Pith. sign in

REVIEW 2 major objections 4 minor 3 cited by

A Partially Massless Superconductor

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read The paper claims that the partially massless graviton on de Sitter space can be Higgsed: coupling it to a fractonic scalar with a dipolar shift symmetry and condensing that scalar gives a fully massive, unitary spin-2 field with mass…

desk verdict First covariant PM Higgs mechanism, with a plausible and explicit core construction, but the no-ghost tuning in the nonstandard log-kinetic EFT is asserted more than proven. read the letter →

arxiv 2507.15932 v1 pith:EJ7BOE5Y submitted 2025-07-21 hep-th cond-mat.supr-congr-qc

Kurt Hinterbichler , Austin Joyce This is my paper
classification hep-thcond-mat.supr-congr-qc
keywords partiallymasslessgravitonHiggsmechanismdeSitterspacefractonsgalileonhigher-formsymmetriesBFtheoryedgemodes
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

The paper aims to show that the partially massless (PM) graviton—a spin-2 gauge field on de Sitter space that sits between massless and massive gravity—can undergo a Higgs transition and become fully massive. To achieve this, the authors construct a covariant relativistic fracton theory whose global dipolar shift symmetry, a phase rotation whose parameter is linear in the spacetime coordinates, is exactly the reducibility of the PM gauge invariance, and they couple this matter to the PM field. When the fractonic matter condenses, the dipole symmetry breaks spontaneously, the Goldstone mode is a galileon, and it combines with the PM graviton to produce a Fierz–Pauli massive spin-2 field with mass squared $(D-2)H^2+\Delta m_h^2$, safely above the unitary bound. At long distances the Higgs phase is captured by a quasi-topological BF-like action that supports gapless edge modes on the future boundary of de Sitter and persistent dipole currents, just as a superconductor supports edge modes and persistent currents. This is the first covariant Higgsing of a relativistic spin-2 gauge field, and it offers a proof-of-principle for what phase transitions in gravitational theories could look like.

What carries the argument

The load-bearing object is a complex scalar $\Phi$ with a linear dipolar shift symmetry $\Phi \mapsto e^{i\alpha(x)}\Phi$, where $\alpha(x)$ obeys $(\nabla_\mu\nabla_\nu+H^2g_{\mu\nu})\alpha(x)=0$—the reducibility condition of the PM gauge symmetry, which also defines the dS galileon shift. To make this symmetry manifest, the paper introduces the covariant derivative $\nabla^2_{\mu\nu}\Phi^2 \equiv \Phi^2(\nabla_\mu\nabla_\nu+H^2g_{\mu\nu})\log(\Phi/f)$, which transforms covariantly by a phase when $\alpha$ satisfies that condition. Coupling to the PM graviton replaces $\nabla^2$ with $D^2$, obtained by adding $-iqM^{(6-D)/2}h_{\mu\nu}\Phi^2$, thereby turning the global symmetry into a gauge symmetry. This covariant derivative, together with the symmetry-breaking potential and the Fierz–Pauli-tuned kinetic terms, produces the quadratic mixing between the Goldstone mode and $h_{\mu\nu}$ that, after fixing $\phi=0$, yields the massive spin-2 with $\Delta m_h^2>0$. In the deep infrared the same symmetries reorganize into a quasi-topological BF action whose gauge structure, boundary anomaly, and induced currents carry the superconductor-like phenomenology.

What would settle it

Take the full action (3.14), expand around $\Phi=v$ without using the hierarchy $H^2/M^2 \ll \lambda \ll 1$, and diagonalize the quadratic sector exactly; if any kinetic eigenvalue changes sign, or if the $\Delta m_h^2$ correction turns negative, the claimed unitary massive phase fails. This finite calculation settles the central claim.

Watch

Extended reading notes

Core claim

The central claim is that the partially massless graviton can be Higgsed: gauging the dipolar shift symmetry of a fracton-like complex scalar on de Sitter, and then condensing that scalar, turns the PM spin-2 gauge field into a fully massive spin-2 field. In the analogue of unitary gauge the quadratic action reduces to a Fierz–Pauli massive graviton with $m_h^2 = (D-2)H^2 + \Delta m_h^2$, where $\Delta m_h^2 = 2q^2\mu^D/(M^{D-2}\lambda^2) > 0$; the positive shift pushes the graviton above the unitary (Higuchi) bound, so the massive phase is consistent at quadratic order. The Goldstone mode of the broken dipole symmetry is a dS galileon with mass $m_\phi^2 = -DH^2$, which serves as the Stückelberg field for the PM gauge symmetry, while the radial mode $\sigma$ has positive mass $m_\sigma^2 = \lambda M^2/4$. After integrating out all massive degrees of freedom, the infrared is a quasi-topological BF-like theory with no local propagating bulk modes; on the future boundary it has a gapless edge mode described by a $\phi \Box^2 \phi$ action, and background PM electric fields induce persistent dipole currents $\langle J_{\alpha\beta}\rangle = -(p/q)E_{\alpha\beta}$. The same construction, with higher-derivative fractonic matter built from the higher-spin reducibility operator $D^s_{\mu_1\cdots\mu_s}$, extends to maximal-depth partially massless fields of every spin.

Load-bearing premise

The argument depends on the fractonic matter action (2.6) being a well-behaved effective field theory that can be expanded around the symmetry-breaking vacuum even though its logarithmic derivative is singular at $\Phi=0$, with no hidden ghosts and with the hierarchy $H^2/M^2 \ll \lambda \ll 1$ keeping the expansion controlled.

Editorial extensions

If this is right

  • The partially massless graviton acquires a Fierz–Pauli mass $m_h^2=(D-2)H^2+\Delta m_h^2$ with $\Delta m_h^2>0$, so the Higgs phase is a fully massive spin-2 theory above the unitary bound, free of ghosts and tachyons at quadratic order.
  • The broken dipole symmetry produces a galileon Goldstone mode, so the PM gauge symmetry is realized à la Stückelberg and can be fixed to $\phi=0$, exactly as in unitary gauge.
  • At long distances all massive degrees of freedom integrate out to a quasi-topological BF-like action with no local propagating bulk modes; the remaining physics lives on the future boundary as a gapless edge mode.
  • The edge mode cancels a mixed 't Hooft anomaly between the fractonic U(1) and the magnetic PM symmetry, and background fields induce persistent dipole currents, analogous to the superconducting phase of electromagnetism.
  • The construction extends to all maximal-depth partially massless higher spins, each Higgsed by a higher-multipole fracton theory and described at long distance by the same kind of BF theory with edge modes.

