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REVIEW 4 major objections 4 minor 46 references

Topology and the Conformal Invariance of Nodal Lines in Two-Dimensional Active Scalar Turbulence

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Pith's one-line read Zero-isoline curves in two-dimensional turbulent inverse cascades are proposed to be domain walls of a time-reversal-broken topological state, with a gapless sector described by a Liouville CFT.

desk verdict A speculative but clearly argued proposal that ties 2D active-scalar nodal lines to a Clebsch patch gas and Liouville CFT; the m>1 numerology works, but the m≤1 branch is retrofitted and one example is internally inconsistent. read the letter →

arxiv 2505.09657 v1 pith:UES3T53Y submitted 2025-05-14 hep-th cond-mat.stat-mechnlin.CD

Christopher Eling This is my paper · ORCID
classification hep-thcond-mat.stat-mechnlin.CD MSC 81T4060J67 PACS 47.27.Gs11.25.Hf
keywords two-dimensionalturbulenceinversecascadeactivescalarmodelsSchramm-LoewnerevolutionconformalinvarianceLiouvillefieldtheoryClebschvariablesKolmogorov-Kraichnanscaling
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

This paper proposes an explanation for a long-standing numerical puzzle: in the inverse cascade of two-dimensional active scalar turbulence, the zero-isoline curves of the transported scalar (or, for negative $m$, of the stream function) are random fractal paths whose statistics match Schramm-Loewner evolution, as if they were critical curves of an equilibrium statistical model. The core claim is that the inverse cascade is a topologically ordered, gapped state in which a local energy (or enstrophy) flux spontaneously breaks time-reversal invariance, and the nodal isolines are domain walls separating regions of opposite flux. On these walls sit gapless, conformally invariant degrees of freedom. Modeling the domains with a two-dimensional Clebsch-scalar effective theory, the paper derives a fractional winding number $|\nu| = (4-m)/3$ from Kolmogorov-Kraichnan scaling and shows that the gapless sector is a Liouville conformal field theory whose central charge reproduces the numerically measured SLE parameters.

What carries the argument

The central object is the complex Clebsch scalar $\Psi=\sqrt{\beta}\,e^{i\gamma}$ with the free two-dimensional Euclidean action $S_{2d}=\int d^2x\, \frac{1}{\pi g_0}\,\partial_\mu\Psi\,\partial^\mu\Psi^\dagger$. Written in Bogomolny form, the action becomes a square plus a topological circulation term, and the self-duality equation has an axially symmetric patch solution $\beta(r)=C r^{\pm 2\nu}$, $\gamma=\nu\phi$, cut off at radius $R$. For fractional $\nu$ this patch reproduces the Kolmogorov-Kraichnan scaling of the hydrodynamic fields. The gapless sector is then described by an imaginary compact Liouville action whose coupling to background curvature makes the screening operator marginal, shifting the central charge from $c=1$ to $c=1-6(1-|\nu|)^2/|\nu|$.

What would settle it

Run a direct numerical simulation of the active scalar equation at $m=3/2$ and measure the SLE diffusion constant of the $\hat{\theta}=0$ isolines; the paper's formula gives $\kappa=12/(4-3/2)=24/5=4.8$ and $c=4/5$, so a measured $\kappa$ clearly outside the neighborhood of $4.8$ would falsify the proposed identification.

Watch

Extended reading notes

Core claim

The fully developed inverse cascade is not a featureless turbulent soup but a scale-invariant gas of 'scalar patches': compact droplets inside which the field $\hat{\theta}$ (or $\hat{\psi}$ for $m<0$) has power-law, Kolmogorov-Kraichnan behavior, while the phase $\gamma$ of the complex Clebsch scalar $\Psi=\sqrt{\beta}\,e^{i\gamma}$ winds by a fractional amount. The winding number $\nu$ is fixed by the condition that the generalized circulation of a patch scales with its area in the same way as the turbulent circulation, giving $|\nu|=(4-m)/3$ for $m>1$ and $|\nu|=1$ for $0<m\le 1$. The zero-isoline curves are the domain walls where the local flux vanishes and the winding number jumps. Coupling the two-dimensional boson to background curvature in the manner of an imaginary compact Liouville theory shifts the conformal weight of the vertex operators so that the Liouville potential is marginal, fixing the central charge $c=1-6(1-|\nu|)^2/|\nu|$ and hence the SLE diffusion constant $\kappa=12/(4-m)$ (with $\kappa'=16/\kappa=4(4-m)/3$ for the outer hulls), in agreement with the numerical results of $[7,10,11]$. Under the $m\to -m$ duality the same formulas apply to the stream-function isolines.

