High Energy Physics - Theory · arXiv:hep-th/0508079 v1 · submitted 11 Aug 2005 · 2 references
On Schrödinger superalgebras
The paper says
“We construct, using the supersymplectic framework of Berezin, Kostant and others, two types of supersymmetric extensions of the Schrödinger algebra (itself a conformal extension of the Galilei algebra).”
more of the abstract
An `I-type' extension exists in any space dimension, and for any pair of integers N+ and N−. It yields an N = N+ + N− superalgebra, which generalizes the N=1 supersymmetry Gauntlett et al. found for a free spin-$\half$ particle, as well as the N=2 supersymmetry of the fermionic oscillator found by Beckers et al. In two space dimensions, new, `exotic' or `IJ-type' extensions arise for each pair of integers ν+ and ν−, yielding an N = 2(ν+ + ν−) superalgebra of the type discovered recently by Leblanc et al. in non relativistic Chern-Simons theory. For the magnetic monopole the symmetry reduces to $\o(3)\times\osp(1/1)$, and for the magnetic vortex it reduces to $\o(2)\times\osp(1/2)$.
from the abstract
Pith says
Machine review · deepseek-v4.1-flash · 13 Nov 2026 · what was read was not recorded · not checked by a person
Reviewed in 2026 with later literature in view.
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Holds up. The canonical construction paper for super-Schrödinger algebras: explicit, checkable, and it correctly identifies the Gauntlett, Beckers, and Leblanc-Lozano-Min algebras as special cases. This line is wrong
8 minor
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8 minor objections
The paper says, at Section 5, Eq. (5.5)
Pith says: minor objection 1 of 8
The exhaustiveness claim for n > 2 rests on the fact that the commutant of o(n) in so(n) is trivial for n > 2 (including n = 4, where so(4) is semisimple); the text states the conclusion without spelling out this fact, and one sentence making it explicit would make the classification claim self-contained.
The paper says, at Section 4, Eq. (4.10)
Pith says: minor objection 2 of 8
The phrase 'we claim that' introduces a result that is derived from (4.9) and (4.11); 'we find' or 'one finds' would be more appropriate for a derived result.
show 6 more
The paper says, at Section 8, Eq. (8.4)
Pith says: minor objection 3 of 8
The text says the supermoment map 'yields now 16 super charges', but the list (8.4) contains 13 generators (or 16 components counting the vector generators G, P and Ξ); '16 generators' would be clearer than '16 super charges', since only six of them are odd.
The paper says, at Section 9, Eq. (9.8)
Pith says: minor objection 4 of 8
The minimal-coupling symplectic form requires dF = 0, which fails at the vortex and monopole singularities; the paper should state explicitly that the analysis is on the punctured plane or space, or that the singular point is excluded.
The paper says, at Reference [15]
Pith says: minor objection 5 of 8
Reference [15] is listed as unpublished ('Tours Preprint 60/93 (unpublished)') and is cited in Section 1 for the vortex osp(1/2) result; since Section 9 provides the derivation, the authors should make clear that Section 9 is the primary support and [15] is cited for priority.
The paper says, at Section 6, Eq. (6.10)
Pith says: minor objection 6 of 8
The phrase 'spin-½N' is ambiguous; 'spin N/2' would be clearer.
The paper says, at Section 5, Eq. (5.3)
Pith says: minor objection 7 of 8
The text discusses only the generic case λ_j ≠ 0; the non-generic case λ_j = 0 gives Ω^j = 0 (since an antisymmetric matrix with square zero vanishes), which is the I-type case, and a sentence noting this would complete the argument.
The paper says, at Abstract
Pith says: minor objection 8 of 8
The abstract's 'two types of supersymmetric extensions' could be read as exhaustive, whereas the body restricts the search to the affine orthosymplectic envelope iosp(n/m) and to the block Ansatz (5.2); a qualifier in the abstract would align it with the body.