Reading between the lines

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

  • Beyond the paper, the same Landau–Ginzburg logic suggests that a Higgs phase for the massless graviton would require matter with a global diffeomorphism-like (vector fracton) symmetry; constructing such covariant matter is the natural next step but is harder because dS Killing vectors control the reducibility.
  • If the construction is right, the gapless edge modes localized at the future boundary of an inflationary universe could leave observable imprints in cosmological correlators even when the massive spin-2 field is too heavy to be produced directly; this is a testable extension the paper raises only as an open question.
  • One could test the superfluid analogy by deriving the finite-temperature or hydrodynamic correlation functions of the persistent dipole current and comparing them with known fracton superfluid hydrodynamics; the paper does not compute these.
  • The preserved magnetic symmetry should support magnetically charged vortex solutions in the Higgs phase; constructing them, or showing they cannot exist, would distinguish this PM superconductor from an ordinary superconductor and is not done in the paper.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper constructs a de Sitter-invariant effective field theory of a complex scalar field with a linearly realized dipolar shift symmetry, couples it to the partially massless graviton by covariantizing the shift symmetry, and studies the spontaneously broken phase. Expanding around a constant vacuum, the angular fluctuation becomes a dS galileon Goldstone mode and the radial mode acquires a positive mass. In unitary gauge the partially massless graviton combines with the Goldstone mode to become a fully massive spin-2 field with Fierz-Pauli mass m_h^2 = (D-2)H^2 + Delta m_h^2 (Eq. (3.20)), with Delta m_h^2 > 0 given in Eq. (3.17). The authors then integrate out the massive fields to obtain a BF-type quasi-topological action, from which they derive a gapless edge mode on the future boundary and persistent dipole currents, and they sketch the generalization to maximal-depth partially massless higher spins.

Significance. Assuming the EFT expansion is legitimate, this is a novel and conceptually interesting result: it provides the first explicit covariant Higgs mechanism for a non-vector gauge field, with transparent formulas for the mass shift, the strong-coupling scale, and Type I/II regimes, and it connects the PM graviton to fracton and galileon physics in a concrete way. The paper is also commendably candid about the non-standard nature of the matter action: it explicitly states that the theory is not defined at Phi = 0, that lambda << 1 is a fine tuning, and in the conclusions that a Fock-space description is obscure. These admissions do not by themselves undermine the calculation, but they sharpen the main correctness question, which is whether the quadratic expansion around the symmetry-breaking vacuum is genuinely ghost-free. The central claim therefore requires an explicit verification of the no-ghost property before the mass formula (3.20) can be regarded as established.

major comments (2)
  1. [Secs. 2.2 and 3.3 (Eqs. (2.13)-(2.17) and (3.16)-(3.20))] The central result (3.20) assumes that the quadratic fluctuations around the symmetry-breaking vacuum (2.10) are ghost-free and that the kinetic normalizations Z_sigma^2 and Z_phi^2 in (2.14)-(2.15) are positive. The paper asserts that the Fierz-Pauli-like tuning in (2.6) makes the equations of motion second order and removes ghosts, but it does not display the full quadratic action before dropping terms suppressed by H^2/(M^2 lambda) and lambda, nor does it compute the principal symbol of the coupled sigma-phi-h_mu_nu system. Because the action has no standard kinetic term and the covariant derivative (2.3) contains log(Phi/f), positivity of the kinetic matrix is not protected by a canonical Kahler metric; it must be verified explicitly under the hierarchy (2.34). I request an appendix containing the complete quadratic expansion of (3.14) around v, the resulting kinetic matrix, and a demonstration of positive definiteness (or a precise statement of the additional conditions required). Without this, Eq. (3.20) is not established.
  2. [Sec. 3.5 (Eqs. (3.29)-(3.31), (3.34), (3.39))] The BF action (3.31), the gapless edge mode (3.34), and the persistent current (3.39) all rely on an additional matter U(1) symmetry with charge p and background field B_mu_nu that is not present in the microscopic action (3.14). The text introduces this structure with the phrase "we could consider e.g. introducing additional fields charged under this symmetry," which makes the advertised long-distance phenomenology conditional on an extra assumption rather than a derived consequence of the fractonic Higgs mechanism. Please either construct this U(1) within the Phi matter sector, show that it emerges in the symmetry-broken phase, or clearly state in the abstract and conclusions that the quasi-topological description and its edge-mode phenomenology apply to an extended theory with an additional U(1).
minor comments (4)
  1. [Sec. 2.1, Eq. (2.3)] The notation nabla^2_mu_nu Phi^2 is confusing: it is not the square of a first-order covariant derivative but a second-order differential operator acting on log(Phi/f) multiplied by Phi^2. Please define an operator symbol (for example D_mu_nu) and state its index symmetries, so that the Fierz-Pauli-like tuning in Eq. (2.6) is unambiguous.
  2. [Sec. 2.2, Eqs. (2.24)-(2.26)] The interaction action (2.24) is written with ellipses that hide terms with additional derivatives. Since the strong-coupling scale (2.29) is extracted from the leading interaction, please state explicitly how the omitted terms scale at the scale Lambda_s and why they are subleading, rather than leaving this to the schematic power counting (2.26).
  3. [Sec. 3.5, Eq. (3.33)] The boundary variation (3.33) is evaluated after imposing that the non-tangential components of the gauge fields and gauge parameters vanish on the boundary; please specify how these boundary conditions are imposed consistently for both gauge fields and large gauge transformations, since the edge mode discussion depends on this choice.
  4. [Sec. 4] The higher-spin generalization is explicitly a sketch: the tuned quadratic contractions in the matter kinetic terms are not written, and the BF action (4.7) is stated without derivation. If this section is intended as a result, it needs a derivation or a reference to one; otherwise it should be labeled as an outlook, and the abstract's mention of the generalization should reflect that status.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the mass shift, Goldstone action, BF description, and edge modes are derived from the stated EFT action with free parameters, not from fitting or self-citation.

full rationale

Walking the derivation chain: the matter action (2.6) is constructed so that its global symmetry matches the reducibility of the PM gauge transformation; this is deliberate gauging, not circular reasoning. The quadratic fluctuation action (2.13) is an honest expansion around the vacuum solution (2.10), with Z_sigma^2, Z_phi^2, m_sigma^2, and m_phi^2 computed from the action, and no quantity is fitted to data. The Higgs-phase action (3.16) is obtained by expanding the gauged action (3.14); Delta m_h^2 and kappa_{h phi} are explicit functions of the free parameters q, mu, M, and lambda and are positive in the stated regime, so (3.20) is a derived consequence rather than an input renamed as a prediction. The BF/edge-mode analysis is self-contained: the boundary action (3.37) is verified to cancel the bulk anomaly by direct comparison of (3.33) and (3.38), and the persistent-current formula (3.40) follows from differentiating the effective action. Self-citations, e.g. [2,5,48,49], are used for background representation theory and for a construction method whose output is explicitly checked in this paper; none is invoked as a uniqueness theorem to forbid alternatives. The main risk, namely whether the log-kinetic EFT around Phi = v is ghost-free beyond leading order, is an internal-consistency or correctness question, not circularity, because the paper's equations do not define the target mass formula in terms of itself. No circular step can be exhibited with a specific reduction, so the appropriate finding is no significant circularity.