Load-bearing premise

The load-bearing premise is that the free two-dimensional Euclidean action for the complex Clebsch field, with a compact phase and fractional winding, is the correct effective description of the fully developed inverse cascade; the paper itself states that deriving this action from the active scalar dynamics is future work.

Editorial extensions

If this is right

  • For $m>1$, zero-isoline curves of $\hat{\theta}$ should be SLE curves with $\kappa=12/(4-m)$, and the outer cluster boundaries with $\kappa'=4(4-m)/3$; for the Euler case $m=2$ this gives $\kappa=6$ and $c=0$, matching the original numerical observation.
  • For $0<m\le 1$ the winding number saturates at $|\nu|=1$, so the curves should have $\kappa=4$ and $c=1$, as seen in the surface quasi-geostrophic and related simulations; the saturation coincides with the change of sign of the velocity and scalar scaling exponents at $m=1$.
  • Under the duality $m\to -m$, the same formulas describe the $\hat{\psi}=0$ isolines, explaining why the stream-function curves in the $m=-2$ case are $\mathrm{SLE}_6$.
  • If the patch gas picture is correct, the bulk turbulent fields are gapped with Kolmogorov-Kraichnan dimensions and conformal invariance lives only in the gapless wall sector; corrections to scaling away from the walls are exponentially small in the wall width.
  • The quantum-Hall analogy assigns each domain a filling-fraction-like parameter $\nu=N\Gamma_0/(2\pi\beta_0)$, giving a microscopic picture of the clusters as collections of minimal-circulation vortex constituents.

Reading between the lines

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

  • The paper does not derive the effective action $S_{2d}$ from the active scalar equations; a natural next step would be to derive it, which would fix the coupling $g_0$ and the patch cut-off in terms of the forcing scale and flux.
  • If the Liouville description is correct, the turbulent state should show boundary-CFT predictions beyond the diffusivity, such as specific crossing probabilities or multi-curve correlation functions, and these could be tested in existing numerical simulations.
  • The quantum-Hall analogy suggests the inverse cascade may possess a topological, dissipationless transport coefficient such as a Hall viscosity under slowly varying strain or metric perturbations; measuring such a response would directly test the gapped topological sector.
  • The fractional charges $\nu=p/q$ map onto Virasoro minimal models, so scanning $m$ through values such as $m=3/2$ and $m=5/2$ in simulations could test whether the CFT remains the same along the line or whether additional operators appear.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a mechanism to explain the observed SLE statistics of nodal lines in two-dimensional active scalar turbulence. It argues that the inverse cascade is a gas of constant-flux domains separated by nodal-line domain walls, with the domains characterized by a Clebsch winding number. The paper introduces a two-dimensional Euclidean action for the complex Clebsch scalar, Eq. (47), obtains power-law patch solutions, Eq. (54), and matches the patch scaling to Kolmogorov-Kraichnan exponents to fix |ν|=(4-m)/3, Eq. (59). It then identifies the nodal-line sector with a Liouville CFT, Eq. (68), whose central charge, Eq. (70), combined with the SLE/CFT dictionary, yields κ=12/(4-m) for m>1, matching numerical results, while a saturation rule |ν|=1 is imposed for 0<m≤1, Eq. (73), to recover κ=4. The paper is explicitly heuristic and states in Section VI that connecting the model to the active scalar equations is future work.

Significance. If the proposed construction were established, it would offer a topological and CFT-based explanation for a striking numerical observation and unify SLE measurements across a family of active scalar models. The paper contains a concrete Bogomolnyi rewriting, explicit patch solutions, and a transparent dictionary relating the model parameter m, the Clebsch winding number ν, and the SLE diffusion constant κ. It is also unusually candid about its heuristic status, which is a strength. However, as it stands the central claim is not a derivation from the fluid dynamics, and several internal inconsistencies—detailed in the major comments—prevent the agreement with numerics from being a genuine prediction of the framework.

