REVIEW
Reduced Conformal Symmetry
High Energy Physics - Theory · Submitted 25 Sep 2020 · 17 references
Pith's verdict: Explicit continuum examples of sub-dimensional conformal symmetry, with a solid central construction; the main soft spot is an asserted, not shown, Lie algebra identification that is fixable.
T0 · AI review
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Referee report
The paper
The paper claims
Four free 2+1-dimensional scalar theories each carry two independent conformal symmetries acting in the two spatial-time planes, which together exhaust or enlarge the spacetime symmetry algebra.
The reading
What carries the argument
The central object is the mode-expansion trick: expand $\varphi(t,x,y)$ in Fourier modes in one spatial direction, writing $\varphi(t,x,y)=\int dk_y\, e^{iyk_y}\varphi_{k_y}(t,x)/\sqrt{2\pi}$, so that the transverse momentum $k_y$ becomes a fixed parameter of a lower-dimensional theory. Each of the four models then reduces to a known free system—a massless relativistic boson, a free non-relativistic particle with space and time interchanged, or a holomorphic fermion—whose conformal or Schrodinger symmetry is known. The sub-dimensional generators are re-expressed in the original coordinates, where the $k_y$-dependent phases become local coordinate transformations in $y$. For the (2,2) model the same argument is recast in light-cone coordinates, where $\partial_x\partial_y$ turns into a sum of squares and the two sub-dimensional Schrodinger symmetries assemble into the full conformal symmetry of a relativistic free boson in 2+1 dimensions.
Extended reading notes
Core claim
On its own terms, the paper establishes that the Lagrangians $|\partial_t\varphi|^2+|\partial_x\partial_y\varphi|^2$, $|\partial_t\varphi|^2+\varphi^*\partial_x\partial_y\varphi$, $i\varphi^*\partial_t\varphi+\varphi^*\partial_x\partial_y\varphi$, and $i\varphi^*\partial_t\varphi+|\partial_x\partial_y\varphi|^2$ each have two sub-dimensional conformal symmetries in the two primary 1+1 planes. For the (2,2) model, the two non-relativistic Schrodinger-type symmetries generate $\mathrm{sp}(4,\mathbb{R})\simeq\mathrm{so}(3,2)$, the same algebra as a relativistic 2+1-dimensional CFT, because that model is a free boson in light-cone coordinates; for the (1,2) model, two relativistic sub-dimensional symmetries generate a modified Schrodinger algebra. For the (2,4) and (1,4) models, the two sub-dimensional algebras do not commute and do not appear to close, hinting at algebras with infinitely many generators. The paper also shows that the dipole-breaking deformation preserves the sub-dimensional conformal symmetry in the (1,4) model while breaking it in the (2,4) model, and that the (1,4) model has an exact ground-state degeneracy that grows with system length rather than volume.
Load-bearing premise
The load-bearing premise is the paper's reliance on the claim from [3] that the free actions include all relevant and marginal terms consistent with the symmetries; if that completeness claim fails, the four models remain valid field theories with the stated symmetries, but they would no longer show that sub-dimensional conformal symmetry is the generic low-energy symmetry of the corresponding fracton lattice models.
Editorial extensions
If this is right
- If the four free actions are the most general low-energy actions consistent with the subsystem symmetries, then sub-dimensional conformal symmetry is a generic property of those fractonic RG fixed points, not a fine-tuned accident.
- The (2,2) model shows that two non-relativistic Schrodinger symmetries can combine into the full relativistic conformal algebra $\mathrm{so}(3,2)$, so observing 2+1 conformal symmetry does not by itself imply underlying Lorentz invariance.
- The (1,4) model has an exact ground-state degeneracy of $2N_x+2N_y+1$ on a lattice, providing a sharp finite-volume signature that cannot be lifted by quantum effects.
- The (1,4) sub-dimensional Schrodinger algebra can be deformed continuously to the standard Schrodinger algebra, suggesting a route to interacting theories by starting with an interacting non-relativistic CFT and deforming.
- In higher spatial dimensions, gradient terms of the form $(\partial_{x_1}\cdots\partial_{x_D}\varphi)^2$ give one sub-dimensional scaling symmetry per direction, which the same Fourier-mode construction should extend to conformal symmetries.
