{"id":"b16e20f5-6cda-43f3-b197-24e1ec01488a","arxiv_id":"2112.00531","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Coordinate-dependent symmetries and elastic dualities provide a unified effective description of gapless fractons, their defects, and their subdiffusive hydrodynamics.","lead":"This review surveys how coordinate-dependent symmetries, especially polynomial shifts and the Heisenberg group, organize gapless fractons, and how elasticity dualities map lattice defects to fracton charges. It is a useful entry point for physicists who want the symmetry-based view of fracton effective theories and hydrodynamics.","discovery_kind":"unclear","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"The abstract's thesis that coordinate-dependent symmetries 'shape the properties of fractons' is qualified by the paper's own Sec. 5, where the same symmetry is realized without immobile excitations and ideal hydrodynamics reduces to an ordinary fluid.","rationale":"This is a review whose cited results - the multipole algebra [5], the fracton-elasticity duality [33], the quasicrystal and Cosserat duals [49,54], and the hydrodynamics [19,87] - are published and quoted faithfully. The Heisenberg-group construction of Sec. 3.2 is imported from the preprint [26], but the review presents it as one geometric approach rather than as a proved result, and Secs. 4-5 do not depend on it, so the reader's weakest_assumption is not where the thesis is most exposed. The genuine soft spot is the gap between the abstract's sufficiency-sounding claim [abstract:s2] and Sec. 5's own demonstrations that the symmetry can be realized without immobile excitations and that ideal fracton hydrodynamics is 'essentially the same as those of an ordinary fluid.' The paper acknowledges both observations in passing but does not fold them into a caveat on the thesis. That is a framing qualification, not an error, and it does not warrant downgrading a review that otherwise meets its stated purpose. Verdict stays ACCEPT; the recommendation is that a revised abstract or a sentence at the head of Sec. 5 state explicitly that the symmetry is an organizing principle, not a sufficient condition for restricted mobility.","tokens_in":23181,"tokens_out":4726,"duration_ms":92088,"concrete_test":"Solve the quadratic action (5.80) for the Goldstone mode phi in the condensed phase and read off its dispersion relation. If it is omega proportional to k^2 (propagating) rather than omega proportional to k^0 (immobile), then the vector global symmetry does not by itself produce fractonic mobility, and the abstract's 'shape the properties of fractons' needs an explicit qualifier distinguishing the symmetric phase of Eqs. (5.76)-(5.78) from the condensed phase of Eq. (5.80). This is a short, self-contained computation from the paper's own equations and would settle whether the thesis is a sufficiency claim or only a necessary-condition heuristic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing sentence is [abstract:s2]. The review's own Sec. 5 supplies two counterexamples to reading it as a sufficiency claim. First, around the trivial vacuum of Eq. (5.75), the vector global symmetry produces an emergent subsystem symmetry and immobile excitations (Eqs. 5.76-5.78); but in the condensed phase (Eq. 5.79), the same symmetry is realized as the polynomial shift (5.81) and the text states there is 'no emergent subsystem symmetry, so there are no immobile fluctuations in the condensed fractonic phase' (after Eq. 5.80). Second, the hydrodynamic section concludes that 'in both cases ... hydrodynamic equations are essentially the same as those of an ordinary fluid, with the higher moment or subsystem charge conservation following from the constitutive relations' (Sec. 5, final paragraph). So the symmetry is neither necessary nor sufficient for restricted mobility in the review's own telling, yet the abstract presents it as the organizing principle without stating the conditions under which it implies fractonicity. The paper acknowledges each piece but does not synthesize them into a caveat on the thesis. Separately, the reader's weakest_assumption targets the Heisenberg embedding (Sec. 3.2), which is imported from the preprint [26] and is not used by the elasticity dualities (Sec. 4) or hydrodynamics (Sec. 5); it is therefore not the load-bearing link for the review's central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"This is a 19-page review (about 100 