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arxiv: 2606.13388 · v1 · pith:7KBICRCUnew · submitted 2026-06-11 · ✦ hep-th · math-ph· math.MP· quant-ph

From 2D Yang-Mills to Calogero-Sutherland via a colored particle

Marcia Tenser , Amilcar R. Queiroz This is my paper

Pith reviewed 2026-06-27 06:09 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MPquant-ph
keywords Yang-Mills theoryCalogero-Sutherlandgauge invariancecylindercolored particlequantum reductionSU(N)
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The pith

Yang-Mills theory coupled to a colored particle on a cylinder reduces to the Calogero-Sutherland model for SU(N).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines two-dimensional Yang-Mills theory on a cylinder coupled to a single particle carrying color charge under the gauge group. Gauge invariance and the compactness of the spatial circle reduce the infinite-dimensional field theory to a finite-dimensional quantum-mechanical system. In the Abelian case this system is equivalent to the Landau problem on a torus. For the non-Abelian group SU(N) the effective dynamics become those of a one-dimensional quantum many-body problem whose particles interact through a singular Calogero-Sutherland potential.

Core claim

We study Yang-Mills theory coupled to a particle on a cylinder, where gauge invariance and compactness reduce the dynamics to a finite dimensional quantum system. In the Abelian case, this yields a model equivalent to the Landau problem on a torus, with a degenerate ground state structure. We generalize this construction to non-Abelian gauge groups and show that, for SU(N), the system reduces to a one dimensional quantum many body problem with a singular Calogero-Sutherland-type interaction.

What carries the argument

The reduction of the coupled Yang-Mills-plus-particle system to a finite-dimensional quantum system via gauge invariance and compactness of the cylinder.

If this is right

  • The Abelian reduction reproduces the Landau problem on a torus with degenerate ground states.
  • For SU(N) the color degrees of freedom of the particle map onto the coordinates of particles in a Calogero-Sutherland chain.
  • The effective Hamiltonian contains the characteristic singular 1/r^2 interaction of the Calogero-Sutherland model.
  • The construction applies to any compact gauge group, producing different many-body interactions depending on the group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mapping supplies a gauge-theory origin for the integrability of the Calogero-Sutherland model.
  • Similar particle-gauge couplings on other topologies might generate other exactly solvable models.
  • Observables in the reduced quantum mechanics could be lifted back to compute Wilson loops or other gauge-invariant quantities in the original 2D theory.

Load-bearing premise

Gauge invariance and the compactness of the cylinder suffice to eliminate all but a finite number of degrees of freedom in the coupled system.

What would settle it

An explicit gauge fixing and mode integration that produces an effective Hamiltonian different from the Calogero-Sutherland form would show the reduction does not hold.

Figures

Figures reproduced from arXiv: 2606.13388 by Amilcar R. Queiroz, Marcia Tenser.

Figure 1
Figure 1. Figure 1: The setup: a cylindrical manifold M = R × S 1 of radius r; a point-particle of mass m and isospin I. Time evolution is in the vertical direction. The internal symmetry group G is a compact, connected Lie group. We focus on the particular case of G = SU(N). Notation and conventions regarding the gauge group structure are outlined in Appendix A. The dynamics of the charged point-particle is described by a se… view at source ↗
Figure 2
Figure 2. Figure 2: Two non-trivial cycles in configuration space: the horizontal cycle is the spatial S 1 factor from the cylindrical manifold and the vertical cycle is a large gauge transformation along Y i of TSU(N) . where ei is a vector representing a unit length translation in the i-th direction. That is, Y + ei stands for {Y 1 , . . . , Y i + 1, . . . , Y N−1}. The boundary conditions lead to non-Abelian charge quantiz… view at source ↗
Figure 3
Figure 3. Figure 3: Potential V (Y ) for G = SU(2) and K = 1. Other values for K simply shift the curves vertically. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: G = SU(3): (a) the 2D lattice from TSU(3)/S3. The fundamental unit cell is 2-simplex, a triangle. The honeycomb lattice is obtained after acting upon its vertices with permutation matrices. Blue and green shifts correspond to a P1 and a P2 action, respectively. The entire plane is covered by copies of this hexagon structure, generated by large gauge transformations. In (b) the potential V (Y 1 , Y 2 ) assu… view at source ↗
read the original abstract

We study Yang-Mills theory coupled to a particle on a cylinder, where gauge invariance and compactness reduce the dynamics to a finite dimensional quantum system. In the Abelian case, this yields a model equivalent to the Landau problem on a torus, with a degenerate ground state structure. We generalize this construction to non-Abelian gauge groups and show that, for SU(N), the system reduces to a one dimensional quantum many body problem with a singular Calogero-Sutherland-type interaction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript studies Yang-Mills theory coupled to a colored particle on a cylinder. Gauge invariance together with compactness of the cylinder is used to reduce the dynamics to a finite-dimensional quantum system. In the Abelian case the reduced system is equivalent to the Landau problem on a torus, exhibiting a degenerate ground-state structure. For SU(N) the construction yields a one-dimensional quantum many-body problem whose effective interaction is of Calogero-Sutherland type.