Assumptions & free parameters 8 free parameters · 7 assumptions · 2 invented entities

The central claim rests on several tunings: small lambda, Fierz-Pauli-tuned derivative terms, zero imaginary part of a coefficient, and an ad hoc additional U(1) for the topological phase. No free parameter is fitted to data, but the construction is not derived from a more fundamental theory.

free parameters (8)
  • M
    Mass scale suppressing derivative terms in (2.6); it enters the graviton mass shift Delta m_h^2 in (3.17).
  • mu
    Mass scale in the potential term; it sets the condensate v = mu^{D/4}/sqrt(lambda) and the graviton mass shift.
  • lambda = lambda << 1, with H^2/M^2 << lambda
    Quartic coupling controlling the radial mode mass m_sigma^2 = lambda M^2/4 and the strong coupling scale (2.29); the smallness is an acknowledged tuning (2.11).
  • q = q^2 << lambda^2 (mu/M)^D
    Dimensionless gauge coupling to the PM field; it controls Delta m_h^2 and the type I/type II ratio in (3.23).
  • f = set to mu^{D/4}/sqrt(lambda)
    Scale inside the logarithm in (2.3); the paper uses field redefinitions to fix it to this value in (2.9).
  • p
    Charge of phi under an additional U(1) introduced in (3.29); it sets the BF coefficient p/q and is not fixed by the microscopic model.
  • Fierz-Pauli tuning of four-derivative terms = chosen to make EOMs second order
    The relative coefficient of the four-derivative terms in (2.6) is tuned to avoid extra ghostly modes; the paper states this tuning explicitly.
  • Imaginary part of two-derivative coefficient = tuned to zero
    The paper notes that an imaginary part would destroy the symmetry-breaking solution (2.8).
assumptions (7)
  • domain assumption The dS representation theory mass value m^2 = (D-2)H^2 gives a unitary PM spin-2 field with gauge invariance (3.3).
    Invoked in Sec. 3.1 and based on refs. [12-23]; not re-derived in this paper.
  • domain assumption The solutions of (nabla_mu nabla_nu + H^2 g_mu_nu) alpha = 0 are the reducibility parameters and also the galileon shift symmetries on dS.
    Used in Sec. 2.1 to identify the matter symmetry; relies on refs. [43-49].
  • ad hoc to paper Perturbative expansion around the symmetry-breaking vacuum v is valid, despite the action being ill-defined at Phi = 0.
    The paper notes the absence of a standard kinetic term and the singularity at Phi = 0, but proceeds with the expansion around (2.10).
  • ad hoc to paper The Fierz-Pauli-like tuning in (2.6) ensures second-order EOMs and no extra ghosts.
    Stated in Sec. 2.2; no explicit proof of absence of ghosts beyond the tuned structure is given.
  • domain assumption The boundary theory with box^2 kinetic term (3.34) provides a valid gapless edge mode.
    The paper relies on refs. [5,113]; the unitarity of this higher-derivative boundary scalar is not discussed.
  • domain assumption Flat-space no-go results for galileon UV completion are evaded on dS.
    Sec. 2.2 and Sec. 3.4 argue that the dS mass and strongly coupled flat limit avoid refs. [74-77]; this is an argument, not a theorem.
  • ad hoc to paper The additional U(1) symmetry and charge p in Sec. 3.5 can be added consistently to the matter sector.
    Introduced so that the BF action (3.29) can be written; it is necessary for the edge-mode and persistent-current results.
invented entities (2)
  • Fractonic complex scalar Phi with mass dimension D/4 and a logarithmic covariant derivative
    purpose: Realizes the linearly realized dipole shift symmetry and provides the condensate that Higgses the PM graviton.
    Postulated in Sec. 2.1; it is a legitimate EFT ingredient but has no observational handle or independent evidence.
  • Additional matter U(1) symmetry with background gauge field B_mu_nu and charge p
    purpose: Used in Sec. 3.5 to derive the BF theory, the anomalous boundary variation, and persistent currents.
    Auxiliary structure introduced ad hoc; the original model does not fix the charge p or the existence of this sector.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Partially Massless Superconductor." pith.science (2026). https://pith.science/paper/EJ7BOE5Y

@misc{pith2026250715932,
  author       = {Pith},
  title        = {Pith review of: A Partially Massless Superconductor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EJ7BOE5Y}},
  note         = {Machine review of arXiv:2507.15932}
}
read the original abstract

We describe a Higgs mechanism for the partially massless graviton. In order to do so, we first construct a covariant fracton-like effective field theory on de Sitter space that linearly realizes a dipolar shift symmetry. The global symmetry of this theory can be gauged by coupling it to a partially massless spin-2 gauge field. When the fractonic matter condenses, the dipole symmetry is spontaneously broken, the resulting Goldstone mode is a galileon and the partially massless graviton combines with this mode and becomes fully massive. At long distances, the phenomenology of this theory is captured by a quasi-topological field theory and displays many features analogous to those of the superconducting phase in electromagnetism, including gapless edge modes and persistent currents. We also describe the generalization to higher spins.

Discussion (0). Sign in to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Thermodynamics of homogeneous Universes: de Sitter, Bonnor-Melvin and static Einstein

    gr-qc 2026-05 unverdicted novelty 5.0 of 10

    Three homogeneous universes with distinct matter content obey the same thermodynamic equation for energy density, implying vanishing cosmological constant in vacuum.

  2. Emergent fracton strings from covariant bi-form gauge field theory

    hep-th 2026-03 unverdicted novelty 5.0 of 10

    A rank-4 tensor gauge theory yields emergent fracton strings with a new generalised dipole conservation law for closed strings and reduces to linearised area-metric gravity in a suitable limit.

  3. Thermodynamics of homogeneous Universes: de Sitter, Bonnor-Melvin and static Einstein

    gr-qc 2026-05 unverdicted novelty 4.0 of 10

    De Sitter, Bonnor-Melvin-Λ and static Einstein universes share the same thermodynamic energy-density equation despite dissimilar matter fields, yielding zero cosmological constant in Minkowski vacuum.

Reference graph

Works this paper leans on

134 extracted references · 24 canonical work pages · cited by 2 Pith papers

  1. [1]

    Generalized symmetries of the graviton,

    V. Benedetti, H. Casini, and J. M. Magan, “Generalized symmetries of the graviton,” JHEP 05 (2022) 045, arXiv:2111.12089 [hep-th]

    arXiv 2022

  2. [2]

    Gravity as a gapless phase and biform symmetries,

    K. Hinterbichler, D. M. Hofman, A. Joyce, and G. Mathys, “Gravity as a gapless phase and biform symmetries,” JHEP 02 (2023) 151, arXiv:2205.12272 [hep-th]

    arXiv 2023

  3. [3]

    Generalized Symmetries for Generalized Gravitons,

    V. Benedetti, P. Bueno, and J. M. Magan, “Generalized Symmetries for Generalized Gravitons,” Phys. Rev. Lett. 131 no. 11, (2023) 111603, arXiv:2305.13361 [hep-th]

    arXiv 2023

  4. [4]

    Covariant generalized conserved charges of General Relativity,

    C. G´ omez-Fayr´ en, P. Meessen, and T. Ort ´ ın, “Covariant generalized conserved charges of General Relativity,” JHEP 09 (2023) 174, arXiv:2307.04041 [gr-qc]

    arXiv 2023

  5. [5]

    Impossible symmetries and conformal gravity,

    K. Hinterbichler, A. Joyce, and G. Mathys, “Impossible symmetries and conformal gravity,” Phys. Rev. D 110 no. 8, (2024) 085003, arXiv:2403.03256 [hep-th]. 24

    arXiv 2024

  6. [6]

    Charges and topology in linearised gravity,

    C. Hull, M. L. Hutt, and U. Lindstr¨ om, “Charges and topology in linearised gravity,” JHEP 07 (2024) 097, arXiv:2401.17361 [hep-th]

    arXiv 2024

  7. [7]

    Generalised symmetries in linear gravity,

    C. Hull, M. L. Hutt, and U. Lindstr¨ om, “Generalised symmetries in linear gravity,” JHEP 04 (2025) 046, arXiv:2409.00178 [hep-th]

    arXiv 2025

  8. [8]