major comments (4)
  1. [Section V, Eq. (66)] Equation (66) states c=(3κ-8)(κ-6)/(2κ), but the standard SLE/CFT relation is c=(3κ-8)(6-κ)/(2κ), as given in the cited review [9]. With the equation as printed, κ=4 gives c=-1, which contradicts the text that immediately follows and states 'For κ=4, c=1'. The sign error propagates into the derivation of the dictionary |ν|=4/κ: with the printed sign, the identity does not hold for general κ, even though it is correct with the standard sign convention. Please correct Eq. (66) and re-verify the subsequent relations.
  2. [Section IV, Eq. (59) and Section V, Eq. (73)] For 0<m<1, Eq. (59) gives |ν|=(4-m)/3>1, which through the dictionary |ν|=4/κ would imply κ=12/(4-m)<4, contradicting the observed κ=4. The saturation rule in Eq. (73) is introduced in Section V without derivation from the action (47) or the patch gas; it is an input chosen to match numerics, not a prediction of the model. Since the range 0<m≤1 is part of the central quantitative claim, this is a load-bearing gap. The paper should either derive the saturation from the action or clearly state that the model applies only for m>1.
  3. [Section IV, after Eq. (59)] The text states 'for m=1/2, |ν|=5/6', but Eq. (59) gives |ν|=(4-1/2)/3=7/6; the value 5/6 corresponds to m=3/2. This arithmetic inconsistency suggests the saturation rule is being applied retroactively when computing examples, and it obscures whether |ν| is taken from Eq. (59) or from the saturated formula (73). Please correct the example and clarify the status of the saturation rule.
  4. [Section VI] The paper explicitly states that connecting the approach to the active scalar equations is future work, and the free action (47) is introduced by fiat as the IR description of the inverse cascade. Consequently, the matching of κ for m>1 is a consistency check of the effective model, not a derivation from the fluid dynamics. If the action (47) does not actually describe the turbulent steady state, the explanatory claim collapses. Please either derive (47) from the Clebsch/gauge-theory formulation (for example, through the mapping in [13]) or reformulate the paper as a phenomenological model with clearly stated assumptions and falsifiable predictions.
minor comments (4)
  1. [Section V] The word 'chordial' should be 'chordal' in the description of SLE curves.
  2. [Throughout] The name 'Kraichnan' is misspelled as 'Kraichan' in several places, including Section V and parts of the introduction; please correct.
  3. [Section IV, Eq. (44)] The notation for the area of a droplet is inconsistent: it appears as AD in Eq. (44) and as A_D elsewhere. Please unify.
  4. [Section V, after Eq. (66)] The statement 'For κ=4, c=1' is inconsistent with the printed Eq. (66) as noted in Major Comment 1; after correcting the sign, please ensure the text matches the corrected formula.

Circularity Check

1 steps flagged · score 6.0 of 10

The 0<m<1 branch is circular: Eq. (73) sets |ν|=1 because κ=4 was observed, making that regime's prediction an input; the m>1 derivation is a consistency mapping, not a circular fit.

  1. fitted input called prediction [Section V, Eq. (73)]
    "The remaining case is when 0 < m <1. In these cases, it has been observed that the ˆθ = 0 lines have κ = 4 and thus are defect lines in a CFT with c = 1. In our picture, this implies that the filling fraction/charge |ν| = 1 for 0 <m ≤ 1 and the “level” appears to be saturated |ν| = 4−m/3, m> 1 |ν| = 1, 0<m ≤ 1 (73)"

    For 0<m<1, the paper's own Eq. (59) gives |ν|=(4-m)/3>1, which through the dictionary |ν|=4/κ (Section V, above Eq. (71)) would predict κ<4, not the observed κ=4. Instead of deriving the saturation |ν|=1 from the action (47) or the patch gas, the paper uses the observed κ=4 to 'imply' |ν|=1, then outputs c=1 and κ=4. The agreement in this range is therefore the target datum inserted as a case rule. The step is further exposed by §IV's m=1/2 example: Eq. (59) gives |ν|=7/6, not the stated 5/6, so the saturation is retrofitted to the numerics rather than following from the model.

