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arxiv: 2604.27184 · v1 · submitted 2026-04-29 · ✦ hep-th · math-ph· math.DG· math.MP

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A Nahm transform for rotating calorons

Josh Cork , Derek Harland
Authors on Pith no claims yet

Pith reviewed 2026-05-07 08:00 UTC · model grok-4.3

classification ✦ hep-th math-phmath.DGmath.MP
keywords rotating caloronsNahm transformanti-self-dual gauge fieldsdelayed-differential equationsnontrivial holonomyglide rotationcharge onequark-gluon plasma
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The pith

A Nahm transform maps rotating calorons to delayed-differential equations and proves an eight-parameter family of charge-one examples exists.

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

Rotating calorons are anti-self-dual gauge fields on four-dimensional Euclidean space that stay the same under a glide rotation combining a spatial rotation with a shift in the time direction. The paper sets up a Nahm transform that converts the search for these fields into the task of solving a delayed-differential equation whose solutions encode the gauge-field data. With this correspondence the authors establish that an eight-parameter family of charge-one rotating calorons with nontrivial holonomy and rotational angle exactly pi can be built explicitly. These examples are obtained and pictured by numerically solving the delayed equation and feeding the results back through the transform. The construction supplies concrete objects that can be used to model aspects of rotating quark-gluon plasmas.

Core claim

Rotating calorons are anti-self-dual gauge fields invariant under a glide rotation. We formulate a Nahm transform which identifies rotating calorons with solutions of a delayed-differential equation. Using this transform, we prove existence of an eight-parameter family of charge 1 rotating calorons with nontrivial holonomy and rotational angle π, which we construct and visualise using a numerical implementation of the Nahm transform.

What carries the argument

The Nahm transform that associates each rotating caloron with a solution of a delayed-differential equation on the dual circle.

If this is right

  • An eight-parameter family of charge-one rotating calorons with nontrivial holonomy and rotational angle pi exists and can be constructed explicitly.
  • Numerical solution of the delayed-differential equation produces explicit examples that can be visualized in three-dimensional space.
  • The transform supplies a practical method for generating further rotating calorons once the delayed equation is solved.
  • The correspondence reduces the original nonlinear gauge-field equations to a linear delayed-differential problem.

Where Pith is reading between the lines

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

  • The same transform may extend to higher topological charge or other rotational angles once the delayed equation is solved for those cases.
  • Analytic study of the delayed-differential equation could produce closed-form expressions for special values of the parameters.
  • These explicit fields offer concrete test cases for numerical simulations of rotating plasmas or for stability analysis under small perturbations.
  • The glide-rotation invariance suggests the construction could be adapted to other periodic or orbifold settings in gauge theory.

Load-bearing premise

The Nahm transform correctly pairs every rotating caloron of the given holonomy and rotational angle with a solution of the delayed-differential equation and preserves the anti-self-duality condition.

What would settle it

A concrete charge-one rotating caloron with rotational angle pi whose Nahm data fails to satisfy the delayed-differential equation, or a solution of the equation whose inverse transform yields a field that violates anti-self-duality or glide invariance.

Figures

Figures reproduced from arXiv: 2604.27184 by Derek Harland, Josh Cork.

Figure 1
Figure 1. Figure 1: The scale λ for the rotating caloron solutions with τ = 0 as a function of (κ1, κ2) in the case θ = 2π and µ = ν = π 2 . The black region indicates K(κ1) ≤ K(κ2)/ρ where we did not seek solutions. 23 view at source ↗
Figure 2
Figure 2. Figure 2: Yang–Mills action density isosurface plots of a monopole-like family within view at source ↗
Figure 3
Figure 3. Figure 3: Yang–Mills action density isosurface plots of an instanton-like configuration view at source ↗
Figure 4
Figure 4. Figure 4: Yang–Mills action density isosurfaces of examples in the view at source ↗
read the original abstract

Rotating calorons were introduced in the context of rotating quark-gluon plasmas. They are anti-self-dual gauge fields on $\mathbb{R}^4$ that are invariant under a glide rotation. We formulate a Nahm transform which identifies rotating calorons with solutions of a delayed-differential equation. Using this transform, we prove existence of an eight-parameter family of charge 1 rotating calorons with nontrivial holonomy and rotational angle $\pi$, which we construct and visualise using a numerical implementation of the Nahm transform.

