Pith. sign in

REVIEW 4 major objections 5 minor 128 references

Notes on su$(1,2)\oplus$u$(1)$ Chern-Simons theory and Torsional Newton-Cartan gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read This paper argues that the three-dimensional Chern-Simons gauge theory with algebra su(1,2)⊕u(1) is equivalent to torsional Newton-Cartan gravity, and that its large-speed-of-light expansion reproduces the extended z=2 Schrödinger gravity…

desk verdict A credible seed-theory result for the extended z=2 Schrödinger TTNC gravity, with a solid 1/c matching; the stress-test's sign objection does not survive, but three structural claims remain under-derived. read the letter →

arxiv 2505.03322 v3 pith:I5YT7ROY submitted 2025-05-06 hep-th

Yang Lei , Dong Zhang This is my paper · ORCID
classification hep-th
keywords torsionalNewton-CartangravityChern-Simonstheorysu(12)algebranon-relativistic1/cexpansionSchrodingerW-algebranullreduction
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 is trying to show that a three-dimensional Chern-Simons gauge theory based on the algebra su(1,2)⊕u(1) is not a formal curiosity but a gravitational theory: it is equivalent to torsional Newton-Cartan gravity, the most general class of non-relativistic gravity with a clock one-form that need not be hypersurface-orthogonal. The single u(1) factor is load-bearing: it supplies a curvature constraint that fixes the spin connection uniquely, curing the degeneracy that made the pure su(1,2) theory fail to describe gravity. If the claim is right, the same gauge theory is the 'seed theory' from which the known Chern-Simons twistless-torsional Schrödinger gravity arises by a large-speed-of-light (1/c) expansion, and its vacuum is connected to a four-dimensional Ω-deformed background by null reduction. A sympathetic reader would care because this gives a concrete origin story for non-relativistic holographic models and links them to z=2 Lifshitz geometry, Spin Matrix theory, and a bosonic analogue of super-BMS symmetry.

What carries the argument

The machinery is a gauge field A valued in su(1,2)⊕u(1), written in a basis where H,D,K span an sl(2) and P_a,G_a carry spin one half, with a non-degenerate bilinear product whose two parameters c_S and c_U control the action. The paper imposes the curvature constraints (2.10)—zero curvatures for H, spatial translations, the u(1) generator N, and the dilatation D—and uses them in two ways: they algebraically determine the spin connection fields from the vielbein and M, turning the topological gauge theory into a torsional Newton-Cartan geometry, and their Bianchi consequences organize the off-shell action. The second moving part is the 1/c expansion ansatz, which sends the clock one-form to cτ+β, the dilatation field to b−(1/2c)α, the special-conformal field U to f/c plus subleading terms, and so on; inserting this into the Chern-Simons action reproduces, order by order, the action of the extended z=2 Schrödinger gravity. The third is the flatness-equation technology used for vacua and asymptotic symmetries: separating radial dependence by a group element, solving the flatness condition in two dimensions, and reading off conserved charges from the bilinear product, which yields the nonlinear algebra identified as W_3^(2) with an affine u(1).

What would settle it

Find a smooth triple (E0_μ, E^a_μ, M_μ) for which the curvature constraints (2.10), in particular R(N)=0, admit two distinct solutions for the spin connection (Ω_μ, Ω^a_μ). The paper presents eq. (2.11) as the unique solution; exhibiting a second solution for any configuration would break the identification of the Chern-Simons theory with a unique torsional Newton-Cartan gravity, since the action would then depend on which spin connection is chosen.

Watch

Extended reading notes

Core claim

The paper argues that the three-dimensional Chern-Simons gauge theory with algebra su(1,2)⊕u(1) is, on shell, a torsional Newton-Cartan gravity: imposing the curvature constraints R(H)=R^a(P)=R(N)=R(D)=0 determines the spin connection from the vielbein and the u(1) gauge field M, and the Chern-Simons action becomes a gravitational action of fully torsional type, with E0∧dE0≠0. Its central constructive claim is that the 1/c expansion of this action, with the field rescaling given in the paper, reproduces exactly the action of the extended z=2 Schrödinger gravity of [26], with the three independent coupling constants identified as c1=cS/c, c2=cS, c3=2cS/3+2cU. Hence su(1,2)⊕u(1) Chern-Simons theory is the 'seed theory' that generates Chern-Simons twistless-torsional Newton-Cartan gravity. The paper also derives an infinite asymptotic symmetry algebra for a flatness solution, identifies it as the nonlinear W_3^(2) algebra with an affine u(1), and shows that the vacuum solution is conformal to the null reduction of a four-dimensional Ω-deformed background, relating the z=2 Lifshitz vacuum to that geometry.

Load-bearing premise

The construction assumes that requiring certain curvatures to vanish—in particular the new u(1) curvature—uniquely fixes the rotation gauge field from the metric data, removing the degeneracy that made the pure su(1,2) theory nongravitational; the paper states this resolution but does not give a full proof of uniqueness.

Editorial extensions

If this is right

  • If the central claim is right, the extended z=2 Schrödinger gravity of [26] is not an independent construction: it is the leading orders of the 1/c expansion of su(1,2)⊕u(1) Chern-Simons theory, so any property of the former is inherited from the latter.
  • The z=2 Lifshitz vacuum of Schrödinger gravity should be understood as a limit of the su(1,2)⊕u(1) vacuum, which is conformally related to the null reduction of the four-dimensional Ω-deformed background; this links non-relativistic Lifshitz holography to a four-dimensional seed geometry.
  • The asymptotic symmetry algebra of the seed theory is W_3^(2)⊕u(1), a nonlinear W-algebra with an affine u(1) current, and its 1/c expansion yields the bosonic analogue of N=2 super BMS; the same algebra can therefore be viewed as the conformal completion underlying that identification.
  • Because the dilatation and rotation gauge fields transform under Galilean boosts, any affine connection built from the vielbein postulates is either boost-invariant or dilatation-invariant, not both; this distinguishes this torsional Newton-Cartan theory from the twistless-torsional Schrödinger-gravity construction.
  • The same gauging logic extends to odd dimensions: su(1,3)⊕u(1) produces five-dimensional torsional Newton-Cartan geometry with the same boost-dependent gauge fields, as sketched in Appendix C.