    Gauging generalised symmetries in linear gravity,

    C. Hull, M. L. Hutt, and U. Lindstr¨ om, “Gauging generalised symmetries in linear gravity,” JHEP 01 (2025) 145, arXiv:2410.08720 [hep-th]

    arXiv 2025

Show all 134 references
  1. [9]

    Scale versus conformal invariance at the IR fixed point of quantum gravity,

    K. Farnsworth, K. Hinterbichler, and O. Hulik, “Scale versus conformal invariance at the IR fixed point of quantum gravity,” Phys. Rev. D 105 no. 6, (2022) 066026, arXiv:2110.10160 [hep-th]

    2022 arXiv

  2. [10]

    Generalized symmetry in dynamical gravity,

    C. Cheung, M. Derda, J.-H. Kim, V. Nevoa, I. Rothstein, and N. Shah, “Generalized symmetry in dynamical gravity,” JHEP 10 (2024) 007, arXiv:2403.01837 [hep-th]

    2024 arXiv

  3. [11]

    Topological Gravity as the Early Phase of Our Universe,

    P. Agrawal, S. Gukov, G. Obied, and C. Vafa, “Topological Gravity as the Early Phase of Our Universe,” arXiv:2009.10077 [hep-th]

    2009 arXiv

  4. [12]

    Anomalous Propagation of Gauge Fields in Conformally Flat Spaces,

    S. Deser and R. I. Nepomechie, “Anomalous Propagation of Gauge Fields in Conformally Flat Spaces,” Phys. Lett. B 132 (1983) 321–324

    1983

  5. [13]

    Gauge Invariance Versus Masslessness in De Sitter Space,

    S. Deser and R. I. Nepomechie, “Gauge Invariance Versus Masslessness in De Sitter Space,” Annals Phys. 154 (1984) 396

    1984

  6. [14]

    Forbidden Mass Range for Spin-2 Field Theory in De Sitter Space-time,

    A. Higuchi, “Forbidden Mass Range for Spin-2 Field Theory in De Sitter Space-time,” Nucl. Phys. B 282 (1987) 397–436

    1987

  7. [15]

    How massless are massless fields in AdS(d),

    L. Brink, R. R. Metsaev, and M. A. Vasiliev, “How massless are massless fields in AdS(d),” Nucl. Phys. B 586 (2000) 183–205, arXiv:hep-th/0005136

    2000 arXiv

  8. [16]

    Gauge invariances and phases of massive higher spins in (A)dS,

    S. Deser and A. Waldron, “Gauge invariances and phases of massive higher spins in (A)dS,” Phys. Rev. Lett. 87 (2001) 031601, arXiv:hep-th/0102166

    2001 arXiv

  9. [17]

    Partial masslessness of higher spins in (A)dS,

    S. Deser and A. Waldron, “Partial masslessness of higher spins in (A)dS,” Nucl. Phys. B 607 (2001) 577–604, arXiv:hep-th/0103198

    2001 arXiv

  10. [18]

    Stability of massive cosmological gravitons,

    S. Deser and A. Waldron, “Stability of massive cosmological gravitons,” Phys. Lett. B 508 (2001) 347–353, arXiv:hep-th/0103255

    2001 arXiv

  11. [19]

    Null propagation of partially massless higher spins in (A)dS and cosmological constant speculations,

    S. Deser and A. Waldron, “Null propagation of partially massless higher spins in (A)dS and cosmological constant speculations,” Phys. Lett. B 513 (2001) 137–141, arXiv:hep-th/0105181

    2001 arXiv

  12. [20]

    On massive high spin particles in AdS,

    Y. M. Zinoviev, “On massive high spin particles in AdS,” arXiv:hep-th/0108192

    arXiv

  13. [21]

    What is mass in de Sitterian physics?,

    T. Garidi, “What is mass in de Sitterian physics?,” arXiv:hep-th/0309104

    arXiv

  14. [22]

    Geometric formulation for partially massless fields,

    E. D. Skvortsov and M. A. Vasiliev, “Geometric formulation for partially massless fields,” Nucl. Phys. B 756 (2006) 117–147, arXiv:hep-th/0601095

    2006 arXiv

  15. [23]

    Gauge fields in (A)dS(d) and Connections of its symmetry algebra,

    E. D. Skvortsov, “Gauge fields in (A)dS(d) and Connections of its symmetry algebra,” J. Phys. A 42 (2009) 385401, arXiv:0904.2919 [hep-th]

    2009 arXiv

  16. [24]

    Partially Massless Spin 2 Electrodynamics,

    S. Deser and A. Waldron, “Partially Massless Spin 2 Electrodynamics,” Phys. Rev. D 74 (2006) 084036, arXiv:hep-th/0609113

    2006 arXiv

  17. [25]

    PM = EM: Partially Massless Duality Invariance,

    S. Deser and A. Waldron, “PM = EM: Partially Massless Duality Invariance,” Phys. Rev. D 87 (2013) 087702, arXiv:1301.2238 [hep-th]

    2013 arXiv

  18. [26]

    Manifest Duality Invariance for the Partially Massless Graviton,

    K. Hinterbichler, “Manifest Duality Invariance for the Partially Massless Graviton,” Phys. Rev. D 91 no. 2, (2015) 026008, arXiv:1409.3565 [hep-th]

    2015 arXiv

  19. [27]

    Fractons,

    R. M. Nandkishore and M. Hermele, “Fractons,” Ann. Rev. Condensed Matter Phys. 10 (2019) 295–313, arXiv:1803.11196 [cond-mat.str-el]. 25

    2019 arXiv

  20. [28]

    Fracton Phases of Matter,

    M. Pretko, X. Chen, and Y. You, “Fracton Phases of Matter,” Int. J. Mod. Phys. A 35 no. 06, (2020) 2030003, arXiv:2001.01722 [cond-mat.str-el]

    2020 arXiv

  21. [29]

    Higher-Rank Tensor Field Theory of Non-Abelian Fracton and Embeddon,

    J. Wang and K. Xu, “Higher-Rank Tensor Field Theory of Non-Abelian Fracton and Embeddon,” Annals Phys. 424 (2021) 168370, arXiv:1909.13879 [hep-th]

    2021 arXiv

  22. [30]

    Maxwell theory of fractons,

    E. Bertolini and N. Maggiore, “Maxwell theory of fractons,” Phys. Rev. D 106 no. 12, (2022) 125008, arXiv:2209.01485 [hep-th]

    2022 arXiv

  23. [31]

    The theory of symmetric tensor field: From fractons to gravitons and back,

    A. Blasi and N. Maggiore, “The theory of symmetric tensor field: From fractons to gravitons and back,” Phys. Lett. B 833 (2022) 137304, arXiv:2207.05956 [hep-th]

    2022 arXiv

  24. [32]

    Fracton gravity from spacetime dipole symmetry,

    E. Afxonidis, A. Caddeo, C. Hoyos, and D. Musso, “Fracton gravity from spacetime dipole symmetry,” Phys. Rev. D 109 no. 6, (2024) 065013, arXiv:2311.01818 [hep-th]

    2024 arXiv

  25. [33]

    Anomalies in covariant fracton theories,

    D. Rovere, “Anomalies in covariant fracton theories,” Phys. Rev. D 110 no. 8, (2024) 085012, arXiv:2406.06686 [hep-th]

    2024 arXiv

  26. [34]