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

2 major / 3 minor

Summary. The manuscript formulates a Nahm transform that identifies rotating calorons (anti-self-dual gauge fields on R^4 invariant under glide rotations) with solutions of a delayed-differential equation. It proves the existence of an eight-parameter family of charge-1 rotating calorons with nontrivial holonomy and rotational angle π, and constructs/visualizes them via a numerical implementation of the transform.

Significance. If the Nahm correspondence is rigorously established, the work provides a valuable extension of the standard Nahm transform to gauge fields with glide-rotational symmetry, relevant to applications in rotating quark-gluon plasmas. The existence proof for a concrete 8-parameter family and the numerical construction are strengths that enable explicit examples and further study, building on prior caloron literature.

major comments (2)
  1. [Section 4] The existence proof (Section 4) constructs solutions to the delayed-differential equation with the stated boundary conditions for nontrivial holonomy and angle π, then asserts that the inverse transform yields rotating calorons. However, it does not explicitly verify that the resulting connection satisfies the anti-self-duality equation after the glide rotation is imposed, nor does it check the monodromy and asymptotic decay properties tied to the delay. This verification is load-bearing for the central claim that every such DDE solution produces a valid ASD caloron.
  2. [Section 3] In the derivation of the Nahm transform (Section 3, around the mapping to the delayed-differential equation), the argument relies on analytic continuation of the Nahm data across the period to accommodate the glide rotation with angle exactly π. The manuscript should supply a detailed argument showing that this continuation preserves anti-self-duality and the required holonomy without introducing singularities or mismatches, as the delay is directly linked to the rotational symmetry.
minor comments (3)
  1. [Section 3] The notation for the delay parameter in the differential equation could be defined more explicitly in terms of the rotational angle to improve readability.
  2. [Section 5] The numerical visualizations (Section 5) would benefit from quantitative error estimates on the ASD violation and glide invariance, beyond qualitative plots.
  3. [Introduction] Ensure the introduction clearly distinguishes the new transform from the standard Nahm transform for calorons, citing relevant prior works on nontrivial holonomy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. We address each major comment below and will revise the manuscript to strengthen the presentation of the Nahm correspondence and the existence proof.

read point-by-point responses

  1. Referee: [Section 4] The existence proof (Section 4) constructs solutions to the delayed-differential equation with the stated boundary conditions for nontrivial holonomy and angle π, then asserts that the inverse transform yields rotating calorons. However, it does not explicitly verify that the resulting connection satisfies the anti-self-duality equation after the glide rotation is imposed, nor does it check the monodromy and asymptotic decay properties tied to the delay. This verification is load-bearing for the central claim that every such DDE solution produces a valid ASD caloron.

    Authors: We agree that an explicit verification step is required to make the argument fully rigorous. In the revised manuscript we will insert a new subsection in Section 4 that carries out this verification in detail. Starting from the solutions of the delayed-differential equation with the prescribed boundary conditions, we will (i) reconstruct the connection via the inverse Nahm transform, (ii) impose the glide rotation with angle π, (iii) compute the curvature two-form and confirm that it is anti-self-dual, and (iv) verify the required monodromy and the asymptotic decay rates dictated by the delay. These steps follow directly from the algebraic properties of the Nahm data and the standard reconstruction formulae; we will spell them out explicitly rather than leaving them implicit. revision: yes

  2. Referee: [Section 3] In the derivation of the Nahm transform (Section 3, around the mapping to the delayed-differential equation), the argument relies on analytic continuation of the Nahm data across the period to accommodate the glide rotation with angle exactly π. The manuscript should supply a detailed argument showing that this continuation preserves anti-self-duality and the required holonomy without introducing singularities or mismatches, as the delay is directly linked to the rotational symmetry.

    Authors: The referee is correct that the analytic continuation step needs a more self-contained justification. In the revised Section 3 we will add a dedicated paragraph (or short appendix) that proves the continuation preserves anti-self-duality. The argument proceeds by noting that the original Nahm data on the fundamental interval satisfy the appropriate differential equations and boundary conditions; the continuation is then defined by the glide-rotation action, which by construction maps solutions to solutions. We will show that the resulting extended data remain holomorphic in the appropriate sense, that the holonomy is unchanged because the boundary values at the identification points are matched by the choice of rotational angle π, and that no singularities arise because the delay is commensurate with the period. This establishes consistency between the delay and the rotational symmetry without additional assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity: Nahm transform derived directly from glide invariance and ASD condition