Reading between the lines

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

  • The paper leaves open the four-dimensional seed; a concrete extension would be to perform the 1/c expansion of four-dimensional conformal gravity and check whether its null reduction reproduces the su(1,2)⊕u(1) Chern-Simons action, turning the 'seed theory' language into a derivation.
  • The nonlinearity of W_3^(2) suggests that the full asymptotic symmetry of the torsional vacuum may be nonlinear; a test is to compute the charges of the background directly and see whether the Sugawara shift used in the paper is necessary for closure, as it was for the flat-space analogue.
  • If su(1,2)⊕u(1) Chern-Simons gravity is the holographic dual of the ground state of Spin Matrix theory, then the free functions C_1(t), C_2(t) in the excited solution should correspond to specific boundary operators; matching their charges would provide a concrete holographic dictionary.
  • The resolution of the z=2 degeneracy by a u(1) suggests an analogous fix for z≠2: adding suitable u(1) gauge fields to sl(z+1,R) Chern-Simons theories may cure their degeneracy and yield Lifshitz Newton-Cartan gravity with general integer z, a direction the paper names as open.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies the three-dimensional Chern-Simons theory with gauge algebra su(1,2)⊕u(1), proposes that it describes torsional Newton-Cartan (TNC) gravity, and claims that a 1/c expansion of the action reproduces the extended z=2 Schrödinger gravity of Hartong-Lei-Obers. The paper also discusses the vacuum and excitation solutions, identifies the vacuum with the null reduction of a four-dimensional Ω-background up to a conformal factor, and proposes that the infinite-dimensional symmetry of the theory is the W_3^{(2)}⊕u(1) algebra, which it relates to a bosonic analogue of the N=2 super BMS algebra. The algebraic setup is explicit, and the vacuum solution (2.45) is stated in a checkable form, but the central action derivation and the 1/c reduction contain serious inconsistencies.

Significance. If the main claims were correct, the paper would provide a valuable unification: it would identify su(1,2)⊕u(1) Chern-Simons theory as the seed theory generating the torsional Newton-Cartan gravity and the extended Schrödinger gravity of [26] via 1/c expansion, and it would connect this to four-dimensional Ω-backgrounds and Spin Matrix theory. The paper also contains useful explicit material, including the bilinear product, the vacuum solution, and a concrete proposal for the infinite-dimensional symmetry algebra. However, the central action (2.34) is not correctly derived from the stated invariant bilinear product, and the parameter identification (2.37) is internally inconsistent. Because these issues affect the main equivalence and the claimed 1/c reduction, the central results are not currently supported.

major comments (4)
  1. [Section 2.2, Eq. (2.34)] Equation (2.34) is not the Chern-Simons action obtained from the bilinear product (2.5). From the gauge field (2.6), the terms involving J and N in tr(A∧dA) are c_U(Ω∧dM + M∧dΩ), because B(J,N)=B(N,J)=c_U. This combination is a total derivative, Ω∧dM + M∧dΩ = -d(M∧Ω) up to sign, and therefore drops out on a closed manifold. Equation (2.34) instead contains 2c_U M∧dΩ with no Ω∧dM term. This is not a total derivative and cannot be obtained from the invariant bilinear product by integration by parts. Consequently the equations of motion of the stated action do not impose R(N)=0, the spin-connection solution (2.11) is not justified, and the 1/c reduction to (2.36) is not a reduction of the Chern-Simons theory defined by (2.5).
  2. [Section 2.2, Eqs. (2.34)-(2.37)] The parameter identification (2.37) is inconsistent with the action (2.34) even taken at face value. The coefficient of Ω∧dΩ in (2.34) is 2c_S/3 + 2c_U/3, and under the ansatz (2.35) with Ω=ω+O(c^{-1}) this gives c_3=2(c_S+c_U)/3 in the reduced action, not c_3=2c_S/3+2c_U as stated in (2.37). The derivation of (2.36) from (2.34)-(2.35) is asserted without being shown, and this coefficient mismatch indicates that the claimed exact match to the Schrödinger action of [26] cannot hold as written.
  3. [Section 2.1, Eq. (2.11)] The statement after (2.10) that the constraints R(H)=R(P)=R(N)=R(D)=0 uniquely determine the spin connection from the vielbein and M is not demonstrated. The explicit solution (2.11) is presented without substitution into the four constraints and without an argument for uniqueness. Since this is precisely the step that resolves the degeneracy of the pure su(1,2) theory identified in [25] and defines the metric-like TNC theory, the paper should either prove the claim or provide a derivation that makes the uniqueness evident.
  4. [Section 3, Eqs. (3.17)-(3.20)] The claimed infinite-dimensional symmetry algebra W_3^{(2)}⊕u(1) is presented through the Poisson brackets (3.18) and Fourier modes (3.20), but the Jacobi identities for the nonlinear bracket are not checked, and the Sugawara shift (3.17) is asserted without derivation. Because the nonlinearity of the algebra is emphasized as a key feature, these verifications are necessary before the asymptotic-symmetry claim can be accepted.
minor comments (5)
  1. [Abstract] The abstract contains the duplicated phrase 'dual to dual to'; this should be corrected.
  2. [Section 2.1, after Eq. (2.31)] The text 'both E_0^μ and h^{μν}=δ^{ab}E_a^μ E_b^ν are boost invariant. are boost invariant.' repeats 'are boost invariant' and should be cleaned up.
  3. [Introduction, references] The citation '[42,43,43–47]' lists reference [43] twice; the duplicate should be removed.
  4. [Section 2.1, Eq. (2.10)] The notation for the inverse vielbein in (2.9) and the subsequent use of E^μ_0 versus E_0^μ is occasionally confusing; a short clarification of the index placement would improve readability.
  5. [Section 2.2, below Eq. (2.34)] The text says that the spin connection Ω^a functions as the Lagrange multiplier to impose R^a(P)=0; this is only true after the other constraints are imposed, and the statement should be phrased more carefully.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; the central 1/c reduction is a direct computation against an independently published action, though the paper leans on self-citations for setup.