What would settle it
Verify the Section 8 dictionary by computing [J_LLM, Q1_LLM] and [J_LLM, Q*1_LLM] explicitly under the dictionary, using (8.6) and (A.5), and checking that the helicities are real and match LLM's stated commutators. If they match, the identification is confirmed; if not, the dictionary needs correction.
from the skeptic's reading
How the review went
4 readings by deepseek-v4.1-flash · 13 Nov 2026
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- Skeptic agrees
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- Referee accept: 8 minor
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The review in full
The review in brief, machine-written
Bottom line: this is the standard reference for super-Schrödinger algebras, and it holds up. The I-type family sch(n/N+,N-) and the planar exotic IJ-type family are constructed explicitly, and the paper shows by an explicit change of generators that the Gauntlett N=1, Beckers N=2, and Leblanc-Lozano-Min algebras sit inside them. That identification is the useful part, and it is done rather than asserted. One thing to know before you read it: this is the 1994 J. Math. Phys. paper (35, 2516), posted to arXiv in 2005. The novelty score of 6 is right relative to the 1993 literature, but nothing here is new now. Read it as the canonical construction, not a recent result. What it does well: the supersymplectic machinery is set up cleanly, from the contact form (4.1) through the affine orthosymplectic ansatz (4.2) and the closure conditions (4.3) to the block decomposition that yields the I-type algebra. The supermoment map (4.10) and the bracket formula (4.11) do the real work, and the resulting commutators (4.12-4.18) are written out in full. Section 8's dictionary to LLM is explicit enough to check by hand, and Section 9's monopole and vortex reductions are a useful payoff. Soft spots, in proportion: the search is confined to iosp(n/m), which the authors state in Section 1; that is a scope choice, not a hidden assumption. The n>2 exhaustiveness claim in Section 5 is conditional on the block ansatz (5.2), again stated. The exotic commutators are relegated to the Appendix as 'rather complicated', which is honest but means the reader either trusts the algebra or grinds it. Reference [15] is an unpublished Tours preprint, but the vortex result is re-derived in Section 9, so it is not load-bearing. Who it is for: anyone working on non-relativistic supersymmetry, Chern-Simons-matter, or dynamical symmetry in magnetic backgrounds. It is dense and specialist; a competent group can re-derive it by hand, but there is no code or formal check. Recommendation: cite it and keep it on the shelf. It is a solid construction paper, and the identifications are the reason to have it.
The strongest objection
I looked for a hidden assumption or an internal inconsistency in the central construction and did not find one. The I-type algebra (4.6-4.8) and the exotic algebra (8.2-8.10) both satisfy the graded Jacobi identity for representative triples I checked (e.g., (Q_j, Q_k, D) and (L, Q, S*) in the exotic case). The identifications with Gauntlett et al. (Section 6), Beckers et al. (Section 7), and LLM (Section 8) are supported by explicit generator dictionaries. The two scope limitations — the restriction to the affine orthosymplectic envelope iosp(n/m) and the block Ansatz (5.2) for the n>2 exhaustiveness claim — are stated in the text (Section 1 and Section 5 respectively), so they are acknowledged rather than hidden. The one place where the paper asserts rather than shows a check is the Section 8 dictionary to LLM, but the dictionary is explicit and a reader can verify it; this is a reproducibility detail, not a load-bearing flaw.
The paper is a careful construction in mathematical physics. The central claim — that two families of super-Schrödinger algebras exist and generalize the known N=1, N=2, and Chern-Simons algebras — is supported by explicit commutation relations, a consistent supermoment map, and concrete generator identifications. I checked representative Jacobi identities in both families and found no inconsistency. The limitations the reader identified (iosp envelope, Ansatz (5.2)) are stated in the text and do not undermine the construction. The only unshown check is the Section 8 dictionary to LLM, which is explicit enough to verify by hand. I would not change the reader's ACCEPT verdict.
The referee's report
The construction is explicit and parameter-free, derived from the supersymplectic two-form (2.3) and the moment-map formula (4.11), with the closure conditions (4.3) and the full supercommutators (4.7-4.8) written out for verification. It unifies several previously scattered results into a single framework and provides the exotic IJ-type family that contains the LLM algebra. The identification of the LLM algebra as the ν=1 exotic extension in Section 8 is a concrete, checkable external comparison supported by an explicit generator dictionary. The paper is a useful reference for the non-relativistic supersymmetry literature.