full rationale

The paper defines rotating calorons as ASD connections on R^4 invariant under a specific glide rotation (rotation by angle π combined with translation). It then formulates the Nahm transform by imposing this symmetry on the standard Nahm data construction, yielding a delayed-differential equation whose solutions are asserted to correspond to the desired gauge fields. This mapping is presented as a direct, first-principles derivation from the invariance and self-duality conditions rather than a redefinition or fit. The existence of the eight-parameter family is obtained by solving the resulting DDE with appropriate boundary conditions for nontrivial holonomy and then applying the inverse transform; the numerical implementation serves only to visualize explicit solutions. No load-bearing step reduces by construction to a fitted parameter, a self-citation chain, or an ansatz imported from prior work by the same authors. The correspondence is claimed to be proven analytically, and the derivation remains self-contained against external benchmarks such as the classical Nahm transform for ordinary calorons. No quoted equation or definition exhibits the self-referential reduction required for a positive circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definition of rotating calorons as anti-self-dual fields with glide invariance and on the new correspondence to delayed-differential equations; no free parameters are fitted to data and no new physical entities are postulated.

axioms (2)
  • domain assumption Rotating calorons are anti-self-dual gauge fields on R^4 invariant under a glide rotation
    This is the defining property stated in the abstract and used to set up the transform.
  • ad hoc to paper The Nahm transform maps these fields to solutions of a delayed-differential equation
    The formulation of this specific transform is the novel step introduced by the paper.

pith-pipeline@v0.9.0 · 5374 in / 1534 out tokens · 59320 ms · 2026-05-07T08:00:00.246194+00:00 · methodology

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Reference graph

Works this paper leans on

66 extracted references · 1 canonical work pages

  1. [1]

    Global Λ hyperon polarization in nuclear collisions.Nature, 548(7665):62–65, 2017

    STAR Collaboration. Global Λ hyperon polarization in nuclear collisions.Nature, 548(7665):62–65, 2017

    2017

  2. [2]

    Lattice QCD in rotating frames.Phys

    A Yamamoto and Y Hirono. Lattice QCD in rotating frames.Phys. Rev. Lett., 111:081601, 2013

    2013

  3. [3]

    Kotov, D D Kuznedelev, and A A Roenko

    V V Braguta, A Yu. Kotov, D D Kuznedelev, and A A Roenko. Study of the con- finement/deconfinement phase transition in rotating lattice SU(3) gluodynamics. JETP Lett., 112(1):6–12, 2020

    2020

  4. [4]

    Influence of rel- ativistic rotation on the confinement-deconfinement transition in gluodynamics

    V V Braguta, A Yu Kotov, D D Kuznedelev, and A A Roenko. Influence of rel- ativistic rotation on the confinement-deconfinement transition in gluodynamics. Phys. Rev. D, 103(9), 2021

    2021

  5. [5]

    Pairing phase transitions of matter under rotation.Phys

    Y Jiang and J Liao. Pairing phase transitions of matter under rotation.Phys. Rev. Lett., 117:192302, 2016

    2016

  6. [6]

    Gluodynamics and deconfinement phase transition under rotation from holography.J

    X Chen, L Zhang, D Li, D Hou, and M Huang. Gluodynamics and deconfinement phase transition under rotation from holography.J. High Energ. Phys., 2021, 2021

    2021

  7. [7]

    Deconfining phase boundary of rapidly rotating hot and dense matter and analysis of moment of inertia.Phys

    Y Fujimoto, K Fukushima, and Y Hidaka. Deconfining phase boundary of rapidly rotating hot and dense matter and analysis of moment of inertia.Phys. Lett. B, 816:136184, 2021

    2021

  8. [8]

    Inhomogeneous confining-deconfining phases in rotating plas- mas.Phys

    M N Chernodub. Inhomogeneous confining-deconfining phases in rotating plas- mas.Phys. Rev. D, 103:054027, 2021

    2021

  9. [9]

    Perturbative confinement in thermal Yang-Mills theories induced by imaginary angular velocity.Phys

    S Chen, K Fukushima, and Y Shimada. Perturbative confinement in thermal Yang-Mills theories induced by imaginary angular velocity.Phys. Rev. Lett., 129:242002, 2022. 31

    2022

  10. [10]

    Quark-meson model under rotation: A functional renormalization group study.Phys

    H L Chen, Z B Zhu, and X G Huang. Quark-meson model under rotation: A functional renormalization group study.Phys. Rev. D, 108:054006, 2023

    2023

  11. [11]

    Chiral symmetry restoration in a rotating medium.Phys

    I I Gaspar, L A Hern´ andez, and R Zamora. Chiral symmetry restoration in a rotating medium.Phys. Rev. D, 108:094020, 2023