    Covariant Fractons and Weitzenb¨ ock Torsion,

    D. Rovere, “Covariant Fractons and Weitzenb¨ ock Torsion,”arXiv:2505.21022 [hep-th]

  27. [35]

    Electric-Magnetic Duality for Symmetric Tensor Gauge Theories and Immobile p-branes,

    R. Makino, S. Sasaki, and K. Shiozawa, “Electric-Magnetic Duality for Symmetric Tensor Gauge Theories and Immobile p-branes,” arXiv:2505.11174 [hep-th]

  28. [36]

    Fractons in curved space,

    A. Jain and K. Jensen, “Fractons in curved space,” SciPost Phys. 12 no. 4, (2022) 142, arXiv:2111.03973 [hep-th]

    2022 arXiv

  29. [37]

    Fractons, dipole symmetries and curved spacetime,

    L. Bidussi, J. Hartong, E. Have, J. Musaeus, and S. Prohazka, “Fractons, dipole symmetries and curved spacetime,” SciPost Phys. 12 no. 6, (2022) 205, arXiv:2111.03668 [hep-th]

    2022 arXiv

  30. [38]

    Fractons, symmetric gauge fields and geometry,

    F. Pe˜ na Benitez, “Fractons, symmetric gauge fields and geometry,” Phys. Rev. Res. 5 no. 1, (2023) 013101, arXiv:2107.13884 [cond-mat.str-el]

    2023 arXiv

  31. [39]

    The Galileon as a local modification of gravity,

    A. Nicolis, R. Rattazzi, and E. Trincherini, “The Galileon as a local modification of gravity,” Phys. Rev. D 79 (2009) 064036, arXiv:0811.2197 [hep-th]

    2009 arXiv

  32. [40]

    On ghosts in theories of self-interacting massive spin-2 particles,

    S. Folkerts, A. Pritzel, and N. Wintergerst, “On ghosts in theories of self-interacting massive spin-2 particles,” arXiv:1107.3157 [hep-th]

    arXiv

  33. [41]

    Ghost-Free Derivative Interactions for a Massive Graviton,

    K. Hinterbichler, “Ghost-Free Derivative Interactions for a Massive Graviton,” JHEP 10 (2013) 102, arXiv:1305.7227 [hep-th]

    2013 arXiv

  34. [42]

    Resummation of Massive Gravity,

    C. de Rham, G. Gabadadze, and A. J. Tolley, “Resummation of Massive Gravity,” Phys. Rev. Lett. 106 (2011) 231101, arXiv:1011.1232 [hep-th]

    2011 arXiv

  35. [43]

    Wave equations in conformal space,

    P. A. M. Dirac, “Wave equations in conformal space,” Annals Math. 37 (1936) 429–442

    1936

  36. [44]

    Singletons and Massless, Integral Spin Fields on de Sitter Space (Elementary Particles in a Curved Space. 7.,

    C. Fronsdal, “Singletons and Massless, Integral Spin Fields on de Sitter Space (Elementary Particles in a Curved Space. 7.,” Phys. Rev. D 20 (1979) 848–856

    1979

  37. [45]

    A New Class of Effective Field Theories from Embedded Branes,

    G. Goon, K. Hinterbichler, and M. Trodden, “A New Class of Effective Field Theories from Embedded Branes,” Phys. Rev. Lett. 106 (2011) 231102, arXiv:1103.6029 [hep-th]

    2011 arXiv

  38. [46]

    Symmetries for Galileons and DBI scalars on curved space,

    G. Goon, K. Hinterbichler, and M. Trodden, “Symmetries for Galileons and DBI scalars on curved space,” JCAP 07 (2011) 017, arXiv:1103.5745 [hep-th]

    2011 arXiv

  39. [47]

    de Sitter Galileon,

    C. Burrage, C. de Rham, and L. Heisenberg, “de Sitter Galileon,” JCAP 05 (2011) 025, arXiv:1104.0155 [hep-th]

    2011 arXiv

  40. [48]

    Shift Symmetries in (Anti) de Sitter Space,

    J. Bonifacio, K. Hinterbichler, A. Joyce, and R. A. Rosen, “Shift Symmetries in (Anti) de Sitter Space,” JHEP 02 (2019) 178, arXiv:1812.08167 [hep-th]

    2019 arXiv

  41. [49]

    Exceptional scalar theories in de Sitter space,

    J. Bonifacio, K. Hinterbichler, A. Joyce, and D. Roest, “Exceptional scalar theories in de Sitter space,” JHEP 04 (2022) 128, arXiv:2112.12151 [hep-th]. 26

    2022 arXiv

  42. [50]

    Fracton topological order from the higgs and partial-confinement mechanisms of rank-two gauge theory,

    H. Ma, M. Hermele, and X. Chen, “Fracton topological order from the higgs and partial-confinement mechanisms of rank-two gauge theory,” Physical Review B 98 no. 3, (2018) , arXiv:1802.10108 [cond-mat.str-el]

    2018 arXiv

  43. [51]

    The Higgs Mechanism in Higher-Rank Symmetric U (1) Gauge Theories,

    D. Bulmash and M. Barkeshli, “The Higgs Mechanism in Higher-Rank Symmetric U (1) Gauge Theories,” Phys. Rev. B 97 no. 23, (2018) 235112, arXiv:1802.10099 [cond-mat.str-el]

    2018 arXiv

  44. [52]

    The Fracton Gauge Principle,

    M. Pretko, “The Fracton Gauge Principle,” Phys. Rev. B 98 no. 11, (2018) 115134, arXiv:1807.11479 [cond-mat.str-el]

    2018 arXiv

  45. [53]

    Effective field theory for hydrodynamics: thermodynamics, and the derivative expansion,

    S. Dubovsky, L. Hui, A. Nicolis, and D. T. Son, “Effective field theory for hydrodynamics: thermodynamics, and the derivative expansion,” Phys. Rev. D 85 (2012) 085029, arXiv:1107.0731 [hep-th]

    2012 arXiv

  46. [54]

    Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics,

    H. Liu and P. Glorioso, “Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics,” PoS T ASI2017(2018) 008, arXiv:1805.09331 [hep-th]

    2018 arXiv

  47. [55]

    General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas,

    D. T. Son and M. Wingate, “General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas,” Annals Phys. 321 (2006) 197–224, arXiv:cond-mat/0509786

    2006 arXiv

  48. [56]

    On the CFT Operator Spectrum at Large Global Charge,

    S. Hellerman, D. Orlando, S. Reffert, and M. Watanabe, “On the CFT Operator Spectrum at Large Global Charge,” JHEP 12 (2015) 071, arXiv:1505.01537 [hep-th]

    2015 arXiv

  49. [57]

    Ghost condensation and a consistent infrared modification of gravity,

    N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, and S. Mukohyama, “Ghost condensation and a consistent infrared modification of gravity,” JHEP 05 (2004) 074, arXiv:hep-th/0312099

    2004 arXiv

  50. [58]

    Scalar tachyons in the de Sitter universe,

    J. Bros, H. Epstein, and U. Moschella, “Scalar tachyons in the de Sitter universe,” Lett. Math. Phys. 93 (2010) 203–211, arXiv:1003.1396 [hep-th]