Pith's reading of the paper (a machine's summary, not the authors')
The Schrödinger algebra is the symmetry algebra of a free non-relativistic quantum particle: it contains translations, rotations, Galilean boosts, plus dilatations and expansions that together form an sl(2,R) 'conformal' piece. Physicists had already found supersymmetric versions of it for a single spinning particle (N=1, Gauntlett et al.) and for the fermionic harmonic oscillator (N=2, Beckers et al.), and a distinct N=2 version in planar Chern-Simons theory (Leblanc, Lozano, Min). This paper asks: what is the general classification? The authors embed the Schrödinger algebra inside the affine orthosymplectic algebra, add Grassmann odd variables, and solve the closure conditions. They obtain an 'I-type' family sch(n/N+,N-) in any spatial dimension n, labelled by two integers, with structure (o(n) x osp(1/N+,N-)) semidirect h(n/N+,N-). In two dimensions a second 'exotic' family appears, where the odd generators can carry an extra antisymmetric piece built from the planar rotation generator; this yields N=2(nu+ + nu-) Q-type and S-type charges but only nu T-type charges. The authors then check that the known N=1, N=2, and Chern-Simons algebras are special cases, and work out the free particle, harmonic oscillator, monopole, and magnetic vortex examples. The results are explicit commutation relations that a specialist can verify by direct computation.
Core claim. The paper constructs two families of super-Schrödinger algebras: an I-type sch(n/N+,N-) = (o(n) x osp(1/N+,N-)) semidirect h(n/N+,N-) existing in any dimension n for any pair of integers (N+,N-), and a two-dimensional exotic IJ-type sche(nu+,nu-) with N=2(nu+ + nu-) Q-type and S-type charges but only nu T-type charges. The exotic family is claimed to contain the Leblanc-Lozano-Min Chern-Simons superalgebra as the special case nu+=1, nu-=0, recovered in Section 8 by an explicit change of generators.
The premise it rests on. The entire search is restricted to supersymmetric extensions of sch(n) that sit inside the affine orthosymplectic algebra iosp(n/m+,m-), as stated in Section 1 ('We look therefore for supersymmetric extensions of sch(n) within iosp(n/m)') and implemented by the contact-transformation Ansatz (4.2). If a physically relevant super-Schrödinger algebra existed outside this affine orthosymplectic envelope, the construction would not find it. The exhaustiveness claim for n>2 in Section 5 is further conditional on the block Ansatz (5.2) and the scalar-coefficient form (4.5).
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The paper
Abstract
We construct, using the supersymplectic framework of Berezin, Kostant and others, two types of supersymmetric extensions of the Schrödinger algebra (itself a conformal extension of the Galilei algebra). An `I-type' extension exists in any space dimension, and for any pair of integers N+ and N−. It yields an N = N+ + N− superalgebra, which generalizes the N=1 supersymmetry Gauntlett et al. found for a free spin-$\half$ particle, as well as the N=2 supersymmetry of the fermionic oscillator found by Beckers et al. In two space dimensions, new, `exotic' or `IJ-type' extensions arise for each pair of integers ν+ and ν−, yielding an N = 2(ν+ + ν−) superalgebra of the type discovered recently by Leblanc et al. in non relativistic Chern-Simons theory. For the magnetic monopole the symmetry reduces to $\o(3)\times\osp(1/1)$, and for the magnetic vortex it reduces to $\o(2)\times\osp(1/2)$.
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. No sentence in this record was written by a model.
abstract compiled · 2026-09-23
The authors' main claim, as the classifier picked it out 0.85
“We construct, using the supersymplectic framework of Berezin, Kostant and others, two types of supersymmetric extensions of the Schr\"odinger algebra (itself a conformal extension of the Galilei algebra).”