full rationale

The central m>1 chain (Eqs. (57), (59), (70), (71)) is not circular under the hard rules: the winding number ν is fixed by matching the patch scaling to the externally established Kolmogorov-Kraichnan exponent, and the SLE diffusion constant is then obtained from the standard Liouville central-charge formula and the SLE/CFT dictionary. This is a parameter-matched consistency argument rather than a fit to the target κ, and it has independent content to the extent the KK scaling law is established separately from the observed SLE parameters. The self-citation [13] is motivational for the Chern-Simons analogy and is not load-bearing for the quantitative κ/c output. The genuine circularity is confined to the 0<m<1 branch. There the paper takes the observed κ=4 as the reason to set |ν|=1 in Eq. (73), and then reports c=1 and κ=4; the target datum is inserted as a case rule. That branch is also internally inconsistent with Eq. (59), which gives |ν|=(4-m)/3>1 (e.g. |ν|=7/6 for m=1/2, not the stated 5/6), so the saturation is an ad hoc override rather than a consequence of the Clebsch action or patch gas. Because this retrofitted branch is one of the paper's central claimed regimes (0<m≤1), the overall score reflects partial circularity, not full equivalence.

Assumptions & free parameters 5 free parameters · 8 assumptions · 4 invented entities

The central claim depends on several parameters and assumptions not derived from the active scalar equations. The fractional winding ν is the most important: it is fixed by matching patch scaling to the empirical Kolmogorov-Kraichnan exponents, then controls the central charge and SLE parameter. The model action and the Liouville sector are introduced ad hoc, and the compact Clebsch ansatz is a modeling assumption. There are no invented entities with independent falsifiable evidence.

free parameters (5)
  • fractional winding number ν = |ν|=(4-m)/3 for m>1; |ν|=1 for 0<m≤1
    Introduced as integer, then analytically continued to rational; fixed by equating patch scaling θ ~ L^{2|ν|-2} with the Kolmogorov-Kraichnan exponent θ ~ L^{(2-2m)/3} (Eqs. 57-59). It then controls c and κ via Eqs. 70-71.
  • coupling g0 in Clebsch action
    Free coupling in S2d (Eq. 47); not fixed by active scalar dynamics; appears in the patch action (Eq. 55) and in the Coulomb gas coupling g.
  • constant C in β = C r^{±2ν} = chosen equal to energy flux E0^{1/3}
    After Eq. (58), the text says 'we have chosen the constant C to be the energy flux'; this sets the patch normalization to the empirical flux.
  • minimal circulation Γ0 and boundary value β0 = Γ0 ~ F Lf tf (Eq. 46), β0 unspecified
    Γ0 is introduced as the basic circulation unit of vortex constituents; β0 is the constant value of the Clebsch field on cluster boundaries and acts as a flux analogous to magnetic flux. Neither is derived from Eq. (7).
  • domain wall width W and patch radius R
    W appears in the domain wall profile (Eq. 28) and R in the patch action (Eq. 55); R is identified with the forcing scale, but no equation fixes either from the fluid dynamics.
assumptions (8)
  • standard math Standard SLE/CFT dictionary c = (3κ-8)(κ-6)/(2κ)
    Used in Eq. (66) to convert the SLE diffusion constant into a central charge.
  • standard math Imaginary Liouville central charge c = 1 - 6(1-|ν|)^2/|ν|
    Standard background-charge formula for Liouville with imaginary coupling, Eq. (70).
  • standard math Kolmogorov-Kraichnan scale-invariant dimensional analysis for active scalar models
    Eqs. (14), (19), (21) are assumed; ν is matched to these exponents in Eq. (59).
  • domain assumption The inverse cascade reaches a steady state with constant localized fluxes and spontaneously broken time reversal, O = ±O0
    Postulated in Section II B; not derived from the active scalar equation (7).
  • ad hoc to paper Clebsch variable γ is compact/multivalued and β is constant on cluster boundaries
    Introduced in Section III, Eqs. (35)-(43), to make generalized circulation a winding number.
  • ad hoc to paper The free Euclidean action S2d = ∫ (1/πg0) ∂Ψ∂Ψ† is the IR description of the inverse cascade
    Eq. (47); introduced without derivation from the active scalar equation; Section VI defers this link.
  • ad hoc to paper Winding number ν may be analytically continued to rational values
    Section IV, after Eq. (56); load-bearing for fractional charges and for matching to scaling.
  • ad hoc to paper The Liouville action (68) with imaginary curvature coupling describes the nodal line sector
    Eq. (68); proposed, not derived; used to compute c and κ.
invented entities (4)
  • Time-reversal-breaking constant-flux domains (±O0)
    purpose: Ground states of the inverse cascade separated by nodal-line domain walls
    Order parameter O = 1/2 θv² is proposed, not measured; no independent falsifiable handle is provided.
  • Fractionally charged Clebsch vortices and patch constituents
    purpose: Microscopic constituents of turbulent domains with fractional winding ν; quasi-particle analog in the quantum Hall picture
    No independent prediction is given; ν is set to the existing scaling exponent.
  • Gapless topological degrees of freedom on nodal lines
    purpose: Provide conformal invariance and SLE behavior of nodal lines
    Proposed mechanism; the observed SLE behavior is the input being explained.
  • Scalar patches (β = C r^{±2ν} droplets)
    purpose: Concrete instanton-like solutions of the Clebsch action representing turbulent domains
    Solutions of the ad hoc model; no direct experimental signature is given.