    2010 arXiv

  51. [59]

    de Sitter tachyons and related topics,

    H. Epstein and U. Moschella, “de Sitter tachyons and related topics,” Commun. Math. Phys. 336 no. 1, (2015) 381–430, arXiv:1403.3319 [hep-th]

    2015 arXiv

  52. [60]

    Fractonic Superfluids,

    J.-K. Yuan, S. A. Chen, and P. Ye, “Fractonic Superfluids,” Phys. Rev. Res. 2 no. 2, (2020) 023267, arXiv:1911.02876 [cond-mat.str-el]

    2020 arXiv

  53. [61]

    Non-Abelian gauged fracton matter field theory: Sigma models, superfluids, and vortices,

    J. Wang and S.-T. Yau, “Non-Abelian gauged fracton matter field theory: Sigma models, superfluids, and vortices,” Phys. Rev. Res. 2 no. 4, (2020) 043219, arXiv:1912.13485 [cond-mat.str-el]

    2020 arXiv

  54. [62]

    Fractonic superfluids. II. Condensing subdimensional particles,

    S. A. Chen, J.-K. Yuan, and P. Ye, “Fractonic superfluids. II. Condensing subdimensional particles,” Phys. Rev. Res. 3 no. 1, (2021) 013226, arXiv:2010.03261 [cond-mat.str-el]

    2021 arXiv

  55. [63]

    Fracton hydrodynamics,

    A. Gromov, A. Lucas, and R. M. Nandkishore, “Fracton hydrodynamics,” Phys. Rev. Res. 2 no. 3, (2020) 033124, arXiv:2003.09429 [cond-mat.str-el]

    2020 arXiv

  56. [64]

    Hydrodynamics of ideal fracton fluids,

    K. T. Grosvenor, C. Hoyos, F. Pe˜ na Ben ´ ıtez, and P. Sur´ owka, “Hydrodynamics of ideal fracton fluids,” Phys. Rev. Res. 3 no. 4, (2021) 043186, arXiv:2105.01084 [cond-mat.str-el]

    2021 arXiv

  57. [65]

    Entanglement in the quantum Hall fluid of dipoles,

    J. R. Fliss, “Entanglement in the quantum Hall fluid of dipoles,” SciPost Phys. 11 no. 3, (2021) 052, arXiv:2105.07448 [cond-mat.str-el]

    2021 arXiv

  58. [66]

    Hydrodynamics of dipole-conserving fluids,

    A. G l´ odkowski, F. Pe˜ na Ben ´ ıtez, and P. Sur´ owka, “Hydrodynamics of dipole-conserving fluids,”Phys. Rev. E 107 no. 3, (2023) 034142, arXiv:2212.06848 [cond-mat.str-el]

    2023 arXiv

  59. [67]

    Hydrodynamics of higher-rank gauge theories,

    M. Qi, O. Hart, A. J. Friedman, R. Nandkishore, and A. Lucas, “Hydrodynamics of higher-rank gauge theories,” SciPost Phys. 14 no. 3, (2023) 029, arXiv:2205.05695 [cond-mat.str-el]

    2023 arXiv

  60. [68]

    Dipole symmetry breaking and fractonic Nambu-Goldstone mode,

    E. Afxonidis, A. Caddeo, C. Hoyos, and D. Musso, “Dipole symmetry breaking and fractonic Nambu-Goldstone mode,” SciPost Phys. Core 6 (2023) 082, arXiv:2304.12911 [hep-th]

    2023 arXiv

  61. [69]

    Goldstone bosons and fluctuating hydrodynamics with dipole and momentum conservation,

    P. Glorioso, X. Huang, J. Guo, J. F. Rodriguez-Nieva, and A. Lucas, “Goldstone bosons and fluctuating hydrodynamics with dipole and momentum conservation,” JHEP 05 (2023) 022, arXiv:2301.02680 [hep-th]. 27

    2023 arXiv

  62. [70]

    Ideal fracton superfluids,

    J. Armas and E. Have, “Ideal fracton superfluids,” SciPost Phys. 16 no. 1, (2024) 039, arXiv:2304.09596 [hep-th]

    2024 arXiv

  63. [71]

    Dipole superfluid hydrodynamics,

    A. Jain, K. Jensen, R. Liu, and E. Mefford, “Dipole superfluid hydrodynamics,” JHEP 09 (2023) 184, arXiv:2304.09852 [hep-th]

    2023 arXiv

  64. [72]

    Fracton superfluid hydrodynamics,

    C. Stahl, M. Qi, P. Glorioso, A. Lucas, and R. Nandkishore, “Fracton superfluid hydrodynamics,” Phys. Rev. B 108 no. 14, (2023) 144509, arXiv:2303.09573 [cond-mat.stat-mech]

    2023 arXiv

  65. [73]

    Fractonic superfluids. III. Hybridizing higher moments,

    H.-X. Wang, S. A. Chen, and P. Ye, “Fractonic superfluids. III. Hybridizing higher moments,” arXiv:2412.10280 [cond-mat.quant-gas]

    arXiv

  66. [74]

    Causality, analyticity and an IR obstruction to UV completion,

    A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and R. Rattazzi, “Causality, analyticity and an IR obstruction to UV completion,” JHEP 10 (2006) 014, arXiv:hep-th/0602178

    2006 arXiv

  67. [75]

    Positive moments for scattering amplitudes,

    B. Bellazzini, J. Elias Mir´ o, R. Rattazzi, M. Riembau, and F. Riva, “Positive moments for scattering amplitudes,” Phys. Rev. D 104 no. 3, (2021) 036006, arXiv:2011.00037 [hep-th]

    2021 arXiv

  68. [76]

    Extremal Effective Field Theories,

    S. Caron-Huot and V. Van Duong, “Extremal Effective Field Theories,” JHEP 05 (2021) 280, arXiv:2011.02957 [hep-th]

    2021 arXiv

  69. [77]

    New positivity bounds from full crossing symmetry,

    A. J. Tolley, Z.-Y. Wang, and S.-Y. Zhou, “New positivity bounds from full crossing symmetry,” JHEP 05 (2021) 255, arXiv:2011.02400 [hep-th]

    2021 arXiv

  70. [78]

    UV properties of Galileons: Spectral Densities,

    L. Keltner and A. J. Tolley, “UV properties of Galileons: Spectral Densities,” arXiv:1502.05706 [hep-th]

    arXiv

  71. [79]

    On unitary representations of the group of de sitter space,

    L. H. Thomas, “On unitary representations of the group of de sitter space,” Annals of Mathematics 42 no. 1, (1941) 113–126

    1941

  72. [80]

    A note on the representations of the de sitter group,

    T. D. Newton, “A note on the representations of the de sitter group,” Annals of Mathematics 51 no. 3, (1950) 730–733

    1950

  73. [81]

    Repr´ esentations int´ egrables du groupe de De Sitter,

    J. Dixmier, “Repr´ esentations int´ egrables du groupe de De Sitter,”Bulletin de la Soci´ et´ e Math´ ematique de France 89 (1961) 9–41

    1961

  74. [82]

    On irreducible representations of the lorentz group of n-th order,

    T. Hirai, “On irreducible representations of the lorentz group of n-th order,” Proceedings of the Japan Academy Series A: Mathematical Sciences 38 no. 6, (Jan, 1962)

    1962

  75. [83]