references) on gapless fractons, organized around the thesis that coordinate-dependent symmetries — polynomial shift symmetries and the associated multipole algebra — provide the organizing principle for these systems. Section 2 traces the polynomial shift symmetries from the Griffin–Grosvenor–Hořava–Yan naturalness work to Gromov's multipole algebra, and shows how a Lagrangian depending on ∂0φ and ∂i∂jφ yields the generalized continuity equation ∂0ρ + ∂i∂jJ^{ij} = 0, hence conserved charge and dipole moment (Eqs. 2.4–2.6). Section 3 recasts the monopole–dipole–momentum algebra [P_i, Q_j] = δ_{ij}Q as the Heisenberg algebra, constructs the group manifold N^{2d+1} with its Maurer–Cartan form (Eqs. 3.13–3.23), embeds physical spacetime in it, imposes the inverse Higgs constraint z_i = −∂iφ, and derives the fracton metric B (Eq. 3.27) and a Born–Infeld action (Eq. 3.29), before discussing the curved-space gauging of the tensor symmetry (Eqs. 3.33–3.37). Section 4 reviews the fracton–elasticity duality for Cauchy elasticity (Eqs. 4.39–4.51), quasicrystal elasticity with phonons and phasons (Eqs. 4.55–4.63), and Cosserat elasticity with an internal rotation (Eqs. 4.64–4.72), in each case identifying the dual tensor gauge theory and the defect-to-charge map. Section 5 reviews fracton superfluids via a vector global symmetry acting on a complex scalar (Eqs. 5.73–5.81) and fracton hydrodynamics with conserved higher moments, including the subdiffUS","tokens_in":23669,"tokens_out":15702,"duration_ms":262628,"significance":"The review's value is as a compact, equation-level synthesis connecting three literatures that are usually presented separately: the effective-field-theory naturalness tradition, the fracton/tensor-gauge-theory literature, and the elasticity-duality literature. The distinctive contribution relative to existing fracton reviews is the Heisenberg-group packaging of Sec. 3, which gives the monopole–dipole–momentum algebra a geometric home and produces the fracton metric and a diffeomorphism-invariant action in a way that is not available in the standard tensor-gauge presentations. The elasticity sections are worked through with explicit dual actions and explicit defect–charge maps, which will be useful to readers entering the field, and the hydrodynamics section gives a concrete, falsifiable dispersion relation (Eq. 5.88) for the subsystem-symmetric case. The Conclusions are candid about what is not understood, in particular the coupling of tensor gauge fields to curved backgrounds and the treatment of fracton defects as singular configurations. The paper is a review and quotes published results by design; that is appropriate to its genre.","major_comments":[{"comment":"The abstract states without qualification that 'in the effective theory novel coordinate-dependent symmetries emerge that shape the properties of fractons', and the Introduction frames the multipole algebra as the guiding principle. Section 5 supplies two explicit qualifications that the abstract does not carry. First, around the trivial vacuum of Eq. (5.75) the vector global symmetry produces an emergent subsystem symmetry and immobile excitations (Eqs. 5.76–5.78), but in the condensed phase of Eq. (5.79) the same symmetry is realized as the polynomial shift of Eq. (5.81) and the text states there is 'no emergent subsystem symmetry, so there are no immobile fluctuations in the condensed fractonic phase'. Second, the final paragraph of §5 concludes that in both the ideal and dissipative cases the hydrodynamic equations 'are essentially the same as those of an ordinary fluid, with the higher moment or subsystem charge conservation following from the constitutive relations'. Taken together, the symmetry is neither necessary nor sufficient for restricted mobility in the review's own telling. Because the abstract is the citable unit of a review, the thesis sentence should be qualified there (for example, by naming the conditions under which the symmetry implies fractonicity, and by pointing forward to §5), and the Introduction should state the same caveat once. The body already contains all the needed material, so the fix is local; I raise it because it concerns how the review's central claim will be quoted.","section":"Abstract (sentence abstract:s2) and §5","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"As written, ρ_disc = ½ ε^{kl} ε^{ij} ∂_i∂_j∂_k u_l^{sing} vanishes identically, because ε^{ij}∂_i∂_j annihilates any smooth field and ∂_k