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Pith. "Pith review of Topology and the Conformal Invariance of Nodal Lines in Two-Dimensional Active Scalar Turbulence." pith.science (2026). https://pith.science/paper/UES3T53Y

@misc{pith2026250509657,
  author       = {Pith},
  title        = {Pith review of: Topology and the Conformal Invariance of Nodal Lines in Two-Dimensional Active Scalar Turbulence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UES3T53Y}},
  note         = {Machine review of arXiv:2505.09657}
}
read the original abstract

The inverse cascade in two-dimensional hydrodynamic turbulence exhibits a mysterious phenomenon. Numerical simulations have shown that the nodal isolines of certain scalars actively transported in the flow (eg, the vorticity in Navier-Stokes theory) obey Schramm-Loewner evolution (SLE), which indicates the presence of conformal invariance. Therefore, these turbulent isolines are somehow in the same class as cluster boundaries in equilibrium statistical mechanical models at criticality, such as critical percolation. In this paper, we propose that the inverse cascade is characterized by a local energy (or in some cases, enstrophy) flux field that spontaneously breaks time reversal invariance. The turbulent state consists of random constant flux domains, with the nodal isolines acting as domain walls where the local flux vanishes. The generalized circulation of the domains is proportional to a topological winding number. We argue that these turbulent states are gapped states, in analogy with quantum Hall systems. The turbulent flow consists of many strongly coupled vortices that are analogous to quasi-particles. The nodal isolines are associated with the gapless topological degrees of freedom in the flow, where scale invariance is enhanced to conformal invariance. We introduce a concrete model of this behavior using a two-dimensional effective theory involving the canonical Clebsch scalars. This theory has patch solutions that exhibit power law scaling. The fractional winding number associated with the patches can be related to the Kolmogorov-Kraichnan scaling dimension of the corresponding fluid theory. We argue that the fully developed inverse cascade is a scale invariant gas of these patches. This theory has a conformally invariant sector described by a Liouville conformal field theory whose central charge is fixed by the fractional winding number.

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Reference graph

Works this paper leans on

46 extracted references · 29 canonical work pages

  1. [9]

    Scaling limits of loop-erased random walks and uniform spanning trees,

    O. Schramm, “Scaling limits of loop-erased random walks and uniform spanning trees,” Isr. J. Math. 118, 221-288 (2000) doi:10.1007/BF02803524 [arXiv:math/9904022 [math.PR]]

    arXiv 2000

  2. [13]

    Conformal invariance in hydrodynamic turbulence,

    G. Falkovich, “Conformal invariance in hydrodynamic turbulence,” Russian Mathematical Surveys 62, no. 3 497 (2007)

    work page 2007

  3. [1]

    Turbulence: the legacy of A. N. Kolmogorov,

    U. Frisch, “Turbulence: the legacy of A. N. Kolmogorov,” Cambridge university press, 1995

    work page 1995

  4. [2]

    constituents

    argued that out of this infinity of conservation laws the two most relevant are quadratic positive definite forms, which for general m are the “energy” E and the enstrophy A: E = 1 2 Z d2x ψθ A = 1 2 Z d2x θ2 (10) If there is a statistical steady state (ie, time independent statistics) then the existence of two conserved quantities implies then there are ...

    work page 1941

  5. [3]

    Inertial ranges in two-dimensional turbulence,

    R. H. Kraichnan, “Inertial ranges in two-dimensional turbulence,” Physics of fluids 10, no. 7 1417 (1967)

    work page 1967

  6. [4]

    Lack of self-similarity in transverse velocity increments and circulation statistics in two-dimensional turbulence,