Reading between the lines
- If the completeness claim from [3] is accepted, then interacting fractonic RG fixed points with these subsystem conformal symmetries may be impossible as exact long-distance limits, and the paper's own deformation idea implies that interactions could only appear away from the free point; that dichotomy is testable in lattice models.
- The non-closure of the two Witt algebras in the (2,4) and (1,4) models suggests the true symmetry is an infinite-dimensional algebra, and a concrete next step would be to identify a known infinite algebra, such as a toroidal or $W$-type extension, inside the commutator closure.
- Because the sub-dimensional generators act locally on the original coordinates only through $k_y$-dependent phase factors, correlation functions with separated $x$ and $y$ insertions should be far less constrained than in ordinary CFTs; a direct check is to compute four-point functions in the (1,4) model and test whether they obey an effective 1+1 conformal bootstrap in each channel.
- The same construction in 3+1 dimensions would yield three mutually non-commuting sub-dimensional conformal algebras, and the Fourier-mode method can settle directly whether any of them close into a larger finite algebra.
Original abstract
We construct field theories in $2+1$ dimensions with multiple conformal symmetries acting on only one of the spatial directions. These can be considered a conformal extension to "subsystem scale invariances", borrowing the language often used for fractons.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We construct field theories in $2+1$ dimensions with multiple conformal symmetries acting on only one of the spatial directions. These can be considered a conformal extension to "subsystem scale invariances", borrowing the language often used for fractons.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We construct field theories in $2+1$ dimensions with multiple conformal symmetries acting on only one of the spatial directions. | contribution 1.00 | 0.94 | 0.34 | asserted 1.00 |
| s1 | These can be considered a conformal extension to "subsystem scale invariances", borrowing the language often used for fractons. | interpretation 0.99 | 0.33 | 0.16 | hedged possibility 0.58 |
What was examined, and what was not
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References
How the literature uses it
15 arXiv papers cite this in 23 sentences
Papers citing it · 14 sentences without a confident label
Compares with it 2 papers
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This model will also have blueuced conformal symmetry as those studied in [ref] .
Hydrodynamics of ideal fracton fluids · Hydrodynamics of chiral fractons · 0.83
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This is reminiscent of the reduced conformal symmetry recently discussed in [ref] .] This emergent symmetry has the effect that taking the limit of the relativistic NS-NS gravity action produces a NR action that leads to one equation of motion less than the ones that one obtains by taking the NR limit of the relativistic NS-NS gravity equations of motion.
A Non-Relativistic Limit of NS-NS Gravity · Conclusions · 0.82
Cites it as background 6 papers
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The theory based on the continuum action [ref] has been analyzed in [ref] .
The low-energy limit of some exotic lattice theories and UV/IR mixing · Introduction · 0.91
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See also [ref] for related discussions.] [ref] S = β_0/2 ∑_τ-link (Δ_τϕ- 2πn_τ)^2 + β/2 ∑_xy-plaq (Δ_xΔ_y ϕ- 2πn_xy)^2 + i∑_cube ϕ^xy (Δ_τn_xy - Δ_xΔ_y n_τ) on a special torus.
Global Dipole Symmetry, Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons · Relation to 2+1d $ϕ$-theory · 0.85
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(See also [ref] for related discussions on this theory.)
A Modified Villain Formulation of Fractons and Other Exotic Theories · Continuum limit · 1.00
What it cites · 15 sentences without a confident label
Builds on 1 paper
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The four models we consider in this paper are the complex version of the scalar field theory from [ref] , as well as three simple generalizations of this theory.
Exotic Symmetries, Duality, and Fractons in 2+1-Dimensional Quantum Field Theory · Introduction · 0.98
Compares with 1 paper
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Unlike in [ref] and the $(2,4)$ model, the ground state degeneracy of the $(1,4)$ model is exact, and cannot be lifted by quantum effects.
Exotic Symmetries, Duality, and Fractons in 2+1-Dimensional Quantum Field Theory · Spectral analysis and comparison of the $(1,4)$ and $(2,4)$ models · 1.00
Cites as support 1 paper
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The existence of an infinite number of generators, along with the fact that this free action includes all relevant and marginal terms consistent with the global symmetries [ref] , may be enough to forbid any interacting theories with the same sub-system conformal symmetry.
Exotic Symmetries, Duality, and Fractons in 2+1-Dimensional Quantum Field Theory · The $(2,4)$ model · 0.98
Discussion
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