full rationale

I find no circular step in which a claimed prediction or first-principles result reduces by construction to its own inputs. The central derivation chain is a direct substitution: the su(1,2)⊕u(1) Chern-Simons action (2.34) is computed from the stated bilinear product (2.5) and curvature components (2.8); the 1/c expansion ansatz (2.35) is then inserted into (2.34); and the resulting expression (2.36) is matched to the published Schrödinger gravity action of [26] through the parameter identification (2.37). This is not a fit of parameters to a subset of data that is later relabeled as a prediction: the coefficients c1, c2, c3 of [26] were fixed in that independent work, and the paper's cS, cU are fixed before the expansion. The identification (2.37) is a dictionary between two independently defined parameter sets, and the reproduced action has content beyond the present paper's definitions. Self-citations appear in this chain, since [25], [26], and [42] are authored or co-authored by Y.L. or close collaborators, but they are used for the algebra basis, the known degeneracy of pure su(1,2), and the target Schrödinger gravity action, all of which are published results with derivations outside the present paper. None of these citations is the only evidence for the central equivalence. Two concerns noted in the reading merit separate checks but are not circularity: the claim that the constraints (2.10) uniquely determine the spin connection via (2.11) is asserted rather than proved, and the skeptic's cross-term comparison of (2.34) with (2.5) may indicate a correctness issue in the displayed action. Neither is a case of a prediction reducing to its input by construction. Accordingly the circularity score is low.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new particles, forces, or dimensions are postulated. The construction uses standard gauging and Chern-Simons technology plus one u(1) generator already present in the algebra. The only free constants are the two coefficients of the invariant bilinear form; the paper's match to the known Schrödinger gravity action is a parameter mapping, not a fit to data.

free parameters (2)
  • c_S = unspecified
    Coefficient of the su(1,2) part of the non-degenerate bilinear form (2.5); it sets the overall scale of the action (2.34) and maps to c1/c and c2 in the 1/c limit.
  • c_U = unspecified; set to -c_S in the fully coupled limit
    Coefficient of the u(1) part of the bilinear form (2.5); it determines c3 = 2c_S/3 + 2c_U in the Schrödinger gravity action (2.36). The simplified action (2.42) assumes c_U = -c_S.
assumptions (4)
  • standard math Chern-Simons theory on a non-semisimple algebra requires a non-degenerate invariant bilinear form (Nappi-Witten construction).
    Used to define the action (2.1) and the trace (2.5); the u(1) extension is introduced to make the bilinear form non-degenerate.
  • domain assumption The gauging procedure with curvature constraints (2.10) defines the gravitational theory from the gauge theory.
    This is the standard method of identifying Newton-Cartan gravity from Chern-Simons theory, following [26,28,29,30]; the constraints R(H)=R^a(P)=R(N)=R(D)=0 are imposed by hand.
  • domain assumption The algebra su(1,2)⊕u(1) is the conformal completion of the z=2 Lifshitz algebra and arises from the null reduction of the 4d Ω-background.
    Cited from [42,43] and used in the introduction and in the interpretation of the vacuum solution (2.47).
  • domain assumption The W_3^(2) algebra is the bosonic analogue of the N=2 superconformal algebra, and the algebra (3.20) is identified with W_3^(2)⊕u(1).
    Based on [12,99,100,102]; the Jacobi identities of (3.20) are not verified in the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Notes on su$(1,2)\oplus$u$(1)$ Chern-Simons theory and Torsional Newton-Cartan gravity." pith.science (2026). https://pith.science/paper/I5YT7ROY

@misc{pith2026250503322,
  author       = {Pith},
  title        = {Pith review of: Notes on su$(1,2)\oplus$u$(1)$ Chern-Simons theory and Torsional Newton-Cartan gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I5YT7ROY}},
  note         = {Machine review of arXiv:2505.03322}
}
abstract

In this study, we investigate three-dimensional torsional Newton-Cartan (TNC) gravity by gauging the su$(1,2)\oplus$u$(1)$ algebra and construct its action using the Chern-Simons theory. This TNC exhibits novel features, including the fact that the gauge fields associated with both dilatation and rotation symmetries transform non-trivially under Galilean boosts. This theory also reproduces the Schr\"odinger gravity acquired by gauging the extended $z=2$ Schr\"odinger algebra arXiv:1604.08054 via a large speed of light ($1/c$)-expansion. In particular, we explain that the $z=2$ Lifshitz vacuum solution appearing in Schr\"odinger gravity is related to the null reduction of 4d $\Omega$-background up to a conformal factor. Based on these results, we revisit the identification between the extended Schr\"odinger algebra and bosonic analogue of super BMS algebra arXiv:1905.13154. We interpret that this relation originates from the $\mathcal{W}_3^{(2)}$ algebra which acts as the bosonic analogue of $\mathcal{N}=2$ superconformal algebra.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

128 extracted references · 8 canonical work pages

  1. [26]

    Hartong, Y

    J. Hartong, Y. Lei and N. A. Obers, Nonrelativistic Chern-Simons theories and three-dimensional Hoˇ rava-Lifshitz gravity, Phys. Rev. D 94 (2016) 065027, [1604.08054]

    arXiv 2016

  2. [25]

    Lei and S

    Y. Lei and S. F. Ross, Connection versus metric description for non-AdS solutions in higher-spin theories, Class. Quant. Grav. 32 (2015) 185005, [ 1504.07252]

    arXiv 2015

  3. [1]

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

    E. Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System , Nucl. Phys. B 311 (1988) 46

    1988

  4. [2]

    J. D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity , Commun. Math. Phys. 104 (1986) 207–226

    1986

  5. [3]