u_l is smooth. The expression obtained from ρ = ∂_iρ^i with ρ^i = ε^{ij}ε^{kl}∂_k∂_l u_j^{sing} is ε^{ij}ε^{kl}∂_i∂_k∂_l u_j^{sing}; the antisymmetrized pair should act on u, not on ∂_k u_l. Please correct the index structure.","section":"Eq. (4.51)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sign of the A_0 gauge transformation is not consistent across the paper: Eq. (3.31) gives δA_0 = −∂0α, while Eqs. (4.48) and (4.59) give δA_0 = +∂0α. With the electric field defined as E^{ij} = ∂0A^{ij} + ∂_i∂_jA_0 in Eq. (3.38), only the minus sign makes E^{ij} invariant. Please fix one convention and use it throughout.","section":"Eqs. (3.31), (3.38), (4.48), (4.59)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The operator written 'iP_i' in Eq. (3.34) and the combination 'D_iP_i dα' below it are not defined anywhere in the text, and Eq. (3.36) is compressed to the point of being unreadable without consulting Ref. [26]. A sentence defining the notation and a short unpacking of the action would make Sec. 3.2 usable on its own.","section":"Eqs. (3.34)–(3.36)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The 2×2 block matrix of elastic moduli is run together in the text as '(u_{ij}w_{ij})(C_{ijkl} R_{ijkl}R'_{ijkl} K_{ijkl})(u_{kl}w_{kl})', which does not display as a matrix. The symmetry statement R'_{klij} = R_{ijkl} also does not match the index order used in the displayed block. Please reformat.","section":"Eq. (4.56)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There is a typographical error, 'averege', in the sentence introducing the average lattice separations l_a and l_b.","section":"§4.2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The geometric construction of Sec. 3.2 — the inverse Higgs constraint, the fracton metric, the Born–Infeld action, and the curved-space gauging — is taken from Ref. [26], which is listed as an arXiv preprint with no journal reference. Since this section is the review's most distinctive synthesis, a parenthetical note on the status of that work would help readers weigh it.","section":"Ref. [26]","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sentence 'the leading one-loop correction to c² in the unbroken phase ... comes from the two-loop diagram' is confusing as phrased; presumably what is meant is the one-loop diagram with two quartic vertices. A few words of clarification would help.","section":"§2, Eq. (2.3)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The author affiliations and the acknowledgements contain corrupted characters ('Wroc/suppress law', 'Marie Sk/suppress lodowska'); these should be corrected at proof stage.","section":"Affiliations and Acknowledgements","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"The review draws substantially on the authors' own prior work, notably Ref. [26] for the Heisenberg-group geometry of Sec. 3 and Ref. [19] for the ideal fracton hydrodynamics of Sec. 5, with Ref. [54] also by an author. The attribution is explicit in each case and the elasticity duality rests on independent work by Pretko, Radzihovsky and Kleinert, so I do not regard this as a problem, but the editor may wish to note the proportion of self-citation for a review of this length."},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"Short version: this is a competent review of the symmetry-first approach to gapless fractons, and it is the one I would hand a new student. It is not a research paper and does not pretend to be, so judge it on coverage, accuracy, and whether the through-line holds. Mostly it does. Section 4 is the strongest part: Cauchy, quasicrystal, and Cosserat elasticity are run through the same Hubbard-Stratonovich template, so you can see exactly what changes when you add phasons or rotational degrees of freedom, and the defect-to-charge mapping is spelled out each time. Section 5 is compact and correct, including the subdiffusive scaling and the anisotropic dispersion in Eq. 5.88. The Heisenberg-group material in Sec. 3 is cleanly presented as geometry rather than as a claim about dynamics, which is the right register for it. One framing soft spot. The abstract's line that coordinate-dependent symmetries 'shape the properties of fractons' reads as a sufficiency claim, and Sec. 5 quietly undercuts it twice: around the trivial vacuum of Eq. 5.75 the vector symmetry gives immobile excitations, but in the condensed phase the same symmetry is realized as a polynomial shift with no subsystem symmetry and no