    2023

  12. [12]

    Deconfinement transition in the revolving bag model.Phys

    K Mameda and K Takizawa. Deconfinement transition in the revolving bag model.Phys. Lett. B, 847:138317, 2023

    2023

  13. [13]

    Inhibition of the splitting of the chiral and deconfinement transition due to rotation in QCD: The phase diagram of the linear sigma model coupled to Polyakov loops.Phys

    P Singha, V E Ambru¸ s, and M N Chernodub. Inhibition of the splitting of the chiral and deconfinement transition due to rotation in QCD: The phase diagram of the linear sigma model coupled to Polyakov loops.Phys. Rev. D, 110:094053, 2024

    2024

  14. [14]

    Vortical effects and the critical end point in the linear sigma model coupled to quark.Phys

    L A Hern´ andez and R Zamora. Vortical effects and the critical end point in the linear sigma model coupled to quark.Phys. Rev. D, 111:036003, 2025

    2025

  15. [15]

    Imaginary rotating gluonic matter at strong coupling.Phys

    K Fukushima and Y Shimada. Imaginary rotating gluonic matter at strong coupling.Phys. Lett. B, 868:139716, 2025

    2025

  16. [16]

    An SU(2) KvBLL caloron gas model and confinement.Nucl

    P Gerhold, E M Ilgenfritz, and M M¨ uller-Preussker. An SU(2) KvBLL caloron gas model and confinement.Nucl. Phys. B, 760(1):1–37, 2007

    2007

  17. [17]

    Confining ensemble of dyons.Phys

    D Diakonov and V Petrov. Confining ensemble of dyons.Phys. Rev. D, 76:056001, 2007

    2007

  18. [18]

    Confining dyon-antidyon Coulomb liquid model

    Y Liu, E Shuryak, and I Zahed. Confining dyon-antidyon Coulomb liquid model. i.Phys. Rev. D, 92:085006, 2015

    2015

  19. [19]

    Instantons in rotating finite-temperature Yang–Mills gas.arXiv preprint arXiv:2208.04808, 2022

    M N Chernodub. Instantons in rotating finite-temperature Yang–Mills gas.arXiv preprint arXiv:2208.04808, 2022

    work page arXiv 2022

  20. [20]

    Monopole constituents insideSU(n) calorons.Phys

    T C Kraan and P van Baal. Monopole constituents insideSU(n) calorons.Phys. Lett. B, 435(3-4):389–395, 1998

    1998

  21. [21]

    K Lee and C Lu.SU(2) calorons and magnetic monopoles.Phys. Rev. D, 58(2):025011, 1998

    1998

  22. [22]

    Rotating SU(2) gluon matter and deconfinement at finite temperature

    Y Jiang. Rotating SU(2) gluon matter and deconfinement at finite temperature. Phys. Lett. B, 853:138655, 2024

    2024

  23. [23]

    Inhomogeneous SU(2) gluon matter under rotation.Phys

    Y Jiang. Inhomogeneous SU(2) gluon matter under rotation.Phys. Rev. D, 110:054047, 2024

    2024

  24. [24]

    Symmetric calorons and the rotation map.J

    J Cork. Symmetric calorons and the rotation map.J. Math. Phys., 59(6):062902, 2018. 32

    2018

  25. [25]

    Construction of instan- tons.Phys

    M F Atiyah, V G Drinfeld, N J Hitchin, and Y I Manin. Construction of instan- tons.Phys. Lett. A, 65(3):185–187, 1978

    1978

  26. [26]

    Springer, 1983

    W Nahm.The algebraic geometry of multimonopoles. Springer, 1983

    1983

  27. [27]

    All self-dual multimonopoles for arbitrary gauge groups

    W Nahm. All self-dual multimonopoles for arbitrary gauge groups. InStructural elements in particle physics and statistical mechanics, pages 301–310. Springer, 1983

    1983

  28. [28]

    Self-dual monopoles and calorons

    W Nahm. Self-dual monopoles and calorons. InGroup theoretical methods in physics, pages 189–200. Springer, 1984

    1984

  29. [29]

    The Nahm transform for calorons.The many facets of geometry: a tribute to Nigel Hitchin, 2010

    B Charbonneau and J Hurtubise. The Nahm transform for calorons.The many facets of geometry: a tribute to Nigel Hitchin, 2010

    2010

  30. [30]

    A survey on Nahm transform.J

    M Jardim. A survey on Nahm transform.J. Geom. Phys., 52(3):313–327, 2004

    2004

  31. [31]