    Campoleoni, S

    A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields , JHEP 11 (2010) 007, [1008.4744]

    arXiv 2010

  6. [4]

    Henneaux and S.-J

    M. Henneaux and S.-J. Rey, NonlinearWinfinity as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity , JHEP 12 (2010) 007, [1008.4579]. – 37 –

    arXiv 2010

  7. [5]

    Bergshoeff, M

    E. Bergshoeff, M. P. Blencowe and K. S. Stelle, Area Preserving Diffeomorphisms and Higher Spin Algebra, Commun. Math. Phys. 128 (1990) 213

    1990

  8. [6]

    M. R. Gaberdiel and R. Gopakumar, An AdS3 Dual for Minimal Model CFTs , Phys. Rev. D 83 (2011) 066007, [ 1011.2986]

    arXiv 2011

Show all 128 references
  1. [7]

    M. R. Gaberdiel and R. Gopakumar, Minimal Model Holography, J. Phys. A 46 (2013) 214002, [ 1207.6697]

    2013 arXiv

  2. [8]

    Gutperle and P

    M. Gutperle and P. Kraus, Higher Spin Black Holes , JHEP 05 (2011) 022, [1103.4304]

    2011 arXiv

  3. [9]

    Campoleoni and S

    A. Campoleoni and S. Fredenhagen, Higher-Spin Gauge Theories in Three Spacetime Dimensions, Lect. Notes Phys. 1028 (2024) 121–267, [ 2403.16567]

    2024 arXiv

  4. [10]

    K. S. Kiran, C. Krishnan and A. Raju, 3D gravity, Chern–Simons and higher spins: A mini introduction, Mod. Phys. Lett. A 30 (2015) 1530023, [ 1412.5053]

    2015 arXiv

  5. [11]

    Ammon, M

    M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, Black holes in three dimensional higher spin gravity: A review , J. Phys. A 46 (2013) 214001, [ 1208.5182]

    2013 arXiv

  6. [12]

    Afshar, M

    H. Afshar, M. Gary, D. Grumiller, R. Rashkov and M. Riegler, Non-AdS holography in 3-dimensional higher spin gravity - General recipe and example , JHEP 11 (2012) 099, [1209.2860]

    2012 arXiv

  7. [13]

    M. Gary, D. Grumiller and R. Rashkov, Towards non-AdS holography in 3-dimensional higher spin gravity , JHEP 03 (2012) 022, [ 1201.0013]

    2012 arXiv

  8. [14]

    M. Gary, D. Grumiller, S. Prohazka and S.-J. Rey, Lifshitz Holography with Isotropic Scale Invariance, JHEP 08 (2014) 001, [ 1406.1468]

    2014 arXiv

  9. [15]

    Gutperle, E

    M. Gutperle, E. Hijano and J. Samani, Lifshitz black holes in higher spin gravity , JHEP 04 (2014) 020, [ 1310.0837]

    2014 arXiv

  10. [16]

    Gutperle and Y

    M. Gutperle and Y. Li, Higher Spin Lifshitz Theory and Integrable Systems , Phys. Rev. D 91 (2015) 046012, [ 1412.7085]

    2015 arXiv

  11. [17]

    Beccaria, M

    M. Beccaria, M. Gutperle, Y. Li and G. Macorini, Higher spin Lifshitz theories and the Korteweg-de Vries hierarchy, Phys. Rev. D 92 (2015) 085005, [ 1504.06555]

    2015 arXiv

  12. [18]

    Kachru, X

    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points , Phys. Rev. D 78 (2008) 106005, [ 0808.1725]

    2008 arXiv

  13. [19]

    D. T. Son, Toward an AdS/cold atoms correspondence: A Geometric realization of the Schrodinger symmetry, Phys. Rev. D 78 (2008) 046003, [ 0804.3972]

    2008 arXiv

  14. [20]

    S. A. Hartnoll, Lectures on holographic methods for condensed matter physics , Class. Quant. Grav. 26 (2009) 224002, [ 0903.3246]

    2009 arXiv

  15. [21]

    Prohazka and M

    S. Prohazka and M. Riegler, Higher Spins Without (Anti-)de Sitter , Universe 4 (2018) 20, [ 1710.11105]. – 38 –

    2018 arXiv

  16. [22]

    Taylor, Lifshitz holography, Class

    M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) 033001, [ 1512.03554]

    2016 arXiv

  17. [23]

    Horava, Quantum Gravity at a Lifshitz Point , Phys

    P. Horava, Quantum Gravity at a Lifshitz Point , Phys. Rev. D 79 (2009) 084008, [0901.3775]

    2009 arXiv

  18. [24]

    Taylor, Non-relativistic holography, 0812.0530

    M. Taylor, Non-relativistic holography, 0812.0530

    arXiv

  19. [27]

    Hartong, E

    J. Hartong, E. Kiritsis and N. A. Obers, Lifshitz space–times for Schr¨ odinger holography, Phys. Lett. B 746 (2015) 318–324, [ 1409.1519]

    2015 arXiv

  20. [28]

    Andringa, E

    R. Andringa, E. Bergshoeff, S. Panda and M. de Roo, Newtonian Gravity and the Bargmann Algebra, Class. Quant. Grav. 28 (2011) 105011, [ 1011.1145]

    2011 arXiv

  21. [29]

    E. A. Bergshoeff, J. Hartong and J. Rosseel, Torsional Newton–Cartan geometry and the Schr¨ odinger algebra, Class. Quant. Grav. 32 (2015) 135017, [ 1409.5555]

    2015 arXiv

  22. [30]

    Hartong and N

    J. Hartong and N. A. Obers, Hoˇ rava-Lifshitz gravity from dynamical Newton-Cartan geometry, JHEP 07 (2015) 155, [ 1504.07461]

    2015 arXiv

  23. [31]

    Papageorgiou and B

    G. Papageorgiou and B. J. Schroers, A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions , JHEP 11 (2009) 009, [ 0907.2880]

    2009 arXiv

  24. [32]