immobile fluctuations, and the hydrodynamic conclusion is that the equations are essentially an ordinary fluid's. So the symmetry is neither necessary nor sufficient for restricted mobility. The paper states each piece, so this is a framing issue rather than an error, but a skimming reader will take away more than the body supports. A second, smaller point: Sec. 3 rests on Ref. [26], still a preprint, so the inverse Higgs reduction is the least independently checkable step here. The stress-test's worry that this is load-bearing for the central claim does not land; Secs. 4 and 5 do not use it, and the reader's weakest_assumption is aimed at the wrong link. Minor: the Cosserat constraint structure around Eqs. 4.67-4.68 goes by fast, and one sentence on why the second constraint is not a conservation law and what that costs would help. Who it is for: condensed matter and hep-th people entering fractons, and anyone working on elasticity dualities who wants the three cases side by side. Worth a serious referee, with instructions to check index conventions in the dualized actions and citation coverage rather than to hunt for originality.","headline":"A solid, well-organized review of the symmetry-first view of gapless fractons; its abstract promises slightly more than the body delivers, but the body is honest about it.","tokens_in":24221,"tokens_out":1412,"would_cite":true,"duration_ms":29091,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"selfhosted/deepseek-v4.1-flash","headline":"The review argues that coordinate-dependent symmetries—polynomial shift symmetries and the multipole algebra—are the organizing principle for gapless fractons, and that elasticity dualizes to a tensor gauge theory whose charges are lattice.","keywords":["fractons","polynomial shift symmetries","multipole algebra","tensor gauge theory","fracton-elasticity duality","Heisenberg group","fracton hydrodynamics","subdiffusion"],"falsifier":"A concrete check would be a microscopic lattice or spin-liquid model with conserved charge and dipole moment in which the low-energy Goldstone mode does not reduce to the single field $��phi$ predicted by the inverse Higgs constraint, or in which the tensor gauge field's coupling to a curved background requires symmetry-breaking terms that the review's construction says are absent. Either observation would undercut the geometric unification.","tokens_in":24630,"feed_emoji":"🧩","tokens_out":7276,"duration_ms":117983,"temperature":0.7,"pith_summary":"This review sets out to show that a class of symmetries whose transformation parameters depend on position—polynomial shift symmetries, equivalently a multipole algebra—is the common organizing principle for gapless fractons, excitations that cannot move freely. The authors argue that conserving charge and dipole moment forces charge to relax through a two-index current, $��partial_0��rho+��partial_i��partial_j J^{ij}=0$, and that gauging the symmetry produces tensor gauge fields $A_0,A_{ij}$ with unusual transformation laws. They then review how ordinary, quasicrystal, and Cosserat elasticity can be dualized into such tensor gauge theories, with lattice defects playing the role of fracton charges, and how the Heisenberg group supplies a geometric picture of the Goldstone mode when monopole, dipole, and momentum symmetries break. Finally they review hydrodynamics: conserved higher moments slow spreading from ordinary diffusion to subdiffusion, $��Delta x��sim t^{1/(n+1)}$. The review is a synthesis rather than a new derivation, and it acknowledges that coupling tensor gauge fields to curved space and the detailed dynamics of individual fractons remain unresolved.","feed_headline":"Fractons are organized by position-dependent symmetries","feed_subtitle":"Charge and dipole conservation, elastic duality, and subdiffusion follow from one symmetry principle.","key_machinery":"The load-bearing object is the multipole algebra, realized as polynomial shift symmetry and geometrically as the Heisenberg group. In the elastic dual, the stress tensor is written in terms of a symmetric tensor gauge field, and the dual gauge transformations are $��delta A_{ij}=��partial_i��partial_j��alpha$ and $��delta A_0=��partial_0��alpha$. The Heisenberg group's Maurer-Cartan form supplies the invariant one-form $v=d��phi+z��cdot dx$ and the symplectic form $J=e^i��wedge��omega_i$, which reproduces $[P_i,Q_j]=��delta_{ij}Q$; embedding physical spacetime into the group manifold and