    N. P. M¨ uller and G. Krstulovic, “Lack of self-similarity in transverse velocity increments and circulation statistics in two-dimensional turbulence,” Phys. Rev. Fluids 10 (1), L012601 (2025)

    work page 2025

  7. [5]

    Conformal Field Theory,

    P. Di Francesco, P. Mathieu and D. Senechal, “Conformal Field Theory,” Springer-Verlag, 1997, ISBN 978-0-387-94785-3, 978-1-4612-7475-9 doi:10.1007/978-1-4612-2256-9 29

    doi:10.1007/978-1-4612-2256-9 1997

  8. [6]

    The Theory of Turbulence in Two Dimensions

    A. M. Polyakov, “The Theory of turbulence in two-dimensions,” Nucl. Phys. B 396, 367-385 (1993) doi:10.1016/0550-3213(93)90656-A [arXiv:hep-th/9212145 [hep-th]]

    work page Pith review arXiv 1993

Show all 46 references
  1. [7]

    Is 2-d turbulence a conformal turbulence?,

    G. Falkovich and A. Hanany, “Is 2-d turbulence a conformal turbulence?,” Phys. Rev. Lett. 71, 3454-3457 (1993) doi:10.1103/PhysRevLett.71.3454 [arXiv:hep-th/9301030 [hep-th]]

    1993 arXiv

  2. [8]

    Conformal invariance in two- dimensional turbulence,

    D. Bernard, G. Boffetta, A. Celani, and G. Falkovich, “Conformal invariance in two- dimensional turbulence,” Nature Physics 2, 2, 124 (2006) [arxiv: nlin/0602017 [nlin.CD]]

    2006 arXiv

  3. [10]

    2D growth processes: SLE and Loewner chains,

    M. Bauer and D. Bernard, “2D growth processes: SLE and Loewner chains,” Phys. Rept. 432, 115-221 (2006) doi:10.1016/j.physrep.2006.06.002 [arXiv:math-ph/0602049 [math-ph]]

    2006 arXiv

  4. [11]

    Inverse turbulent cascades and confor- mally invariant curves,

    D. Bernard, G. Boffetta, A. Celani, and G. Falkovich, “Inverse turbulent cascades and confor- mally invariant curves,” Phys. Rev. Lett. 98, 2 (2007) 024501 [arxiv: nlin/0609069 [nlin.CD]]

    2007 arXiv

  5. [12]

    Conformal invariance in inverse turbulent cas- cades

    G. Falkovich and S. Musacchio, “Conformal invariance in inverse turbulent cas- cades” [arXiv:1012.3868 [cond-mat.stat-mech]]

    arXiv

  6. [14]

    A gauge theory for the 2+1 dimensional incompressible Euler equations,

    C. Eling, “A gauge theory for the 2+1 dimensional incompressible Euler equations,” [arXiv:2305.04394 [hep-th]]

    arXiv

  7. [15]

    Lectures on the Quantum Hall Effect,

    D. Tong, “Lectures on the Quantum Hall Effect,” [arXiv:1606.06687 [hep-th]]

    arXiv

  8. [16]

    Surface quasi-geostrophic dynamics,

    I. M. Held, R. T. Pierrehumbert, S. T. Garner, and K. L. Swanson, “Surface quasi-geostrophic dynamics,” J. Fluid Mech. 282 (1995) 1-20

    1995

  9. [17]

    Weakly decaying turbulence in an equivalent barotropic fluid,

    V. Larichev and J. McWilliams, “Weakly decaying turbulence in an equivalent barotropic fluid,” Phys. Fluids, A 3:5 (1991) 938

    1991

  10. [18]

    Spectra of local and nonlocal two- dimensional turbulence,

    R. T. Pierrehumbert, I. M. Held, and K. L. Swanson, “Spectra of local and nonlocal two- dimensional turbulence,” Chaos, Solitons and Fractals 4, no. 6 (1994): 1111-1116

    1994

  11. [19]

    A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,

    A. N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,” J. Fluid Mech., 13 82, (1962); A. M. Obukhov, “Some specific features of atmospheric turbulence,” J. Fluid Mech., 13 ...