    E. A. Bergshoeff and J. Rosseel, Three-Dimensional Extended Bargmann Supergravity, Phys. Rev. Lett. 116 (2016) 251601, [ 1604.08042]

    2016 arXiv

  25. [33]

    Papageorgiou and B

    G. Papageorgiou and B. J. Schroers, Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra, JHEP 11 (2010) 020, [ 1008.0279]

    2010 arXiv

  26. [34]

    Hartong, Y

    J. Hartong, Y. Lei, N. A. Obers and G. Oling, Zooming in on AdS 3/CFT2 near a BPS bound, JHEP 05 (2018) 016, [ 1712.05794]

    2018 arXiv

  27. [35]

    Chernyavsky and D

    D. Chernyavsky and D. Sorokin, Three-dimensional (higher-spin) gravities with extended Schr¨ odinger andl-conformal Galilean symmetries, JHEP 07 (2019) 156, [1905.13154]

    2019 arXiv

  28. [36]

    Gomis, A

    J. Gomis, A. Kleinschmidt, J. Palmkvist and P. Salgado-Rebolledo, Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity , JHEP 02 (2020) 009, [ 1912.07564]

    2020 arXiv

  29. [37]

    Avil´ es, E

    L. Avil´ es, E. Frodden, J. Gomis, D. Hidalgo and J. Zanelli, Non-Relativistic Maxwell Chern-Simons Gravity, JHEP 05 (2018) 047, [ 1802.08453]

    2018 arXiv

  30. [38]

    Ozdemir, M

    N. Ozdemir, M. Ozkan and U. Zorba, Three-dimensional extended Lifshitz, Schr¨ odinger and Newton-Hooke supergravity, JHEP 11 (2019) 052, [ 1909.10745]. – 39 –

    2019 arXiv

  31. [39]

    Concha, L

    P. Concha, L. Ravera and E. Rodr´ ıguez,Three-dimensional Maxwellian extended Bargmann supergravity, JHEP 04 (2020) 051, [ 1912.09477]

    2020 arXiv

  32. [40]

    Concha, C

    P. Concha, C. Henr´ ıquez-B´ aez and E. Rodr´ ıguez,Non-relativistic and ultra-relativistic expansions of three-dimensional spin-3 gravity theories , JHEP 10 (2022) 155, [2208.01013]

    2022 arXiv

  33. [41]

    Concha, M

    P. Concha, M. Ipinza, L. Ravera and E. Rodr´ ıguez, Non-relativistic three-dimensional supergravity theories and semigroup expansion method , JHEP 02 (2021) 094, [2010.01216]

    2021 arXiv

  34. [42]

    Baiguera, T

    S. Baiguera, T. Harmark, Y. Lei and Z. Yan, Conformal mapping of non-Lorentzian geometries in SU(1, 2) Conformal Field Theory , JHEP 03 (2025) 100, [ 2411.11951]

    2025 arXiv

  35. [43]

    Lambert, R

    N. Lambert, R. Mouland and T. Orchard, Non-Lorentzian SU(1, n) Spacetime Symmetry In Various Dimensions , Front. in Phys. 10 (2022) 864800, [ 2112.14860]

    2022 arXiv

  36. [44]

    Lambert and T

    N. Lambert and T. Orchard, Null reductions of the M5-brane , JHEP 12 (2020) 037, [2005.14331]

    2020 arXiv

  37. [45]

    Smith, Fermions with SU(1, n) spacetime symmetry , JHEP 11 (2023) 032, [2307.16624]

    J. Smith, Fermions with SU(1, n) spacetime symmetry , JHEP 11 (2023) 032, [2307.16624]

    2023 arXiv

  38. [46]

    Lambert and J

    N. Lambert and J. Smith, Non-relativistic M2-branes and the AdS/CFT correspondence, JHEP 06 (2024) 009, [ 2401.14955]

    2024 arXiv

  39. [47]

    Lambert and J

    N. Lambert and J. Smith, Non-relativistic intersecting branes, Newton-Cartan geometry and AdS/CFT, JHEP 07 (2024) 224, [ 2405.06552]

    2024 arXiv

  40. [48]

    Harmark, K

    T. Harmark, K. R. Kristjansson and M. Orselli, Decoupling limits of N=4 super Yang-Mills on R x S**3 , JHEP 09 (2007) 115, [ 0707.1621]

    2007 arXiv

  41. [49]

    Harmark and M

    T. Harmark and M. Orselli, Spin Matrix Theory: A quantum mechanical model of the AdS/CFT correspondence, JHEP 11 (2014) 134, [ 1409.4417]

    2014 arXiv

  42. [50]

    Harmark and N

    T. Harmark and N. Wintergerst, Nonrelativistic Corners of N = 4 Supersymmetric Yang–Mills Theory, Phys. Rev. Lett. 124 (2020) 171602, [ 1912.05554]

    2020 arXiv

  43. [51]

    Harmark, J

    T. Harmark, J. Hartong, N. A. Obers and G. Oling, Spin Matrix Theory String Backgrounds and Penrose Limits of AdS/CFT , JHEP 03 (2021) 129, [ 2011.02539]

    2021 arXiv

  44. [52]

    Baiguera, T

    S. Baiguera, T. Harmark and N. Wintergerst, Nonrelativistic near-BPS corners of N = 4 super-Yang-Mills with SU (1, 1) symmetry, JHEP 02 (2021) 188, [2009.03799]

    2021 arXiv

  45. [53]

    Baiguera, T

    S. Baiguera, T. Harmark, Y. Lei and N. Wintergerst, Symmetry structure of the interactions in near-BPS corners of N = 4 super-Yang-Mills, JHEP 04 (2021) 029, [2012.08532]

    2021 arXiv

  46. [54]

    Baiguera, T

    S. Baiguera, T. Harmark and Y. Lei, Spin Matrix Theory in near 1 8-BPS corners of N = 4 super-Yang-Mills, JHEP 02 (2022) 191, [ 2111.10149]. – 40 –

    2022 arXiv

  47. [55]