Significance. If the reduction is correctly derived, the work supplies an explicit bridge between 2D Yang-Mills and the Calogero-Sutherland model, an integrable system whose spectrum and wave-functions are known in closed form. The Abelian limit recovers the Landau problem on the torus, providing an immediate consistency check. Such a gauge-theoretic origin for the Calogero-Sutherland interaction could be useful for constructing new integrable deformations or for studying the spectrum of 2D gauge theories in a controlled finite-dimensional setting.

minor comments (2)
  1. The precise form of the reduced Hamiltonian for SU(N) (including the coefficient of the 1/sin² interaction and any overall constant) should be displayed explicitly, together with the Hilbert space on which it acts, so that the Calogero-Sutherland identification can be verified by direct comparison with the standard literature.
  2. A short paragraph or appendix comparing the Abelian (Landau) and non-Abelian (Calogero-Sutherland) reduced Hamiltonians would help the reader see how the non-Abelian structure constants enter the effective interaction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our work, as well as the recommendation for minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point rebuttal or manuscript changes at this stage.

Circularity Check

0 steps flagged

No significant circularity; reduction presented as consequence of gauge invariance

full rationale

The abstract and provided context describe a reduction of 2D Yang-Mills plus colored particle on a cylinder to a finite-dimensional quantum system (Abelian case maps to Landau problem; non-Abelian SU(N) to Calogero-Sutherland) via gauge invariance and compactness. No equations, self-citations, fitted parameters, or ansatze are quoted that would allow any step to reduce to its own inputs by construction. The central claim is framed as a physical consequence rather than a self-referential fit or renamed result. This is the normal case of a self-contained derivation with no detectable circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no information on free parameters, background axioms, or new postulated entities; all lists left empty.

pith-pipeline@v0.9.1-grok · 5610 in / 1034 out tokens · 17511 ms · 2026-06-27T06:09:24.035631+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

47 extracted references · 17 linked inside Pith

  1. [1]

    Rajeev,Yang-Mills Theory on a Cylinder,Phys

    S.G. Rajeev,Yang-Mills Theory on a Cylinder,Phys. Lett. B212(1988) 203

    1988

  2. [2]

    Hetrick and Y

    J.E. Hetrick and Y. Hosotani,Yang-Mills Theory on a Circle,Phys. Lett. B230(1989) 88

    1989

  3. [3]

    Witten,On quantum gauge theories in two-dimensions,Commun

    E. Witten,On quantum gauge theories in two-dimensions,Commun. Math. Phys.141(1991) 153

    1991

  4. [4]

    Cordes, G.W

    S. Cordes, G.W. Moore and S. Ramgoolam,Lectures on 2-d Yang-Mills theory, equivariant cohomology and topological field theories,Nucl. Phys. B Proc. Suppl.41(1995) 184 [hep-th/9411210]

    Pith/arXiv arXiv 1995

  5. [5]

    Ares, A.R

    F. Ares, A.R. De Queiroz and M.R. Tenser,Entanglement in a Maxwell Theory coupled to a non-relativistic particle,JHEP07(2020) 079 [1911.11264]

    arXiv 2020

  6. [6]

    Thouless, M

    D.J. Thouless, M. Kohmoto, M.P. Nightingale and M. den Nijs,Quantized hall conductance in a two-dimensional periodic potential,Physical Review Letters49(1982) 405

    1982

  7. [7]

    Carvalho, A.M

    J. Carvalho, A.M. de M. Carvalho and C. Furtado,Quantum influence of topological defects in g¨ odel-type space-times,European Physical Journal C74(2014) 2935

    2014

  8. [8]

    Manton,The Schwinger Model and Its Axial Anomaly,Annals Phys.159(1985) 220

    N.S. Manton,The Schwinger Model and Its Axial Anomaly,Annals Phys.159(1985) 220

    1985

  9. [9]

    Esteve,Anomalies in Conservation Laws in the Hamiltonian Formalism,Phys

    J.G. Esteve,Anomalies in Conservation Laws in the Hamiltonian Formalism,Phys. Rev. D34 (1986) 674

    1986

  10. [10]