    1962

  12. [20]

    Scaling exponents saturate in three- dimensional isotropic turbulence,

    K. P. Iyer, K. R. Sreenivasan, and P. K. Yeung, “Scaling exponents saturate in three- dimensional isotropic turbulence,” Physical Review Fluids 5, 054605 (2020). 30

    2020

  13. [21]

    Model for a Quantum Hall Effect without Landau Levels: Condensed- Matter Realization of the ’Parity Anomaly’,

    F. D. M. Haldane, “Model for a Quantum Hall Effect without Landau Levels: Condensed- Matter Realization of the ’Parity Anomaly’,” Phys. Rev. Lett. 61, 2015-2018 (1988) doi:10.1103/PhysRevLett.61.2015

    1988 doi

  14. [22]

    Fractional Quantum Hall States at Zero Magnetic Field,

    T. Neupert, L. Santos, C. Chamon and C. Mudry, “Fractional Quantum Hall States at Zero Magnetic Field,” Phys. Rev. Lett. 106, 23, 236804 (2011) doi:10.1103/PhysRevLett.106.236804 [arXiv:1012.4723 [cond-mat.str-el]]

    2011 arXiv

  15. [23]

    Fractional quantum Hall effect in the ab- sence of Landau levels,

    D. N. Sheng, Z. C. Gu, K. Sun and L. Sheng, “Fractional quantum Hall effect in the ab- sence of Landau levels,” Nature Communications 2, 389 (2011) doi:10.1038/ncomms1380 [arXiv:1102.2658 [cond-mat.str-el]]

    2011 arXiv

  16. [24]

    Fractional Chern insulator,

    N. Regnault and B. A. Bernevig, “Fractional Chern insulator,” Phys. Rev. X 1, 021014 (2011) doi:10.1103/PhysRevX.1.021014 [arXiv:1105.4867 [cond-mat.str-el]]

    2011 arXiv

  17. [25]

    Statistical equilibrium of circulating fluids,

    A. Migdal, “Statistical equilibrium of circulating fluids,” Phys. Rept. 1011, 1-117 (2023) doi:10.1016/j.physrep.2023.02.001 [arXiv:2209.12312 [physics.flu-dyn]]

    2023 arXiv

  18. [26]

    SUPERFLUID DYNAMICS OF THE FRACTIONAL QUANTUM HALL STATE,

    M. Stone, “SUPERFLUID DYNAMICS OF THE FRACTIONAL QUANTUM HALL STATE,” Phys. Rev. B 42, no.1, 212 (1990) doi:10.1103/PhysRevB.42.212

    1990 doi

  19. [27]

    On the effective hydrodynamics of the fractional quantum Hall effect,

    A. G. Abanov, “On the effective hydrodynamics of the fractional quantum Hall effect,” J. Phys. A 46, 292001 (2013) doi:10.1088/1751-8113/46/29/292001 [arXiv:1212.0461 [cond-mat.str-el]]

    2013 arXiv

  20. [28]

    Topological fluids with boundaries and frac- tional quantum Hall edge dynamics: A fluid dynamics derivation of the chiral boson action,

    G. M. Monteiro, V. P. Nair and S. Ganeshan, “Topological fluids with boundaries and frac- tional quantum Hall edge dynamics: A fluid dynamics derivation of the chiral boson action,” Phys. Rev. B 109, no.17, 174525 (2024) doi:10.1103/PhysRevB.109.174525 [arXiv:2203.06516 [cond-...

    2024 arXiv

  21. [29]

    Integer quantum Hall transition: An alternative approach and exact results

    A. W. W Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, “Integer quantum Hall transition: An alternative approach and exact results.” Physical Review B 50, no. 11, 7526 (1994)

    1994

  22. [30]

    Vortices as fractons

    D. Doshi and A. Gromov, “Vortices as fractons.” Communications Physics 4, no. 1, 44 (2021)

    2021

  23. [31]

    Percolation, quantum tunnelling and the integer Hall effect

    J. T. Chalker and P. D. Coddington, “Percolation, quantum tunnelling and the integer Hall effect.” Journal of Physics C: Solid State Physics 21, no. 14, 2665 (1988)

    1988

  24. [32]

    The Confining Phase of Superstrings and Axionic Strings,

    S. J. Rey, “The Confining Phase of Superstrings and Axionic Strings,” Phys. Rev. D 43, 526-538 (1991) doi:10.1103/PhysRevD.43.526

    1991 doi

  25. [33]