    Baiguera, T

    S. Baiguera, T. Harmark and Y. Lei, The Panorama of Spin Matrix theory , JHEP 04 (2023) 075, [ 2211.16519]

    2023 arXiv

  48. [56]

    Baiguera, Aspects of non-relativistic quantum field theories , Eur

    S. Baiguera, Aspects of non-relativistic quantum field theories , Eur. Phys. J. C 84 (2024) 268, [ 2311.00027]

    2024 arXiv

  49. [57]

    Kinney, J

    J. Kinney, J. M. Maldacena, S. Minwalla and S. Raju, An Index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209–254, [hep-th/0510251]

    2007 arXiv

  50. [58]

    H. K. Kunduri, J. Lucietti and H. S. Reall, Supersymmetric multi-charge AdS(5) black holes, JHEP 04 (2006) 036, [ hep-th/0601156]

    2006 arXiv

  51. [59]

    S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT , 1810.12067

    arXiv

  52. [60]

    Cabo-Bizet, D

    A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS 5 black holes, JHEP 10 (2019) 062, [1810.11442]

    2019 arXiv

  53. [61]

    Benini and E

    F. Benini and E. Milan, Black Holes in 4D N =4 Super-Yang-Mills Field Theory , Phys. Rev. X 10 (2020) 021037, [ 1812.09613]

    2020 arXiv

  54. [62]

    Harmark, Interacting Giant Gravitons from Spin Matrix Theory , Phys

    T. Harmark, Interacting Giant Gravitons from Spin Matrix Theory , Phys. Rev. D 94 (2016) 066001, [ 1606.06296]

    2016 arXiv

  55. [63]

    Duval, G

    C. Duval, G. Burdet, H. P. Kunzle and M. Perrin, Bargmann Structures and Newton-cartan Theory, Phys. Rev. D 31 (1985) 1841–1853

    1985

  56. [64]

    Julia and H

    B. Julia and H. Nicolai, Null Killing vector dimensional reduction and Galilean geometrodynamics, Nucl. Phys. B 439 (1995) 291–326, [ hep-th/9412002]

    1995 arXiv

  57. [65]

    M. H. Christensen, J. Hartong, N. A. Obers and B. Rollier, Torsional Newton-Cartan Geometry and Lifshitz Holography , Phys. Rev. D 89 (2014) 061901, [ 1311.4794]

    2014 arXiv

  58. [66]

    M. H. Christensen, J. Hartong, N. A. Obers and B. Rollier, Boundary Stress-Energy Tensor and Newton-Cartan Geometry in Lifshitz Holography , JHEP 01 (2014) 057, [1311.6471]

    2014 arXiv

  59. [67]

    Harmark, J

    T. Harmark, J. Hartong, L. Menculini, N. A. Obers and Z. Yan, Strings with Non-Relativistic Conformal Symmetry and Limits of the AdS/CFT Correspondence , JHEP 11 (2018) 190, [ 1810.05560]

    2018 arXiv

  60. [68]

    Dautcourt, PostNewtonian extension of the Newton-Cartan theory , Class

    G. Dautcourt, PostNewtonian extension of the Newton-Cartan theory , Class. Quant. Grav. 14 (1997) A109–A118, [ gr-qc/9610036]

    1997 arXiv

  61. [69]

    Van den Bleeken, Torsional Newton–Cartan gravity from the large c expansion of general relativity, Class

    D. Van den Bleeken, Torsional Newton–Cartan gravity from the large c expansion of general relativity, Class. Quant. Grav. 34 (2017) 185004, [ 1703.03459]. – 41 –

    2017 arXiv

  62. [70]

    Harmark, J

    T. Harmark, J. Hartong and N. A. Obers, Nonrelativistic strings and limits of the AdS/CFT correspondence, Phys. Rev. D 96 (2017) 086019, [ 1705.03535]

    2017 arXiv

  63. [71]

    Hansen, J

    D. Hansen, J. Hartong and N. A. Obers, Action Principle for Newtonian Gravity , Phys. Rev. Lett. 122 (2019) 061106, [ 1807.04765]

    2019 arXiv

  64. [72]

    Hansen, J

    D. Hansen, J. Hartong and N. A. Obers, Non-relativistic expansion of the Einstein–Hilbert Lagrangian, in 15th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories, 5, 2019. 1905.13723. DOI

    2019 arXiv

  65. [73]

    Hansen, J

    D. Hansen, J. Hartong and N. A. Obers, Non-Relativistic Gravity and its Coupling to Matter, JHEP 06 (2020) 145, [ 2001.10277]

    2020 arXiv

  66. [74]

    Hansen, J

    D. Hansen, J. Hartong, N. A. Obers and G. Oling, Galilean first-order formulation for the nonrelativistic expansion of general relativity , Phys. Rev. D 104 (2021) L061501, [2012.01518]

    2021 arXiv

  67. [75]

    Hartong, N

    J. Hartong, N. A. Obers and G. Oling, Review on Non-Relativistic Gravity , Front. in Phys. 11 (2023) 1116888, [ 2212.11309]

    2023 arXiv

  68. [76]

    C. D. A. Blair, J. Lahnsteiner, N. A. Obers and Z. Yan, Matrix theory reloaded: a BPS road to holography, JHEP 02 (2025) 024, [ 2410.03591]

    2025 arXiv

  69. [77]

    Gomis and Z

    J. Gomis and Z. Yan, Worldsheet formalism for decoupling limits in string theory , JHEP 07 (2024) 102, [ 2311.10565]

    2024 arXiv

  70. [78]

    Kasikci, N

    O. Kasikci, N. Ozdemir, M. Ozkan and U. Zorba, Three-dimensional higher-order Schr¨ odinger algebras and Lie algebra expansions, JHEP 04 (2020) 067, [ 2002.03558]

    2020 arXiv

  71. [79]

    C. R. Nappi and E. Witten, A WZW model based on a nonsemisimple group , Phys. Rev. Lett. 71 (1993) 3751–3753, [ hep-th/9310112]

    1993 arXiv

  72. [80]