    On irreducible representations of the Lorentz group of n-th order,

    T. Hirai, “On irreducible representations of the Lorentz group of n-th order,” Proceedings of the Japan Academy 38 no. 6, (1962) 258 – 262

    1962

  76. [84]

    Sur les repr´ esentations unitaires des groupes de Lorentz g´ en´ eralis´ es,

    R. Takahashi, “Sur les repr´ esentations unitaires des groupes de Lorentz g´ en´ eralis´ es,”Bulletin de la Soci´ et´ e Math´ ematique de France91 (1963) 289–433

    1963

  77. [85]

    Mixed-symmetry fields in de Sitter space: a group theoretical glance,

    T. Basile, X. Bekaert, and N. Boulanger, “Mixed-symmetry fields in de Sitter space: a group theoretical glance,” JHEP 05 (2017) 081, arXiv:1612.08166 [hep-th]

    2017 arXiv

  78. [86]

    A note on the representations of SO(1,d + 1),

    Z. Sun, “A note on the representations of SO(1,d + 1),” Rev. Math. Phys. 37 no. 01, (2025) 2430007, arXiv:2111.04591 [hep-th]

    2025 arXiv

  79. [87]

    On relativistic wave equations for particles of arbitrary spin in an electromagnetic field,

    M. Fierz and W. Pauli, “On relativistic wave equations for particles of arbitrary spin in an electromagnetic field,” Proc. Roy. Soc. Lond. A 173 (1939) 211–232

    1939

  80. [88]

    Scale and conformal invariance on (A)dS spacetimes,

    K. Farnsworth, K. Hinterbichler, and O. Hulik, “Scale and conformal invariance on (A)dS spacetimes,” Phys. Rev. D 110 no. 4, (2024) 045011, arXiv:2402.12430 [hep-th]

    2024 arXiv

  81. [89]

    Manifest Duality for Partially Massless Higher Spins,

    K. Hinterbichler and A. Joyce, “Manifest Duality for Partially Massless Higher Spins,” JHEP 09 (2016) 141, arXiv:1608.04385 [hep-th]

    2016 arXiv

  82. [90]

    Partially Massless Monopoles and Charges,

    K. Hinterbichler and R. A. Rosen, “Partially Massless Monopoles and Charges,” Phys. Rev. D 92 no. 10, (2015) 105019, arXiv:1507.00355 [hep-th]

    2015 arXiv

  83. [91]

    On the (A)dS Decoupling Limits of Massive Gravity,

    C. De Rham, K. Hinterbichler, and L. A. Johnson, “On the (A)dS Decoupling Limits of Massive Gravity,” JHEP 09 (2018) 154, arXiv:1807.08754 [hep-th]. 28

    2018 arXiv

  84. [92]

    On the cubic interactions of massive and partially-massless higher spins in (A)dS,

    E. Joung, L. Lopez, and M. Taronna, “On the cubic interactions of massive and partially-massless higher spins in (A)dS,” JHEP 07 (2012) 041, arXiv:1203.6578 [hep-th]

    2012 arXiv

  85. [93]

    Generating functions of (partially-)massless higher-spin cubic interactions,

    E. Joung, L. Lopez, and M. Taronna, “Generating functions of (partially-)massless higher-spin cubic interactions,” JHEP 01 (2013) 168, arXiv:1211.5912 [hep-th]

    2013 arXiv

  86. [94]

    On the consistency of (partially-)massless matter couplings in de Sitter space,

    C. Sleight and M. Taronna, “On the consistency of (partially-)massless matter couplings in de Sitter space,” JHEP 10 (2021) 156, arXiv:2106.00366 [hep-th]

    2021 arXiv

  87. [95]

    Pseudolinear spin-2 interactions,

    J. Bonifacio, K. Hinterbichler, and L. A. Johnson, “Pseudolinear spin-2 interactions,” Phys. Rev. D 99 no. 2, (2019) 024037, arXiv:1806.00483 [hep-th]

    2019 arXiv

  88. [96]

    Constraints on a gravitational Higgs mechanism,

    J. Bonifacio, K. Hinterbichler, and R. A. Rosen, “Constraints on a gravitational Higgs mechanism,” Phys. Rev. D 100 no. 8, (2019) 084017, arXiv:1903.09643 [hep-th]

    2019 arXiv

  89. [97]

    Masslessness of photon and Goldstone theorem,

    B. Rosenstein and A. Kovner, “Masslessness of photon and Goldstone theorem,” Int. J. Mod. Phys. A 6 (1991) 3559–3570

    1991

  90. [98]

    New look at QED in four-dimensions: The Photon as a Goldstone boson and the topological interpretation of electric charge,

    A. Kovner and B. Rosenstein, “New look at QED in four-dimensions: The Photon as a Goldstone boson and the topological interpretation of electric charge,” Phys. Rev. D 49 (1994) 5571–5581, arXiv:hep-th/9210154

    1994 arXiv

  91. [99]

    Generalized Global Symmetries,

    D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized Global Symmetries,” JHEP 02 (2015) 172, arXiv:1412.5148 [hep-th]

    2015 arXiv

  92. [100]

    D-brane charges in five-brane backgrounds,

    J. M. Maldacena, G. W. Moore, and N. Seiberg, “D-brane charges in five-brane backgrounds,” JHEP 10 (2001) 005, arXiv:hep-th/0108152

    2001 arXiv

  93. [101]

    Superconductors are topologically ordered,

    T. H. Hansson, V. Oganesyan, and S. L. Sondhi, “Superconductors are topologically ordered,” Annals Phys. 313 no. 2, (2004) 497–538, arXiv:cond-mat/0404327

    2004 arXiv

  94. [102]

    Symmetries and Strings in Field Theory and Gravity,

    T. Banks and N. Seiberg, “Symmetries and Strings in Field Theory and Gravity,” Phys. Rev. D 83 (2011) 084019, arXiv:1011.5120 [hep-th]

    2011 arXiv

  95. [103]

    Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories,

    S. Gukov and A. Kapustin, “Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories,” arXiv:1307.4793 [hep-th]

    arXiv

  96. [104]

    Higgs Condensates are Symmetry-Protected Topological Phases: II. U (1) Gauge Theory and Superconductors,

    R. Thorngren, T. Rakovszky, R. Verresen, and A. Vishwanath, “Higgs Condensates are Symmetry-Protected Topological Phases: II. U (1) Gauge Theory and Superconductors,” arXiv:2303.08136 [cond-mat.str-el]

    arXiv

  97. [105]

    Exactly Soluble Diffeomorphism Invariant Theories,

    G. T. Horowitz, “Exactly Soluble Diffeomorphism Invariant Theories,” Commun. Math. Phys. 125 (1989) 417

    1989

  98. [106]

    Covariant fracton gauge theory with boundary,

    E. Bertolini, N. Maggiore, and G. Palumbo, “Covariant fracton gauge theory with boundary,” Phys. Rev. D 108 no. 2, (2023) 025009, arXiv:2306.13883 [hep-th]

    2023 arXiv

  99. [107]

    A Chern-Simons theory for dipole symmetry,

    X. Huang, “A Chern-Simons theory for dipole symmetry,” SciPost Phys. 15 no. 4, (2023) 153, arXiv:2305.02492 [cond-mat.str-el]

    2023 arXiv

  100. [108]