    Spatially periodic instantons: Nahm transform and moduli.Commun

    B Charbonneau and J Hurtubise. Spatially periodic instantons: Nahm transform and moduli.Commun. Math. Phys., 365(1):255–293, 2019

    2019

  32. [32]

    Yang–Mills instantons on ALE gravitational instantons.Mathematische Annalen, 288(1):263–307, 1990

    P B Kronheimer and H Nakajima. Yang–Mills instantons on ALE gravitational instantons.Mathematische Annalen, 288(1):263–307, 1990

    1990

  33. [33]

    Instantons on gravitons.Commun

    S A Cherkis. Instantons on gravitons.Commun. Math. Phys., 306(2):449–483, 2011

    2011

  34. [34]

    Instantons on multi-Taub-NUT spaces II: bow construction.J

    S A Cherkis, A Larrain-Hubach, and M Stern. Instantons on multi-Taub-NUT spaces II: bow construction.J. Differ. Geom., 127(2):433–503, 2024

    2024

  35. [35]

    On the construction of monopoles with arbitrary symmetry breaking.Trans

    B Charbonneau and ´A Nagy. On the construction of monopoles with arbitrary symmetry breaking.Trans. Am. Math. Soc., In Press 2026

    2026

  36. [36]

    Periodic instantons with non-trivial holonomy.Nucl

    T C Kraan and P van Baal. Periodic instantons with non-trivial holonomy.Nucl. Phys. B, 533(1):627–659, 1998

    1998

  37. [37]

    Continuous symmetries of differential-difference equations: the Kac-van Moerbeke equation and Painlev´ e reduction.Phys

    G R W Quispel, H W Capel, and R Sahadevan. Continuous symmetries of differential-difference equations: the Kac-van Moerbeke equation and Painlev´ e reduction.Phys. Lett. A, 170(5):379–383, 1992

    1992

  38. [38]

    Symmetries and conditional symmetries of differential- difference equations.J

    D Levi and P Winternitz. Symmetries and conditional symmetries of differential- difference equations.J. Math. Phys., 34(8):3713–3730, 1993

    1993

  39. [39]

    Delay-differential equations and the Painlev´ e transcendents.Phys

    B Grammaticos, A Ramani, and I C Moreira. Delay-differential equations and the Painlev´ e transcendents.Phys. A, 196(4):574–590, 1993. 33

    1993

  40. [40]

    Painlev´ e analysis and sin- gularity confinement: the ultimate conjecture.J

    A Ramani, B Grammaticos, and K M Tamizhmani. Painlev´ e analysis and sin- gularity confinement: the ultimate conjecture.J. Phys. A, 26(2):L53–L58, 1993

    1993

  41. [41]

    On the method of solution of the differential-delay Toda equation.Phys

    J Villarroel and M J Ablowitz. On the method of solution of the differential-delay Toda equation.Phys. Lett. A, 180(6):413–418, 1993

    1993

  42. [42]

    Direct ‘delay’ reductions of the Toda equation.J

    N Joshi. Direct ‘delay’ reductions of the Toda equation.J. Phys. A, 42(2):022001, 8, 2009

    2009

  43. [43]

    Direct “delay”’ reductions of the Toda hierarchy.J

    N Joshi and P E Spicer. Direct “delay”’ reductions of the Toda hierarchy.J. Phys. Soc. Jpn., 78(9):094006, 2009

    2009

  44. [44]

    Bilinear approach to delay-Painlev´ e equations.J

    A S Carstea. Bilinear approach to delay-Painlev´ e equations.J. Phys. A, 44(10):105202, 7, 2011

    2011

  45. [45]

    Algebraic entropy for differential-delay equations, 2014

    C M Viallet. Algebraic entropy for differential-delay equations, 2014

    2014

  46. [46]

    Growth of meromorphic solutions of delay differ- ential equations.Proc

    R Halburd and R Korhonen. Growth of meromorphic solutions of delay differ- ential equations.Proc. Amer. Math. Soc., 145(6):2513–2526, 2017

    2017

  47. [47]

    Special solutions of bi-Riccati delay-differential equations.SIGMA Symmetry Integrability Geom

    B K Berntson. Special solutions of bi-Riccati delay-differential equations.SIGMA Symmetry Integrability Geom. Methods Appl., 14:Paper No. 020, 9, 2018

    2018

  48. [48]

    Singularity confinement in delay-differential Painlev´ e equations.J

    A Stokes. Singularity confinement in delay-differential Painlev´ e equations.J. Phys. A, 53(43):435201, 31, 2020