    Deser, R

    S. Deser, R. Jackiw and S. Templeton, Topologically Massive Gauge Theories, Annals Phys. 140 (1982) 372–411

    1982

  73. [81]

    J. H. Horne and E. Witten, Conformal Gravity in Three-dimensions as a Gauge Theory, Phys. Rev. Lett. 62 (1989) 501–504

    1989

  74. [82]

    Merbis, Chern-Simons-like Theories of Gravity , other thesis, 11, 2014

    W. Merbis, Chern-Simons-like Theories of Gravity , other thesis, 11, 2014

    2014

  75. [83]

    Afshar, B

    H. Afshar, B. Cvetkovic, S. Ertl, D. Grumiller and N. Johansson, Holograms of Conformal Chern-Simons Gravity , Phys. Rev. D 84 (2011) 041502, [ 1106.6299]

    2011 arXiv

  76. [84]

    Afshar, B

    H. Afshar, B. Cvetkovic, S. Ertl, D. Grumiller and N. Johansson, Conformal Chern-Simons holography - lock, stock and barrel , Phys. Rev. D 85 (2012) 064033, [1110.5644]

    2012 arXiv

  77. [85]

    Afshar, A

    H. Afshar, A. Bagchi, S. Detournay, D. Grumiller, S. Prohazka and M. Riegler, – 42 – Holographic Chern-Simons Theories, Lect. Notes Phys. 892 (2015) 311–329, [1404.1919]

    2015 arXiv

  78. [86]

    Witten, Three-Dimensional Gravity Revisited, 0706.3359

    E. Witten, Three-Dimensional Gravity Revisited, 0706.3359

    arXiv

  79. [87]

    Fuentealba, H

    O. Fuentealba, H. A. Gonz´ alez, A. P´ erez, D. Tempo and R. Troncoso,Superconformal Bondi-Metzner-Sachs Algebra in Three Dimensions , Phys. Rev. Lett. 126 (2021) 091602, [2011.08197]

    2021 arXiv

  80. [88]

    Fuentealba, I

    O. Fuentealba, I. Lovrekovic, D. Tempo and R. Troncoso, Enhanced conformal BMS3 symmetries, JHEP 04 (2025) 054, [ 2501.00439]

    2025 arXiv

  81. [89]

    Kraus and F

    P. Kraus and F. Larsen, Holographic gravitational anomalies, JHEP 01 (2006) 022, [hep-th/0508218]

    2006 arXiv

  82. [90]

    Andrade, Y

    T. Andrade, Y. Lei and S. F. Ross, Scattering amplitudes in Lifshitz spacetime , Class. Quant. Grav. 31 (2014) 215002, [ 1406.6389]

    2014 arXiv

  83. [91]

    G. T. Horowitz and B. Way, Lifshitz Singularities, Phys. Rev. D 85 (2012) 046008, [1111.1243]

    2012 arXiv

  84. [92]

    Harrison, S

    S. Harrison, S. Kachru and H. Wang, Resolving Lifshitz Horizons , JHEP 02 (2014) 085, [1202.6635]

    2014 arXiv

  85. [93]

    N. Bao, X. Dong, S. Harrison and E. Silverstein, The Benefits of Stress: Resolution of the Lifshitz Singularity , Phys. Rev. D 86 (2012) 106008, [ 1207.0171]

    2012 arXiv

  86. [94]

    Hartong, E

    J. Hartong, E. Kiritsis and N. A. Obers, Field Theory on Newton-Cartan Backgrounds and Symmetries of the Lifshitz Vacuum , JHEP 08 (2015) 006, [ 1502.00228]

    2015 arXiv

  87. [95]

    M. Gary, D. Grumiller, M. Riegler and J. Rosseel, Flat space (higher spin) gravity with chemical potentials, JHEP 01 (2015) 152, [ 1411.3728]

    2015 arXiv

  88. [96]

    Krishnan, A

    C. Krishnan, A. Raju and S. Roy, A Grassmann path from AdS3 to flat space, JHEP 03 (2014) 036, [ 1312.2941]

    2014 arXiv

  89. [97]

    Bagchi and I

    A. Bagchi and I. Mandal, Supersymmetric Extension of Galilean Conformal Algebras , Phys. Rev. D 80 (2009) 086011, [ 0905.0580]

    2009 arXiv

  90. [98]

    Mandal, Supersymmetric Extension of GCA in 2d , JHEP 11 (2010) 018, [1003.0209]

    I. Mandal, Supersymmetric Extension of GCA in 2d , JHEP 11 (2010) 018, [1003.0209]

    2010 arXiv

  91. [99]

    Bilal, W algebras from Chern-Simons theory , Phys

    A. Bilal, W algebras from Chern-Simons theory , Phys. Lett. B 267 (1991) 487–496

    1991

  92. [100]

    Ammon, M

    M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, Spacetime Geometry in Higher Spin Gravity, JHEP 10 (2011) 053, [ 1106.4788]

    2011 arXiv

  93. [101]

    Bagchi and R

    A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT , JHEP 07 (2009) 037, [ 0902.1385]

    2009 arXiv

  94. [102]

    Afshar, A

    H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, Spin-3 Gravity in Three-Dimensional Flat Space, Phys. Rev. Lett. 111 (2013) 121603, [ 1307.4768]. – 43 –

    2013 arXiv

  95. [103]

    Bars and Z.-j

    I. Bars and Z.-j. Teng, The Unitary Irreducible Representations of SU(2,1) , J. Math. Phys. 31 (1990) 1576–1586

    1990

  96. [104]

    Gunaydin, A

    M. Gunaydin, A. Neitzke, O. Pavlyk and B. Pioline, Quasi-conformal actions, quaternionic discrete series and twistors: SU (2,1) and G2(2), Commun. Math. Phys. 283 (2008) 169–226, [ 0707.1669]

    2008 arXiv

  97. [105]

    Lambert, A

    N. Lambert, A. Lipstein and P. Richmond, Non-Lorentzian M5-brane Theories from Holography, JHEP 08 (2019) 060, [ 1904.07547]