    Aguado, M

    M. Aguado, M. Asorey and J.G. Esteve,Vacuum nodes and anomalies in quantum theories, Communications in Mathematical Physics218(2001) 233

    2001

  11. [11]

    Langmann and G.W

    E. Langmann and G.W. Semenoff,Gauge theories on a cylinder,Phys. Lett. B296(1992) 117 [hep-th/9210011]

    Pith/arXiv arXiv 1992

  12. [12]

    Langmann and G.W

    E. Langmann and G.W. Semenoff,Gribov ambiguity and nontrivial vacuum structure of gauge theories on a cylinder,Phys. Lett. B303(1993) 303 [hep-th/9212038]

    Pith/arXiv arXiv 1993

  13. [13]

    Hetrick,Gauge fixing and Gribov copies in pure Yang-Mills on a circle,Nucl

    J.E. Hetrick,Gauge fixing and Gribov copies in pure Yang-Mills on a circle,Nucl. Phys. B Proc. Suppl.30(1993) 228 [hep-lat/9212005]

    Pith/arXiv arXiv 1993

  14. [14]

    Gupta, R.J

    K.S. Gupta, R.J. Henderson, S.G. Rajeev and O.T. Turgut,Yang-Mills theory on a cylinder coupled to point particles,J. Math. Phys.35(1994) 3845 [hep-th/9311064]

    Pith/arXiv arXiv 1994

  15. [15]

    Calogero,Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials,J

    F. Calogero,Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials,J. Math. Phys.12(1971) 419

    1971

  16. [16]

    Sutherland,Exact results for a quantum many body problem in one-dimension,Phys

    B. Sutherland,Exact results for a quantum many body problem in one-dimension,Phys. Rev. A 4(1971) 2019

    1971

  17. [17]

    Sutherland,Exact results for a quantum many-body problem in one dimension

    B. Sutherland,Exact results for a quantum many-body problem in one dimension. II,Phys. Rev. A5(1972) 1372. 23

    1972

  18. [18]

    Olshanetsky and A.M

    M.A. Olshanetsky and A.M. Perelomov,Quantum integrable systems related to lie algebras, Phys. Rept.94(1983) 313

    1983

  19. [19]

    Minahan and A.P

    J.A. Minahan and A.P. Polychronakos,Equivalence of two-dimensional QCD and thec= 1 matrix model,Phys. Lett. B312(1993) 155 [hep-th/9303153]

    Pith/arXiv arXiv 1993

  20. [20]

    Minahan and A.P

    J.A. Minahan and A.P. Polychronakos,Interacting fermion systems from two-dimensional QCD, Phys. Lett. B326(1994) 288 [hep-th/9309044]

    Pith/arXiv arXiv 1994

  21. [21]

    Gorsky and N

    A. Gorsky and N. Nekrasov,Hamiltonian systems of calogero type and two-dimensional Yang-Mills theory,Nucl. Phys. B414(1994) 213 [hep-th/9304047]

    Pith/arXiv arXiv 1994

  22. [22]

    Wong,Field and particle equations for the classical Yang-Mills field and particles with isotopic spin,Nuovo Cim

    S.K. Wong,Field and particle equations for the classical Yang-Mills field and particles with isotopic spin,Nuovo Cim. A65(1970) 689

    1970

  23. [23]

    Balachandran, S

    A.P. Balachandran, S. Borchardt and A. Stern,Lagrangian and Hamiltonian Descriptions of Yang-Mills Particles,Phys. Rev. D17(1978) 3247

    1978

  24. [24]

    Dixon, J.A

    L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten,Strings on Orbifolds. 2.,Nucl. Phys. B274 (1986) 285

    1986

  25. [25]

    Asorey and A

    M. Asorey and A. Santagata,Singular potentials: Confinement and riemann hypothesis,Il Nuovo Cimento C36(2013) 15

    2013

  26. [26]

    de Alfaro, S

    V. de Alfaro, S. Fubini and G. Furlan,Conformal Invariance in Quantum Mechanics,Nuovo Cim. A34(1976) 569

    1976

  27. [27]

    Camblong, L.N

    H.E. Camblong, L.N. Epele, H. Fanchiotti and C.A. Garcia Canal,Renormalization of the inverse square potential,Phys. Rev. Lett.85(2000) 1590 [hep-th/0003014]

    Pith/arXiv arXiv 2000

  28. [28]

    Beane, P.F

    S.R. Beane, P.F. Bedaque, L. Childress, A. Kryjevski, J. McGuire and U. van Kolck,Singular potentials and limit cycles,Phys. Rev. A64(2001) 042103 [quant-ph/0010073]

    Pith/arXiv arXiv 2001

  29. [29]