How the main claim is stated: asserted 1.00
What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.45 · compares with prior results 0.94 · mentions code or data availability 0.01
The abstract, sentence by sentence
This table labels the abstract only (arXiv listing abstract v1). The number is the probability the classifier gave the label.
| Sentence | Role | The authors' own work | States the main conclusion | Strength |
|---|---|---|---|---|
| We construct, using the supersymplectic framework of Berezin, Kostant and others, two types of supersymmetric extensions of the Schr\"odinger algebra (itself a conformal extension of the Galilei algebra). | contribution 0.99 | 0.94 | 0.39 | asserted 1.00 |
| An `$I$-type' extension exists in any space dimension, and for any pair of integers $N_+$ and $N_-$. | result 0.86 | 0.84 | 0.17 | asserted 1.00 |
| It yields an $N=N_++N_-$ superalgebra, which generalizes the N=1 supersymmetry Gauntlett et al. found for a free spin-$\half$ particle, as well as the N=2 supersymmetry of the fermionic oscillator found by Beckers et al. In two space dimensions, new, `exotic' or `$IJ$-type' extensions arise for each pair of integers $\nu_+$ and $\nu_-$, yielding an $N=2(\nu_++\nu_-)$ superalgebra of the type discovered recently by Leblanc et al. in non relativistic Chern-Simons theory. | result 0.88 | 0.67 | 0.32 | asserted 1.00 |
| For the magnetic monopole the symmetry reduces to $\o(3)\times\osp(1/1)$, and for the magnetic vortex it reduces to $\o(2)\times\osp(1/2)$. | result 0.99 | 0.68 | 0.10 | asserted 1.00 |
What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00
What was examined, and what was not
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Levels run
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- S sections
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- C citations
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How the labels were made
Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,945 input tokens, $0.0002 reported.
Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. This record as JSON
References (2)
- [1] 1 M. Leblanc, G. Lozano and H. Min, Extended superconformal Galilean symmetry in Chern-Simons matter systems , Ann. Phys. (N.Y.) 219, 328 (1992). 2 R. Jackiw and So-Young Pi, Classical and quantal nonrelativistic Chern-Simons theor y, Phys. Rev. D42, 3500 (1990); G. V. Dunne, ... on Pith
- [2] graded Lie algebras , J. Math. Phys. 18, 155 (1977). 17 G. Marmo, G. Morandi, A. Simoni and E. C. G. Sudarshan, Quasi-invariance and central extensions, Phys. Rev. 37, 2196 (1988). 18 R. Puzalowski, Galilean supersymmetry, Acta Phys. Austriaca 50, 45 (1978); J. A. de Azc´ arra... on Pith
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How the literature uses it
32 arXiv papers cite this in 45 sentences
Papers citing it · 19 sentences without a confident label
Compares with it 3 papers
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Indeed, this extra ingredient is what distinguishes the construction in [ref] from several earlier ones [ref] .
Submaximal conformal symmetry superalgebras for Lorentzian manifolds of low dimension · Introduction · 0.92
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In the superconformal case the embedded algebra may be enhanced to a supersymmetric version of the algebra[ Although some supersymmetric extensions of algebra are discussed in [ref] , there would be no overlap with our result.
Super Schrodinger algebra in AdS/CFT · Introduction · 0.91
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However, this supersymmetric structure does not involve the introduction of other degrees of freedom such as Clifford or Grassmann variables as in e.g. [ref] .
Higher-spin symmetries of the free Schrodinger equation · Introduction · 0.91
Cites it as support 1 paper
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Thus the $N=2$ extension exists only in the planar case, in accord with [ref] .
Non relativistic SUSY in variants of the planar L\'evy-Leblond equation · Remarks, discussion · 0.96
Cites it as background 21 papers
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The super Schrödinger algebra [ref] consists of bosonic Schrödinger algebra ([ref]), kinematical superalgebra ([ref]), and the dynamical superalgebra ([ref]),([ref]),([ref]).
Enhanced Supersymmetry of Nonrelativistic ABJM Theory · Dynamical superalgebra · 0.88
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The commutative Landau problem for a non-relativistic electron with spin $$ has an $N=2$ supersymmetry [ref] , see also [ref] , and the conformal $so(2,1)$ symmetry is extended into a superconformal $osp(2|2)$ symmetry, with the angular momentum as central charge [ref] .
Super-extended noncommutative Landau problem and conformal symmetry · Introduction · 0.85
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SUSY extensions of the Schr�dinger symmetry were considered previously [ref] [ref] [ref] [ref] [ref] .
Non relativistic SUSY in variants of the planar L\'evy-Leblond equation · Introduction · 0.99
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