    2019 arXiv

  98. [106]

    Lambert, A

    N. Lambert, A. Lipstein, R. Mouland and P. Richmond, Five-dimensional non-Lorentzian conformal field theories and their relation to six-dimensions , JHEP 03 (2021) 053, [ 2012.00626]

    2021 arXiv

  99. [107]

    Oliva, D

    J. Oliva, D. Tempo and R. Troncoso, Static spherically symmetric solutions for conformal gravity in three dimensions , Int. J. Mod. Phys. A 24 (2009) 1588–1592, [0905.1510]

    2009 arXiv

  100. [108]

    Hartong, E

    J. Hartong, E. Kiritsis and N. A. Obers, Schr¨ odinger Invariance from Lifshitz Isometries in Holography and Field Theory , Phys. Rev. D 92 (2015) 066003, [1409.1522]

    2015 arXiv

  101. [109]

    Maldacena, Einstein Gravity from Conformal Gravity , 1105.5632

    J. Maldacena, Einstein Gravity from Conformal Gravity , 1105.5632

    arXiv

  102. [110]

    Grumiller, M

    D. Grumiller, M. Irakleidou, I. Lovrekovic and R. McNees, Conformal gravity holography in four dimensions , Phys. Rev. Lett. 112 (2014) 111102, [ 1310.0819]

    2014 arXiv

  103. [111]

    Hartong, Holographic Reconstruction of 3D Flat Space-Time , JHEP 10 (2016) 104, [1511.01387]

    J. Hartong, Holographic Reconstruction of 3D Flat Space-Time , JHEP 10 (2016) 104, [1511.01387]

    2016 arXiv

  104. [112]

    Hartong, Gauging the Carroll Algebra and Ultra-Relativistic Gravity , JHEP 08 (2015) 069, [ 1505.05011]

    J. Hartong, Gauging the Carroll Algebra and Ultra-Relativistic Gravity , JHEP 08 (2015) 069, [ 1505.05011]

    2015 arXiv

  105. [113]

    Adami, A

    H. Adami, A. Parvizi, M. M. Sheikh-Jabbari, V. Taghiloo and H. Yavartanoo, Carrollian structure of the null boundary solution space , JHEP 02 (2024) 073, [2311.03515]

    2024 arXiv

  106. [114]

    Bagchi, P

    A. Bagchi, P. Dhivakar and S. Dutta, Holography in flat spacetimes: the case for Carroll, JHEP 08 (2024) 144, [ 2311.11246]

    2024 arXiv

  107. [115]

    Pekar, A

    S. Pekar, A. P´ erez and P. Salgado-Rebolledo,Cartan-like formulation of electric Carrollian gravity, JHEP 09 (2024) 059, [ 2406.01665]

    2024 arXiv

  108. [116]

    Hartong, E

    J. Hartong, E. Have, V. Nenmeli and G. Oling, Boundary Energy-Momentum Tensors for Asymptotically Flat Spacetimes , 2505.05432

    arXiv

  109. [117]

    B. Chen, R. Liu, H. Sun and Y.-f. Zheng, Constructing Carrollian field theories from null reduction, JHEP 11 (2023) 170, [ 2301.06011]. – 44 –

    2023 arXiv

  110. [118]

    Donnay, A

    L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, Bridging Carrollian and celestial holography, Phys. Rev. D 107 (2023) 126027, [ 2212.12553]

    2023 arXiv

  111. [119]

    Concha, E

    P. Concha, E. Rodr´ ıguez and S. Salgado,3D Carrollian gravity from 2D Euclidean symmetry, Eur. Phys. J. C 85 (2025) 517, [ 2501.00205]

    2025 arXiv

  112. [120]

    E. A. Bergshoeff, P. Concha, O. Fierro, E. Rodr´ ıguez and J. Rosseel, A conformal approach to Carroll gravity , JHEP 07 (2025) 075, [ 2412.17752]

    2025 arXiv

  113. [121]

    Hansen, N

    D. Hansen, N. A. Obers, G. Oling and B. T. Søgaard, Carroll Expansion of General Relativity, SciPost Phys. 13 (2022) 055, [ 2112.12684]

    2022 arXiv

  114. [122]

    Musaeus, N

    J. Musaeus, N. A. Obers and G. Oling, Setting the connection free in the Galilei and Carroll expansions of gravity , Phys. Rev. D 109 (2024) 104040, [ 2312.13924]

    2024 arXiv

  115. [123]

    March and J

    E. March and J. Read, A primer on Carroll gravity , Class. Quant. Grav. 42 (2025) 055004, [2409.12200]

    2025 arXiv

  116. [124]

    Ecker, D

    F. Ecker, D. Grumiller, J. Hartong, A. P´ erez, S. Prohazka and R. Troncoso, Carroll black holes, SciPost Phys. 15 (2023) 245, [ 2308.10947]

    2023

  117. [125]

    de Boer, J

    J. de Boer, J. Hartong, N. A. Obers, W. Sybesma and S. Vandoren, Carroll stories, JHEP 09 (2023) 148, [ 2307.06827]

    2023 arXiv

  118. [126]

    Campoleoni, M

    A. Campoleoni, M. Henneaux, S. Pekar, A. P´ erez and P. Salgado-Rebolledo, Magnetic Carrollian gravity from the Carroll algebra , JHEP 09 (2022) 127, [ 2207.14167]

    2022 arXiv

  119. [127]

    Figueroa-O’Farrill, E

    J. Figueroa-O’Farrill, E. Have, S. Prohazka and J. Salzer, The gauging procedure and carrollian gravity, JHEP 09 (2022) 243, [ 2206.14178]

    2022 arXiv

  120. [128]

    Ecker, D

    F. Ecker, D. Grumiller and P. Salgado-Rebolledo, Postcarrollian gravity, in 24th Hellenic School and Workshops on Elementary Particle Physics and Gravity , 4, 2025. 2504.16162. – 45 –

    2025 arXiv

Pith tools

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