    Kaplan, J.-W

    D.B. Kaplan, J.-W. Lee, D.T. Son and M.A. Stephanov,Conformality Lost,Phys. Rev. D80 (2009) 125005 [0905.4752]

    Pith/arXiv arXiv 2009

  30. [30]

    da Silva, C.F.S

    U.C. da Silva, C.F.S. Pereira and A. Alves Lima,Renormalization group and spectra of the generalized P¨ oschl–Teller potential,Annals Phys.460(2024) 169549 [2308.04596]

    arXiv 2024

  31. [31]

    da Silva,Renormalization of Schr¨ odinger equation for potentials with inverse-square singularities: generalized trigonometric P¨ oschl–Teller model,J

    U.C. da Silva,Renormalization of Schr¨ odinger equation for potentials with inverse-square singularities: generalized trigonometric P¨ oschl–Teller model,J. Phys. A58(2025) 505201 [2503.12715]

    arXiv 2025

  32. [32]

    Case,Singular potentials,Phys

    K.M. Case,Singular potentials,Phys. Rev.80(1950) 797

    1950

  33. [33]

    Gupta and S.G

    K.S. Gupta and S.G. Rajeev,Renormalization in quantum mechanics,Phys. Rev. D48(1993) 5940 [hep-th/9305052]

    Pith/arXiv arXiv 1993

  34. [35]

    Kitaev and J

    A. Kitaev and J. Preskill,Topological entanglement entropy,Phys. Rev. Lett.96(2006) 110404 [hep-th/0510092]

    Pith/arXiv arXiv 2006

  35. [36]

    Levin and X.-G

    M. Levin and X.-G. Wen,Detecting Topological Order in a Ground State Wave Function,Phys. Rev. Lett.96(2006) 110405 [cond-mat/0510613]

    Pith/arXiv arXiv 2006

  36. [37]

    Chen, Z.C

    X. Chen, Z.C. Gu and X.G. Wen,Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order,Phys. Rev. B82(2010) 155138 [1004.3835]

    Pith/arXiv arXiv 2010

  37. [38]

    Balachandran, A.R

    A.P. Balachandran, A.R. de Queiroz and S. Vaidya,Quantum entropic ambiguities: Ethylene, Phys. Rev. D88(2013) 025001

    2013

  38. [39]

    Balachandran, A.R

    A.P. Balachandran, A.R. de Queiroz and S. Vaidya,Entropy of quantum states: Ambiguities, Eur. Phys. J. Plus128(2013) 112

    2013

  39. [40]

    Balachandran, S

    A.P. Balachandran, S. Vaidya and A.R. de Queiroz,A matrix model for QCD,Mod. Phys. Lett. A30(2015) 1550080

    2015

  40. [41]

    Balachandran, A.R

    A.P. Balachandran, A.R. de Queiroz and S. Vaidya,A matrix model for QCD: QCD color is mixed,Int. J. Mod. Phys. A30(2015) 1550064

    2015

  41. [42]

    Balachandran, T.R

    A.P. Balachandran, T.R. Govindarajan, A.R. de Queiroz and A.F. Reyes-Lega,Entanglement and particle identity: A unifying approach,Phys. Rev. Lett.110(2013) 080503

    2013

  42. [43]

    Balachandran, T.R

    A.P. Balachandran, T.R. Govindarajan, A.R. de Queiroz and A.F. Reyes-Lega,Algebraic approach to entanglement and entropy,Phys. Rev. A88(2013) 022301

    2013

  43. [44]

    Van Raamsdonk,Building up spacetime with quantum entanglement,Gen

    M. Van Raamsdonk,Building up spacetime with quantum entanglement,Gen. Rel. Grav.42 (2010) 2323 [1005.3035]

    arXiv 2010

  44. [45]

    Maldacena and L

    J. Maldacena and L. Susskind,Cool horizons for entangled black holes,Fortsch. Phys.61(2013) 781 [1306.0533]

    Pith/arXiv arXiv 2013

  45. [46]

    Melnikov, J.T

    D. Melnikov, J.T. Oliveira, V. Peixoto and M. Tenser,States of 2D Yang-Mills and Large-Volume Entanglement,2603.10171

    arXiv

  46. [47]

    Balachandran and A.R

    A.P. Balachandran and A.R. de Queiroz,Mixed States from Anomalies,Physical Review D85 (2012) 025017 [1108.3898]

    Pith/arXiv arXiv 2012

  47. [48]

    Georgi,Lie algebras in particle physics, vol

    H. Georgi,Lie algebras in particle physics, vol. 54, Perseus Books, Reading, MA, 2nd ed. ed. (1999). 25

    1999