deformations of the ModMax theory · Concluding comments · 0.86

Uses its method or tool 9 papers
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