    A Mechanism for Quark Confinement,

    C. G. Callan, Jr., R. F. Dashen and D. J. Gross, “A Mechanism for Quark Confinement,” Phys. Lett. B 66, 375-381 (1977) doi:10.1016/0370-2693(77)90019-3 31

    1977 doi

  26. [34]

    Toward a Theory of the Strong Interactions,

    C. G. Callan, Jr., R. F. Dashen and D. J. Gross, “Toward a Theory of the Strong Interactions,” Phys. Rev. D 17, 2717 (1978) doi:10.1103/PhysRevD.17.2717

    1978 doi

  27. [35]

    Meron Configurations in the Two-Dimensional O(3) Sigma Model,

    D. J. Gross, “Meron Configurations in the Two-Dimensional O(3) Sigma Model,” Nucl. Phys. B 132, 439-456 (1978) doi:10.1016/0550-3213(78)90470-4

    1978 doi

  28. [36]

    The emergence of isolated coherent vortices in turbulent flow,

    J. M. McWilliams, “The emergence of isolated coherent vortices in turbulent flow,” J. Fluid Mech. 146 21 (1984)

    1984

  29. [37]

    Self-organized criticality: an explanation of 1 /f noise,

    P. Bak, C. Tang, and K. Wisenfeld, “Self-organized criticality: an explanation of 1 /f noise,” Phys. Rev. Lett 59, 381 (1987)

    1987

  30. [38]

    Liouville field theory of fluctuating loops,

    J. Kondev, “Liouville field theory of fluctuating loops,” Phys. Rev. Lett. 78, 4320-4323 (1997) doi:10.1103/PhysRevLett.78.4320 [arXiv:cond-mat/9703113 [cond-mat]]

    1997 arXiv

  31. [39]

    On harmonic measure of critical curves,

    E. Bettelheim, I. Rushkin, I. A. Gruzberg and P. Wiegmann, “On harmonic measure of critical curves,” Phys. Rev. Lett. 95, 170602 (2005) doi:10.1103/PhysRevLett.95.170602 [arXiv:hep- th/0507115 [hep-th]]

    2005

  32. [40]

    Critical curves in conformally invariant statistical systems,

    I. Rushkin, E. Bettelheim, I. A. Gruzberg, and P. Wiegmann, “Critical curves in conformally invariant statistical systems,” Journal of Physics A: Mathematical and Theoretical40, 9, 2165 (2007)

    2007

  33. [41]

    Compactified Imaginary Liouville Theory,

    C. Guillarmou, A. Kupiainen and R. Rhodes, “Compactified Imaginary Liouville Theory,” [arXiv:2310.18226 [math-ph]]

    arXiv

  34. [42]

    Boundary conformal field theory approach to the criti- cal two-dimensional Ising model with a defect line,

    M. Oshikawa and I. Affleck, “Boundary conformal field theory approach to the criti- cal two-dimensional Ising model with a defect line,” Nucl. Phys. B 495, 533-582 (1997) doi:10.1016/S0550-3213(97)00219-8 [arXiv:cond-mat/9612187 [cond-mat]]

    1997 arXiv

  35. [43]

    Fractional quantum Hall effect and M-theory,

    C. Vafa, “Fractional quantum Hall effect and M-theory,” Adv. Theor. Math. Phys. 27, no.1, 1-36 (2023) doi:10.4310/ATMP.2023.v27.n1.a1 [arXiv:1511.03372 [cond-mat.mes-hall]]

    2023 arXiv

  36. [44]

    (2+1)-Dimensional Gravity as an Exactly Soluble System,

    E. Witten, “(2+1)-Dimensional Gravity as an Exactly Soluble System,” Nucl. Phys. B 311, 46 (1988) doi:10.1016/0550-3213(88)90143-5

    1988 doi

  37. [45]

    Conformal invariance of 2D quantum turbulence in an exciton-polariton fluid of light,

    R. Panico, A. S. Lanotte, D. Trypogeorgos, G. Gigli, M. De Giorgi, D. Sanvitto, and D. Bal- larini, “Conformal invariance of 2D quantum turbulence in an exciton-polariton fluid of light,” Applied Physics Reviews 10, no. 4 (2023)

    2023

  38. [46]

    Turbulence as statistics of vortex cells,

    A. A. Migdal, “Turbulence as statistics of vortex cells,” [arXiv:hep-th/9306152 [hep-th]]. 32

    arXiv

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Reviewed August 15, 2026 · model on record in the stance chip above.