imposing the inverse Higgs constraint $z_i=-��partial_i��phi$ reduces the Goldstone mode to one field $��phi$ and produces the fracton metric $B=(��partial_0��phi��,dx^0)^2+(��partial_0��partial_i��phi��,dx^0+��partial_i��partial_j��phi��,dx^j)^2$. This machinery is what lets the review treat ordinary elasticity, quasicrystal elasticity, and Cosserat elasticity as instances of one symmetry principle.","core_discovery":"The central claim is that fracton behavior is not an assortment of lattice accidents but the consequence of coordinate-dependent symmetries. In the simplest case, invariance under $��phi��to��phi+��alpha+��beta_i x^i$ yields conservation of total charge and dipole moment; the continuity equation uses a symmetric tensor current, and a gauged version uses a scalar $A_0$ and symmetric tensor $A_{ij}$ transforming as $��delta A_0=-��partial_0��alpha$ and $��delta A_{ij}=��partial_i��partial_j��alpha$. The review's main exhibit is the fracton-elasticity duality: in two dimensions the elastic stress tensor is resolved by tensor gauge fields, $T^{i��mu}=��epsilon^{��mu��nu��rho}��partial_��nu A^i{}_{��rho}$, so dislocations and disclinations become charged sources in a tensor gauge theory. The Heisenberg group encodes the non-Abelian algebra between dipole charge and translations, $[P_i,Q_j]=��delta_{ij}Q$, and gives the Goldstone mode a geometric interpretation as a breathing mode of an enlarged space. The review then argues that hydrodynamics with conserved multipole moments produces subdiffusion rather than ordinary diffusion.","pith_inferences":["The review's unification suggests that the right experimental target is a transport exponent: measuring $��Delta x��sim t^{1/(n+1)}$ in a dipole-conserving cold-atom or circuit platform would test the multipole symmetry more directly than observing a single immobile defect.","If the Heisenberg-group geometry is more than a parametrization, the number of Goldstone modes in a dipole-symmetric phase should be fixed by the inverse Higgs constraint; a microscopic model with a different count would be evidence against the geometric picture.","The quasicrystal and Cosserat extensions imply that materials with internal rotational or phason degrees of freedom are natural places to look for fractonic defect charges, even when the underlying lattice is not a simple periodic crystal.","The unresolved curved-space coupling suggests a test: put a dipole-conserving system on a curved or strained background and look for symmetry-breaking terms that vanish in flat space; their coefficient would measure how seriously the tensor gauge symmetry is a genuine gauge symmetry rather than a flat-space accident."],"forward_implications":["If the symmetry principle is right, the fracton-elasticity duality gives a dictionary in which dislocation and disclination densities are the charge and current sources of a tensor gauge theory, so defect dynamics can be computed in the dual language.","Quasicrystal elasticity adds a phason field, and its matching faults become vector charges in the dual theory, extending the same dictionary beyond ordinary crystals.","Cosserat elasticity with independent rotations dualizes to a tensor gauge theory coupled to an ordinary $U(1)$ gauge field, with defect charges receiving contributions from both displacement and orientation singularities.","Conserved multipole moments make density spreading subdiffusive, $��Delta x��sim t^{1/(n+1)}$ for the highest conserved $n$-multipole, which is a concrete transport signature.","The Heisenberg-group construction predicts that a dipole-symmetric system on an arbitrary curved background generically spontaneously breaks the monopole-dipole-momentum symmetry, which would constrain any curved-space realization."],"supporting_citations":[{"why":"Introduces symmetric tensor gauge theories as models of gapless fractons, the framework the review builds on.","marker":"[1]"},{"why":"Supplies the generalized electromagnetism with scalar and symmetric tensor gauge fields $A_0,A_{ij}$ and the fracton charge story.","marker":"[2]"},{"why":"Introduces the multipole algebra and connects it to polynomial shift symmetries, the review's organizing principle.","marker":"[5]"},{"why":"Traces polynomial shift symmetries to naturalness and RG in nonrelativistic scalar theories.","marker":"[6]"},{"why":"Gives the systematic scalar-field framework for polynomial shift symmetries.","marker":"[7]"},{"why":"Supplies the Heisenberg-group