A partially massless (PM) field is a free symmetric spin-ss tensor field on a constant-curvature background whose mass squared is fixed to a discrete value in units of the cosmological constant, at which the theory acquires a gauge invariance that removes some — but not all — helicity states of a massive field. Dolan, Nappi and Witten (JHEP 0110:016, 2001) matched such fields in anti-de Sitter space to boundary 'partially conserved' conformal operators of dimension d−1d-1 whose first descendant has negative norm, and posed three open questions: whether a fully nonlinear interacting theory of PM fields exists in which the linearized gauge invariance survives beyond linear order (its linear proof uses the background Einstein equation that the PM field's own stress-energy invalidates); whether the AdS negative-norm problem can be cured by global boundary conditions; and what the physical viability and interpretation of PM fields in de Sitter space are. The second question is settled negatively — PM representations in AdS are intrinsically non-unitary — while the free bosonic PM fields are unitary in de Sitter. No-go results (f-g massive gravity, the conformal-gravity route, Yang-Mills deformations, bimetric theories, de Sitter scalar-matter couplings, PM-gravity coupling in dS4) and loophole constructions (infinite PM towers, extra-field SO(1,5) models, wrong-sign multi-PM theories) delimit which routes are already closed. The precise surviving open core, stated as still unclear in 2025 literature, is whether PM fields — taking the spin-2 field with scalar gauge parameter as the smallest case and generalizing to arbitrary spin and gauge-parameter rank, to all orders in the fields — can participate in any consistent interacting unitary theory, gravity necessarily included, and what the would-be de Sitter boundary theory physically means. No-go results for specific cases — the spin-2 prototype coupled to gravity in four-dimensional de Sitter, and the complete cubic-order classification — settle those cases but do not resolve the general statement.

Background

In four-dimensional Minkowski space a massless particle of spin ss carries only the helicities ±s\pm s, while a massive one carries all helicities −s,…,s-s,\dots,s. In a spacetime of constant curvature the dichotomy is richer. Let the cosmological constant be defined through the background Einstein equation Rμν=ΛgμνR_{\mu\nu}=\Lambda g_{\mu\nu}, with Λ<0\Lambda<0 for anti-de Sitter space (AdS) and Λ>0\Lambda>0 for de Sitter space (dS). At discrete values of the mass squared, namely (in four bulk dimensions, the case treated in the source paper)

m2=Λ3(s(s−1)−n(n+1)),m^2=\frac{\Lambda}{3}\bigl(s(s-1)-n(n+1)\bigr),

a free symmetric spin-ss tensor field on such a background gains a gauge invariance whose parameter is a symmetric rank-nn tensor acted on by s−ns-n covariant derivatives. For n<s−1n<s-1 this removes some but not all of the helicity states of a massive field: the field is called partially massless (PM), a construction due to Deser and Waldron building on earlier work of Higuchi and of Deser and Nepomechie. The prototype, and the case studied in the source paper, is s=2s=2, n=0n=0: a symmetric tensor ϕμν\phi_{\mu\nu} whose mass squared sits at the partially massless value (m2=2Λ/3m^2=2\Lambda/3 in four bulk dimensions, m2=(D−2)Λ/(D−1)m^2=(D-2)\Lambda/(D-1) in general DD) and whose gauge transformation is δϕμν=∇μ∇νξ+ΛD−1gμνξ\delta\phi_{\mu\nu}=\nabla_\mu\nabla_\nu\xi+\frac{\Lambda}{D-1}g_{\mu\nu}\xi with a scalar parameter ξ\xi; the helicity-0 mode of the massive spin-2 field is gauged away, leaving helicities ±2,±1\pm 2,\pm 1. In de Sitter, where Λ=3H2\Lambda=3H^2, this value m2=2H2m^2=2H^2 coincides with the Higuchi bound — the lower edge of the only mass window in which a massive spin-2 field is unitary — and at the bound the helicity-0 norm degenerates and becomes pure gauge. The AdS/CFT correspondence relates a field in (d+1)(d+1)-dimensional AdS to a conformal operator on the dd-dimensional conformal boundary of AdS; a massless spin-ss gauge field corresponds to a conserved boundary current. Dolan, Nappi and Witten (2001) established the analogous statement for PM fields: the AdS spin-2 PM field corresponds to a boundary symmetric traceless tensor LijL_{ij} of conformal dimension d−1d-1 obeying the conformally invariant 'partial conservation law'