    Hall-like behaviour of higher rank Chern-Simons theory of fractons,

    E. Bertolini, A. Blasi, N. Maggiore, and D. S. Shaikh, “Hall-like behaviour of higher rank Chern-Simons theory of fractons,” JHEP 10 (2024) 232, arXiv:2405.19446 [hep-th]

    2024 arXiv

  101. [109]

    Quasi-topological fractons: a 3D dipolar gauge theory,

    E. Bertolini, A. Blasi, and N. Maggiore, “Quasi-topological fractons: a 3D dipolar gauge theory,” Eur. Phys. J. C 85 no. 1, (2025) 68, arXiv:2501.01944 [hep-th]

    2025 arXiv

  102. [110]

    Fractons from covariant higher-rank three-dimensional BF theory,

    E. Bertolini, A. Blasi, M. Carrega, N. Maggiore, and D. S. Shaikh, “Fractons from covariant higher-rank three-dimensional BF theory,” Phys. Rev. B 111 no. 8, (2025) 085126, arXiv:2501.19154 [cond-mat.str-el]

    2025 arXiv

  103. [111]

    A gauge theory generalization of the fermion-doubling theorem,

    S. M. Kravec and J. McGreevy, “A gauge theory generalization of the fermion-doubling theorem,” Phys. Rev. Lett. 111 (2013) 161603, arXiv:1306.3992 [hep-th]. 29

    2013 arXiv

  104. [112]

    Entanglement in BF theory II: Edge-modes,

    J. R. Fliss and S. Vitouladitis, “Entanglement in BF theory II: Edge-modes,” arXiv:2310.18391 [hep-th]

    arXiv

  105. [113]

    Free □k scalar conformal field theory,

    C. Brust and K. Hinterbichler, “Free □k scalar conformal field theory,” JHEP 02 (2017) 066, arXiv:1607.07439 [hep-th]

    2017 arXiv

  106. [114]

    Galilean currents and charges,

    A. Nicolis, “Galilean currents and charges,” Phys. Rev. D 85 (2012) 085026, arXiv:1011.3057 [hep-th]

    2012 arXiv

  107. [115]

    Critical drag as a mechanism for resistivity,

    D. V. Else and T. Senthil, “Critical drag as a mechanism for resistivity,” Phys. Rev. B 104 (Nov, 2021) 205132

    2021

  108. [116]

    Towards classification of Fracton phases: the multipole algebra,

    A. Gromov, “Towards classification of Fracton phases: the multipole algebra,” Phys. Rev. X 9 no. 3, (2019) 031035, arXiv:1812.05104 [cond-mat.str-el]

    2019 arXiv

  109. [117]

    Spontaneous breaking of multipole symmetries,

    C. Stahl, E. Lake, and R. Nandkishore, “Spontaneous breaking of multipole symmetries,” Phys. Rev. B 105 no. 15, (2022) 155107, arXiv:2111.08041 [cond-mat.stat-mech]

    2022 arXiv

  110. [118]

    Lagrangian formulation for arbitrary spin. 1. The boson case,

    L. P. S. Singh and C. R. Hagen, “Lagrangian formulation for arbitrary spin. 1. The boson case,” Phys. Rev. D 9 (1974) 898–909

    1974

  111. [119]

    Constant curvature algebras and higher spin action generating functions,

    K. Hallowell and A. Waldron, “Constant curvature algebras and higher spin action generating functions,” Nucl. Phys. B 724 (2005) 453–486, arXiv:hep-th/0505255

    2005 arXiv

  112. [120]

    Gauge Fields as Rings of Glue,

    A. M. Polyakov, “Gauge Fields as Rings of Glue,” Nucl. Phys. B 164 (1980) 171–188

    1980

  113. [121]

    Mean string field theory: Landau-Ginzburg theory for 1-form symmetries,

    N. Iqbal and J. McGreevy, “Mean string field theory: Landau-Ginzburg theory for 1-form symmetries,” SciPost Phys. 13 (2022) 114, arXiv:2106.12610 [hep-th]

    2022 arXiv

  114. [122]

    The discreet charm of the discrete series in dS2,

    D. Anninos, T. Anous, B. Pethybridge, and G. S ¸eng¨ or, “The discreet charm of the discrete series in dS2,” J. Phys. A 57 no. 2, (2024) 025401, arXiv:2307.15832 [hep-th]

    2024 arXiv

  115. [123]

    Higgs phenomenon for 4-D gravity in anti-de Sitter space,

    M. Porrati, “Higgs phenomenon for 4-D gravity in anti-de Sitter space,” JHEP 04 (2002) 058, arXiv:hep-th/0112166

    2002 arXiv

  116. [124]

    Mass and gauge invariance 4. Holography for the Karch-Randall model,

    M. Porrati, “Mass and gauge invariance 4. Holography for the Karch-Randall model,” Phys. Rev. D 65 (2002) 044015, arXiv:hep-th/0109017

    2002 arXiv

  117. [125]

    Higgs phenomenon for the graviton in ADS space,

    M. Porrati, “Higgs phenomenon for the graviton in ADS space,” Mod. Phys. Lett. A 18 (2003) 1793–1802, arXiv:hep-th/0306253

    2003 arXiv

  118. [126]

    Plasmons, Gauge Invariance, and Mass,

    P. W. Anderson, “Plasmons, Gauge Invariance, and Mass,” Phys. Rev. 130 (1963) 439–442

    1963

  119. [127]

    Solidity without inhomogeneity: Perfectly homogeneous, weakly coupled, UV-complete solids,

    A. Esposito, R. Krichevsky, and A. Nicolis, “Solidity without inhomogeneity: Perfectly homogeneous, weakly coupled, UV-complete solids,” JHEP 11 (2020) 021, arXiv:2004.11386 [hep-th]

    2020 arXiv

  120. [128]

    Cosmological Collider Physics,

    N. Arkani-Hamed and J. Maldacena, “Cosmological Collider Physics,” arXiv:1503.08043 [hep-th]

    arXiv

  121. [129]

    The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities,

    N. Arkani-Hamed, D. Baumann, H. Lee, and G. L. Pimentel, “The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities,” JHEP 04 (2020) 105, arXiv:1811.00024 [hep-th]

    2020 arXiv

  122. [130]

    The cosmological bootstrap: weight-shifting operators and scalar seeds,

    D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, “The cosmological bootstrap: weight-shifting operators and scalar seeds,” JHEP 12 (2020) 204, arXiv:1910.14051 [hep-th]

    2020 arXiv

  123. [131]

    Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions,

    D. Anninos, F. Denef, Y. T. A. Law, and Z. Sun, “Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions,” JHEP 01 (2022) 088, arXiv:2009.12464 [hep-th]

    2022 arXiv

  124. [132]

    A compendium of sphere path integrals,

    Y. T. A. Law, “A compendium of sphere path integrals,” JHEP 12 (2021) 213, arXiv:2012.06345 [hep-th]

    2021 arXiv

  125. [133]

    Dynamical edge modes and entanglement in Maxwell theory,

    A. Ball, Y. T. A. Law, and G. Wong, “Dynamical edge modes and entanglement in Maxwell theory,” JHEP 09 (2024) 032, arXiv:2403.14542 [hep-th]

    2024 arXiv

  126. [134]

    De Sitter Horizon Edge Partition Functions,

    Y. T. A. Law, “De Sitter Horizon Edge Partition Functions,” arXiv:2501.17912 [hep-th]. 30

    arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.