embedding, inverse Higgs constraint, and curved-space construction used for the geometric unification.","marker":"[26]"},{"why":"Establishes the fracton-elasticity duality in two dimensions, the review's main physical exhibit.","marker":"[33]"},{"why":"Provides the earlier elastic-duality construction that the fracton-elasticity duality extends.","marker":"[45]"},{"why":"Introduces fracton hydrodynamics and the subdiffusive spreading law for conserved multipole moments.","marker":"[87]"},{"why":"Develops ideal fracton hydrodynamics with dipole and momentum conservation, which the review cites for propagating modes.","marker":"[19]"}],"fun_headline_variants":["Review ties fractons to coordinate-dependent symmetries","Fracton hydrodynamics yields subdiffusion from multipole conservation","Elastic duality maps dislocations to charges in tensor gauge theory","Heisenberg group links dipole charge and translations in fracton models","Fractons follow from position-dependent symmetries, not lattice accidents"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The load-bearing premise is that the Heisenberg-group embedding with the inverse Higgs constraint is the correct geometric description of spontaneously broken monopole-dipole-momentum symmetry, rather than merely a convenient parametrization, and that the resulting tensor gauge symmetry survives coupling to curved space.","fun_headline_variants_meta":{"raw":{"variants":["Review ties fractons to coordinate-dependent symmetries","Fracton hydrodynamics yields subdiffusion from multipole conservation","Elastic duality maps dislocations to charges in tensor gauge theory","Heisenberg group links dipole charge and translations in fracton models","Fractons follow from position-dependent symmetries, not lattice accidents"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":3099,"prompt_tokens":841,"completion_tokens":2258,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":2186}},"tokens_in":841,"tokens_out":2258,"duration_ms":34479,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":2186,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-30T23:17:35.085944+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"falsifier":"A concrete check would be a microscopic lattice or spin-liquid model with conserved charge and dipole moment in which the low-energy Goldstone mode does not reduce to the single field $��phi$ predicted by the inverse Higgs constraint, or in which the tensor gauge field's coupling to a curved background requires symmetry-breaking terms that the review's construction says are absent. Either observation would undercut the geometric unification.","supporting_citations":[{"cited_title":"Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids","cited_arxiv_id":"1604.05329","evidence_quote":"Introduces symmetric tensor gauge theories as models of gapless fractons, the framework the review builds on."},{"cited_title":"Generalized Electromagnetism of Subdimensional Particles: A Spin Liquid Story","cited_arxiv_id":"1606.08857","evidence_quote":"Supplies the generalized electromagnetism with scalar and symmetric tensor gauge fields $A_0,A_{ij}$ and the fracton charge story."},{"cited_title":"Multicritical Symmetry Breaking and Naturalness of Slow Nambu-Goldstone Bosons","cited_arxiv_id":"1308.5967","evidence_quote":"Traces polynomial shift symmetries to naturalness and RG in nonrelativistic scalar theories."},{"cited_title":"Scalar Field Theories with Polynomial Shift Symmetries","cited_arxiv_id":"1412.1046","evidence_quote":"Gives the systematic scalar-field framework for polynomial shift symmetries."},{"cited_title":"Fractons, symmetric gauge fields and geometry","cited_arxiv_id":"2107.13884","evidence_quote":"Supplies the Heisenberg-group embedding, inverse Higgs constraint, and curved-space construction used for the geometric unification."},{"cited_title":"Fracton-Elasticity Duality","cited_arxiv_id":"1711.11044","evidence_quote":"Establishes the fracton-elasticity duality in two dimensions, the review's main physical exhibit."},{"cited_title":"Fracton hydrodynamics","cited_arxiv_id":"2003.09429","evidence_quote":"Introduces fracton hydrodynamics and the subdiffusive spreading law for conserved multipole moments."},{"cited_title":"Hydrodynamics of ideal fracton fluids","cited_arxiv_id":"2105.01084","evidence_quote":"Develops ideal fracton hydrodynamics with dipole and momentum conservation, which the review cites for propagating modes."}],"review_version":1}