DiDjLij+1d−2R~ijLij=0,D_iD_jL^{ij}+\frac{1}{d-2}\tilde R_{ij}L^{ij}=0,

where R~ij\tilde R_{ij} is the Ricci tensor of the boundary conformal metric; this equation is one of the conformally invariant differential operators introduced by Eastwood and Rice. Through the operator–state correspondence of conformal field theory — in which each local operator defines a state in radial quantization, momentum generators PiP_i create descendant states, and a conserved current is a representation with a null descendant Pi∣T⟩=0P_i|T\rangle=0 — the partial conservation law says that a level-2 descendant of the state ∣L⟩|L\rangle is null. Direct norm computations in the conformal algebra then give ∥Pi∣Lij⟩∥2=−⟨Ljs∣Lsj⟩\|P_i|L^{ij}\rangle\|^2=-\langle L^{js}|L_{sj}\rangle: the first descendant of LL has negative norm, so the boundary representation, and with it the AdS PM field, is non-unitary. The authors note that in de Sitter, where the mass formula has the more intuitive sign and where the would-be boundary theory (in the sense of the dS/CFT correspondence proposed in the same period by Strominger, and Witten's de Sitter quantum-gravity notes) is non-unitary anyway, this particular obstruction disappears. The paper explicitly leaves three issues open. First, the gauge invariance of the linearized PM equation was proven using the background Einstein equation; since the PM field itself carries stress-energy, that equation is corrected by terms quadratic in ϕ\phi, and it was unknown whether the field equation, the gauge transformation, and possibly the metric transformation δg∼ϵϕ2\delta g\sim\epsilon\phi^2 can be corrected order by order so that exact gauge invariance survives — no nonlinear interacting theory of PM fields was known. Second, whether the AdS negative-norm problem could be cured by a choice of global boundary conditions was described as a delicate question. Third, the physical viability and interpretation of PM fields in de Sitter space were left open. Later work settled the second question negatively and the free-level part of the third affirmatively (ref 7); the first question and the interacting version of the third remain active research fronts (refs 26, 30).

Problem Statement

Formulation of the question left open by Dolan, Nappi and Witten (JHEP 0110:016, 2001): can partially massless fields be realized as degrees of freedom of a consistent physical theory? The source states three linked sub-questions; the second is now settled and the first and third open, so the surviving core is stated together with the settled part for completeness. (1) Nonlinear gauge-invariant completion (open). Construct, or rule out, a fully nonlinear interacting theory containing partially massless fields — the prototype being the spin-2 field with scalar-parameter gauge invariance δϕμν=∇μ∇νξ+ΛD−1gμνξ\delta\phi_{\mu\nu}=\nabla_\mu\nabla_\nu\xi+\frac{\Lambda}{D-1}g_{\mu\nu}\xi at the partially massless mass value m2=(D−2)Λ/(D−1)m^2=(D-2)\Lambda/(D-1) (that is, 2Λ/32\Lambda/3 at D=4D=4), with arbitrary spin and gauge-parameter rank as generalizations — in which the PM gauge invariance is maintained exactly beyond linear order. Since the PM field carries stress-energy, interactions — including, unavoidably, the coupling to the dynamical metric that its backreaction forces — must be part of the construction: the corrections to the field equation, to the gauge transformation, and possibly to the metric transformation δg∼ϵϕ2\delta g\sim\epsilon\phi^2 must be shown to exist (or to be impossible) simultaneously, order by order in the fields. (2) AdS unitarity (settled negatively by later work, recorded for completeness): whether a positive-norm quantization of PM fields in anti-de Sitter space can be achieved by a suitable choice of global boundary conditions. (3) De Sitter viability and interpretation (interacting core open): determine whether PM fields can participate in consistent local, unitary interacting theories in de Sitter space — in particular coupled to gravity, where the free theory is unitary — and what the corresponding boundary structure in the sense of dS/CFT physically means. The common scientific objective of these sub-questions is whether the linearized PM gauge principle survives as a genuine physical gauge principle. The open core is the conjunction of (1) and the interacting part of (3): does there exist a consistent interacting theory of PM fields with the PM gauge symmetry intact beyond linear order and with an acceptable unitarity and interpretation status — or is there a general no-go? Constructions that keep the PM symmetry only by changing the field content (infinite PM towers, extra vector fields, wrong-sign or conformal-gravity-like kinetic terms) or by leaving the dynamical metric out must explicitly confront the unitarity, backreaction, and viability requirements above, since exactly these loopholes are where the question now lives. The question covers the general PM spectrum and exact (all-orders) gauge invariance, not a single fixed background, a single spin, or a fixed interaction order.

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