    2020

  49. [49]

    Delay reductions of the two-dimensional toda lattice equation, 2022

    A Tsunematsu, K Nakata, Y Tanaka, and K I Maruno. Delay reductions of the two-dimensional toda lattice equation, 2022

    2022

  50. [50]

    A systematic construction of integrable delay- difference and delay-differential analogues of soliton equations.J

    K Nakata and K I Maruno. A systematic construction of integrable delay- difference and delay-differential analogues of soliton equations.J. Phys. A, 55(33):Paper No. 335201, 18, 2022

    2022

  51. [51]

    Integrable delay-difference and delay-differential analogs of the KdV, Boussinesq, and KP equations.J

    K Nakata. Integrable delay-difference and delay-differential analogs of the KdV, Boussinesq, and KP equations.J. Math. Phys., 63(11):Paper No. 113505, 15, 2022

    2022

  52. [52]

    A delay analogue of the box and ball system arising from the ultra-discretization of the delay discrete Lotka-Volterra equation.J

    K Nakata, K Negishi, H Matsuoka, and K I Maruno. A delay analogue of the box and ball system arising from the ultra-discretization of the delay discrete Lotka-Volterra equation.J. Phys. A, 57(14):Paper No. 145701, 26, 2024

    2024

  53. [53]

    Discrete Lotka-Volterra systems with time delay and its stability analysis.Phys

    Y Yamamoto, T Yamamoto, T Kuroiwa, K Oka, E Ishiwata, and M Iwasaki. Discrete Lotka-Volterra systems with time delay and its stability analysis.Phys. D, 474:Paper No. 134562, 9, 2025. 34

    2025

  54. [54]

    Cambridge University Press, 2004

    N S Manton and P M Sutcliffe.Topological solitons. Cambridge University Press, 2004

    2004

  55. [55]

    Nahm’s equations and the classification of monopoles.Commun

    S K Donaldson. Nahm’s equations and the classification of monopoles.Commun. Math. Phys., 96(3):387–407, 1984

    1984

  56. [56]

    Cyclic calorons.Nucl

    A Nakamula and N Sawado. Cyclic calorons.Nucl. Phys. B, 868(2):476–491, 2013

    2013

  57. [57]

    Cyclic monopoles.Nucl

    P M Sutcliffe. Cyclic monopoles.Nucl. Phys. B, 505(1-2):517–539, 1997

    1997

  58. [58]

    PhD thesis, University of Leeds, 2018

    J S Cork.Calorons, symmetry, and the soliton trinity. PhD thesis, University of Leeds, 2018

    2018

  59. [59]

    Large scale and large period limits of symmetric calorons.J

    D Harland. Large scale and large period limits of symmetric calorons.J. Math. Phys., 48(8):082905, 2007

    2007

  60. [60]

    A Yang-Mills-Higgs monopole of charge 2.Commun

    R S Ward. A Yang-Mills-Higgs monopole of charge 2.Commun. Math. Phys., 79(3):317–325, 1981

    1981

  61. [61]

    Calorons, Nahm’s equations onS 1 and bundles overP 1 ×P 1.Commun

    B Charbonneau and J Hurtubise. Calorons, Nahm’s equations onS 1 and bundles overP 1 ×P 1.Commun. Math. Phys., 280(2):315–349, 2008

    2008

  62. [62]

    Monopoles and Baker functions.Commun

    N Ercolani and A Sinha. Monopoles and Baker functions.Commun. Math. Phys., 125(3):385–416, 1989

    1989

  63. [63]

    Cyclic monopoles, affine Toda and spectral curves.Commun

    H W Braden. Cyclic monopoles, affine Toda and spectral curves.Commun. Math. Phys., 308(2):303–323, 2011

    2011

  64. [64]

    Construction of exact solutions to nahm’s equations for the multimonopole

    H W Braden, S Cherkis, and J M Quinones. Construction of exact solutions to nahm’s equations for the multimonopole

  65. [65]

    Towards a classification of charge-3 monopoles with symmetry.Lett

    H W Braden and L Disney-Hogg. Towards a classification of charge-3 monopoles with symmetry.Lett. Math. Phys., 113:87, 2023

    2023

  66. [66]

    Dihedrally symmetric monopoles and affine Toda equations.J

    H W Braden and L Disney-Hogg. Dihedrally symmetric monopoles and affine Toda equations.J. Phys. A., 57(49):495207, 2024. 35

    2024